essays on macroeconomics and credit risk · sudarshan p. gururaj the first chapter of this...
TRANSCRIPT
ESSAYS ON MACROECONOMICS AND CREDIT RISK
SUDARSHAN P. GURURAJ
Submitted in partial fulfillment of the
requirements for the degree
of Doctor of Philosophy
under the Executive Committee
of the Graduate School of Arts and Sciences
COLUMBIA UNIVERSITY
2009
UMI Number: 3348432
INFORMATION TO USERS
The quality of this reproduction is dependent upon the quality of the copy
submitted. Broken or indistinct print, colored or poor quality illustrations and
photographs, print bleed-through, substandard margins, and improper
alignment can adversely affect reproduction.
In the unlikely event that the author did not send a complete manuscript
and there are missing pages, these will be noted. Also, if unauthorized
copyright material had to be removed, a note will indicate the deletion.
®
UMI UMI Microform 3348432
Copyright 2009 by ProQuest LLC.
All rights reserved. This microform edition is protected against
unauthorized copying under Title 17, United States Code.
ProQuest LLC 789 E. Eisenhower Parkway
PO Box 1346 Ann Arbor, Ml 48106-1346
©2009
SUDARSHAN P. GURURAJ
ALL RIGHTS RESERVED
Abstract
Essays on Macroeconomics and Credit Risk
Sudarshan P. Gururaj
The first chapter of this dissertation empirically examines the impact of macroe-
conomic conditions on credit risk, particularly under shifting regimes. The second
chapter links a new Keynesian macroeconomic model with a model of credit risk to
demonstrate how macroeconomic conditions, namely output growth and inflation af
fect the credit risk of firms, as measured by credit spreads. In the third chapter, we
capture more features of the empirical behavior of credit spreads by including time-
varying preferences in the new Keynesian macroeconomic model with time-varying
preferences, which allow for the regime changes we find empirically.
Contents
1 Macroeconomic Determinants of Credit Spreads 1
1.1 Introduction 1
1.2 How Should Macroeconomic Conditions Affect Credit Spreads? . . . . 3
1.3 Data Description 7
1.3.1 Macroeconomic Data 7
1.3.2 Credit Spread Data 9
1.4 Empirical Properties of Credit Spreads 12
1.4.1 Tests of Persistence 13
1.4.2 Structural Break Tests and Inference from Markov Regime Switch
ing 16
1.4.3 Difference from Previous Empirical Studies 20
1.5 Model Estimation 21
1.5.1 OLS Estimation Results 22
1.5.2 Markov Regime-Switching Results 24
1.5.3 Robustness Check with First Differences 26
1.6 Discussion of Results 28
1.6.1 Regime Switching in Credit Spreads 28
1.6.2 The Impact of Macroeconomic Variables 30
1.7 Conclusions and Future Research 33
2 Credit Spreads in a New Keynesian Macro Model 35
i
2.1 Introduction 35
2.2 An Intuitive Framework 41
2.3 The Model 42
2.3.1 The Macroeconomy 43
2.3.2 Risk-Free and Risky Yields 50
2.4 Model Calibration and Simulation 54
2.4.1 The Macroeconomic Model 54
2.4.2 Calibrating and Simulating Credit Spreads 56
2.5 Results 58
2.5.1 Properties of Model-Generated Credit Spreads and Default Prob
abilities 59
2.5.2 Macroeconomic Factors of Credit Spreads: Contemporaneous
Relationships 60
2.6 Impulse Response Functions and Comparative Statics 61
2.6.1 Impulse Response Functions 62
2.6.2 Correlation with Output Growth, p 63
2.6.3 Idiosyncratic Cash Flow Growth and Volatility, (£, oK) . . . . 64
2.6.4 Relative Risk Aversion, 7 65
2.6.5 Alternative Monetary Policy, (x,wy) 65
2.7 Conclusion 67
3 Credit Spreads in a New Keynesian Macro Model with Habit Per
sistence 71
3.1 Introduction 71
3.2 The Model 73
3.2.1 The Macroeconomy 74
3.2.2 Risk-Free and Risky Yields 81
3.3 Model Calibration and Simulation 85
3.3.1 The Macroeconomic Model 85
11
3.3.2 Calibrating and Simulating Credit Spreads 87
3.4 Results 89
3.4.1 Properties of Model-Generated Credit Spreads and Default Prob
abilities 89
3.4.2 Macroeconomic Factors of Credit Spreads: Contemporaneous
Relationships 91
3.4.3 Macroeconomic Factors of Credit Spreads: Different Regimes . 92
3.5 Conclusion 93
Bibliography 95
A Chap. 1 Tables and Figures 100
A.l Unit Root Tests 100
A.2 Structural Break Tests 101
A.3 OLS Estimation 107
A.4 Markov Regime Switching Estimation I l l
A.5 Robustness Check: Tests with Moody's Data 117
A.6 Figures 118
B Chap. 2 Proofs, Tables, and Figures 122
B.l Credit Spread Expression 122
B.2 The Market Price of Risk and the Risk-Neutral (Q) Measure 123
B.3 Regression Test Coefficients 125
B.4 Impulse Response Functions 137
C Chap. 3 Proofs and Tables 146
C.l Market Price of Risk 146
C.2 Regression Test Coefficients 148
m
List of Figures
A.l 5-year credit spreads and possible break dates, May 1994 to June 2007 119
A.2 VIX, May 1994 to June 2007 120
A.3 Smoothed regime probabilities, 5 year A credit spreads 121
B. 1 Impulse response of macroeconomic conditions to adverse technology /productivity
shock 138
B.2 Impulse response of 1 year credit spreads to adverse technology/productivity
shock 139
B.3 Impulse response of 4 year credit spreads to adverse technology/productivity
shock 140
B.4 Impulse response of 10 year credit spreads to adverse technology/productivity
shock 141
B.5 Impulse response of macroeconomic conditions to positive monetary
policy shock 142
B.6 Impulse response of 1 year credit spreads to positive monetary policy
shock 143
B.7 Impulse response of 4 year credit spreads to positive monetary policy
shock 144
B.8 Impulse response of 10 year credit spreads to positive monetary policy
shock 145
IV
List of Tables
1.1 Autocorrelation function values for single A credit spreads, May 1994
to June 2007 14
1.2 Autocorrelation function values for 5-year credit spreads, May 1994 to
June 2007 14
1.3 Autocorrelation function values for first differences in single A credit
spreads, May 1994 to June 2007 15
1.4 Autocorrelation function values for first differences in 5-year credit
spreads, May 1994 to June 2007 15
1.5 Bai and Perron (1998) structural break test dates and confidence in
tervals for single A credit spreads 18
1.6 Bai and Perron (1998) structural break test dates and confidence in
tervals for 5-year credit spreads 19
1.7 OLS regression coefficients for single A credit spreads on regressors . 23
1.8 OLS regression coefficients for 5-year credit spreads on regressors . . 23
1.9 Markov regime switching model coefficients for single A credit spreads
on regressors 25
1.10 Markov regime switching model coefficients for 5-year credit spreads
on regressors 26
1.11 Markov regime switching model coefficients for first differences in 5-
year credit spreads on regressors 28
2.1 Macroeconomic model coefficients based on Ravenna and Seppala (2006) 55
v
2.2 Selected variable volatilities and correlations, model vs. historical,
1952-2006 56
2.3 Moments and correlations of output growth quarterly, inflation quar
terly, and interest rates 57
2.4 Firm-specific cash flow parameters, based on Longstaff and Piazzesi
(2004) and recovery rate in Huang and Huang (2003) 57
2.5 Moody's default probabilities and recovery rates (Huang and Huang,
2003 58
2.6 Average credit spread levels, model generated vs. historical and literature 60
2.7 Selected variable volatilities and correlations, model with relative risk
aversion of 10 vs. historical, 1952-2006 65
2.8 Model moments and correlations of output growth quarterly, inflation
quarterly, and interest rates where relative risk aversion is 10 66
2.9 Selected variable volatilities and correlations, model with relative risk
aversion of 25 vs. historical, 1952-2006 66
2.10 Moments and correlations of output growth quarterly, inflation quar
terly, and interest rates with relative risk aversion is 25 66
2.11 Selected variable volatilities and correlations, model with degree of
monetary policy smoothing of 0.9 vs historical, 1952-2006 67
2.12 Moments and correlations of output growth quarterly, inflation quar
terly, and interest rates with degree of monetary policy smoothing of
0.9 68
2.13 Selected variable volatilities and correlations, model with Taylor coef
ficient of output of 0.1 vs historical, 1952-2006 68
2.14 Moments and correlations of output growth quarterly, inflation quar
terly, and interest rates with Taylor coefficient on output of 0.1 . . . . 68
3.1 Moments and correlations of output growth quarterly, inflation quar
terly, and interest rates 86
vi
3.2 Selected variable volatilities and correlations, model with internal habit
preference vs. historical, 1952-2006 87
3.3 Moments and correlations of output growth quarterly, inflation quar
terly, and interest rates of internal habit persistence model 87
3.4 Firm-specific cash flow parameters, based on Longstaff and Piazzesi
(2004) and recovery rate in Huang and Huang (2003) 88
3.5 Moody's default probabilities and recovery rates (Huang and Huang,
2003 89
3.6 Average credit spread levels from the model, historical data, and om-
parable models 90
A.l Phillips-Perron test results for single A credit spreads 100
A.2 Phillips-Perron test results for 5-year credit spreads 100
A.3 Bai and Perron (1998) structural break test dates and confidence in
tervals for AAA credit spreads 101
A.4 Bai and Perron (1998) structural break test dates and confidence in
tervals for AA credit spreads 102
A.5 Bai and Perron (1998) structural break test dates and confidence in
tervals for A credit spreads 103
A.6 Bai and Perron (1998) structural break test dates and confidence in
tervals for BBB credit spreads 104
A.7 Bai and Perron (1998) structural break test dates and confidence in
tervals for BB credit spreads 105
A.8 Bai and Perron (1998) structural break test dates and confidence in
tervals for B credit spreads 106
A.9 OLS regression coefficients for AAA credit spreads on regressors . . . 107
A. 10 OLS regression coefficients for AA credit spreads on regressors . . . . 108
A. 11 OLS regression coefficients for A credit spreads on regressors 108
A. 12 OLS regression coefficients for BBB credit spreads on regressors . . . 109
vii
A. 13 OLS regression coefficients for BB credit spreads on regressors . . . . 109
A. 14 OLS regression coefficients for B credit spreads on regressors 110
A. 15 Markov regime switching model coefficients for AAA credit spreads on
regressors I l l
A. 16 Markov regime switching model coefficients for AA credit spreads on
regressors 112
A. 17 Markov regime switching model coefficients for A credit spreads on
regressors 113
A. 18 Markov regime switching model coefficients for BBB credit spreads on
regressors 114
A. 19 Markov regime switching model coefficients for BB credit spreads on
regressors 115
A.20 Markov regime switching model coefficients for B credit spreads on
regressors 116
A.21 Phillips-Perron test results for Moody's Corporate Credit Spread series 117
A.22 OLS model coefficients for Moody's corporate credit spread series . . 117
A.23 Markov regime-switching model coefficients for Moody's corporate credit
spread series 117
B.l Regression of forward-looking default probabilities on contemporane
ous output growth and inflation 125
B.2 Regression of credit spreads on one-quarter lagged credit spreads and
contemporaneous output growth and inflation 126
B.3 Model credit spread regression, output growth = 0.4 127
B.4 Model credit spread regression, output growth = 0.8 128
B.5 Model credit spread regression, mean of idiosyncratic firm cash flow
growth = -2% 129
B.6 Model credit spread regression, mean of idiosyncratic firm cash flow
growth = 2% 130
viii
B.7 Model credit spread regression, vol of idiosyncratic firm cash flow
growth =10% 131
B.8 Model credit spread regression, vol of idiosyncratic firm cash flow
growth = 40% 132
B.9 Model credit spread regression, coefficient of relative risk aversion = 10 133
B.10 Model credit spread regression, coefficient of relative risk aversion = 25 134
B.ll Model credit spread regression, coefficient of monetary policy smooth
ing = 0.9 135
B.12 Model credit spread regression, Taylor coefficient of output = 0.1 . . 136
C.l Regression of forward-looking default probabilities on contemporane
ous output growth and inflation 149
C.2 Regression of credit spreads on one-quarter lagged credit spreads and
contemporaneous output growth and inflation 150
C.3 Regression of A A A-A credit spreads with regime switching on one-
quarter lagged credit spreads and contemporaneous output growth and
inflation 151
C.4 Regression of BBB-B credit spreads with regime switching on one-
quarter lagged credit spreads and contemporaneous output growth and
inflation 152
IX
Acknowledgements
I am deeply indebted to my advisor, Prof. Marc Giannoni, who ushered me through
the long and winding road that is the process of writing a dissertation. I am grateful
for his guidance, insight, and encouragement over the past three years.
I would especially like to thank Prof. John Donaldson, my dissertation committee
chair, for his valuable comments and advice about my research. I also want to thank
Profs. Robert Hodrick, Patrick Bolton, and Martin Lettau and Dr. Bobby Porn-
rojnangkool for their detailed comments before and during the dissertation defense.
I would also like to mention my classmates Binu Balachandran, Sam Cheung, Yael
Eisenthal, Hagit Levy, Jorge Murillo, and Simeon Tsonev, along with the seminar
participants at Columbia Business School and at Platinum Grove Asset Management
who provided me with fruitful discussion and thought that led to these papers. The
empirical part of this paper would not have been possible without the assistance of
Drs. Wai Lee and Bobby Pornrojnangkool in obtaining data from Lehman Brothers.
Finally, the focus of my research is, ultimately, the value of debt. However, I owe
an invaluable debt to my family. I am very thankful to my parents, Drs. Pandu and
Sujatha Gururaj, and my brother Gautam for their unconditional support and love.
I would like to thank my wife Mrinalini for her encouragement during the final stages
of this work.
x
To my parents, Drs. Pandu and Sujatha Gururaj
XI
Chapter 1
Macroeconomic Determinants of
Credit Spreads
1.1 Introduction
Ever since Merton (1974) proposed studying credit risk by considering a firm as a
portfolio of contingent claims, many credit risk studies model the firm's debt as a
put option on the firm's assets. These structural credit models suggest that only
firm-specific variables affect claims on the firm's assets and, therefore, credit risk.
However, empirical studies that use credit spreads, the difference between the yields
of defaultable bonds and risk-free yields of the same maturity, as proxy for credit risk
seem to refute the applicability of such structural models. These papers have reached
the common conclusion that 1) that variables implied by structural models cannot
explain a great deal of the variation in credit spreads and 2) some common, unknown
factor exists that may be able to account for the unexplained variation left over.
Structural credit risk models, therefore, have little practical application due to
their focus on firm specifics alone. Corporate borrowers and lenders are interested in
determining the timing of their financing decisions in the context of wider macroe
conomic risk. Government policymakers want to study the impact of their actions
2
on borrowers and lenders. A more comprehensive empirical study of credit risk is
necessary to explain a greater amount of the variation in credit spreads and relate
that variation to aggregate macroeconomic factors. This paper empirically analyzes
the effects of macroeconomic variables, particularly real activity and inflation which
are considered to affect monetary policy, on credit spread changes
We begin our study by first examining the time-series properties of a new credit
spread dataset obtained from Lehman Brothers, spanning from May 1994 to June
2007. We then estimate a contemporaneous OLS regression of credit spreads on pos
sible factors. Based on the regime shifting behavior of credit spread data established
through econometric tests, we also estimate a simple Markov regime switching model
of the determinants of credit spreads. We find that real activity and inflation have
significant contemporaneous impact on credit spreads, particularly within certain
regimes.
We find, through the course of estimating OLS and Markov regime-switching
models, that output growth and inflation both have significant power in explaining the
variation in credit spreads, particularly in certain regimes and for shorter maturities
up to 10 years. Output generally has positive co-movement on bonds with credit
ratings from AAA and a negative co-movement on lower-rated bonds. Inflation has
a generally positive correlation on all credit spreads. This result provides motivation
for introducing monetary policy into standard structural credit models. The effects
of output growth and inflation on credit spreads can be seen through their effect on
the discounting of cashflows, in particular coupon payments made by the firm to meet
its debt obligations. Based on the "good beta, bad beta" explanation of Campbell
and Vuolteenaho (2004), output has both an effect on the future cashflows of the
firm, as well as the future discount rates the firm will face. Higher output means
both better cashflows to pay off debt, but also possibly higher discount rates in the
future. Smaller and lower credit quality firms have higher sensitivity to cash flow
risk. Inflation has a uniform effect on all credit spreads, as it affects only the discount
3
rate, common to all firms.
Our findings suggest that comprehensive credit risk models should consider ag
gregate, as well as firm-specific, factors and should model the firm's capital structure
in the context of the larger macroeconomy. Indeed, recent credit risk models, termed
"structural equilibrium" models 1 , connect credit risk with macroeconomic condi
tions, although output and inflation are exogenously determined in these models The
results presented in this paper provide motivation for a theoretical model of credit
that endogenously incorporates realistic features of macroeconomic risks. We will
explore this topic further in subsequent papers.
The remainder of the paper is organized as follows. Section 2 examines the empiri
cal implications of structural equilibrium models on the relationship between macroe
conomic conditions and credit risk. Section 3 describes our data, while Section 4
describes the empirical tests we run for the properties of credit spreads. Section 5
presents our results and an analysis of our results follows in Section 6. Sections 7
concludes and suggests further extensions.
1.2 How Should Macroeconomic Condit ions Af
fect Credit Spreads?
Traditional structural models of default, based on Merton (1974) , specify a particular
firm value process and assume that default occurs when firm value falls below some
explicit threshold. The firm's value can be modeled with a risk-neutral process such
as
— = (r- S)dt + adzQ + X(dqQ - p dt)
where V is firm value, r is the spot rate, S is the firm payout rate, a is the firm value
volatility, A is the size of the a firm-value jump, and p is the risk-neutral probability
1 This terminology was first used in Bhamra, Kuehn, and Sterbulaev (2007)
4
of a jump. Default occurs the first time that firm value reaches a threshold K. As
suggested by Longstaff and Schwartz (1995) , a higher spot rate should increase the
risk-neutral drift of the firm value process. A higher drift should reduce the incidence
of default, thereby reducing credit spreads.
This historical approach in the credit literature does not explicitly consider how
macroeconomic conditions should affect firm value and credit spreads. If we impose
the Taylor rule on how a monetary policy agency relates current output and inflation
conditions into a future risk-free rate, we can make an implicit connection between
current changes in output, inflation, and credit spreads. One version of the Taylor
rule is as follows:
r = 4>n7r + (/)yy + 6
where r is the interest rate, TT is the current rate of inflation, and y is current output
growth. The coefficients 0„. and 0y have been empirically estimated to be positive
in the monetary policy literature. Therefore, the risk-free rate is positively related
to output and inflation and, in turn, increases in output and inflation should reduce
credit spreads. Yet, this approach does not also consider how the firm's default
probability or payout ratio may be affected by changes in macroeconomic conditions.
In addition to a lack of connection between credit risk and macroeconomic con
ditions, structural credit risk models also generate lower credit spreads and higher
leverage that is empirically observed. Newer structural equilibrium models of credit
generate these empirically observed features embedding the contingent claim idea of
the firm with a firm value or cash flow process correlated with macroeconomic con
ditions. Examples of such models include Hackbarth, Miao, and Morellec (2006) and
Chen (2007) . They generate higher credit spreads than previous structural credit
models by having a positive correlation of firm cash flow and default recovery rate
with macroeconomic output. In so doing, these models imply that firms can meet
their debt obligations more often and default less in periods of economic boom than
in periods of recession. This, in turn, generates countercyclical behavior of credit
5
spreads with economy-wide output, implying a negative correlation between credit
spreads and GDP growth.
Structural equilibrium models like Hackbarth, Miao, and Morellec (2006) and
Chen (2007) directly incorporate output into credit risk, but they do not explicitly
incorporate inflation risk in their credit models. David (2007) incorporates output and
inflation as factors affecting the risk-free rate, which is a component of the dynamics
of the pricing kernel. He suggests that macroeconomic risks raise asset volatilities,
thereby depressing asset values. However, he does not explicitly incorporate macroe
conomic risks into bond pricing. Furthermore, all structural equilibrium models do
not specify how macroeconomic conditions evolve, instead describing output growth
and inflation as exogenously given stochastic processes.
Unlike the above modelling approaches, we explain our findings by postulating a
direct mechanism by which firms are affected by both output growth and inflation,
which, along with interest rates, are endogenously specified jointly in macroeconomic
models. For example, macroeconomic models postulate a VAR that describes the
evolution of output growth, inflation, and interest rates.
9t+i
7T*+1
. Tt+1 .
= A(L)
9t
7T*
. r< .
The VAR jointly specifies how output growth, inflation, and interest rates affect
each other in reduced-form. If we then postulate that output growth is a systematic
component of the firm's cash flow growth, then the firm's bond price, which reflects
the discounted stream of coupon payments made to the bondholder, is affected by
both output and inflation. Output growth affects the firm's cash flow growth, and
the discounting rate is impacted by both output and inflation given by the structural
VAR above that simulates the dynamics of the macroeconomy.
We propose that macroeconomic risks, both output and inflation, directly affect
6
credit spreads in the following manner. Output growth is correlated with firm cashflow
growth as suggested by structural equilibrium models, but output growth and inflation
also enter into the discount rate via the Taylor rule. As seen above, output and
inflation are positively related to the risk-free rate; therefore, an increase in either
quantity increases the discount and reduces asset values. While an increase in inflation
should uniformly increase credit spreads of all ratings and maturities, an increase in
output raises both cashflows and discount rates. Therefore, the impact of an increase
in output should be determined by the net effects on cashflows and the discount
rate. If the bond has little chance of defaulting, then output increases the discount
rate thereby increasing credit spreads for highly-rated bonds. For lower-rated bonds,
output increases also raise cashflow for the firm to pay off debt and offset discount
rate increases. We use this intuitive model of credit risk to explain our results and
propose this as a model to be explored in a future paper.
Each of the mechanisms described above has a different implication for the ef
fect of macroeconomic conditions on credit spreads. After we empirically test the
relationship between macroeconomic variables and credit spreads, we compare our
empirical results with the theoretical predictions of each mechanism to evaluate the
applicability of each in modelling credit risk.
For testing the effects of macroeonomic variables on credit spreads, we use as
controls the following variables as suggested by the structural credit models descibed
above.
1. Short Rate
The short rate represents the risk-neutral drift in structural credit models, as
shown above, and is the basic discount rate faced by all firms in the economy.
As suggested above, the short rate might impact the risk-neutral drift of asset
values, thereby impacting credit spreads.
2. Slope of the Yield Curve
7
The term premium, as captured by the slope of the term structure, might
capture future movements in the short rate. Using logic from Longstaff and
Schwartz (1995), if the short rate is expected to converge to the long rate,
then an increase in the slope of the treasury curve increases expected future
short rates. Other studies have noted that the slope of the term structure also
forecasts future economic output.
3. Equity Returns
Equity returns represent an aggregate measure of leverage in the economy and
also another indicator of business climate. As equity values increase, the firm's
debt-to-equity ratio must necessarily decrease. Furthermore, as equity values
rise, the recovery rate, the amount returned to debtholders upon default of the
firm, should also increase.
4. Market Volatility
Structural models of default imply the firm's debt is equivalent to being short a
put option on the firm's assets. Therefore, the volatility of the firm's assets are
key input into the valuation of risky debt. As suggested by Vassalou and Xing
(2004) , equity volatility can be a good proxy for firm asset volatility under
most normal conditions.
Given the above intuition from structural credit model, we will test the impact of
output and inflation on credit risk, controlling for the above significant factors.
1.3 Data Description
1.3.1 Macroeconomic Data
In this section, we describe the data that we use to proxy for factors of credit risk.
Since our credit risk data series extends from May 1994 to June 2007, we must try to
8
find proxies at the monthly frequency, particularly for macroeconomic series, which
are typically measured quarterly.
The real activity (REAL) factor describes overall economic growth, and we cal
culate the series as the month-on-month percentage change in the non-farm payroll
data. The real activity series calculated in this manner shows a high degree of cor
relation with GDP growth, but is available on a monthly frequency. The inflation
factor (INFL) is the month-on-month percentage change in the personal consumption
expenditure price index. Unlike other forms of inflation measures, such as changes in
the CPI, it is also available on a monthly frequency. Both of these time series are ob
tained from data from the St. Louis Federal Reserve (FRED) Database. Since, for a
given month, both non-farm payroll and personal consumption expenditure informa
tion are announced during the middle of the following month, we lag both series by a
month to reflect when the information was known. This also mitigates the possibility
of simultaneity bias in the coefficients.
The federal funds rate (FFR) factor is the market-traded effective federal funds
rate, taken as the 30-day average of the daily quote. The daily effective federal funds
rate is a weighted average of rates on brokered trades. In addition to the level of
interest rates, as measured by the effective federal funds rates, we include the slope
of the term structure to capture term premia. We measure the slope of the term
structure (SLOPE) as the difference of the constant maturity ten-year and two-year
yields on the Treasury curve, also obtained from the FRED database. These are par
yields, consistent with the par credit spreads that we obtained from Lehman Brothers.
For our measure of the equity market's returns, we use the MKT factors from
Kenneth French's website. The CRSP value-weighted market return (MKT) is the
return of the CRSP value-weighted index in excess of the risk-free rate. We could
also include the other Fama-French factors, including value (HML), size (SMB), and
momentum (UMD), as a robustness check as other factors that explain equity market
returns in future work. Elton, Gruber, et. al, find the Fama-French factors have
9
significant effect in explaining credit spread changes.
We use the VIX index as our proxy for market volatility as it is the fair value of
volatility of S & P 500 index options. Just like the federal funds rate, VIX is similar
to a market price series. As a result, we take the average of the previous month's
VIX values to get an end-of-the-month average measure for VIX. The VIX is often
considered a measure of expected future volatility and represents an easily accessible
forward-looking measure of volatility. Furthermore, the VIX is currently measured as
the fair value of volatility. This method of calculation takes into account the volatility
skew of options, which measures the relative price of insuring against market crashes.
Therefore, it is also a measure of market "fear" or risk aversion.
1.3.2 Credit Spread Data
We use credit spread data obtained from Lehman Brothers for this study. The credit
spread data is calculated from bonds that comprise the Lehman Brothers Credit
Indices. The credit spread quotes have credit ratings from AAA to B and with tenor
from 1 year to 10 years. The credit ratings of the series reflect the average of ratings
of three agencies, S & P, Moody's, and Fitch. We generate monthly credit spreads
quotes from May 1994 to June 2007 by taking the average of credit spread quotes
during throughout the month.
Unlike the data used in previous studies, these data consist of credit spreads
calculated from individual corporate bond prices and aggregated to the credit rating
and maturity level. As a result, this study does not suffer the lack of clarity of results
from using Moody's corporate bond index data. We also reflect more correctly market
perception of credit risk than those studies that use individual corporate bonds, at
the cost of losing information at the firm level.
The Lehman Brothers credit spread data is derived by calculating a survival curve
by fitting the curve to individual bond prices. The price of the bond is related to the
10
survival curve by the following equation:
N C N
PV — 2_^ -JZlibor(ti)Q(ti) + Zubor(tN)Q(tN) + 2_^ RZlibor{ti)(Q(ti-l) ~ Q(ti))
i=l •* i= l
where PV is the bond's dirty price, U and N are coupon times, Q(t) is the survival
probability until time t, C is the coupon rate, / is the coupon frequency, Znbor is the
LIBOR discount rate at time t, and R is the assumed recovery of the bond.
The first two terms reflect the present value of scheduled payments (coupon plus
principal) by discounting by LIBOR and the probability of survival. The third term
accounts for the payment of a recovery payment upon default of the bond, weighted
appropriately by the default probability.
The survival curve Q(t) can be fitted to an individual bond or groups of bonds,
using the exponential spline methodology of Vasicek and Fong. We chose data at the
aggregate credit ratings level.
Based on the fitted survival curves, we can calculate a par-spread term structure
for a hypothetical set of bonds of a particular credit rating or industry class. The
par coupon is defined as the coupon of a hypothetical bond of a given maturity
which would trade at par if evaluated using the group's fitted survival probability
term structure. The par spread is defined by subtracting the fitted par yield of the
risk-free bond of the same maturity from the fitted par coupon of the hypothetical
credit-risky bond.
If the hypothetical bond has frequency / and an integer number of payment peri
ods until maturity t^ = 4 , then the par coupon term structure is defined by finding
the coupons which are the solution to the following fundamental pricing equation.
nvaru \ _ A ~ Q(tN)ZlAb(rr(tN) ~ Rj2i=l(Q(tj-l) ~ Q(U))ZLiber(U)
2-a=l Q\ti)^Libcn-\U)
11
The par yield of the risk-free bond is defined in a similar fashion as
vpar i. \ _ j 1 — ZBasejtN) XBase\lN) — J „ j v , .
The par spread to the base curve, in this case the Treasury curve, can then be
derived by subtracting the par base yields from the par risky coupons of the same
maturities.
sBaL(t) = c^(t)-YZrJt)
The above calculated spread measures the spread of a hypothetical bond of a
particular maturity of a given grouping, whether credit rating or industry class. The
fitting procedure used to derive the survival probability curve reduces the impact of
idiosyncratic risk of the individual bonds and provides a better measure of aggregate
credit risk.
The lack of consistency in empirical facts in other papers about credit spreads
results not only from the different possible econometric properties of the data, but
also from the data used to calculate credit spreads. As detailed above, most em
pirical studies about credit spreads use spreads calculated from credit indices, such
as Moody's corporate bond index and Merrill Lynch's credit indices. Other studies
use individual corporate bond prices to calculate risky yields from spreads can be
calculated. Both data sources have features that make calculated credit spreads not
representative of true credit risk in the market.
The most popular source of credit spread data are the Moody's seasoned corporate
bond indices, which are constructed from an equally weighted sample of yields on 75
to 100 non-financial corporate bonds. The maturity of the bonds that comprise the
indices may be anywhere from 10 to 30 years, making inference of risk premia in
credits of different maturities unclear. In fact, many studies arbitrarily subtract the
10-year Treasury yield from the index yield to calculate credit spread, which leads to
incorrect inference.
12
Furthermore, the Moody's bond indices may have issues with the bonds that com
prise the index and how they are weighted within the index. The Moody's bond index
includes callable bonds with optionality that gets reflected in the average corporate
yield. Additionally, the index must be purged of index rebalancing effects that change
the time series.
Credit spreads derived from individual bond prices have their own idiosyncracies.
Firstly, corporate bonds historically are traded rather infrequently, so many data
sources use matrix pricing, interpolating prices between transaction prices. While this
method completes the data set, it also may not accurately reflect changes in credit
conditions. Secondly, individual bond prices are subject to their own idiosyncracies,
due to unique features in the bond convenants and to market valuation of the issuing
firm. Again, credit spreads derived from bond prices may not reflect aggregate credit
condition, which is the focus of our research.
The features of the data used to measure credit risk may have contributed to the
wide variation of results extant in the literature. As we explain below, the dataset
we propose to use from Lehman Brothers should avoid the idiosyncracies found in
individual bond prices and the induced features found in credit indices. However, as
a robustness test of our empirical findings, we apply the same econometric analysis to
Moody's seasoned Aaa and Baa corporate bond indices in Appendix C. We find that
the results are similar to the results we find for our Lehman Brothers credit spread
series.
1.4 Empirical Properties of Credit Spreads
In this section, we propose a plan to study the impact of macroeconomic conditions
on credit risk empirically. Previous empirical credit risk studies imposed arbitrary
econometric specifications without considering the properties of the time series of
credit spreads when choosing the appropriate econometric model. This may have lead
to inconclusive results regarding the nature of the determinants of credit spreads.
13
We propose to document the time series properties of credit spreads and select
an appropriate economic framework, based on those properties. We find that credit
spreads exhibit persistence and regime shifts through unit root and structural break
tests. Previous studies in the literature account for persistence in spreads empirically
by regressing with first differences in credit spreads. We believe that credit spreads
appear to be nonstationary and cannot be rejected for the presence of a unit root
through conventional econometric tests, because credit spreads exhibit regime shift
ing behavior that violates the stationarity assumption. The standard approach in the
literature of taking first differences to handle persistence removes information from
the time series that can be valuable for studying the properties of credit spreads.
Therefore, we perform our study as a regression in levels with lags of credit spreads
among the regressors to account for persistence in the data, while retaining the infor
mation found in the levels, but not in first differences. Our positive confirmation of
structural breaks via Bai and Perron tests (1998) suggest that we should empirically
test the determinants of credit spreads in the context of a regime shifting model.
1.4.1 Tests of Persistence
To select an appropriate econometric framework in which to study the determinants
of credit spreads, we first examine the existence of persistence or non-stationarity
in credit spreads. Several studies have documented the persistence of interest rates
(Fama, 1976, 1977; Rose, 1998 ) and credit spreads (Pedrosa and Roll, 1998 ), par
ticularly measured on frequencies higher than monthly. We, therefore, test our data
for non-stationarity by looking at the autocorrelation functions for different credit
spreads, as well as applying standard unit root tests to our data.
Tables 1.1 through 1.4 show the autocorrelation function of selected credit spread
series from our dataset. We find strong serial correlation in each credit spread series
across different credit ratings and maturities. When the autocorrelation function is
tested on first difference of the monthly credit spread series, we find low autocorre-
14
lation, although not exactly equal to zero. In tables A.l and A.2, we test each credit
spread series for persistence via unit root tests such as augmented Dickey-Fuller and
Phillips-Perron. We also ran augmented Dickey-Fuller tests, as well, with similar re
sults, so we do not present them here. Conforming with our autocorrelation functions
and previous tests in the literature, we cannot reject the credit spread series for the
presence of a unit root.
Maturity lYr 2Yr 3Yr 5Yr 7Yr
lOYr
Autocorrelation Functions Lag 1 0.94 0.96 0.96 0.96 0.96 0.95
Lag 2 0.90 0.93 0.93 0.93 0.92 0.91
Lag 3 0.85 0.88 0.89 0.88 0.88 0.87
Table 1.1: Autocorrelation function values for single A credit spreads, May 1994 to June 2007
Maturity AAA
AA A
BBB BB
B
Autocorrelation Functions Lag 1 0.94 0.94 0.96 0.97 0.92 0.96
Lag 2 0.89 0.89 0.93 0.94 0.88 0.90
Lag 3 0.84 0.83 0.88 0.91 0.82 0.84
Table 1.2: Autocorrelation function values for 5-year credit spreads, May 1994 to June 2007
15
Maturity lYr 2Yr 3Yr 5Yr 7Yr
lOYr
Autocorrelation Functions Lag 1 -0.06 0.04 0.07 0.05 0.04 0.02
Lag 2 0.09 0.11 0.13 0.09 0.06 0.03
Lag 3 -0.04 -0.06 -0.06 -0.08 -0.07 -0.06
Table 1.3: Autocorrelation function values for first differences in single A credit spreads, May 1994 to June 2007
Maturity AAA
AA A
BBB BB
B
Autocorrelation Lag 1 -0.12 0.02 0.05 0.11 -0.17 0.32
Lag 2 0.10 0.04 0.09 0.17 0.08 0.02
Functions Lag 3 -0.11 -0.14 -0.08 0.06 0.03 -0.03
Table 1.4: Autocorrelation function values for first differences in 5-year credit spreads, May 1994 to June 2007
Although we confirm the presence of strong persistence in the credit spread series,
we have several reasons why we choose not to model credit spreads as a unit root or
take first differences in our econometric tests. Firstly, the standard unit root tests,
such as Dickey-Fuller and Phillips-Perron, have low power, not rejecting those cases
that may not be unit root. Furthermore, we have a relatively short sample of data,
consisting of monthly credit spreads between May 1994 and June 2007, which may
not be sufficient to determine the time series properties of credit spreads. Indeed,
interest rates and credit spreads are often modeled in the theoretical literature as
stationary processes, although they empirically exhibit persistence or close to unit
root behavior.
The inability to reject the presence of a unit root in each of the credit spreads
along with the subsamples of each series suggests we should correct for persistence in
our choice of econometric specification. Unlike the specification used by most of the
16
empirical literature testing first differences in credit spreads, we test the determinants
of credit spreads in levels, including lagged credit spreads to account for persistence.
Therefore, in the structural breaks tests and models we estimate below, we employ
the following specification:
yt = a + y't^p + xti + tt
where yt is a single credit spread series. The vector
xt = [REALt INFLt FFRt SLOPEt MKTt VIXt]
contains changes in real activity (REAL), inflation (INFL), the fed funds rate (FFR),
the slope of the term structure (SLOPE), and VIX. We also include the market excess
return (MKT).
1.4.2 Structural Break Tests and Inference from Markov Regime
Switching
A contingent-claim view implies that the factors of credit risk have a non-linear
impact on credit spreads. A simple way to adapt the traditional, linear econometric
specification, such as the one suggested above, to a non-linear phenomenon would be
to introduce changes of regime. Such regime shifts in the model, though, would only
be warranted if we can document that the model experiences structural breaks.
To determine the prescence of structural breaks in credit spreads, we use a battery
of tests developed in Bai and Perron (1998, 2003) . 2 We summarize the purpose
of each test here and postpone their detailed econometric description to Appendix
A. The Bai and Perron methodology first chooses the exact break dates through
the least-squares principle. In a multiple linear regression model, the partitions are
2 Gauss code to implement structural break tests can be found at Pierre Perron's website
17
chosen by minimizing the sum of square residuals. The partition that minimizes the
sum of square residuals objective are the break dates. There are two types of tests to
determine whether there is a structural change: a supFr(k) test that tests the null
of no breaks versus the alternative of k breaks and the double maximum test that
tests the null of no breaks versus the alternative of an unknown number of breaks.
The method to determine the number of breaks consists of sequentially applying the
supFT(l + l\l) test with / breaks versus the alternative of / + 1 breaks starting with
1 = 1. One concludes for a rejection in favor of a model with (I + 1) breaks if the
overall minimal value of the sum of squared residuals is sufficiently smaller than the
sum of squared residuals from the / breaks model.
To test for the existence, number, and location of structural breaks, we applied the
above Bai and Perron (1998, 2003) structural change econometric procedure to the
common econometric specification given earlier. Given the graphs of different credit
spread series as shown in Figure 1, we choose the maximum number of breaks in the
test to be m = 3. We clearly see two breaks in most of the series in the middle of 1998
and again towards the end of 2002 to the middle of 2003. When applying the Bai and
Perron tests, we also account for potential serial correlation and heteroscedasticity.
The results from the structural break tests show some common results across all
credit ratings and maturities. As shown in Tables 1.5 and 1.6, most of the regressions
show a structural break around the summer of 1998 and again around the end of
2002 and the beginning of 2003, with relatively tight confidence interval within those
periods. Bold numbers denote significance at the 1 percent level. The values of
the supF test statistic which test the null of no break versus the alternative of 1
to 3 breaks are generally significant, with the test statistic almost always significant
for two breaks. Similarly, the values of the UDmax and the WDmax statistics (not
shown in the table) which test for the null of no break versus the alternative of an
unknown number of breaks, are also highly significant at the 1 percent level for all
ratings and maturities. From the strong significance of the above three statistics, we
18
can conclusively state that credit spread do exhibit structural breaks in the period
between May 1994 and June 2007.
With m = 3, the maximum number of breaks, we proceed to estimate the break
dates and their 95% confidence intervals. Under global minimization of the sum of
square residuals, we find the first break date for credit spreads from AAA to BB to be
around August 1998 with relatively tight confidence interval between July 1998 and
February 1999. The second break date for most credit spreads occurs around October
2002, although there is much higher variation in the specific break date, depending
on the credit rating and maturity, with wider confidence intervals between May 2002
and September 2004. The relationship for B credit spreads exhibit different breaks
with the first break date in the series occuring around the beginning of 2000 with the
exception of 5 year and 7 year tenors, which correspond to the late 1998 break date.
The second break date for B credit spreads almost uniformly falls in the beginning of
2004.
The break dates we find in all the credit spread series correspond to key financial
and macroeconomic events, giving further credence to existence of actual breaks in
Maturity lYr
2Yr
3Yr
5Yr
7Yr
lOYr
Break date 07/1998 03/2003
08/1998 04/2003
08/1998 04/2003
08/1998 04/2003
08/1998 05/2003
08/1998 06/2003
95% C.I. 02/1998 03/2003
07/1998 03/2003
07/1998 07/2002
08/1998 07/2002
08/1998 07/2002
02/1998 07/2002
08/1998 11/2003
11/1998 12/2003
09/1998 05/2003
11/1998 03/2003
11/1998 11/2003
12/1998 04/2004
1 731.13
5442.18
6009.19
25.90
85.04
805.78
supF test 2
7018.15
58233.22
52040.66
214820.02
37406.52
29824.46
3 287281.06
643382.28
309221.33
41417.50
44353.14
525900.69
supF(i+l || i) i = 1 i =2
181.49
579.13
949.42
3239.84
261.73
153.38
65.17
30.54
29.83
30.29
50.41
18.99
Table 1.5: Bai and Perron (1998) structural break test dates and confidence intervals for single A credit spreads
19
Rating AAA
AA
A
BBB
BB
B
Break date 08/1998 02/2003
08/1998 03/2003
08/1998 04/2003
08/1998 04/2003
08/1998 06/2003
02/2000 04/2003
95% C.I. 04/1998 04/2002
01/1998 07/2002
08/1998 07/2002
08/1998 12/2001
04/1998 03/2003
08/1999 03/2003
11/1998 02/2003
11/1998 04/2003
11/1998 03/2003
08/1998 09/2003
12/1998 07/2003
08/2000 02/2004
1 254.12
892.15
25.90
124.57
6.72
38.19
supF test 2
18751.25
55627.44
214820.02
101708.45
14011.47
97037.08
3 46436.70
491577.71
41417.50
751472.02
73961.83
285133.30
supF(i+l || i) i = 1 i =2
35.42 5.77
26.23 22.37
3239.84 30.29
174.02 26.85
43.86 13.45
109.09 8.64
Table 1.6: Bai and Perron (1998) structural break test dates and confidence intervals for 5-year credit spreads
the data. The first break date for most of the series, November 1998, corresponds to
the failure of LTCM around the middle to end of that year. Prior to the failure of the
hedge fund, a series of international financial crises occurred in Asia in late 1997 and
in Russia around the middle of 1998. The impact of these financial crises, followed by
the failure of one of the world's largest hedge funds, may have increased risk aversion
in the markets, causing a widening of credit spreads for the following five or six years.
The second break date in the end of 2002 or the beginning of 2003 may correspond to
the gradual recovery of the markets from September 11 and the corporate accounting
scandals that followed in 2002, coupled with the Federal Reserve's easing of interest
rates. Additionally, global uncertainty regarding the possibility of war in the Mid
dle East may also have been realized, reducing risk aversion and the corresponding
volatility in credit spreads around this time. Single B credits, which behave more like
equity than like debt, may respond more to equity market factors and, as a result,
the first regime shift in our sample occurs around the beginning of 2000 during the
bursting of the dot-com bubble.
When we consider the graph of VIX (Figure 2) over the sample period between
20
May 1994 and June 2007, we also find that VIX experiences a change in mean over
the period corresponding roughly to the breaks found in most of the credit spread
series. VIX remains consistently above the 17% mark in the period between the
middle of 1998 and the middle of 2003. Therefore, we generally describe our credit
spread regimes as high- and low-volatility regimes. The period from the beginning of
our sample until the fall of 1998 (spring of 2000 for single B) and the period from the
spring of 2003 until the end of our sample corresponds to regime 1, the low-volatility
regime. The period between fall 1998 (spring 2000 for single B) and the spring of
2003 corresponds to our regime 2, the high-volatility regime.
1.4.3 Difference from Previous Empirical Studies
As we mentioned in the introduction, the literature contains a number of empirical
studies on the determinants of credit spreads. Unlike our paper, these studies gener
ally arrive at the conclusion that the variation in credit spreads is difficult to explain
with simple factors, but that a common, unknown factor exists that may explain the
majority of the variation. Our results differ as we test credit spreads in levels and
allow for regime shifts.
Studies, including Collin-Dufresne, Goldstein, and Martin , employ ordinary least
squares on first differences in credit spreads as their econometric framework for an
alyzing credit spreads. 4 As we noted above, such studies cannot explain a great
deal of the variation in spreads. This lack of explanatory power may be attributable
to the imposition of an arbitrary econometric framework, testing first differences in
spreads to account for persistence or the exclusion of the structural break properties
we find here in credit spreads.
Other papers, such as Morris, Neal, and Rolph (1998) , also determine unit root
behavior in credit spreads and impose a cointegration framework on credit spreads
4 Other papers in a similar vein include Elton, Gruber, Aggarwal and Mann (2001) , Huang and Kong (2003) , and Alessandrini (1999)
21
with possible aggregate factors. As we stated earlier, we do not believe that credit
spreads are truly unit root in nature, and therefore a cointegration framework may not
be approprate for analyzing determinants of credit spreads. Furthermore, estimating
an error-correction model, as they do, to correct for cointegration does not truly
capture the non-linearity inherent in credit risk.
Our approach differs from existing papers in the literature by testing for the
time series properties of credit spreads and imposing a regime switching model on
a linear model of credit risk. In so doing, we find that simple factors, including
macroeconomic conditions, do explain credit spreads in certain regimes and provides
empirical motivation for theoretical model that link credit risk and macroeconomics.
1.5 Model Estimation
Once we establish the presence and number of structural breaks via the Bai and Per
ron (1998) tests, we should incorporate structural breaks into our model estimation.
Although the structural break tests probabilistically determine the timing of breaks,
we cannot identify the cause for the break. Therefore, we estimate the above linear
specification with a Markov regime switching model where the regimes shift based
on information in all the determinants we chose to study. The coefficients a, (3, and
7 are allowed to vary over some subsamples of the data, determined endogenously
with fixed volatility for the error term. We estimate the regime switching model us
ing Hamilton's (1990) Markov approach with constant transition probabilities. The
model maximizes the likelihood function of the linear model in a number of states
using the EM algorithm of Dempster and Laird (1994).3
In this section, we first estimate the model using OLS regression techniques. Our
results are similar to those found in previous papers of little explanatory power. We
then present our estimation of the model allowing for regime shifts, which increases
3 We estimated the model using the hmarkov_em code from James LeSage's Econometrics toolbox, written in Matlab
22
the explanatory power of the model and the significance of certain factors.
1.5.1 OLS Estimation Results
We first estimate our econometri specification using OLS with no structural breaks.
As we mentioned earlier, previous studies that estimated linear regressions of changes
in spreads on changes in possible factors resulted in low explanatory power and in
significant coefficients for a variety of macroeconomic and firm-specific factors. In
contrast, using our specification in levels with lagged credit spreads, we find a great
deal of explanatory power with R2 around 90% with a few macroeconomic factors
explaining much of the variation. Table 1.7 below shows sample regression results for
single A credit spreads of different maturities, while Table 3b shows sample regression
results for 5-year credit spreads of different ratings. For each maturity or rating, the
first row shows the regression coefficient and the second row shows the t-statistic with
the t-stats significant at the 5% level in bold.
Four common macroeconomic factors seem to explain credit spreads: previous
period credit spreads, real activity, inflation, and VIX. Consistent with the results
of our persistence tests, we find that lagged credit spreads have strongly significant
impact on the current level of credit spreads. Output growth generally has a negative
effect on credit spreads, although it also has a significant and positive effect on AAA
credit spreads. Inflation has a positive effect on credit spreads of all maturities and
ratings, as does VIX, the measure of market volatility and risk aversion.
23
Maturity
lYr
2Yr
3Yr
5Yr
7Yr
lOYr
Adj. R2
0.9172
0.9442
0.9489
0.9471
0.9440
0.9337
Const
-0.0006 -0.7400
-0.0006 -0.8944
-0.0007 -0.9573
-0.0005 -0.6172
-0.0002
-0.2239
0.0001 0.1364
cs^ 0.8059
20.5299
0.8613 27.0646
0.8542
27.0828
0.8484 26.2869
0.8444
26.0583
0.8344 24.4374
REAL
-0.0380 -2.9959
-0.0255 -2.3485
-0.0291 -2.6000
-0.0343 -2.7695
-0.0339 -2.7674
-0.0312 -2.6294
INPL
0.0263 1.9783
0.0259 2.0429
0.0306 2.3709
0.0365 2.6423
0.0374
2.7393
0.0376 2.8047
FFR 0.0109 1.1151
0.0050 0.5794
0.0058 0.6588
0.0046 0.4962
0.0023 0.2557
0.0019 0.2140
SLOPE
-0.0289 -1.3693
-0.0284 -1.5214
-0.0316 -1.6860
-0.0423 -2.0765
-0.0512
-2.5214
-0.0561 -2.8227
MKT 0.0003 0.1881
-0.0008
-0.4988
-0.0006 -0.3977
-0.0002 -0.1052
-0.0004
-0.2238
-0.0006 -0.3709
VIX 0.0080
4.9879
0.0073 5.0122
0.0081 5.3877
0.0087 5.5214
0.0085 5.5743
0.0079 5.3160
Table 1.7: OLS regression coefficients for single A credit spreads on regressors
Rating
AAA
AA
A
BBB
BB
B
Adj. R2
0.9032
0.9183
0.9471
0.9615
0.8839
0.9340
Const
0.0004
0.6438
-0.0002 -0.2438
-0.0005 -0.6172
-0.0006
-0.5477
-0.0065 -1.5809
-0.0233 -2.2378
CSt^
0.8138
20.6120
0.8161 22.5977
0.8484 26.2869
0.8702
29.3963
0.7637 17.0164
0.8508 22.2116
REAL
0.0156 2.3954
-0.0245 -2.3646
-0.0343 -2.7695
-0.0535 -3.0759
-0.1663 -2.7529
-0.3153 -2.0596
INFL
0.0199 2.8117
0.0336 2.6176
0.0365 2.6423
0.0361 2.8813
0.1955 2.5010
0.3869 1.9919
FFR -0.0038 -0.5312
-0.0013 -0.1541
0.0046 0.4962
0.0093
0.7138
0.0790 1.4665
0.2872 2.0663
SLOPE
-0.0372
-2.3119
-0.0437 -2.2713
-0.0423 -2.0765
-0.0527
-1.8973
0.0278 0.2452
0.1006 0.3691
MKT 0.0001
0.0820
-0.0003 -0.1984
-0.0002 -0.1052
0.0005 0.2001
-0.0029 -0.3087
-0.0285 -1.2741
VIX 0.0055
4.3548
0.0078 5.3469
0.0087 5.5214
0.0117 5.2629
0.0427 4.9171
0.0875 3.9030
Table 1.8: OLS regression coefficients for 5-year credit spreads on regressors
24
1.5.2 Markov Regime-Switching Results
Our intent in this paper is to attribute some of the variation in credit spreads to
macroeconomic factors through an appropriate, yet parsimonious econometric model.
Based on the results presented above and the graph of different credit spreads, we esti
mate the general specification presented above with a 2-state Markov regime switching
model. While we do not pre-specify the determinants of the states, we find that they
usually correspond to the high- and low-volatility regimes discussed earlier in our
structural break results. Tables 1.9 and 1.10 present our Markov regime switching
estimation results. In each row corresponding to a credit rating/maturity pair, the
first two rows present the coefficients and t-statistics for the low volatility regime,
and the second two rows present the corresponding coefficients and t-statistics for the
high-volatility regime.
By estimating the relationship between credit spread changes and the macroeco
nomic and financial determinants as a regime-switching model, we find that impact of
macroeconomic variables have significant impact on in the high volatility regime. In
the low volatility regime, it seems that only the previous level of credit spreads is rel
evant for the current level of spreads, consistent with the persistence we cannot reject
in the credit spread series. While Collin-Dufresne, Goldstein, and Martin (2001) and
others find low explanatory power and conclude the existence of a common, unidenti
fied systematic factor, we find that, by estimating a model taking non-linearities into
account, that certain common factors explain credit spreads over certain periods in
the sample
Furthermore, the smoothed probabilities of the Markov regime-switching models
for the different states for each model correspond to the break dates we find through
the global minimization procedure. Figure A.3 shows smoothed probabiities for the
regime-switching model for 5-year A credit spreads. The smoothed probability for
regime 2 (high volatility) increases around the end of 1998 and falls around 2003.
This gives further evidence for the existence of structural breaks, particularly during
25
the periods we find through the Bai and Perron (1998) tests.
In the high volatility regime, the coefficients of the regression on macroeconomic
factors, such as real activity and inflation, are consistent with the signs we find under
OLS. We find the inflation is positively related to credit spreads across all maturities
and ratings classes, although mostly in the high volatility regime. Real activity or
output also affect spreads only in the high volatility regime, but its effect varies with
credit rating. For AAA and AA bonds, an increase in output also increases credit
spreads. For all other credit spreads, particularly lower than investment grade bonds,
output is negatively correlated with credit spreads.
Maturity lYr
2Yr
3Yr
5Yr
7Yr
lOYr
Regime 1
2
1
2
1
2
1
2
1
2
1
2
Adj. R2
0.9598
0.9783
0.9832
0.9840
0.9819
0.9708
Const 0.0008 1.4659
-0.0011 -0.1853
0.0000 0.8565
-0.0033 -6.5662
0.0007 1.5904
-0.0019 -3.8528
0.0002 0.2194
-0.0078 -0.8457
0.0003 0.6377
-0.0069 -2.7133
-0.0009 -1.3659 0.0000
-0.0036
CSt-i 0.9500
21.9782 0.3628
3.8695
0.8923 35.5516
0.7833 6.5453
0.8530 25.3093
0.7440 5.5315
0.9892 9.8172 0.3569 1.7984
0.9678 21.2988
0.3572 4.3204
1.0415 8.8436 0.4389
4.6397
REAL 0.0066 0.4103
-0.1931 -2.7541
-0.0108 -1.1287 -0.0424 3.1243
-0.0216 -2.1349 -0.0221
-2.0736
0.0128 0.3227
-0.1134 -2.8605
0.0064 0.4540
-0.1100 -3.2511
0.0169 0.6125
-0.1176 -3.5198
INFL -0.0116 -1.1512 0.1093
2.5765
0.0142 3.5285 0.0644
1.9755
0.0130 1.7369 0.0332
3.7207
-0.0079 -0.6706 0.0637
2.1150
-0.0025 -0.3853 0.2762
5.7875
0.0104 0.5797 0.0903
3.4604
FFR -0.0145
-2.1267 0.0488 0.7578
0.0034 0.9323 0.0245 0.8565
0.0012 0.2026 0.0120 0.4810
-0.0085 -1.4099 0.0830 1.5016
-0.0078 -1.8441 0.0722
3.3233
-0.0043 -0.3808 0.0475 1.1596
SLOPE -0.0430
-2.7907 -0.0885 -0.4723
-0.0153 -2.5890 -0.0049 -0.3403
-0.0273 -2.4790 -0.0205 -0.6321
-0.0220 -1.2697 -0.0039 -0.0831
-0.0238 -1.8675 -0.0577
-1.9741
-0.0133 -0.6654 -0.0935
-4.4543
MKT -0.0004 -0.2350 0.0008 0.1127
-0.0004 -0.3432 0.0039 1.2111
-0.0009 -0.6608 -0.0026 -0.9063
0.0004 0.1538
-0.0002 -0.0446
-0.0002 -0.1875 0.0023 0.6040
-0.0012 -0.5090 0.0009 0.2824
VIX 0.0026 1.7996 0.0222
3.9386
0.0017 1.6078 0.0197
6.0123
0.0021 1.9439 0.0203
9.6121
0.0016 1.3355 0.0347
3.6194
0.0020 1.8758 0.0349
7.4425
0.0023 1.3876 0.0205
3.7003
Table 1.9: Markov regime switching model coefficients for single A credit spreads on regressors
26
Rating AAA
AA
A
BBB
BB
B
Regime 1
2
1
2
1
2
1
2
1
2
1
2
Adj. R2
0.9553
0.9732
0.9840
0.9901
0.9524
0.9709
Const 0.0010 0.0003
-0.0010 0.0006
-0.0014 -3.1226
0.0008 8.4425
0.0002 0.2194
-0.0078 -0.8457
-0.0007 -2.5062 0.0078 4.9930
-0.0245 -2.1327 0.0014 0.4979
-0.1954 -9.5151 -0.0100 -0.8703
CS,_i 0:8556
5.9474 0.7819
22.6640
0.8477 9.1639 0.8038
30.2014
0.9892 9.8172 0.3569
3.7984
1.0122 43.7808
0.4776 7.9258
0.8275 5.9326 0.7558
12.2385
0.9759 5.4522 0.9329
20.8274
REAL 0.0121 0.9272 0.0240
3.2327
0.0124 0.7992
-0.0214 -2.7477
0.0128 0.3227
-0.1134 -2.8605
0.0138 0.9823
-0.4388 -12.2073
-0.4160 -3.1525 -0.0841
-4.3992
-0.6050 -0.7562 -1.0260
-3.2106
INFL 0.0250 1.9572 0.0160
2.3349
0.0110 0.6229 0.0246
3.4463
-0.0079 -0.6706 0.0637
2.1150
0.0002 0.0225 0.0548
2.6205
0.4566 3.2595 0.0777
3.6482
3.5731 4.0761 1.1537 0.7742
FFR -0.0015 -0.3529 -0.0223 -1.4887
-0.0085 -0.6612 -0.0065 -1.4782
-0.0085 -1.4099 0.0830 1.5016
-0.0059 -0.9087 -0.0113 -0.4658
0.2120 0.6424 0.0165 1.5225
1.2886 2.0295 0.1007 0.7984
SLOPE -0.0226
-2.4924 -0.0792
-3.2184
-0.0296 -1.1994 -0.0404
-4.2189
-0.0220 -1.2697 -0.0039 -0.0831
-0.0226 -1.8616 -0.5926
-8.7880
-0.0210 -0.0570 0.0007 0.0138
1.0215 1.1842
-0.0838 -0.3556
MKT -0.0011 -0.8552 -0.0003 -0.0968
0.0002 0.0683
-0.0003 -0.2593
0.0004 0.1538
-0.0002 -0.0446
0.0020 1.0679
-0.0001 -0.0103
-0.0529 -3.0180 -0.0011 -0.1427
-0.3860 -4.4033
0.0138 0.7372
VIX 0.0022
2.0973 0.0154
6.2167
0.0181 8.5420 0.0036
3.4920
0.0016 1.3355 0.0347
3.6194
0.0037 2.8265 0.0390
10.6517
0.1042 6.8452 0.0185
3.2424
0.0346 1.2830 0.4443
5.7224
Table 1.10: Markov regime switching model coefficients for 5-year credit spreads on regressors
1.5.3 Robustness Check with First Differences
Although we find strong evidence for the impact of output growth and inflation on
credit spreads, even while controlling for persistence, the large R2 and significant
coefficients may be due to persistence in the credit spread time series. To check the
robustness of our results, we test the impact of our macroeconomic factors on credit
spread changes. This approach is akin to assuming that credit spreads are, in fact,
unit root, a notion that we reject from our theoretical knowledge of interest rates and
credit spreads.
We employ the following specification:
Ayt = a + x't(3 + et
27
where yt is a single credit spread series. Running a regression in first differences
in credit spreads removes persistence and avoids spurious regression estimates. The
vector xt = [AREALt AINFLt AFFRt ASLOPEt MKT AVIXt] contains changes
in real activity (REAL), inflation (INFL), the fed funds rate (FFR), the slope of the
term structure (SLOPE), and VIX. We also include the market excess return (MKT).
All changes in the regression are taken as the one-month difference in the variable.
Like many previous studies, we find that OLS estimation cannot capture the
impact of macroeconomic factors on credit spread changes. Our estimation, not shown
here, of a simple linear model results in low explanatory power and insignificance of
most factors, except VIX, similar to the results in the the literature.
Estimating the specification in differences using Markov regime switching confirms
the results we obtain from the regression in levels. We find that, in certain regimes,
changes in output growth and inflation do impact changes in credit spreads. The
regression exhibits structural breaks at the same points as the specification in levels.
Estimating the model with two regimes also increases the explanatory power of the
model. Furthermore, changes in output growth have a negative effect and inflation
have a positive effect on changes in credit spreads.
28
Rating AAA
AA
A
BBB
BB
B
Regime 1
2
1
2
1
2
1
2
1
2
1
2
Adj. R2
0.5436
0.6218
0.6748
0.6057
0.4884
0.5990
Const 0.0000 0.1450 0.0001 0.5987
0.0000 -0.5237 0.0003 0.6509
0.0000 -0.5119 0.0002 0.9337
-0.0001 -1.2306 -0.0013 0.0000
-0.0009 -2.7047
0.0048 2.8037
-0.0002 -0.2976 -0.0035 -1.0803
REAL 0.0040 0.1222 0.0355
2.5260
-0.0298 -0.7897 -0.4750 -1.5799
-0.0330 -0.8897 -0.4447
-3.5807
-0.0355 -0.6618 -0.6541
-5.7734
-0.2297 -0.9751 -2.2798
-3.0210
-0.7676 -1.6403 -4.8168
-3.8020
INFL -0.0040 -0.2016 0.3965
3.7518
0.0235 0.9331 0.2359
2.3599
0.0126 0.4723 0.0452
2.7224
-0.0096 -0.2605 0.0148
3.0567
0.0780 0.5092 1.4439 1.7933
-0.3338 -1.0352 4.5114
3.0835
A F F R -0.0531 -1.8547 0.4780
6.0867
-0.0160 -0.4237 0.4870
5.4227
-0.0294 -0.8826 0.4414
5.9739
0.0147 0.3051 0.2335
2.1430
-0.9489 -1.0524 0.5736
2.7679
0.7321 1.7232 0.9718 0.4574
A SLOPE -0.0258 -0.6684 0.1205 0.5596
-0.0532 -1.2326 0.6602
2.4753
-0.0322 -0.7511 0.4545
2.5419
0.0017 0.0280
-1.2083 0.0000
-0.2605 -0.8940 2.5843
2.6166
0.3730 0.6155
-3.3120 -1.0403
MKT 0.0001 0.0617
-0.0077 -3.4580
0.0002 0.1451
-0.0056 -3.2154
0.0007 0.5452
-0.0051 -1.9713
0.0036 1.8546
-0.0292 3.2349
-0.0003 -0.0311 -0.0776
-2.7101
0.0158 0.9160
-0.2871 -4.3243
A V1X 0.0004 0.2878 0.0200
5.7959
-0.0007 -0.4036 0.0264
7.3939
-0.0033 -1.4848 0.0298
10.3452
-0.0010 -0.5366 0.0746
4.0230
0.0081 0.8111 0.1266
4.5887
-0.0185 -0.8095 0.5951
10.6494
Table 1.11: Markov regime switching model coefficients for first differences in 5-year credit spreads on regressors
1.6 Discussion of Results
1.6.1 Regime Switching in Credit Spreads
As shown previously, the time series of credit spreads tested here between May 1994
and June 2007 exhibit regime shifts at least twice in the sample. These results are
consistent with new models of credit risk and with empirical evidence
Both Hackbarth, Miao, and Morellec (2006) and David (2007) incorporate regime
switching into their theoretical models of credit risk and, in so doing, relate macroeco-
nomic conditions with credit spreads. Hackbarth, Miao, and Morellec (2006) observe
that firms adapt their default and financing policies to the position of the economy in
the business cycle phase. Based on this concept, they find that credit spreads change
29
with aggregate economic shocks, due to changes in the firms' behaviors with regards
to their capital structures. Hackbarth, Miao, and Morellec (2006) and Chen (2007)
use the ideas that firms change capital structure under different macroeconomic con
ditions to answer many puzzles extant in both the theoretical and empirical credit
risk literature.
David (2007), on the other hand, specifies that current inflation and output signal
future conditions and affect asset valuations. Asset valuation ratios and volatilities
vary over time as investors update their beliefs about the hidden states of fundamen
tals and reasses the prospects of future real growth in fundamentals. He, therefore,
models the regime shifts as an exogenous change in fundamentals.
Among the empirical literature, Alessandrini (1999) observes that credit spreads
are more sensitive to macroeconomic risks during recessions than during expansions.
Although he tests credit spread factors ony with OLS, his observation of different
behaviors in subsamples provides some empirical evidence for regime shifts.
While we arrive at the same conclusions as the authors mentioned above, our
rationale about the cause of regime shifting in credit spreads is different. We do find,
based on the Bai and Perron (1998) tests and Markov regime-switching models, that
most credit spreads change regime once around the summer/fall of 1998 and again
around the spring of 2003. Single B credits, however, exhibit their first regime change
around the early part of 2000, reflecting that they behave more like equities. Based
on such evidence, we believe that credit spreads do exhibit regime shifts not based
on macroeconomic fundamentals, but rather based on market risk aversion.
We base this on the similarities in behavior between credit spreads of all ratings
and the VIX, the measure of the fair value of volatility in the market. VIX, measured
as the fair value of volatility of S&P 500 index options, reflects not only the present
level of expected volatility in the market, but also is a measure of risk aversion as
it incorporates information about the index volatility skew. The volatility smile on
index options reflects how risk averse investors are, because it incorporate information
30
from out-of-the-money puts, which investors purchase and bid up to protect against
market downturns.
The key regime shift dates of fall 1998, spring 2000, and spring 2003 are marked
not by revelations about macroeconomic fundamentals, but rather exogenous asset
market events. Risk aversion and credit spreads appear to have jumped in the fall
of 1998 during the Russian default and the collapse of LTCM and again during the
2000 market crash. Neither events had direct connection with macroeconomic funda
mentals. In the case of the collapse of LTCM, macroeconomic fundamentals remained
strong and asset valuations recovered quickly; yet, credit spreads remained high there
after. The moderation of credit spreads during the early part of 2003 also had little
basis in macroeconomic events or even asset valuations. That date seems to reflect
the quick resolution of the military conflict in Iraq, after which credit spreads seemed
to decline continuously until the end of our sample.
Furthermore, we find that the impact of many of the factors we have chosen for
credit spreads seem only to have impact in the high-volatility state. This may explain
lack of explanatory power in other work when testing for credit risk factors with first
differences, particularly with a simple linear regression.
1.6.2 The Impact of Macroeconomic Variables
As we observed earlier, inflation in the high volatility regime increases credit spreads
across all ratings and maturities. Increased output in the high volatility regime, on
the other hand, increases spreads for higher rated bonds and decreases spreads for
lower rated bonds. This result contradicts the predictions of most structural credit
models and confirms the predictions of structural equilibrium models.
As mentioned previously, structural models of default take the risk-free rate as
given and not determined endogenously. In such a model, an increase in output and
in inflation, which increases the risk-free rate via the Taylor rule, would cause the
risk-neutral drift of the firm value process to increase. This, in turn, reduces credit
31
spreads as the firm hits its default boundary less frequently. While this prediction
matches what we find empirically with lower-rated firms, it does not conform with
our results for an increase in output for AAA and AA rated firms nor does it predict
the correct sign for the sensitivity of credit spreads with inflation.
Structural equilibrium models which embed credit risk within a general equilib
rium framework allow the firm value and risk-free rate to be determined endogenously.
Therefore, papers like Hackbarth, Miao, and Morellec (2006) and David (2007) pre
dict that increased output should decrease credit spreads. Again, this result matches
what we find for lower-rated firms and contradicts what we find for higher-quality
firms. Furthermore, David (2007) finds that inflation increases asset volatilities and
reduces valuations, thereby increasing spreads. This is consistent with the effect we
find for inflation for all classes of credit spreads.
To gain better intuition of the possible drivers behind our empirical results, we
propose the following intuition. Suppose the payout of the bond at each period is
git) = cX{t < T)X(t <r) + FS(t - T)X{t < r) + uF8{t - r)X(t < T)
where c is the coupon, T is the maturity, K(t) is the firm's cash flow at time t, F is
the face value of the bond, r represents the first time Kit) < c, and UJ represents the
recovery rate when the firm defaults. The function x(.) is the indicator function, and
the function 5(.) is the Dirac delta function.
The price of the bond at time 0 is therefore
P(0, T) = EQ
where Q represents the risk-neutral measure.
When realized macroeconomic output or real activity increases, it has two effects
on the bond pricing equation. If we assume that an individual firm's cash flow growth
is directly correlated with GDP growth, then, as GDP growth increases, market par-
g(t)
i + rty
32
ticipants would expect that future cash flows of the firm would increase, thereby
increasing the probability of the firm meeting its coupon obligation in every period
until maturity. However, the imposition of a Taylor rule would also imply the mon
etary policy agency would increase interest rates, thereby increasing the discount
factor, and reducing future cash flows. As a result, the impact of increasing real ac
tivity on firms differs by credit rating, as shown in the empirical results. Firms with
higher credit ratings already are generally able to meet their coupon obligations, so
that cash flow impact of increased GDP growth is minimal. They, however, are im
pacted more by an increase in interest rates, which lowers the price of their bonds and
subsequently increase credit spreads. In contrast, firms with lower credit quality see
an increase in their cash flows and their ability to meet debt obligations with GDP
growth increases. This positive effect offsets the discounting effects of the Taylor rule
and reduces the spreads on lower-rated bonds. Furthermore, the recovery rate can be
assumed to be positively correlated with GDP growth, contributing to lower spreads
for firms that default more often. Bondholders can expect to recover more of their
investment in case of default when macroeconomic conditions are good.
The effects described here parallel the "good beta, bad beta" theory, proposed
by Campbell and Vuolteenaho (2004) . They separate the beta of a stock with the
market portfolio into two components, one reflecting news about the market's future
cash flows and one reflecting news about the market's future discount rate. We
propose a similar effect with credit spreads here through this simple bond-pricing
equation. The change in GDP growth reflects information about future cash flows
and discount rates for both the macroeconomy and the firm. Firms with higher
credit ratings have greater sensitivity to discount rate information or "bad beta" and
perform like large-cap or growth stocks. Similarly, firms with lower credit ratings
have greater sensitivity to cash flow information or "good beta" and perform like
small-cap or value stocks.
The effects of an increase in inflation are far clearer and reflected in the strong
33
positive relationship between inflation and credit spreads in the results. In the context
of the bond-pricing equation above, inflation increases interest rates via the Taylor
rule, which, in turn, reduces discounted expectations of future cash flows.
1.7 Conclusions and Future Research
In this paper, we intend to study the impact of macroeconomic variables on credit
spreads as a measure of credit risk. First, we study the time series properties of credit
spreads as proxied by a relatively new dataset from Lehman Brothers. This dataset
avoids the pitfalls associated with other time series used to calculate credit spreads
and, therefore, provides a cleaner series for econometric inference. We find that credit
spreads between May 1994 and June 2007 exhibit regime shifts around key market
events, including the collapse of LTCM, the NASDAQ market collapse of early 2000,
and the end of the military phase of the war in Iraq in 2003.
Based on the above time series properties of credit spreads, we test the explanatory
of a set of factors, including output and inflation, on credit spreads in a Markov
regime-switching linear model. We find that such a model explains up to 70 % of
the variations in credit spreads and, in the high-volatility regime between 1998 and
2003, a number of factors are significant in the explanation of changes in credit
spreads. Based on a simple bond-pricing equation and the Taylor rule, we explain
the significant effects of output and inflation on credit spreads in terms of the future
cash flow and discount rate effects.
The results presented in this paper help resolve some of the ambiguity found in
previous empirical studies of credit spreads. The conclusive test of regime shifts in
the series may help explain the assertion in Collin-Dufresne, Goldstein, and Martin
(2001) of the existence of an unknown common factor affecting all credit spreads. Fur
thermore, the cash flow/discount rate explanation of the impact of macroeconomic
variables on credit spreads could be extended in a theoretical model. Currently, theo
retical models only consider the firm's capital structure behavior in isolation without
34
macroeconomic effects. Our future work will focus only explaining the behavior of
credit spreads in the face of changing macroeconomic conditions by embedding a
model of credit risk within a monetary economy model.
35
Chapter 2
Credit Spreads in a New
Keynesian Macro Model
2.1 Introduction
Valuation of credit risk is central to corporate financing decisions. Most of the models
in the credit risk literature, following Merton (1974), value the firm's risky debt as
a short put option on the firm's assets. This structural approach takes as given the
dynamics of the risk-free rate and the dynamics of firm value with default occuring
when the firm's asset value falls below a pre-specified boundary. Corporate bonds
are, therefore, valued as claims on the firm's asset value contingent on default oc
curence. This approach, however successful in characterizing firm behavior, ignores
how aggregate economic conditions affect the dynamics of the firm value and risk-free
interest rate processes.
The above structural credit approach also does not address several empirical ob
servations about credit spreads, a measure of credit risk. Fama and French (1989)
were among the first to document that credit spreads widen when economic conditions
are weak. Collin-Dufresne, Goldstein, and Martin (2001) and Elton, et. al, (2001)
both document that firm-specific default risk factors " account for a surprisingly small
36
fraction" of credit spreads. Alessandrini (1999) finds greater sensitivity of credit risk
to aggregate and firm-specific factors during recessions than during expansions. As
suggested by Collin-Dufresne, et. al (2001), the literature "suggests the need for fur
ther work on the interaction between market risk and credit risk - that is, general
equilibrium models embedding default risk."
Two recent empirical papers directly verify the impact of macroeconomic con
ditions on credit spreads. In a previous paper, we examine the direct relationship
between credit spread changes and macroeconomic conditions, as summarized by
output and inflation, in a linear model. We find that increasing inflation raises credit
spreads uniformly, while greater output growth raises higher-rated credit spreads and
lowers the spreads on lower-rated bonds. When the linear model is then estimated
with two possible regimes, we find that output growth and inflation impact credit
spreads in the manner described earlier in periods of high volatility or risk aversion in
the market. This result concurs with an earlier study done by Wu and Zhang (2008)
who arrive at the same conclusion when valuing credit spreads in a no-arbitrage term
structure model.
Recently, several structural equilibrium models have been developed in the liter
ature to explain the connection between macroeconomic conditions and credit risk.
1 Such structural equilibrium models explicitly link macroeconomic dynamics with
the firm value or cash flow process to generate macroeconomic implications for credit
risk. Hackbarth, Miao, and Morellec (2006) and Chen (2007) , for example, suggest
firms adjust their capital structure optimally in response to changes in aggregate out
put and find counter-cyclicality of credit spreads with economic growth as a result.
David (2007) prices corporate debt in a model where expected earnings growth rates
and expected inflation are unobservable, but follow a Markov regime switching pro
cess. Tang and Yan (2006) link macroeconomic output to the cash flow process of
the firm, which is then used to price risky debt in a contigent claims fashion. Al-
1 This terminology "structural equilibrium" was first used by Bhamra, Kuehn, and Sterbulaev to describe equilibrium models that embed credit risk
37
though each of these studies link macroeconomic conditions to the firm's default, all
these studies model the economy with exogenous stochastic processes, rather than
in a structural framework. Furthermore, with the exception of David (2007), most
structural equilibrium models can only show how output affects credit risk, ignoring
inflation altogether. Therefore, these models do not depict a true picture of how
credit risk respond to aggregate macroeconomic risk.
This paper rectifies this gap in the present literature by employing a simple new
Keynesian macroeconomic model and linking the time series of output, inflation,
and interest rates generated by the model to the firm's cash flow process. Real
aggregate output growth comprises part of the systematic component of the firm's
overall real cash flow growth. The firm issues a bond with finite maturity and a
fixed coupon to be serviced each period by the cash flow until maturity. Default
occurs when the firm's cash flow cannot meet its coupon obligation. Upon default,
the bondholders receive a percentage of the principal loaned called the recovery rate,
which is positively correlated with the prevailing output growth of the economy. The
risky bond is thus valued as a contingent claim on the firm's discounted cash flows.
Inflation, as well as output growth, impacts the discounting of the bond's cash flow,
because new Keynesian models impose a Taylor rule to relate the risk-free rate and
the term structure to macroeconomic conditions.
While previous structural credit models consider macroeconomic conditions as
exogenous and existing structural equilibrium models exogenously specify the dy
namics of the macroeconomic condition, we explicitly consider the dynamics of the
macroeconomy in a model where output growth, inflation, and the risk-free rate are
determined endogenously. Treating macroeconomic variables in reduced form, as is
done by other papers, may mask the determinants and dynamics of credit spreads.
We also model the firm's cash flow rather than its asset values, because it allows a
period-by-period view of the impact of default and liquidity risk.
We first simulate a standard new Keynesian macroeconomic model, such as those
38
postulated by Woodford (2003) and King (2000), using calibrated parameters that
match the features of historical US data. The results of this macroeconomic model
match the observed empirical correlations of output, inflation, and the risk-free term
structure. We then relate the time series properties of the marginal firm's cash flow
growth to the generated output growth of the macroeconomy, simulating a number
of possible cash flow growth paths for the firm. To define the initial credit risk of the
firm, we calibrate the realized default probability along simulation paths to actual
default probabilities provided in Huang and Huang (2003) and Leland (2004) , thereby
choosing initial coverage ratio for each credit rating class. We then price risky bonds
and calculate credit spreads based on simulated discounted cash flow paths for each
time period in the macroeconomy. This procedure generates a time series of credit
spreads for different credit ratings contemporaneous with the output, inflation, and
term structures generated by the macroeconomic model.
After simulating the model, we test the model-generated credit spreads by re
gressing credit spread levels on contemporaneous output growth and inflation. Since
inflation is positively related to the nominal term structure used in discounting via
the Taylor rule and no-arbitrage properties of the macroeconomic model, we expect
that an increase in inflation should increase future discount rates, thereby reducing
bond prices and increasing credit spreads. Output growth positively affects firm-level
cash growth and the nominal term structure; therefore, its effect depends upon the
credit rating of the considered firm. A firm with high credit rating and low initial
probability of default will have a negligible effect on ability to meet its coupon obli
gations, but will have its coupon payments discounted more when output growth
increases. This, in turn, will lower its bond price and raise its spread to the Treasury
rate. Conversely, a firm with low credit rating will lower its probability of default
upon an increase of output growth to counteract the rise in the discount rate. As a
result, its bond price should rise and its credit spread should fall.
To model the impact of macroeconomic conditions on credit spreads, we make
39
a necessary deviation from the traditional New Keynesian macroeconomic model.
Unlike most micro-founded macroeconomic models, we assume that industries, not
firms, are the optimizing productive agents in the economy. Firms are passive units in
our model that produce as much as is demanded by the particular industry in which
they reside and hire the amount of labor necessary for that output amount. While
this assumption may differ from the common practice in the literature, it allows us to
model the impact of the macroeconomy on the credit risk of firms without the more
intractable problem of modeling the impact of firm default on the macroeconomy. If
we model the macroeconomy with firms as optimizing agents that default, then the
macroeconomy would not have complete markets and would be much more intractable
to model using standard numerical techniques. Furthermore, this assumption allows
us to model the direct impact of the macroeconomy on credit risk, a feature not found
in either the macroeconomics or credit risk literature to date.
This model not only generates realistic interactions between macroeconomic vari
ables and credit spreads, but also generates credit spread properties observed in data
which are not generated by existing structural credit models. For example, the credit
spread volatility puzzle, as termed in the literature, is the inability of structural credit
models to generate the high levels of credit spread volatility observed empirically.
The model we propose generates larger credit spread volatility by correlating states
in which defaults are more prevalent with macroeconomically "bad" states, times of
high marginal utility and low recovery rates. In such periods, both the probability of
default and the recovery rate upon default fluctuate with the state of the economy,
thereby generating greater fluctations in credit spreads. Therefore, this approach
taken in this paper should be able to match higher levels of credit spread volatility
found in the data.
Another counterfactual result generated by many structual credit models that
incorporate asset pricing models are procyclical default probabilities. For example,
Chen, Collin-Dufresne, and Goldstein (2005) attempt to explain the equity premium
40
puzzle and credit spread puzzle jointly from asset pricing models, such as Camp
bell and Cochrane (1999) and Bansal and Yaron (2004) . While they find that the
larger time-varying risk premia generate credit spreads larger than those generated
by existing structural models, they also find that their approach generates procyclical
default probabilities without the imposition of a counter-cyclical default boundary.
In contrast, because our model separates the pricing of corporate bonds from the
default condition of the firm, we can generate counter-cyclical default probabilities
observed in reality. Our model can allow for alternative preference specification that
can generate larger credit spreads, while maintaining procyclical default probabilities.
The connection between credit spreads and macroeconomic conditions is of great
interest beyond the need to explain the empirical properties of credit spreads. While
the finance literature focuses on asset pricing issues such as the size and volatility
of credit spreads, the macroeconomics literature has focused on how the availability
of credit affects the transmission of monetary policy and its impact on the macroe-
conomy. Bernanke, Gertler, and Gilchrist (2000) model a framework in which en
dogenous developments in the credit markets amplify and propagate shocks in the
macroeconomy, including monetary policy. Stiglitz and Greenwald (2003) suggest,
in their book, that the entire purpose of monetary policy is to control the supply of
credit and, therefore, liquidity in the economy, not the money supply as is conven
tionally believed. Regardless of the interpretation of the purpose of monetary policy,
this paper provides a framework in which to study the relationship between monetary
policy and credit.
The remainder of this paper is as follows. Section 2 explores a simple, intuitive
mechanism of how credit spreads are related to macroeconomic conditions. Section 3
describes the theoretical setup of the full model. Section 4 describes the calibration
coefficients for the macroeconomic sub-model, the calibration approach for the credit
part of the model, and the resulting simulation. Section 5 describes the results gen
erated by simulation of the model and compare with empirical observations. Section
41
6 describes some comparative statics of the model results and credit spread impulse
responses generated by the model. Section 7 concludes the papers and discusses areas
of future research.
2.2 An Intuitive Framework
Bhamra, Kuehn, and Strebulaev (2007) observe that the price of a one-period risky
bond B can be written as the payoff P of the bond divided by the risk-free rate
r plus the credit spread s as in equation 2.1. Alternatively, in a contingent claims
setting, the price of the bond B is the price of a risk-free bond — times the Arrow-
Debreu probability of survival qp plus the recovery value of the bond A times the
Arrow-Debreu price of default as in equation 2.2.
£ = — (2.1)
B = -(l-qD) + qDA (2.2) r
Setting the two bond pricing equations equal to each other, they find that credit
spread on default risky debt can be written as
where r is the risk-free rate, I is the loss ratio of the bond which gives the proportional
value loss if default occurs, and q^ is the Arrow-Debreu security price, which pays
out 1 unit of consumption upon default. A more detailed derivation of this result is
presented in Appendix A. Furthermore, they derive that the Arrow-Debreu security
price can be further decomposed into three factors
qD = TKpD (2.4)
42
where pr> is the actual default probability, T is the discounting for the time value of
money, and 1Z is an adjustment for risk.
In the context of our model, we can relate each of the terms in the credit spread
equation to output growth and inflation, which define our macroeconomy. Via the
Taylor rule, output growth and inflation directly impact the risk-free rate, as the mon
etary policy agency sets risk-free rates as a linear function of the two macroeconomic
conditions. Again, via the Taylor rule, output growth and inflation impact T, the
time adjustment, because future discounting is performed through the term structure
of interest rates, which, in turn, is affected by the Taylor rule through no-arbitrage
relationships.
While both the risk-free rate and the discount factor positively relate output
growth and inflation to credit spreads in a linear context, higher output growth also
lower the loss given default and the probability of default in the context of our model.
2.3 The Model
The model we present here consists of two parts: the macroeconomy and individual
firms. The macroeconomic model we employ is a standard new Keynesian macroe
conomic model with pricing rigidities. Households optimize their consumption and
labor decisions, subject to a period-by-period budget constraint. Industries exhibit
monopolistic competition in the goods that they produce. Furthermore, they exhibit
Calvo (1983) type pricing rigidities that allow them to optimize their product prices
only at random times. Industries, when they are allowed to, optimize prices subject
to meeting all demand. This pricing rigidity induces inflation with effects on real
variables into the model. Finally, a monetary policy agency sets nominal one-period
rates, according to a contemporaneous Taylor rule, which makes both discount rates
depend on economic output growth and inflation.
The resulting output growth, inflation, and risk-free term structure that evolve
from the macroeconomic model are then inputs into pricing risky corporate debt. We
43
assume that the optimizing productive agents in the macroeconomic model summa
rized above are industries that set product prices and employee wages. In our model,
the marginal firm is a measure-zero productive unit that takes the price of goods and
wages as given by the industry to which they belong, rather than as the output of an
optimization. In this construct, the default of the firm does not impact the industries'
ability to meet aggregate consumption. Output growth affects the cash flow of the
marginal firm, while the current term structure of interest rates includes information
about future output growth and inflation risks. The marginal firm uses its cash flow
to meet its coupon obligations, and the price of a risky bond is the sum of discounted
coupon payments plus either the principal or fraction of the principal recovered upon
default, discounted appropriately. We then can calculate credit spreads from the
difference between the yields of risky bonds and their risk-free counterparts.
2.3.1 The Macroeconomy
To model the macroeconomy explicitly, we employ a standard new Keynesian macroe
conomic model that includes a monetary policy rule and nominal price rigidities to
generate inflation and monetary policy with real effects. The macroeconomic litera
ture has many papers on this type of model, including Woodford (2003), Rotemberg
and Woodford (1997), Lubik and Schorfheide (2004) , Ravenna and Seppala (2006)
, and others. This type of model is standard in the macroeconomics literature and
allows us to link real features in the model and in empirical studies with credit spreads.
Households
The economy consists of a continuum of infinitely lived households, indexed by
j E [0,1]. Consumers demand differentiated consumption goods, choosing from a
continuum of goods, indexed by z E [0,1]. Therefore, C\{z) indicates consumption
from household j at time t of the good produced by firm z. As we shall explain
further, an economy with differentiated goods allows optimizing producers to set the
44
price of these goods differently and at different time, thereby causing inflation with
real effects. Households' preferences over the basket of differentiated goods are defined
by the aggregator:
d = Jo
dz ,9> 1 (2.5)
where 9 is the elasticity of demand.
Household j chooses (C^+i, N(+i, B{+j) where Nt denotes the labor supply, and
Bt are bond holdings to maximize a power utility function with disutility of labor.
Ut = Et Y,(3i{u(Cl+i,Dt)-v(N^t+i)) i-Q
= Et Y,?[ (c?+i)1~7A i=0 1 - 7
t i v j , t+i
1+V
In addition to choosing period-by-period consumption C°t+i) the agent also chooses
hours of labor N^+i in a particular industry for a particular good. His hours con
tributed to labor detract from his overall utility.
Household j maximizes the above utility subject to the aggregator in (2.5) and
the budget constraint
C{(z)Pt(z)dz = WtN> + E> - pt(B> - BU) (2.6)
where Wt is the nominal wage rate, n t is the share of the agent's profit from firms, pt
is a vector of asset prices, and Bt is the agent's holding of the corresponding assets
at time t. Therefore, Pt(Bj — B3t_x) represents the nominal value of household j ' s net
asset holdings.
Since asset markets are assumed to be complete, we do not need to designate
individual assets, but we can assume that every agent holds the complete market
through shares of a index fund that includes all assets in the market, including risk-
free bonds and firm shares. As stated in Woodford (2003), the above specification
need not refer to the quantity held of some specific asset. Since we assume complete
markets, households must be able to hold a wide selection of instruments with state-
45
contingent returns. Furthermore, there exists a set of spanning assets with returns
that can span any possible state-contingent returns. Therefore, we do not need to
refer to any particular asset, but know the household can achieve any state-contingent
return it wants.
The choice of capital structure, the composition of a firm's market securities, is de
termined by the individual firms' initial capital structure and not by market demand.
Consuming agents simply invest in the market portfolio and make no individual se
curity selection decisions.
The household j determines its demand for individual good z among the differen
tiated goods with the following condition:
ci(z) = Pt{z)
CI (2.7)
where Pt is the associated price index measuring the minimum expenditure on differ
entiated goods that will buy a unit of the consumption index
Pt.= Pt{zf-edz (2-
Since all households solve an identical optimization problem and face the same
aggregate variables, we can omit the index j in the above optimization problem. The
other optimal conditions for the individual are
MUCt = ^ - = Et A d
(2.9)
~P~t ~ MUCt (2.10)
Et P^Rt l MUCt
where MUCt is the marginal utility of consumption.
(2.11)
46
Industry Price Sett ing and Symmetr ic Equil ibrium Solution
Different industries produce differentiated goods, indexed by z, and optimize linear
production by controlling labor choosing the wage. The production function of the
industry producing good z is Yt(z) — AtNt(z) where Nt(z) is the labor allocated to
the production of differentiated good z and At is an aggregate productivity shock.
Each industry maximizes its profit function
Ut(z) = Pt(z)Yt(z) - Wt(z)Nt(z)
Every industry has a fixed capital stock. Furthermore, each industry hires as much
labor as is necessary in each period and the labor stock is common to all industries.
Industries take the wage demanded as given as an input and hire as much labor as
necessary to optimize the production function.
Industries have monopolistic competition in their particular good, but can only
set prices at particular periods in time. In each period, an industry will be able to
adjust its price with constant probability (1 — 9p), regardless of past history.
The problem of the industry setting the price at time t consists of choosing Pt(z)
to maximize their expected discounted stream of profits, as follows:
Et £0W Mua t+i
j = 0 MUC*
Pt(z)
Pt Yt t,t+i\
t+i
MC™Y (z) —5 rt,t+i{z)
subject to
Yt,t+i(z) = Pt(z)
Pt t+i Y, t+i
where Ytjt+i(z) is the industry's demand for its output at time t + i condition on the
prices set at time t, Pt(z). This optimization constraint is given by market clearance,
the supply of a good equal to its demand (Yt = Ct).
Solving the model with a symmetric equilibrium implies that C\ = Ct and MUC\ —
MUCt- Given that all industries are able to purchase the same labor service bundle
47
and are charged the same aggregate wage, they face the same marginal cost. The lin
ear production technology ensures that the marginal cost is equal across industries,
whether or not they update prices, regardless of the level of production.
Industries are heterogeneous, because only a fraction (1— 6P) of firms can optimally
choose the price charged at time t. In equilibrium, each producer that chooses a new
price Pt(z) in period t will choose the same new price Pt(z) and the same level of
output. The dynamics of the consumption-based price index will follow
Pt = [ < V t i ' + (i - epWz)1-*] ^ (2.12)
We assume that, within each industry, there exists a continuum of firms that take
prices and wages as set by the industry and hire a measure-zero portion of the labor re
quired by the total industry. Since firms are measure zero in their particular industry
and the entire economy, an individual firm's behavior does not impact the aggregate
economy. By abstracting firm behavior from the aggregate behavior of the indus
try, we isolate the firm's default and, as a result, the price of its default-risky bonds
without the firm default affect the industry's or economy's output. Furthermore, a
default is an unhedgeable event and, as a result, the solution to the equilibrium in the
macroeconomy becomes much more complex as financial markets are now incomplete.
In this model, we deviate from the standard specification of traditional micro-
founded macroeconomic models that assume that firms are optimizing agents that
decide product prices and wages. Instead, we assume that infinitessimally small,
passive firms comprise industries that perform the actual optimizing decision over
prices and wages. We are only interested in the impact of macroeconomic conditions
on credit spreads as a measure of credit risk. Therefore, we abstract default able
firms from the industries that help set the market equilibrium. A firm produces an
infinitesimally small portion of the output for its industry and the economy. If it
should default, another firm in the industry can replace its productive capacity and
thereby not affect the equilibrium solution. Furthermore, since the firm does not
48
affect the equilibrium, we can specify its cash flow behavior arbitrarily as the rest of
the firms in the industry will close the economy to produce the equilibrium solution.
Monetary Policy
The monetary policy authority follows a Taylor rule, as follows:
'l + Rt,t+l\ , / l+7Tt \ , , / Yt loS ( -, , n .<? ) = u* loS I i , ) + ^y l o§ y l + Rss J - b V l + W " \Yss
Furthermore, we assume the central bank assigns positive weight to an interest
rate smooth objective so that the domestic short-term interest rate at time t is set
according to
(1 + Rttl) = [(1 + i W ] 1 _ X [ l + fl*-i,i]x (2-13)
The nominal discount rate changes based on the deviations of current output
and inflation from steady-state trend. The Taylor rule and no-arbitrage that applies
through the stochastic discount factor link macroeconomic conditions, output and
inflation, to asset prices. The consumer Euler equation, 2.11, provides the connection
between the monetary policy agency rate-setting rule and the consumer's investment
decisions. Other rates of return, riskless or risky, are tied to monetary policy via the
pricing kernel.
While this is the standard approach used in the no-arbitrage term structure lit
erature, I apply this approach to the evaluation of credit risk. This model does not
include all the possible factors for credit spreads, but suggests how macroeconomic
conditions in particular affect credit spreads.
Reduced-Form Macro Model
Several papers in the macroeconomics literature, including Woodford (2003) and
Goodfried and King (1997), solve new Keynesian macroeconomic model with a first-
order log-linear approximation, resulting in a linear state space model of the following
49
form:
AEt Vt+i
h+i = B
Vt
h + Cet (2.14)
where y is a vector of non pre-determined variables and k is a vector of pre-determined
variables. We can further decompose the state-space representation into
yt = Dkt + Fet
kt+i = Gkt + Het
This representation arises from linearizing the first-order optimality conditions of
the consumer and firm problems described above. In the case of the model above,
the vector y contains [Y it R] where Y is output, IT is inflation, R is the one-period
nominal interest rate, and x represent the log deviation from steady state of the
variable x.
The resulting state-space model can generate a time series of real output growth,
inflation, and real interest rates that are structurally related to each other via the
dynamics suggested by the above macroeconomic model. The real and nominal term
structures are related to all three variables by the Taylor rule and the no-arbitrage
imposition of the stochastic discount factor. If the systematic portion of a firm's
real cash flow growth, then the price of a firm's risky bonds and the resulting credit
spreads are also functions of output growth, as well as the interest rate and inflation
via the term structure used to discount the firm's cash flow.
We solve the New Keynesian macroeconomic model describes above using Dynare++,
a C+-(-/Matlab package that takes the first-order conditions of the model and per
forms simulations to solve the model.2
2 We thank Juha Seppala for his code, which was used as a template to solve the macroeconomic model using Dynare++ as described in this paper.
50
2.3.2 Risk-Free and Risky Yields
Risk-Free Term Structure
In this section, we use the results of the macroeconomic model to calculate prices and
yields for the real risk-free term structure.
Using the marginal utility of consumption described earlier, the real stochastic
discount factor is given by re-arranging the Euler condition.
_ MUCt+l qt+1 ~ P^fucT
The price of an n-period zero-coupon real bond is
Pn,t = Et
The yields for real bonds are,
n &+j b'=i
= -Etfe+lPn-l.t+l]
1
rn,t = --log(pbnit)
The Firm and Its Risky Bonds
Now that we have characterized the entire economy including the behavior of house
holds and industries, we now focus on the specifics of firm behavior necessary to
generate risky bond prices. As we assumed before, firms are passive and receive
prices and wages as given from the optimal choices of their respective industry. The
marginal firm has an initial capital structure, consisting of some amount of financing
coming from the issue of a risky bond.
Furthermore, we assume that the marginal firm has real cash flow that is a measure
zero portion of the aggregate output and whose growth is given by
gK{t) = Q9t + & + P<?K€? + ( W l - A f (2-15)
51
where gt represents the growth in aggregate output, gx(t) is the growth of the
marginal firm's cash flow, g is the sensitivity of the firm's cash flow growth to aggre
gate output growth, £t is the mean of firm-specific cash flow growth, p is the correla
tion of output growth and firm-specific cash flow growth, <TK is volatility of firm cash
flow growth, and ef, ef ~ N(0,1) independent of each other. The sensitivity of firm
growth to economic growth,
g = cov(gt,g?)/var(gt) = p —
can be thought of as the cash flow beta in Campbell and Vuolteenaho (2004). The
variable oc represents the unconditional volatility of consumption growth.
We base this approach on arguments presented in Longstaff and Piazzesi (2004)
and Tang and Yan (2006) . Longstaff and Piazzesi (2004) suggest that modeling
corporate cash flows, which are more volatile than aggregate output, may resolve the
equity premium puzzle. Cash flows vary with aggregate consumption, but are more
volatile. They, therefore, believe that corporate cash flows are the primitive process
that should be modeled to resolve puzzles in asset pricing. We base our expression for
firm-level cash flow growth on Tang and Yan (2006), who assume that firm-level cash
flow growth has economy-wide and firm-specific components. This assumed form for
cash flow growth allows for a direct correlation with aggregate output while allowing
for a higher degree of volatility.
Similar to Bakshi and Chen (1997), we assume that the rest of the industry and
economy follow a process that allows the sum of the individual firm's cash flow and the
output of the rest of the industry and the economy sum to total aggregate industry
and economy-wide output. Therefore, the assumption of the form of the marginal
firm's cash flow process does not change the path of aggregate output and does not
contradict the equilibrium given by the New Keynesian macro model described earlier.
Our model abstracts away from the firm's capital structure decisions. Implicitly,
therefore, we assume that the Modigliani-Miller theorem holds such that a firm's
52
financing decisions do not impact its value and, therefore, its cash flow available
to stock and bond investors. We assume this approach to identify clearly the rela
tionships between macroeconomic conditions and credit risk without introducing the
complexities of optimal capital structure issues, addressed in other papers such as
Hackbarth, Miao, and Morellec (2006) and Chen (2007). In fact, these papers take
the opposite approach and consider abstract specifications of the macroeconomy to
analyze the impact of optimal capital structure decisions on a firm's credit risk. Like
these models, we could introduce the complexity of optimal capital structure decisions
into our model and show how changes in capital structure affect the firm's cash flow
and credit risk.
Furthermore, the cash flow specification we have chosen above is not limited to
just firms, but can also explain the behavior of any agent in the economy. With
this approach, we hope to describe the sensitivity of any kind of credit risk, not just
corporate credit risk, to macroeconomic conditions, although most of the literature
and empirical observations made by us and others in literature usually pertain to
corporate credit.
Once we define the firm's cash flow, we can also value any contingent claim written
on the cash flow. If the marginal firm issues a bond with face value F and coupon c,
then, in each period, the firm's real payout to the defaultable bond is
kB(t) = ci(t < T)i(t <T) + FS(t - T)i(t <T)+ u(ih)F5(t - r)i(t < T) (2.16)
where r = inf(£ : K(t) < c) is the first passage time of default, l is the indicator
function, and S(t — r ) is the Kronecker delta.
The value us represents the percentage of the face value of the debt that can be
reclaimed upon default. The recovery rate also varies positively with the economy in
the following fashion:
ut = w(gt) =a + bgt
53
We choose to vary the recovery rate with aggregate economic output growth,
due to a number of empirical observations about recovery after default. Altman
and Kishore (1996) show that recovery rates are time-varying, while Collin-Dufresne,
Goldstein, and Martin (2001) suggest that "even if the probability of default remains
constant for a firm, changes in credit spreads can occur due to changes in the expected
recovery rate. The expected recovery rate should be a function of the overall business
climate." Shleifer and Vishny (1992) find that a firm's liquidation value is lower when
its competitors are experiencing cash flow problems. Other empirical papers in the
literature with similar conclusions include Thorburn (2000), Gupton and Stein (2002),
Altman et al. (2002) and Acharya et al (2003). Acharya et al. (2003) also show
that default risk and the recovery rate are only weakly linked by the same factors.
We then assume that they are essentially independent, which makes our simulation
easier. Furthermore, since the actual empirical correlation is slightly positive, the
independence assumption decreases expected loss and biases our model-generated
credit spreads downward slightly.
The value of the risky debt at t = 0 is
The rates r(0, t) represents the real term structure generated by the macro model.
To adjust the firm's cash flow growth to the risk neutral measure, which simplifies
bond pricing, we must reduce the drift of the firm cash flow growth process by its
market price of risk. Since the market price of risk for an economy is simply the
product of relative risk aversion and the volatility of consumption growth, the above
macroeconomy produces a constant market price of risk, since power utility admits
a constant relative risk aversion 7 and the first-order approximation we use to solve
the model admits only a constant volatility of consumption growth equal to the
unconditional volatility of consumption growth of the sample. Therefore, under the
54
risk-neutral measure Q, the cash flow dynamics are
9x{t) = ggt + & - 'yp(rC(7K + pcr^ef + cr^V1 - A f (2.17)
where uc? is the volatility of consumption growth. We explore the issue of calculating
the market price of risk to adjust to represent the firm's cash flow process under the
Q measure in Appendix B.
The risk-free term structure results from the macroeconomic model; therefore we
must adjust the firm's cash flow growth to the risk-neutral measure.
The yield-to-maturity of the risky bond is the solution Y to the equation
DV = ± + (F-±)(I-Y)T
The yield-to-maturity R of a risk-free bond with the same payment structure
The resulting credit yield spread is, therefore, just Y — R. The assumption of this
form for the bond payout and recovery rate allow for current economic conditions as
represented by current output growth and inflation to impact the firm's cash flow and
how its bond's coupon payments are discounted.
2.4 Model Calibration and Simulation
2.4.1 The Macroeconomic Model
To calibrate the new Keynesian macroeconomy in the first part of the model, we
choose calibrated parameters and the estimation technique found in Ravenna and
Seppala (2006). Using parameters chosen from related papers in the monetary policy
literature, they estimate the standard new Keynesian monetary policy model using
55
Dynare++ to match historical correlations of output and inflation as observed from
1952 to present. Table 1 lists the parameters we employ for the macroeconomic
model.3
The preference and technology exogenous shocks follow an AR(1) process:
logZt = (1 - Pz)logZ + pzlogZt^ + ef, ef ~ N(Q, a\) (2.18)
where Z is the steady state value of the variable
This calibration produces correlations between model output, inflation, and real
interest rates that match historical empirical correlations. This match between model
and empirical correlations between macroceonomic conditions and risk-free and risky
yields is one of the primary goals of this paper and differentiates this model from
others in the credit literature that specify output growth and inflation processes
exogenously, not structurally. Table 2.2 compares the model's second moments and
correlations with output with U.S. post-war sample data. The model matches the
3 The literature on parametrizing New Keynesian models to match historical data is extensive. These include Ravenna and Seppala (2006), Ravenna (2006), Christiano, Eichenbaum, and Evans (2005), Ireland (2001), Woodford (2003), and Rabanal and Rubio-Ramirez (2005)
Symbol
7 £ P 9P
it 9
UJy
U*
X Pa
Pd
<?a 4
Description Relative risk aversion
Inverse of labor supply elasticity Discount factor
Price stickiness probability Steady state inflation (quarterly)
Demand elasticity Taylor coefficient (output)
Taylor coefficient (inflation) Monetary policy smoothing parameter Autocorrelation of technology shock Autocorrelation of production shock
Volatility of technology shock Volatility of production shock
Value 2.5 0.5 0.99 0.75 0.75 11
0.01 1.5 0
0.9 0.95
0.0035 0.008
Table 2 .1 : Macroeconomic model coefficients based on Ravenna and Seppala (2006), calibrated to historical correlations of output, inflation, and interest rates
56
volatility and correlations of output and inflation, but increases the volatility and
output correlation of interest rates. We can improve the fit of the model to historical
data by including a number of modifications, including habit persistence, higher-order
approximation, and sticky wages. The model, however, provides a basic framework
linking macroeconomic conditions jointly and allowing us to test their behavior versus
model-generated credit spreads. Table 2.3 shows means, standard deviations, and
correlations for output growth, inflation, and real interest rates.
2.4.2 Calibrating and Simulating Credit Spreads
Once we estimate the macroeconomic model, we simulate the firm's cash flow process
with equation 2.15. We follow Longstaff and Piazzesi (2004) to choose parameters for
the firm's cash flow process. Longstaff and Piazzesi (2004) note that corporate cash
flows are highly volatile, strongly procyclical, and could answer part of the equity
premium puzzle by generating higher volatility for corporate cash flow that underlie
firm-issued securities. They find that the correlation of firm's cash flow with total
output growth to be around p ~ 0.6 — 0.7, while firm cash flow growth has volatility
around 20%.
For each quarter t, we assume that output growth, inflation, and the real term
structure for the given quarter are fixed and that bondholders use the fixed macroe
conomic conditions to forecast firm cash flows and price bonds. We choose 10,000
Variable Yt
7T*
Rt n
Standard Deviation Model 2.02 3.28 4.94 4.59
US Data 1.59 3.00 2.82 2.32
Correlation with Output Model
1.00 0.21 0.32 0.30
US Data 1.00 0.19 0.17 -0.13
Table 2.2: Selected variable volatilities and correlations, model vs. historical 1952-2006. Standard deviation measured in percent. Output series is logged and Hodrick-Prescott filtered. U.S. data: Yt
is real GDP, irt is annualized CPI inflation, Rt is 3-month nominal T-bill rate, and rt is the 3-month real rate in annual terms.
57
Variables 9t
r3
r 12 11 11
r120 11
Mean Std. Dev 0.01 0.89 2.97 3.28 2.75 4.59 3.26 3.90 3.83 2.13 4.00 0.34
Correlation coefficients 9t 7T* r* rf rf r\w
1.00 0.13 0.64 0.64 0.65 0.66 0.13 1.00 0.27 0.24 0.19 0.18 0.64 0.27 1.00 1.00 0.99 0.99 0.64 0.24 1.00 1.00 1.00 0.99 0.65 0.19 0.99 1.00 1.00 1.00 0.66 0.18 0.99 0.99 1.00 1.00
Table 2.3: Moments and correlations of output growth quarterly, inflation quarterly, and interest rates
paths for cash flow growth each quarter and calculate simulated firm cash flows and
payments to bondholders. The average of the sum of payments to bondholders dis
counted by the prevailing term structure gives us the bond price, from which we can
calculate risky yields and spreads.
After we generate a realistic set of time series for output growth, inflation, and
interest rates, we calibrate the model to generate credit spreads to match historical
default probabilities. Taking the parameters from above, we can fix the coupon of the
bond issued by the firm to a value, 5%, and calibrate inital cash flow K0 for each credit
rating category to match historical default probabilities in simulation. Huang and
Huang (2003) provide the following default probabilities for different credit ratings.
The calibration of initial cash flow to historical default probabilities to generate
different credit ratings categories uses the assumption that the rating agencies adopt
the "through the business cycle" approach to assign firms to a rating category. The
representative firm we model is assigned its rating based on its average probability of
Symbol
P £
&K
a b
Description Correlation of macro growth and firm c.f. growth
Mean of firm-specific cash flow growth Standard deviation of firm cash flow growth
Recovery rate constant Recovery rate amplifier
Value 0.6 0%
20% 51.31%
2.5
Table 2.4: Firm-specific cash flow parameters, based on Longstaff and Piazzesi (2004) and recovery rate in Huang and Huang (2003), assuming average of zero economic growth and recovery rate of 51.31% for all bond ratings and maturities
58
Rating
AAA AA
A BBB
BB B
Cumulative default prob. (%) 1 yr. 0.00 0.03 0.10 0.12 1.29 6.47
4 yrs. 0.04 0.23 0.35 1.24 8.51
23.32
10 yrs. 0.77 0.99 1.55 4.39
20.63 43.91
Recovery rate
(%) 51.31 51.31 51.31 51.31 51.31 51.31
Table 2.5: Moody's default probabilities and recovery rates (Huang and Huang, 2003)
default throughout the business cycle we model, rather than on the basis of current
conditions. This assumption allows us to vary the firm's default probability and
credit risk within the same rating category. Again, we consider our firm to be a
representative firm of a particular ratings category throughout the business cycle.
After generating the initial credit spread results based on the values for firm
specific growth mean and variance, we can then perform a sensitivity analysis of the
results by varying the firm-specific properties of cash flow growth. Finally, we conduct
a comparative static analysis to gauge the effects of firm specific characteristics on
credit spread levels and volatilities generated by the model and their relationship with
output growth and inflation.
2.5 Results
In this section, we discuss the properties of the credit spreads generated by the model
we describe above. First, we describe the empirical properties of the credit spread
time series generated by the model. We then discuss the contemporaneous relation
ship between credit spreads and model-generated macroeconomic conditions, namely
output growth and inflation.
59
2.5.1 Properties of Model-Generated Credit Spreads and De
fault Probabilities
Our model produces credit spreads and volatilities of the order of magnitude found
in the data and comparable with other structural credit models. Unfortunately, the
credit spread levels produced by the model are lower than those found in the data
and in structural models. The inability to produce the levels seen in data could be
due to the lack of large premia in the model. Chen, Collin-Dufresne, and Goldstein
(2005) and others have suggested that the equity risk premium puzzle and the credit
spreads level puzzle are linked and that an asset pricing model that captures one also
captures the other.
Furthermore, it has been well documented that asset pricing models based on
production economies have greater difficulty in explaining asset pricing anomalies
than exchange economies. The macroeconomic model that we employ does not have
elements that could solve the equity premium puzzle, such as a time-varying risk aver
sion parameter. Although the credit spreads generated by model depend upon firm
cash flows that are more volatile than consumption growth in the model, Longstaff
and Piazzesi (2004) suggest that this may not entirely resolve the equity premium
puzzle and, additionally, the credit spreads level puzzle.
In constrast with the credit spread level results, the model generates credit spread
volatilities closer to empirical values than other structural credit models. Chen,
Collin-Dufresne, and Goldstein (2005) state structural credit models generate credit
spreads whose volatilities are too low relative to actual data. Most models gener
ate a volatility of the spread between BBB and AAA of around 35 bps per annum,
while the actual volatility is around 75 bps per annum for the entire history of the
Moody's corporate bond index and 56 bps for the post-war period. Our model gener
ates a BBB-AAA spread volatility of around 52 bps, closer than other models to the
full sample volatility and close to the post-war period value. In general, the model
generates volatilities close to the credit spread volatilities observed in the data.
60
Horizon and Maturity 4-year A
4-year BBB 4-year B
10-year A 10-year BBB
10-year B
Model 2.0 6.7
121.0 3.4 8.4
103.0
Historical 96.0
158.0 470.0 123.0 194.0 470.0
TY 13.2 56.6
240.5 48.1
109.6 318.5
LS 7.5
25.4 406.0
14.5 38.6
341.9
LT CDG 9.9
- 31.1 - 435.3
38.5 22.5 59.5 52.3
408.4 371.6
Table 2.6: Average credit spread levels, model generated vs. historical and literature. Historical data taken from Huang and Huang (2003). Comparable models including Tang and Yan (2006) (TY), Longstaff and Schwartz (1995) (LS), Leland and Toft (1996) (LT), and Collin-Dufresne and Goldstein (2005) (CDG)
Our simulation methodology, in addition to providing us with credit spreads, also
finds forward-looking default probabilities. Many previous credit risk models, partic
ularly those that link consumption growth with credit spreads, generate procyclical
default probabilities, an obviously counterfactual property. As shown in Table B.l,
we regress model-generated default probabilities on contemporaneous output growth
and inflation and find a strongly significant countercyclical relationship of default
probabilities and output growth, but no connection with inflation. Both regression co
efficients conform with empirical data that default probabilities rise as output growth
falls, but have inflation has little impact on actual defaults.
2.5.2 Macroeconomic Factors of Credit Spreads: Contempo
raneous Relationships
Given the model developed above, we expect inflation to be positively related to credit
spreads, as an increase in inflation should increase the discount rate, depressing bond
prices and increasing yield spreads. Furthermore, we expect higher output growth to
have a positive impact on credit spreads of AAA and AA and a negative effect on
credit spreads of A rating or below. For the higher rated bonds, increased output
growth should increase the discount rate applied to coupon payments, but should not
substantially decrease the probability of default for the bond. Therefore, the bond
price should be depressed and the credit spread should rise. For the lower rated
61
bonds, the increase in output growth should increase the firm's cash flow, helping
it meet cash flow obligations and reducing the probability of default. This, in turn,
should counteract the effect of an increase in the discount rate and should decrease
the credit spread.
The model produces results close to our intuition and what we observe empirically.
To evaluate the impact of output growth and inflation on credit spread level, we
perform regression on with the following specification:
st = a + f3sst_i + (3ggt + (3vixt (2.19)
As seen in Table B.2, inflation varies positively with credit spread levels, as (3n > 0
for all ratings and maturities. This result is consistent with our empirical findings in
our earlier research. Furthermore, it matches our intuition that higher inflation raises
the risk-free real rate demanded by investors, which correspondingly reduces demands
for risky securities or raise the yield required of risky debt securities to compensate
investors to purchase them.
The coefficient on output growth, however, does not exactly match what we find in
our previous empirical study. Output growth varies negatively with credit spreads of
all maturites and ratings in the model. However, while output growth has a positive
coefficient for higher-rated credit spreads in the empirical data, the model generates
higher-rated credit spreads that vary negatively with output growth, although not
significantly.
2.6 Impulse Response Functions and Comparative
Statics
In addition to replicating the empirical sensitivities of credit risk to macroeconomic
conditions, the model we present here also has several features that could be im-
62
portant for those interested in the behavior of credit markets, particularly monetary
policymakers. In this section, we present impulse response functions of the various
credit spread series of different ratings and maturities to shocks in the macroecon-
omy. Unlike standard credit risk models found in the literature, our ablity to connect
a New Keynesiam macroeconomic model with a model of credit risk allows us to
test the impact of shocks in the macroeconomy on credit spreads. Furthermore, we
also present comparative statics of the model, varying the calibrated parameters to
represent different macroeconomic and firm conditions, including different monetary
policy rules and firm characteristics.
2.6.1 Impulse Response Functions
The macroeconomic model employed in this paper has three sources of uncertainty:
technology, preference, and monetary policy shocks. Our proposed link between the
macroeconomic and credit risk models allows not only to test the sensitivity of credit
spreads to macroeconomic conditions, but also to determine the impact of macroe
conomic shocks on credit risk. We do not show our impulse responses of the credit
spreads in our model to preference shocks, as the results are not entirely clear and
are of too small a magnitude. Figures B.l and B.4 show the responses of macroeco
nomic conditions, namely consumption growth, inflation, and the short-term interest
rate, to an adverse technology shock and to a positive monetary policy shock of 1%,
respectively.
Constructing the impulse response to credit spreads is not as simple as construct
ing the impulse responses to macroeconomic variables, because bond pricing as a con
tingent claim is essentially non-linear. To construct credit spread impulse response,
we first find the impulse response of the individual macroeconomic series. We then
convert the impulse responses of the individual series, which are in log deviations,
back to actual time series of consumption growth, inflation, and interest rates, which
are then used to calculate risky bond spreads. We then calculate bond spreads for
63
each credit category of bond assuming that the macroeconomy is at the steady state
and subtract these spreads from the spreads given by the macroeconomic time series
derived from the impulse response functions.
As seen in figures B.2-B.4, credit spreads respond strongly to an adverse technol
ogy shock in the economy. If we interpret a negative technology shock as a drastic
loss of productivity of almost 3% per quarter, say during a time of war, the impulse
responses for all credit spreads increase dramatically in initial response and decline
over the next couple of years, but do not return to their steady-state levels.
Figures B.6-B.8 shows the response of credit spreads to a positive monetary policy
shock. A positive monetary policy shock reduces consumption growth and inflation,
but they more quickly return to their steady-state levels. A positive monetary policy
shock also increases in credit spreads, which also recover to their original levels within
12 quarters, if not earlier.
2.6.2 Correlation with Output Growth, p
The coefficient p and the resulting sensitivity g determine how connected firm cash
flow growth is with the aggregate macroeconomy. The correlation with the macroe
conomy plays an important role in determining credit spreads. The size of the cor
relation p can proxy for different industries, as well, based on elasticity of demand
faced by the industry. For example, a low correlation with output can correspond to
the utility industry, while a high correlation could correspond to the technology or
financial sector.
Tables B.3 and B.4 show the results of regressing current credit spreads on lagged
spreads and current macroeconomic conditions as described in equation 2.19. Not
surprisingly, the greater the correlation of firm cash flow growth with total output
growth, the larger and more significant the regression coefficients in the regression of
credit spreads on output growth and inflation. Furthermore, the explanatory power
of regression, as measured by the adjusted R2, also increases with increasing p.
64
2.6.3 Idiosyncratic Cash Flow Growth and Volatility, (£, OK)
The firm's cash flow growth process has two components: a systematic component
that varies with the macroeconomy and an idiosyncratic component that is firm-
specific. As we shall demonstrate, the firm's idiosyncratic component also affects
the impact of macroeconomic factors on its credit spread. As we stated earlier, we
assume that firms in the economy obey the Modigliani-Miller theorem, which implies
that their capital structure decisions do not impact the firm's value or cash flow
processes. The idiosyncratic components of the firm's cash flow (£, OK) could be used
in future research to link the firm's optimal capital structure decision to the firm's
cash flow process.
The mean of the firm's idiosyncratic cash flow growth affects its default probability,
as a firm with greater cash flow growth will not default as frequently. Table B.5
show the regression test results for credit spreads generated by firms with mean
idiosyncratic growth of -2% per annum, while table B.6 show results for firms with
idiosyncratic growth of 2% per annum. We notice that output growth and inflation
have greater explanatory power for firms with negative idiosyncratic growth. The
coefficient on output growth, /3g, is larger in magnitude for all ratings, while /3n is
larger for higher-rated credits and smaller for lower rated credits.
A firm with greater idiosyncratic cash flow volatility will have greater variability
of its cash flow and, correspondingly, will default with greater probability and have
credit spreads that are more sensitive to macroeconomic conditions. The impact of
idiosyncratic cash flow volatility on credit spreads and their sensitivity to macroeco
nomic conditions is clearly seen in Tables B.7 and B.8. The larger the idiosyncratic
volatility, the greater the explanatory power of macroeconomic variables and the
larger the magnitude and significance of the regression coefficients on output growth
and inflation.
65
2.6.4 Relative Risk Aversion, 7
Several recent papers in the term structure and credit risk literature have found
evidence of regime switching behavior in the time series of credit spreads. As we
mentioned earlier, Alessandrini (1999) finds varying sensitivity to macroeconomic
conditions during recessions and expansions. In our previous empirical study, we
showed that credit spreads exhibit regime switching with respect to macroeconomic
factors and that regime switches occur during specific, identifiable financial market
events, such as the failure of LTCM and the cessation of hostilities in the Iraq War in
the beginning of 2004. This evidence points to changes in risk aversion changing the
sensitivity of credit risk to macroeconomic conditions. Although the model we employ
in this paper only allows for a constant risk aversion, we re-ran the simulations with
different relative risk aversion parameters to illustrate the change in sensitivity.
As seen in tables B.9 and B.10, increasing risk aversion increases the sensitivity
of credit spreads to macroeconomic conditions. This provides us an impetus for
employing a model with changing risk aversion or curvature of the utility function that
generates time varying risk premia and changing sensitivity to the macroeconomy.
2.6.5 Alternative Monetary Policy, (x^y)
In addition to changing the firm characteristics or the preferences in the economy,
this model is also useful for testing the impact of changes in the monetary policy rule.
Variable Yt
TTi
Rt
n
Standard Deviation Model 2.01 3.32 5.05 4.79
US Data 1.59 3.00 2.82 2.32
Correlation with Output Model
1.00 0.13 0.08 -0.06
US Data 1.00 0.19 0.17 -0.13
Table 2.7: Selected variable volatilities and correlations, 1952-2006. Standard deviation measured in percent. Output series is logged and Hodrick-Prescott filtered. U.S. data: Yt is real GDP, nt is CPI inflation, Rt is 3-month nominal T-bill rate, and rt is the 3-month real rate in annual terms. Coefficient of relative risk aversion is 10
66
Variables gt (quarterly) 7rt (quarterly)
r 3
r 12
' t r120 11
Mean Std. Dev 0.00 0.85 2.66 3.32 2.32 4.79 3.56 2.51 3.89 0.77 3.95 0.46
Correlation coefficients gt 7rt rf r]2 rt
60 r\20
1.00 0.06 -0.10 -0.14 -0.09 -0.02 0.06 1.00 -0.95 -0.93 -0.73 -0.59
-0.10 -0.95 1.00 0.98 0.87 0.75 -0.14 -0.93 0.98 1.00 0.90 0.79 -0.09 -0.73 0.87 0.90 1.00 0.97 -0.02 -0.59 0.75 0.79 0.97 1.00
Table 2.8: Moments and correlations of output growth quarterly, inflation quarterly, and interest rates. Relative risk aversion is 10.
Variable Yt
n Rt rt
Standard Deviation Model 3.87 3.52 5.44 3.79
US Data 1.59 3.00 2.82 2.32
Correlation with Output Model
1.00 0.37 0.06 -0.27
US Data 1.00 0.19 0.17 -0.13
Table 2.9: Selected variable volatilities and correlations, 1952-2006. Standard deviation measured in percent. Output series is logged and Hodrick-Prescott filtered. U.S. data: Yt is real GDP, wt is CPI inflation, Rt is 3-month nominal T-bill rate, and rt is the 3-month real rate in annual terms. Coefficient of relative risk aversion is 25
Variables gt (quarterly) nt (quarterly)
r 3
r 12
-60 ' t
r120 11
Mean Std. Dev 0.00 0.78 2.64 3.52 3.91 3.79 3.96 2.47 4.00 0.41 4.02 0.25
Correlation coefficients gt TTt rf r\2 rf° r t
120
1.00 0.06 -0.10 -0.14 -0.09 -0.02 0.06 1.00 -0.95 -0.93 -0.73 -0.59
-0.10 -0.95 1.00 0.98 0.87 0.75 -0.14 -0.93 0.98 1.00 0.90 0.79 -0.09 -0.73 0.87 0.90 1.00 0.97 -0.02 -0.59 0.75 0.79 0.97 1.00
Table 2.10: Moments and correlations of output growth quarterly, inflation quarterly, and interest rates. Relative risk aversion is 25.
67
We believe this is the first model that allows a direct connection between policy rules
used by the monetary policy agency and credit risk, as measured by credit spreads.
While this is just a first pass attempt at such a model, it provides the natural building
block for policy analysis.
As shown in Tables B.ll and B.12, increasing the smoothing of the monetary
policy function or increasing the Taylor coefficient on output make the impact of
output growth on credit spreads stronger, while reducing the impact of inflation
changes on credit spreads.
2.7 Conclusion
In this paper, we explore a model that combines realistic macroeconomic dynamics
with credit risk. The model generates realistic dynamics of macroeconomic variables
using a simple New Keynesian model and empirically observed correlations of credit
spreads with macroeconomic risk, namely output growth and inflation, using Monte
Carlo simulations.
The first contribution of our paper is methodological. We show how to connect a
New Keynesian model that generates realistic dynamics for macroeconomic variables
with an asset pricing model. Previous papers on the relationship between macroeco
nomic and credit risk exogenously specify either the behavior of the macroeconomy,
such as Hackbarth, Miao, and Morellec (2006) and Chen (2007), or how macroeco-
Variable Yt
Kt
Rt rt
Standard Deviation Model 3.86 3.22 1.54 3.71
US Data 1.59 3.00 2.82 2.32
Correlation with Output Model
1.00 0.38 -0.08 -0.28
US Data 1.00 0.19 0.17 -0.13
Table 2.11: Selected variable volatilities and correlations, 1952-2006. Standard deviation measured in percent. Output series is logged and Hodrick-Prescott filtered. U.S. data: Yt is real GDP, irt is CPI inflation, Rt is 3-month nominal T-bill rate, and rt is the 3-month real rate in annual terms. Degree of monetary policy smoothing is 0.9.
68
Variables gt (quarterly) nt (quarterly)
r 3
r 12 ' t
' t „120 11
Mean Std. Dev 0.00 3.08 2.98 3.25 3.81 3.71 3.87 2.46 3.95 0.76 3.98 0.44
Correlation coefficients gt irt rf r\2 r®° rt
120
1.00 0.22 -0.20 -0.19 -0.11 -0.07 0.22 1.00 -0.97 -0.94 -0.71 -0.58
-0.20 -0.97 1.00 1.00 0.86 0.77 -0.19 -0.94 1.00 1.00 0.91 0.82 -0.11 -0.71 0.86 0.91 1.00 0.99 -0.07 -0.58 0.77 0.82 0.99 1.00
Table 2.12: Moments and correlations of output growth quarterly, inflation quarterly, and interest rates. Degree of monetary policy smoothing is 0.9.
Variable Yt
nt
Rt n
Standard Deviation Model 3.58 4.51 3.67 1.48
US Data 1.59 3.00 2.82 2.32
Correlation with Output Model
1.00 -0.52 -0.46 0.36
US Data 1.00 0.19 0.17 -0.13
Table 2.13: Selected variable volatilities and correlations, 1952-2006. Standard deviation measured in percent. Output series is logged and Hodrick-Prescott filtered. U.S. data: Yt is real GDP, nt is CPI inflation, Rt is 3-month nominal T-bill rate, and rt is the 3-month real rate in annual terms. Taylor coefficient of output is 0.1.
Variables gt (quarterly) TTt (quarterly)
r 3
r 12 11
rm 11
-120 11
Mean Std. Dev 0.00 2.76 3.11 4.59 3.91 1.48 3.93 0.86 3.97 0.48 3.99 0.31
Correlation coefficients gt 7Ti rf r\2 rf° r]20
1.00 -0.16 0.15 0.23 0.20 0.19 -0.16 1.00 -0.56 -0.82 -0.88 -0.90 0.15 -0.56 1.00 0.85 0.72 0.69 0.23 -0.82 0.85 1.00 0.97 0.96 0.20 -0.88 0.72 0.97 1.00 1.00 0.19 -0.90 0.69 0.96 1.00 1.00
Table 2.14: Moments and correlations of output growth quarterly, inflation quarterly, and interest rates. Taylor coefficient of output is 0.1.
69
nomic conditions impact credit risk, as in Bernanke, Gertler, and Gilchrist (1996).
We structurally specify both the macroeconomy and credit risk in our model in the
simplest fashion possible. To connect macroeconomic dynamics with credit risk, we
assume that industries, not firms, are the optimizing agents in the macroeconomy,
while firms are passive in their optimization behavior. This assumption allows us to
investigate the unidirectional impact of macroeconomic conditions on credit risk. This
approach also allows a method to analyze the impact of macroeconomic conditions
and different forms of monetary policy on credit risk.
Our key result is that credit spreads generated by our model exhibit a negative
relationship with output growth and a positive relationship with inflation that is
found in the data. Furthermore, we can generate credit spreads that display higher
volatility than what results from existing structual credit models and closer to what
is observed in data. The model-generated spreads also exhibit a negative relationship
with future model-generated output growth and inflation, which matches empirical
observations in other papers.
Our new methodology also allows us to study credit spreads in a new way, namely
with macroeconomic shocks and with alternatively macroeconomic specifications,
such as different monetary policy rules. This approach is valuable for policymak
ers who would like to study the impact of their policies on financial markets. If we
follow the arguments of Stiglitz and Greenwald in their book, monetary policy is a
method for the government to increase or decrease the amount of credit in the econ
omy, not just the money supply. The model we present here can be used to simulate
and verify their proposition.
We have quite a few avenues of research to improve upon the model and match
more empirical observations on credit spreads. First, the credit risk literature has
focused much effort on trying to explain the observed size of credit spreads, which
cannot be replicated by standard structural credit risk models that consider the im
pact of the capital structure of the firm. Some recent papers that employ " structural
70
equilibrium" models try to explain the level of credit spreads by incorporating varia
tion of default risk with the macroeconomy, as we do in this paper. However, we do
not incorporate any features in our economy that could explain the credit spread puz
zle. As shown by Chen, Collin-Dufresne, and Goldstein (2005), any model that can
explain the size of credit spreads should also be able to explain the equity premium
puzzle. We do not incorporate any such feature that could explain either puzzle,
such as time-varying risk premia. In a later paper, we intend to propose a model
with preferences defined by habit persistence, an approach which has been shown to
resolve the equity premium puzzle.
Chapter 3
Credit Spreads in a New
Keynesian Macro Model with
Habit Persistence
3.1 Introduction
In our previous research, we established an empirical link between macroeconomic
conditions, such as output growth and inflation, and credit risk, as measured by corpo
rate credit spreads. Credit spreads generally increase with increasing inflation, while
increasing output growth causes higher-rated credit spreads to widen and lower-rated
credit spreads to narrow. Furthermore, the magnitude of the effect of macroeconomic
conditions on credit spreads is higher in periods of uncertainty in financial markets,
such as during the crash of LTCM or during the lead-up to the second Gulf War.
In periods of relative financial calm, credit spreads, just like interest rates, strongly
behave like a random walk with little or no variation arising from macroeconomic
conditions.
Upon empirically establishing the properties of credit spreads and their link to
the macroeconomy, we built a model linking a new Keynesian macroeconomic model
71
72
with a model of credit risk. By assuming a unidirectional connection between the
macroeconomy and the growth of the marginal firm, we show that prices on contingent
claims on the firm's cash flow, such as corporate bonds, vary with output growth and
inflation. With this model, we show that corporate credit spreads vary positively
with inflation and negatively with output growth, even when controlling for lagged
credit spreads. Furthermore, this model generates a high volatility for credit spreads
closer to what is observed in the data than other credit risk models, although the
model does not generate a high enough level of credit spreads to match empirical
observations.
Our previous model also does not duplicate the regime shifting behavior that we
found empirically with the relationship between macroeconoic conditions and credit
spreads. We notice in our empirical work that the relationship between credit spreads
and macroeconomic conditions becomes more significant during periods of financial
uncertainty. To duplicate this empirical observation and generate credit spreads that
match empirically observed levels, we propose explicitly allowing modeling agents'
preferences in the macroeconomic model to be time-varying. To model preferences,
we use an internal habit persistence specification, as first described by Constantinides
(1990) , where agents' utility is based on how much current consumption differs from
consumption in the previous period.
Chen, Collin-Dufresne, and Goldstein (2005) and Bhamra, Kuehn, and Strebulaev
(2007) both postulate a connection between the equity premium puzzle, the inability
of existing asset pricing models to explain the level and volatility of historical equity
returns, and the credit spreads level puzzle , the inability of structural credit mod
els based on current asset pricing techniques to explain the historical level of credit
spreads. Chen, Collin-Dufresne, and Goldstein (2006) find that preference specifica
tions that resolve the equity premium puzzle, such as those presented in Campbell
and Cochrane (1999) and Bansal and Yaron (2004), also resolve the credit spread
puzzle, but generate pro-cyclical default probabilities. As shown in our previous
73
paper, by design, our model ensures counter-cyclical default probabilities as is empir
ically observed, while generating higher credit spreads with time-varying sensitivities
to the macroeconomy. For our preference specification, we chose an internal habit
persistence model, as it allows for time-varying elasticity of utility with respect to
consumption, while maintaining relative risk aversion at levels that are empirically
realistic.
The rest of this paper is organized as follows. Section 2 summarizes the model we
proposed in our previous paper with the addition of time-varying preference modeled
with internal habit persistence. Section 3 describes our calibration and simulation
methodology, Section 4 discusses our simulation results, and Section 5 provides our
discussion and conclusion.
3.2 The Model
In this section, we summarize the model we employ to link macroeconomic conditions
and credit spreads. Since, with the exception of the preference specification, the
model is the same as our previous paper, we relegate specific details of the model
specification to our earlier paper.
The model consists of two parts: the macroeconomy and individual firms. The
macroeconomic model we employ is a standard new Keynesian macroeconomic model
with pricing rigidities. Households optimize their consumption over which they have
time-varying preferences and labor decisions, subject to a period-by-period budget
constraint. Industries exhibit monopolistic competition in the goods that they pro
duce. Furthermore, they exhibit Calvo (1983) type pricing rigidities that allow them
to optimize their product prices only at random times. Industries, when they are
allowed to, optimize prices subject to meeting all demand. This pricing rigidity in
duces inflation with effects on real variables into the model. Finally, a monetary policy
agency sets nominal one-period rates, according to a contemporaneous Taylor rule,
which makes discount rate depend on both economic output growth and inflation.
74
The resulting output growth, inflation, and risk-free term structure that evolve
from the macroeconomic model are then inputs into pricing risky corporate debt. We
assume that the optimizing productive agents in the macroeconomic model summa
rized above are industries that set product prices and employee wages. In our model,
the marginal firm is a measure-zero productive unit that takes the price of goods and
wages as given by the industry to which they belong, rather than as the output of an
optimization. In this construct, the default of the firm does not impact the industries'
ability to meet aggregate consumption. Output growth affects the cash flow of the
marginal firm, while the current term structure of interest rates includes information
about future output growth and inflation risks. The marginal firm uses its cash flow
to meet its coupon obligations, and the price of a risky bond are discounted coupon
payment plus either the principal or fraction of the principal recovered upon default,
discounted appropriately. We then can calculate credit spreads from the difference of
the yields of risky bonds and their risk-free counterparts. Since households' prefer
ences vary with time, so does the pricing kernel and the sensitivity of credit spreads
to macroeconomic conditions.
3.2.1 The Macro economy
To model the macroeconomy explicitly, we employ a standard new Keynesian macroe
conomic model that includes a monetary policy rule and nominal price rigidities to
generate inflation and monetary policy with real effects. The macroeconomic litera
ture has many papers on this type of model, including Woodford (2003), Rotemberg
and Woodford (1997), Lubik and Schorfheide (2004) , Ravenna and Seppala (2006),
and others. This type of model is standard in the macroeconomics literature and al
lows us to link real features in the model and in empirical studies with credit spreads.
75
Households
The economy consists of a continuum of infinitely lived households, indexed by
j G [0,1]. Consumers demand differentiated consumption goods, choosing from a
continuum of goods, indexed by z G [0,1]. Therefore, C{{z) indicates consumption
from household j at time t of the good produced by firm z. Households' preferences
over the basket of differentiated goods are defined by the aggregator:
C\ = C\{zp-dz ,9> 1 (3.1;
where 9 is the elasticity of demand.
Household j chooses (CJ+i, N
3+i, Bj+i)i_Q where Nt denotes the labor supply, and
Bt are bond holdings to maximize an internal habit persistent utility function with
disutility of labor.
Ut = Et f2^nci+i,Dt)-v(^t+i)) i=0
Et
i=0
n t+i ^ci+i_x \ l - 7
-A 7 l+ri
In addition to choosing period-by-period consumption C3t+i, the agent also chooses
hours of labor N3+i in a particular industry for a particular good. His hours con
tributed to labor detract from his overall utility.
Household j maximizes the above utility subject to the aggregator in (3.1) and
the budget constraint
Jo C3
t(z)Pt(z)dz = WtNi + Ul -pt(Bi - BU) (3.2)
where Wt is the nominal wage rate, Tlt is the share of the agent's profit from firms, pt
is a vector of asset prices, and Bt is the agent's holding of the corresponding assets
at time t. Therefore, Pt{B3t — B3
t_^) represents the nominal value of household j ' s
net asset holdings.We assume that asset markets are complete and that agents invest
only in the market portfolio to hedge their consumption risk.
76
The household j determines its demand for individual good z among the differen
tiated goods with the following condition:
ci(z) = Pt(z) Pt
CI (3.3)
where Pt is the associated price index measuring the minimum expenditure on differ
entiated goods that will buy a unit of the consumption index
Pt f, Pt(zf-edz (3.4)
Since all households solve an identical optimization problem and face the same
aggregate variables, we can omit the index j in the above optimization problem. The
other optimal conditions for the individual are
MUCt dU
dCt
= Et ci (3.5)
Wt _ IW
~P ~ MUCt
(3.6)
E, 'p^^Rt = l MUCt
where MUCt is the marginal utility of consumption.
(3.7)
Industry Price Sett ing and Symmetr ic Equilibrium Solution
Different industries produce differentiated goods, indexed by z, and optimize linear
production by controlling labor choosing the wage. The production function of the
industry producing good z is Yt(z) = AtNt(z) where A^(^) is the labor allocated to
the production of differentiated good z and At is an aggregate productivity shock.
77
Each industry maximizes its profit function
Ut(z) = Pt{z)Yt(z) - Wt(z)Nt(z)
Every industry has a fixed capital stock. Furthermore, each industry hires as much
labor as is necessary in each period and the labor stock is common to all industries.
Industries take the wage demanded as given as an input and hire as much labor as
necessary to optimize the production function.
Industries have monopolistic competition in their particular good, but can only
set prices at particular periods in time. In each period, an industry will be able to
adjust its price with constant probability (1 — 9P), regardless of past history.
The problem of the industry setting the price at time t consists of choosing Pt(z)
to maximize their expected discounted stream of profits, as follows:
E, Y,w) MUG t+i
i=0 MUCt
Pt{z)y , MC?+i ~T> rt,t+i\z) ^
t L *t+i t+i Yt,t+i(z)
subject to
Yt,tu{z) = Pt(*)' R t+i
Yt+i
where Yt}t+i(z) is the industry's demand for its output at time t + i condition on the
prices set at time t, Pt(z). This optimization constraint is given by market clearance,
the supply of a good equal to its demand (Yt = Ct).
Solving the model with a symmetric equilibrium implies that C\ = Ct and MUC\ =
MUCt- Given that all industries are able to purchase the same labor service bundle
and are charged the same aggregate wage, they face the same marginal cost. The lin
ear production technology ensures that the marginal cost is equal across industries,
whether or not they update prices, regardless of the level of production.
Industries are heterogeneous, because only a fraction (1—9P) of firms can optimally
choose the price charged at time t. In equilibrium, each producer that chooses a new
78
price Pt(z) in period t will choose the same new price Pt(z) and the same level of
output. The dynamics of the consumption-based price index will follow
Pt = [OpPil? + (1 - Op)Pt(zf-e} ^ (3.8)
We assume that, within each industry, there exists a continuum of firms that take
prices and wages as set by the industry and hire a measure-zero portion of the labor re
quired by the total industry. Since firms are measure zero in their particular industry
and the entire economy, an individual firm's behavior does not impact the aggregate
economy. By abstracting firm behavior from the aggregate behavior of the indus
try, we isolate the firm's default and, as a result, the price of its default-risky bonds
without the firm default affect the industry's or economy's output. Furthermore, a
default is an unhedgeable event and, as a result, the solution to the equilibrium in the
macroeconomy becomes much more complex as financial markets are now incomplete.
In this model, we are only interested in the impact of macroeconomic conditions
on credit spreads as a measure of credit risk. Therefore, we abstract defaultable
firms from the industries that help set the market equilibrium. A firm produces an
infinitesimally small portion of the output for its industry and the economy. If it
should default, another firm in the industry can replace its productive capacity and
thereby not affect the equilibrium solution. Furthermore, since the firm does not
affect the equilibrium, we can specify its cash flow behavior independently as the
rest of the firms in the industry will close the economy to produce the equilibrium
solution.
Monetary Policy
The monetary policy authority follows a Taylor rule, as follows:
log (TT^1) = <"•log (ITS) + •+log Gl)
79
Furthermore, we assume the central bank assigns positive weight to an interest
rate smooth objective so that the domestic short-term interest rate at time t is set
according to
(1 + i?t;1) = [(1 + Rt,t+i}^x[l + Rt-i,i]x (3.9)
The nominal discount rate changes based on the deviations of current output
and inflation from steady-state trend. The Taylor rule and no-arbitrage that applies
through the stochastic discount factor link macroeconomic conditions, output and
inflation, to asset prices. The consumer Euler equation, 3.7, provides the connection
between the monetary policy agency rate-setting rule and the consumer's investment
decisions. Other rates of return, riskless or risky, are tied to monetary policy via the
pricing kernel.
While this is the standard approach used in the no-arbitrage term structure liter
ature, we apply this approach to the evaluation of credit risk. This model does not
include all the possible factors for credit spreads, but suggests how macroeconomic
conditions in particular affect credit spreads.
Reduced-Form Macro Model
Several papers in the macroeconomics literature, including Woodford (2003) and
Goodfried and King (1997), solve new Keynesian macroeconomic model with a first-
order log-linear approximation, resulting in a linear state space model of the following
form:
AEt Vt+i
. kt+1 .
= B Vt
_kt_ + Cet (3.10)
where y is a vector of non pre-determined variables and A; is a vector of pre-determined
variables. We can further decompose the state-space representation into
yt = Dkt + Fet
80
kt+1 = Gkt + Het
This representation arises from linearizing the first-order optimality conditions of
the consumer and firm problems described above. In the case of the model above,
the vector y contains [Y n R] where Y is output, n is inflation, R is the one-period
nominal interest rate, and x represent the log deviation from steady state of the
variable x.
The resulting state-space model can generate a time series of real output growth,
inflation, and real interest rates that are structurally related to each other via the
dynamics suggested by the above macroeconomic model. The real and nominal term
structures are related to all three variables by the Taylor rule and the no-arbitrage
imposition of the stochastic discount factor. If the systematic portion of a firm's real
cash flow growth is a function of economy-wide output growth, then the price of a
firm's risky bonds and the resulting credit spreads are also functions of output growth,
as well as the interest rate and inflation via the term structure used to discount the
firm's cash flow.
We solve the New Keynesian macroeconomic model describes above using Dynare-|—f-,
a C++/Matlab package that takes the first-order conditions of the model and per
forms simulations to solve the model.2
In addition to simplifying the solution of the macroeconomy, the loglinear ap
proximation we derive above is useful for obtaining the market price of risk when
simulating to calculate bond prices and yields. As stated in Lettau and Uhlig (2002),
if all the relevant random variables in an economy are lognormal and the pricing kernel
exhibits loglinear behavior, then the market price of risk has a closed-form solution
based on the preference specification. The loglinear approximation of the solution to
the macroeconomy allows us to make the above assumptions when we evaluate bond
yields. Boldrin, Christiano, and Fisher (1997) and Lettau (2003) use this approach
2 We thank Juha Seppala for his code, which was used as a template to solve the macroeconomic model using Dynare++ as described in this paper.
81
when studying the asset pricing implications of habit persistence for exchange and
real business cycle economies, respectively.
3.2.2 Risk-Free and Risky Yields
Risk-Free Term Structure
In this section, we use the results of the macroeconomic model to calculate prices and
yields for the real risk-free term structure.
Using the marginal utility of consumption described earlier, the real stochastic
discount factor is given by re-arranging the Euler condition.
_ „MUCt+l qt+1 ~ P~MUC7
The price of an n-period zero-coupon real bond is
The yields for real bonds are,
nq^ = Et[qt+iPb„-h t+u
rn,t = --fog(pbn,t)
The Firm and Its Risky Bonds
Now that we have characterized the entire economy including the behavior of house
holds and industries, we now focus on the specifics of firm behavior necessary to
generate risky bond prices. As we assumed before, firms are passive and receive
prices and wages as given from the optimal choices of their respective industry. The
marginal firm has an initial capital structure, consisting of some amount of financing
coming from the issue of a risky bond.
Furthermore, we assume that the marginal firm has real cash flow that is a measure
82
zero portion of the aggregate output and whose growth is given by
9K(t) = ggt + & + paKef + oK^J\ - p2ef
where gt represents the growth in aggregate output, gxit) is the growth of the
marginal firm's cash flow, g is the sensitivity of the firm's cash flow growth to aggre
gate output growth, £t is the mean of firm-specific cash flow growth, p is the correla
tion of output growth and firm-specific cash flow growth, &K ls volatility of firm cash
flow growth, and ef, ef ~ iV(0,1) independent of each other. The sensitivity of firm
growth to economic growth,
Q = cov(gt, g?)/var(gt) = p—
can be thought of as the cash flow beta in Campbell and Vuolteenaho (2004). The
variable oc represents the unconditional volatility of consumption growth.
Similar to Bakshi and Chen (1997) , we assume that the rest of the industry and
economy follow a process that allows the sum of the individual firm's cash flow and the
output of the rest of the industry and the economy sum to total aggregate industry
and economy-wide output. Therefore, the assumption of the form of the marginal
firm's cash flow process does not change the path of aggregate output and does not
contradict the equilibrium given by the New Keynesian macro model described earlier.
Our model abstracts away from the firm's capital structure decisions. Implicitly,
therefore, we assume that the Modigliani-Miller theorem holds such that a firm's
financing decisions do not impact its value and, therefore, its cash flow available
to stock and bond investors. We assume this approach to identify clearly the rela
tionships between macroeconomic conditions and credit risk without introducing the
complexities of optimal capital structure issues, addressed in other papers such as
Hackbarth, Miao, and Morellec (2006) and Chen (2007).
Furthermore, the cash flow specification we have chosen above is not limited to
83
just firms, but can also explain the behavior of any agent in the economy. With
this approach, we hope to describe the sensitivity of any kind of credit risk, not just
corporate credit risk, to macroeconomic conditions, although most of the literature
and empirical observations made by us and others in literature usually pertain to
corporate credit.
Once we define the firm's cash flow, we can also value any contingent claim written
on the cash flow. If the marginal firm issues a bond with face value F and coupon c,
then, in each period, the firm's real payout to the defaultable bond is
kB(t) = ci(t < T)i(t <T) + F5(t - T)i(t < r) + Lo(^t)F5{t - r ) i ( t < T) (3.11)
where r = inf(£ : K{t) < c) is the first passage time of default, 1 is the indicator
function, and S(t — T) is the Kronecker delta.
The value UJ represents the percentage of the face value of the debt that can be
reclaimed upon default. The recovery rate also varies positively with the economy in
the following fashion:
ut = u(gt) = a + bgt
We choose to vary the recovery rate with aggregate economic output growth,
due to a number of empirical observations about recovery after default. Altman
and Kishore (1996) show that recovery rates are time-varying, while Collin-Dufresne,
Goldstein, and Martin (2001) suggest that "even if the probability of default remains
constant for a firm, changes in credit spreads can occur due to changes in the expected
recovery rate. The expected recovery rate should be a function of the overall business
climate." Shleifer and Vishny (1992) find that a firm's liquidation value is lower when
its competitors are experiencing cash flow problems. Other empirical papers in the
literature with similar conclusions include Thorburn (2000), Gupton and Stein (2002),
Altman et al. (2002) and Acharya et al (2003). Acharya et al. (2003) also show
that default risk and the recovery rate are only weakly linked by the same factors.
84
We then assume that they are essentially independent, which makes our simulation
easier. Furthermore, since the actual empirical correlation is slightly positive, the
independence assumption decreases expected loss and biases our model-generated
credit spreads downward slightly.
The value of the risky debt at t = 0 is
DV = y r kB(t)
^(l+r(0,t))*\
The rates r(0,t) represents the real term structure generated by the macro model.
To adjust the firm's cash flow growth to the risk neutral measure, which simplifies
bond pricing, we must reduce the drift of the firm cash flow growth process by its
market price of risk. Under the risk-neutral measure Q, the cash flow dynamics are
gK(t) = Q9t + & - n7ecP°c°K + (XTK<$ + °KV1 - A f (3.12)
where oc is the volatility of consumption growth and r^hec reflects the elasticity of
the pricing kernel to innovations in consumption growth. Under power utility, this
elasticity is just the coefficient of risk aversion, which corresponds to 7. However,
under a utility function that does not exhibit time separability, a series of assumptions,
including our loglinear assumption as described earlier, makes the above expression
a first-order approximation to the market price of risk, which is sufficient for our
purposes. Furthermore, since our agents' prefences are dependent on the previous
period's consumption, the elasticity ?7™ec is time-varying, which, in turn, makes our
market price of risk also time-varying. This is the key effect that we obtain by using
an internal habit persistence specification for our utility function, as it allows us to
have time-varying sensitivities to macroeconomic conditions not found in our previous
model and it allows for the possibility of larger risk premia than allowed by a power
utility specification, thereby resulting in higher credit spreads.
The risk-free term structure results from the macroeconomic model; therefore we
85
must adjust the firm's cash flow growth to the risk-neutral measure.
The yield-to-maturity of the risky bond is the solution Y to the equation
OV = ^ + (F~P)(l~Yf
The yield-to-maturity R of a risk-free bond with the same payment structure is
given by
The resulting credit yield spread is, therefore, just Y — R. The assumption of this
form for the bond payout and recovery rate allow for current economic conditions as
represented by current output growth and inflation to impact the firm's cash flow and
how its bond's coupon payments are discounted.
3.3 Model Calibration and Simulation
3.3.1 The Macroeconomic Model
To calibrate the new Keynesian macroeconomy in the first part of the model, we
choose calibrated parameters and the estimation technique found in Ravenna and
Seppala (2006). Using parameters chosen from related papers in the monetary policy
literature, they estimate the standard new Keynesian monetary policy model using
Dynare++ to match historical correlations of output and inflation as observed from
1952 to present. Table 1 lists the parameters we employ for the macroeconomic
model.3
The preference and technology exogenous shocks follow an AR(1) process:
logZt = (1 - pz)logZ + pzlogZt^ + ef, ef ~ N(Q, a\) (3.13)
3 The literature on parametrizing New Keynesian models to match historical data is extensive. These include Ravenna and Seppala (2006), Ravenna (2006), Christiano, Eichenbaum, and Evans (2005), Ireland (2001), Woodford (2003), and Rabanal and Rubio-Ramirez (2005)
86
where 2 is the steady state value of the variable
This calibration produces correlations between model output, inflation, and real
interest rates that match historical empirical correlations. This match between model
and empirical correlations between macroceonomic conditions and risk-free and risky
yields is one of the primary goals of this paper and differentiates this model from
others in the credit literature that specify output growth and inflation processes
exogenously, not structurally. The habit perisistence coefficient, ip, is the standard
value used by Constantinides (1990) and other similar papers. Table 3.2 compares
the model's second moments and correlations with output with U.S. post-war sample
data. The model matches the volatility and correlations of output and inflation,
but increases the volatility and output correlation of interest rates. Table 3.3 shows
means, standard deviations, and correlations for output growth, inflation, and real
interest rates.
Symbol
7 1> £ P 8p
TX
e LVy
UJn
X Pa
Pd
%
4
Description Relative risk aversion
Habit persistence coefficient Inverse of labor supply elasticity
Discount factor Price stickiness probability
Steady state inflation (quarterly) Demand elasticity
Taylor coefficient (output) Taylor coefficient (inflation)
Monetary policy smoothing parameter Autocorrelation of technology shock Autocorrelation of production shock
Volatility of technology shock Volatility of production shock
Value 2.5 0.8 0.5
0.99 0.75 0.75 11
0.01 1.5 0
0.9 0.95
0.0035 0.008
Table 3.1: Macroeconomic model coefficients. Values based on Ravenna and Seppala (2006), calibrated to historical correlations of output, inflation, and interest rates
87
Variable Yt
nt
Rt n
Standard Deviation Model 2.01 3.32 1.52 3.74
US Data 1.59 3.00 2.82 2.32
Correlation with Output Model 1.00 0.13 0.08 -0.06
US Data 1.00 0.19 0.17 -0.13
Table 3.2: Selected variable volatilities and correlations, 1952-2006. Standard deviation measured in percent. Output series is logged and Hodrick-Prescott filtered. U.S. data: Yt is real GDP, irt
is annualized CPI inflation, Rt is 3-month nominal T-bill rate, and rt is the 3-month real rate in annual terms.
Variables gt (quarterly) nt (quarterly)
r 3
r 12
„60 ' t
-120 11
Mean Std. Dev 0.00 0.78 2.66 3.32 2.30 3.74 3.56 2.50 3.89 0.77 3.94 0.45
Correlation coefficients gt nt r\ r]2 rf° r]20
1.00 0.06 -0.10 -0.14 -0.09 -0.02 0.06 1.00 -0.94 -0.92 -0.73 -0.59
-0.10 -0.94 1.00 1.00 0.88 0.76 -0.14 -0.92 1.00 1.00 0.90 0.79 -0.09 -0.73 0.88 0.90 1.00 0.98 -0.02 -0.59 0.76 0.79 0.98 1.00
Table 3.3: Moments and correlations of output growth quarterly, inflation quarterly, and interest rates
3.3.2 Calibrating and Simulating Credit Spreads
Once we estimate the macroeconomic model, we simulate the firm's cash flow process
with equation 3.3. We follow Longstaff and Piazzesi (2004) to choose parameters for
the firm's cash flow process. Longstaff and Piazzesi (2004) note that corporate cash
flows are highly volatile, strongly procyclical, and could answer part of the equity
premium puzzle by generating higher volatility for corporate cash flow that underlie
firm-issued securities. They find that the correlation of firm's cash flow with total
output growth to be around p fa 0.6 — 0.7, while firm cash flow growth has volatility
around 20%.
For each quarter t, we assume that output growth, inflation, and the real term
structure for the given quarter are fixed and that bondholders use the fixed macroe
conomic conditions to forecast firm cash flows and price bonds. We choose 10,000
paths for cash flow growth each quarter and calculate simulated firm cash flows and
88
payments to bondholders. The average of the sum of payments to bondholders dis
counted by the prevailing term structure gives us the bond price, from which we can
calculate risky yields and spreads.
After we generate a realistic set of time series for output growth, inflation, and
interest rates, we calibrate the model to generate credit spreads to match historical
default probabilities. Taking the parameters from above, we can fix the coupon of the
bond issued by the firm to a value, 5%, and calibrate inital cash flow KQ for each credit
rating category to match historical default probabilities in simulation. Huang and
Huang (2003) provide the following default probabilities for different credit ratings.
The calibration of initial cash flow to historical default probabilities to generate
different credit ratings categories uses the assumption that the rating agencies adopt
the "through the business cycle" approach to assign firms to a rating category. The
representative firm we model is assigned its rating based on its average probability of
default throughout the business cycle we model, rather than on the basis of current
conditions. This assumption allows us to vary the firm's default probability and
credit risk within the same rating category. Again, we consider our firm to be a
representative firm of a particular ratings category throughout the business cycle.
After generating the initial credit spread results based on the values for firm
specific growth mean and variance, we can then perform a sensitivity analysis of the
results by varying the firm-specific properties of cash flow growth. Finally, we conduct
a comparative static analysis to gauge the effects of firm specific characteristics on
credit spread levels and volatilities generated by the model and their relationship with
Symbol
P £
OK
a b
Description Correlation of macro growth and firm c.f. growth
Mean of firm-specific cash flow growth Standard deviation of firm cash flow growth
Recovery rate constant Recovery rate amplifier
Value 0.6 0%
20% 51.31%
2.5
Table 3.4: Firm-specific cash flow parameters, based on Longstaff and Piazzesi (2004) and recovery rate in Huang and Huang (2003), assuming average of zero economic growth and recovery rate of 51.31% for all bond ratings and maturities
89
Rating
AAA AA
A BBB
BB B
Cumulative default prob. (%) 1 yr. 0.00 0.03 0.10 0.12 1.29 6.47
4 yrs. 0.04 0.23 0.35 1.24 8.51
23.32
10 yrs. 0.77 0.99 1.55 4.39
20.63 43.91
Recovery rate
(%) 51.31 51.31 51.31 51.31 51.31 51.31
Table 3.5: Moody's default probabilities and recovery rates (Huang and Huang, 2003)
output growth and inflation.
3.4 Results
In this section, we discuss the properties of the credit spreads generated by the model
we describe above. First, we describe the empirical properties of the credit spread
time series generated by the model. We then discuss the contemporaneous relation
ship between credit spreads and model-generated macroeconomic conditions, namely
output growth and inflation.
3.4.1 Properties of Model-Generated Credit Spreads and De
fault Probabilities
Our model produces credit spreads and volatilities of the order of magnitude found
in the data and comparable with other structural credit models. Unfortunately, the
credit spread levels produced by original model are lower than those found in the
data and in structural models. When internal habit persistence is the preference
specification of the model, we produce substantially higher credit spreads on par with
other structural credit models in the literature, but not as high the levels observed in
the data. The inability to produce the levels seen in data could be due to the reduced
90
consumption growth volatility that naturally occurs with higher elasticity of income
substitution that an internal habit persistent model generates. Chen, Collin-Dufresne,
and Goldstein (2005) and others have suggested that the equity risk premium puzzle
and the credit spreads level puzzle are linked and that an asset pricing model that
captures one also captures the other. Therefore, any model that does not produce
high elasticity of income substitution with low relative risk aversion for agents in the
economy cannot generate high enough equity or credit risk premia, because either the
volatility of consumption growth or the elasticity of the pricing kernel to consumption
growth innovations are not large enough to justify a large maximal Sharpe ratio.
In constrast with the credit spread level results, the model generates credit spread
volatilities closer to empirical values than other structural credit models. Chen,
Collin-Dufresne, and Goldstein (2005) state structural credit models generate credit
spreads whose volatilities are too low relative to actual data. Most models generate
a volatility of the spread between BBB and AAA of around 35 bps per annum, while
the actual volatility is around 75 bps per annum for the entire history of the Moody's
corporate bond index and 56 bps for the post-war period. Our model with power
utility generates a BBB-AAA spread volatility of around 52 bps and generates an
annual volatility for the BBB-AAA spread of 64 bps, close enough with the actual
observed values.
Our simulation methodology, in addition to providing us with credit spreads, also
finds forward-looking default probabilities. Many previous credit risk models, partic-
Horizon and Maturity 4-year A
4-year BBB 4-year B
10-year A 10-year BBB
10-year B
Power 2.0 6.7
121.0 3.4 8.4
103.0
Habit 25.2 86.5
142.0 44.7 75.6 217
Historical 96.0
158.0 470.0 123.0 194.0 470.0
TY 13.2 56.6
240.5 48.1
109.6 318.5
LS 7.5
25.4 406.0
14.5 38.6
341.9
LT CDG 9.9
- 31.1 - 435.3
38.5 22.5 59.5 52.3
408.4 371.6
Table 3.6: Average credit spread levels. Historical data taken from Huang and Huang (2003). Comparable models including Tang and Yan (2006) (TY), Longstaff and Schwartz (1995) (LS), Leland and Toft (1996) (LT), and Collin-Dufresne and Goldstein (2005) (CDG)
91
ularly those that link consumption growth with credit spreads, generate procyclical
default probabilities, an obviously counterfactual property. As shown in Table C.l,
we regress model-generated default probabilities on contemporaneous output growth
and inflation and find a strongly significant countercyclical relationship of default
probabilities and output growth, but no connection with inflation. Both regression co
efficients conform with empirical data that default probabilities rise as output growth
falls, but have inflation has little impact on actual defaults.
3.4.2 Macroeconomic Factors of Credit Spreads: Contempo
raneous Relationships
Given the results of our previous model, we expect inflation to be positively related
to credit spreads, as an increase in inflation should increase the discount rate, de
pressing bond prices and increasing yield spreads. Furthermore, we expect higher
output growth to have a positive impact on credit spreads of AAA and AA and a
negative effect on credit spreads of A rating or below. For the higher rated bonds,
increased output growth should increase the discount rate applied to coupon pay
ments, but should not substantially decrease the probability of default for the bond.
Therefore, the bond price should be depressed and the credit spread should rise. For
the lower rated bonds, the increase in output growth should increase the firm's cash
flow, helping it meet cash flow obligations and reducing the probability of default.
This, in turn, should counteract the effect of an increase in the discount rate and
should decrease the credit spread.
The model produces results close to our intuition and what is seen empirically. To
evaluate the impact of output growth and inflation on credit spread level, we perform
regression on with the following specification:
st = a + psst-i + Pggt + /3n7Tt (3.14)
92
As seen in Table C.2, inflation varies positively with credit spread levels, as /3n > 0.
This result is generally consistent with our empirical findings in our earlier research.
Furthermore, it matches our intuition that higher inflation raises the risk-free real rate
demanded by investors, which correspondingly reduces demands for risky securities
or raise the yield required of risky debt securities to compensate investors to purchase
them.
The coefficient on output growth, however, does not exactly match what we find in
our previous empirical study. Output growth varies negatively with credit spreads of
all maturites and ratings in the model. However, while output growth has a positive
coefficient for higher-rated credit spreads in the empirical data, the model generates
higher-rated credit spreads that vary negatively with output growth, although not
significantly or with large magnitude. This result is present in both the power utility
and habit persistence versions of our model.
3.4.3 Macro economic Factors of Credit Spreads: Different
Regimes
In addition to testing the magnitude and volatility of credit spreads generated by
the model as well the relationship with macroeconomic conditions, we also wanted to
produce credit spreads whose relationship with macroeconomic variables has different
sensitivities in different periods. In particular, we developed a model to replicate
the phenomenon that macroeconomic conditions primarily impact credit spreads in
periods of financial uncertainty. To acheive this objective, we chose a preference
specification with time variation, namely internal habit persistence.
With habit persistence, the elasticity of the pricing kernel with consumption
growth innovation is time-varying, in the range of 10 to 22. In contrast, under power
utility, the elasticity of the pricing kernel is equal to the relative risk aversion of 2.5.
To test if our model produces different regimes, we divided our time series of credit
spreads into regimes of high and low pricing kernel elasticity and performed the above
93
regression exclusively under the regimes.
As seen in Tables C.3 and C.4, credit spreads demonstrate little sensitivity and
cannot be explained by macroeconomic conditions under low elasticity regimes. Under
regimes of high pricing kernel elasticity, credit spreads are related to output growth
in a manner consistent with the whole sample. This is exactly the behavior exhibited
empirically in our credit spread data. Inflation impacts credit spreads positively and
significantly in the high elasticity regime only for BBB-B spreads, but seems not
to affect AAA-A rated spreads. Although we can replicate the qualitative signs of
the relationships between credit spreads and macroeconomic conditions, we cannot
exactly replicate the perisistence of credit spreads, particularly in the low elasticity
regime, where empirically the coefficients on lagged credit spreads are close to one.
3.5 Conclusion
In this paper, we present a variation of the macroeconomic-credit model that we pre
sented earlier with agents' utility specified by an internal habit specification. We chose
this specification for the utility function, because it is time-varying and generates a
time-varying elasticity of the pricing kernel to consumption innovations. The time-
varying elasticity allows us to replicate the changing sensitivities of credit spreads to
macroeconomic conditions and generates higher levels and volatilities of credit spreads
than our previous model. As stated in other research, the size of credit spreads and
the equity premium are related and, therefore, any model specification, such as habit
persistence, that helps to generate an empirically-plausible equity premium also can
generate realistic credit spreads.
The credit spreads generated by the model presented in this paper do conform
with the empirical properties observed in the data better than our previous model.
The model-generated credit spreads demonstrate greater sensitivity to macroeconomic
conditions during periods of lower elasticity of the pricing kernel to consumption, a
feature that we found in our empirical results. Furthermore, we generate larger credit
94
spreads on the order of those found in other structural credit models and the volatility
of the credit spreads we find are close to those observed empirically and much higher
than those generated by any other credit risk model.
While this model comes closer to matching empirical observations about credit
spreads and their relationship with macroeconomic variables, there are further av
enues we can explore. In this and our previous papers, we abstracted away from the
optimal capital structure decisions the firm can make to take advantage of macroeco
nomic conditions. Optimal capital structure decisions are a relatively new avenue in
the credit risk literature and maybe, we can incorporate such a feature in our model
of the macroeconomy with credit risk.
Bibliography
[1] V.V. Acharya, S.T. Bharath, and A. Srinivasan. Understanding the recovery
rates on defaulted securities. Working paper, LBS, Michigan, and Georgia, 2003.
[2] Fabio Alessandrini. Credit risk, interest rate risk, and the business cycle. The
Journal of Fixed Income, 9(2):2177-2207, Dec 1999.
[3] E. Altman, B. Brady, A. Resti, and A. Sironi. The link between default and
recovery rates: implications for credit risk models nad procyclicality. Working
paper, NYU, 2002.
[4] E. Altman and V.M. Kishore. Almost everything you wanted to know about
recoveries on defaulted bonds. Financial Analysts Journal, 526:57-64, 1996.
[5] Jushan Bai and Pierre Perron. Estimating and testing linear models with mul
tiple structural changes. Econometrica, 66:47-78, 1998.
[6] Jushan Bai and Pierre Perron. Computation and analysis of multiple structural
change models. Journal of Applied Econometrics, 18:1-22, 2003.
[7] G. Bakshi and Z. Chen. An alternative valuation model for contingent claims.
Journal of Financial Economics, 44:123-165, 1997.
[8] Ravi Bansal and Amir Yaron. Risks for the long run: A potential resolution of
asset pricing puzzles. The Journal of Finance, 59(4): 1481-1509, August 2004.
95
96
[9] Ben Bernanke, Mark Gertler, and Simon Gilchrist. The financial accelerator
and the flight to quality. The Review of Economics and Statstics, 78(1):1—15,
February 1996.
[10] Ben Bernanke, Mark Gertler, and Simon Gilchrist. The financial accelerator in
a quantitative business cycle framework. Handbook of Macroeconomics, pages
1341-93, 2000.
[11] Harjoat Bhamra, Lars-Alexander Kuehn, and Ilya Strebulaev. The levered equity
risk premium and credit spreads: A unified framework. University of British
Columbia and Stanford University, Oct 2007.
[12] M. Boldrin, L. Christiano, and J.D.M. Fisher. Habit persistence and asset returns
in an exchange economy. Macroeconomic Dynamics, pages 312-332, 1997.
[13] Guillermo Calvo. Staggered pricing in a utility-maximizing framework. Journal
of Monetary Economics, 12(3):383-98, 1983.
[14] John Campbell and Tuomo Vuolteenaho. Bad beta, good beta. The American
Economic Review, 94(5):1249-1275, December 2004.
[15] J.Y. Campbell and J.H. Cochrane. By force of habit: A consumption-based
explanation of aggregate stock market behavior. Journal of Political Economy,
107:205-251, 1999.
[16] Hui Chen. Macroeconomic conditions and the puzzles of credit spreads and
capital structure. University of Chicago GSB Working Paper, Jan 2007.
[17] Long Chen, Pierre Collin-Dufresne, and Robert S. Goldstein. On the relation
between the credit spread puzzle and the equity premium puzzle. Presented at
the Second Credit Risk Conference, London, England, May, 2005, Mar 2006.
97
[18] Pierre Collin-Dufresne, Robert S. Goldstein, and J. Spencer Martin. The deter
minants of credit spread changes. The Journal of Finance, 56(6):2177-2207, Dec
2001.
[19] George Constantinides. Habit formation: A resolution of the equity premium
puzzle. Journal of Political Economy, 98:519-543, 1990.
[20] Alexander David. Inflation uncertainty, asset valuations, and the credit spread
puzzles. Forthcoming: Review of Financial Studies, 2007.
[21] Edwin Elton, Martin Gruber, Deepak Agarwal, and Christopher Mann. Ex
plaining the rate spread on corporate bonds. Journal of Finance, 41(l):247-278,
February 2001.
[22] Paul Glasserman. Monte Carlo Methods in Financial Engineering. Springer
Verlag, 2004.
[23] Dirk Hackbarth, Jianjun Miao, and Erwan Morellec. Capital structure, credit
risk, and macroeconomic conditions. Journal of Financial Economics, 82(3) :519-
550, December 2006.
[24] J. Huang and W. Kong. Explaining credit spread changes: New evidence from
option-adjusted bond indexes. The Journal of Derivatives, pages 30-42, 2003.
[25] Hayne Leland. Predictions of default probabilities in structural models of debt.
Journal of Investment Management, pages 5-20, 2004.
[26] M. Lettau and H. Uhlig. The sharpe ratio and preferences: A parametric ap
proach. Macroeconomic Dynamics, pages 242-265, 2002.
[27] Martin Lettau. Inspecting the mechanism: Closed-form solutions for asset prices
in real business cycle models. The Economic Journal, pages 550-575, July 2003.
98
[28] George Li. Time-varying risk aversion and asset prices. Journal of Banking and
Finance, 31(1), Jan 2007.
[29] Francis Longstaff and Monika Piazzesi. Corporate earnings and the equity pre
mium. Journal of Financial Economics, 74:401-421, 2004.
[30] Francis Longstaff and Eduardo Schwartz. A simple approach to valuing risky
fixed and floating rate debt. Journal of Finance, 50:789-819, 1995.
[31] Thomas Lubik and Frank Schorfheide. Testing for indeterminacy: An application
to u.s. monetary policy. American Economic Review, 94(1):190-217, 2004.
[32] R. Mashal and M. Naldi. Default-adjusted credit curves and bond analytics
on lehmanlive. Lehman Brothers Fixed Income Research Quantitative Credit
Strategies, September 2005.
[33] R. C. Merton. On the pricing of corporate debt: The risk structure of interest
rates. Journal of Finance, pages 449-470, 1974.
[34] C. Morris, R. Neal, and D. Rolph. Credit spreads and interest rates: A cointe-
gration approach. Federal Reserve Bank of Kansas City, 1998.
[35] M. Pedrosa and R. Roll. Systematic risk in corporate bond credit spreads. Jour
nal of Fixed Income, Dec 1998.
[36] Federico Ravenna and Juha Seppala. Monetary policy and the rejections of the
expectations hypothesis. Working paper, Aug 2006.
[37] Andrew Rose. Is the real interest rate stable. Journal of Finance, 43:1095-112,
1988.
[38] A. Shleifer and R. Vishny. Liquidation values and debt capacity: A market
equilibrium approach. The Journal of Finance, 47:1343-1366, 1992.
99
[39] Joseph Stiglitz and Bruce Greenwald. Towards a New Paradigm in Monetary
Economics. Cambridge University Press, 2003.
[40] Dragon Yongjun Tang and Hong Yan. Macroeconomic conditions, firm charac
teristics, and credit spreads. Journal of Financial Services Research, 29:311-344,
Apr 2006.
[41] Maria Vassalou and Yuhang Xing. Default risk in equity returns. Journal of
Finance, 59(2):831-868, Aug 2004.
[42] Jing zhi Huang and Ming Huang. How much of the corporate-treasury yield
spread is due to credit risk? Penn State University and Stanford University,
May 2003.
Appendix A
Chap. 1 Tables and Figures
A. l Unit Root Tests
Period Full Sample
May 1994 - Aug 1998 Sep 1998 - Mar 2003 Apr 2003 - Jun 2007
1 Yr -2.0148 -0.5959 -2.3081
-4.9418
2 Yr -1.7000 -1.8881 -1.9823
-3.2817
3 Yr -1.7026 -0.4323 -1.5556
-7.6593
5 Yr -1.7162 -0.0039 -1.6578
-9.0780
7Yr -1.7424 -0.2186 -1.7961
-5.1038
10 Yr -1.8222 -1.0940 -1.8936
-3.8597
Table A . l : Phillips-Perron test results for single A credit spreads
Period Full Sample
May 1994 - Aug 1998 Sep 1998 - Mar 2003 Apr 2003 - Jun 2007
AAA -2.1264 1.7604
-2.4991 -4.1803
AA -2.0849 1.7577
-2.5629 -2.7661
A -1.7162 -0.0039 -1.6578
-9.0780
BBB -1.6050 -0.3675 -0.8276
-10.8433
BB -2.1860 -1.2582 -1.3030
-6.0400
B -2.3163 -1.1323 -2.3888 -2.2181
Table A.2: Phillips-Perron test results for 5-year credit spreads
100
101
A.2 Structural Break Tests
Maturity
lYr
2Yr
3Yr
5Yr
7Yr
lOYr
Break date
08/1998
03/2003
08/1998 03/2003
08/1998 03/2003
08/1998 02/2003
08/1998
02/2003
08/1998 02/2003
95% C.I.
05/1998
12/2001
05/1998 12/2001
05/1998 12/2001
04/1998 04/2002
08/1998 02/2003
04/1998 01/2003
02/1999
05/2003
02/1999 06/2003
01/1999 05/2003
11/1998
02/2003
10/1998 04/2003
01/1999 06/2003
1 621.31
71.49
112.23
254.12
2395.04
121.52
supF test
2
8780.78
23562.32
45254.88
18751.25
31290.68
26640.68
3 24795.86
82791.26
68309.23
46436.70
111695.89
77551.85
supF(i+l || i)
i = 1 i =2
21.62 12.34
75.45 24.00
36.45 3.61
35.42 5.77
42.77 12.50
114.63 1.51
Table A.3 : Bai and Perron (1998) structural break test dates and confidence intervals for AAA credit spreads
Maturity
lYr
2Yr
3Yr
5Yr
7Yr
lOYr
Break date
08/1998 01/2003
08/1998 01/2003
08/1998
01/2003
08/1998 03/2003
08/1998 03/2003
08/1998 12/2002
95% CI.
04/1998 01/2003
11/1997 01/2003
07/1998 07/2002
01/1998 07/2002
08/1998 07/2002
01/1998 06/2002
03/1999 05/2003
09/1998 12/2003
12/1998
03/2003
11/1998 04/2003
05/1999 03/2003
01/1999 07/2003
1 573.33
648.87
63.87
892.15
909.38
228.60
supF test 2
52403.16
18723.58
94448.78
55627.44
155117.04
8319.77
3 476589.82
21686.91
52793.13
491577.71
149751.62
68649.57
supF(i+l || i)
i = 1 i =2
337.05 35.24
121.83 82.62
586.09 57.57
26.23 22.37
61.44 10.03
162.81 5.93
Table A.4: Bai and Perron (1998) structural break test dates and confidence intervals for AA credit spreads
Maturity
lYr
2Yr
3Yr
5Yr
7Yr
lOYr
Break date
07/1998 03/2003
08/1998
04/2003
08/1998
04/2003
08/1998 04/2003
08/1998
05/2003
08/1998
06/2003
95% C.I.
02/1998 03/2003
07/1998
03/2003
07/1998 07/2002
08/1998 07/2002
08/1998 07/2002
02/1998
07/2002
08/1998 11/2003
11/1998
12/2003
09/1998 05/2003
11/1998 03/2003
11/1998 11/2003
12/1998 04/2004
1 731.13
5442.18
6009.19
25.90
85.04
805.78
supF test 2
7018.15
58233.22
52040.66
214820.02
37406.52
29824.46
3 287281.06
643382.28
309221.33
41417.50
44353.14
525900.69
supF(i+l || i) i = 1 i =2
181.49 65.17
579.13 30.54
949.42 29.83
3239.84 30.29
261.73 50.41
153.38 18.99
Table A.5: Bai and Perron (1998) structural break test dates and confidence intervals for A credit spreads
Maturity
lYr
2Yr
3Yr
5Yr
7Yr
lOYr
Break date
08/1998 03/2003
08/1998 04/2003
08/1998 04/2003
08/1998 04/2003
08/1998 04/2003
08/1998 04/2003
95% C.I.
05/1998 03/2003
08/1998 09/2002
01/1998 12/2001
08/1998
12/2001
07/1998 12/2001
01/1998 12/2001
11/1998 04/2004
12/1999 08/2003
11/1998 09/2003
08/1998
09/2003
11/1998 07/2003
10/1998 08/2003
1 58.92
707.29
357.01
124.57
132.79
326.35
supF test 2
30619.86
21319.92
39080.97
101708.45
387344.74
44555.87
3 364333.35
1959487.32
620366.28
751472.02
680318.31
636413.12
supF(i+l || i) i = 1 i=2
362.44 18.28
342.34 117.16
366.58 45.95
174.02 26.85
60.13 37.34
4808.33 14.90
Table A.6: Bai and Perron (1998) structural break test dates and confidence intervals for BBB credit spreads
Maturity
lYr
2Yr
3Yr
5Yr
7Yr
lOYr
Break date
08/1998 03/2003
08/1998 03/2003
08/1998
03/2003
08/1998 06/2003
08/1998 06/2003
08/1998 06/2003
95% C.I.
05/1998 11/2002
11/1997 08/2002
01/1998
01/2003
04/1998 03/2003
04/1998 03/2003
04/1998 03/2003
12/1998 04/2003
03/1999 04/2003
11/1998 01/2004
12/1998 07/2003
12/1998 08/2003
12/1998 09/2003
1 27.20
5.90
4.29
6.72
6.93
18.72
supF test 2
1441.19
415.51
91.02
14011.47
8339.88
7662.66
3 18042.79
28912.14
81660.83
73961.83
47409.48
51620.18
supF(i+l || i)
i = 1 i =2
386.04
456.46
167.89
43.86
69.59
159.97
11.87
14.64
13.48
13.45
19.35
80.46
Table A.7: Bai and Perron (1998) structural break test dates and confidence intervals for BB credit spreads
Maturity
lYr
2Yr
3Yr
5Yr
7Yr
lOYr
Break date
02/2000 03/2003
02/2000 03/2003
03/2000 03/2003
02/2000 04/2003
03/2000
03/2003
03/2000
03/2003
95% C.I. 03/1998 09/2002
08/1998
01/2003
10/1999 09/2003
08/1999 03/2003
05/1998
03/2003
05/1998
03/2003
08/2000 03/2003
08/2000
09/2003
08/2000 09/2002
08/2000 02/2004
04/2000 12/2003
04/2000
12/2003
1 73.09
90.40
84.10
38.19
10.69
4.90
supF test 2
332.36
460.47
292.82
97037.08
435.67
650.68
3 310189.92
54992.04
27833.24
285133.30
42795.27
69522.23
supF(i+l 1 i) i = 1 i =2
96.72 16.38
107.03 75.21
110.51 15.14
109.09 8.64
95.94 22.57
212.14 58.14
Table A.8: Bai and Perron (1998) structural break test dates and confidence intervals for B credit spreads
A.3 OLS Est imation
Maturity
lYr
2Yr
3Yr
5Yr
7Yr
lOYr
Adj. R2
0.7550
0.8887
0.9071
0.9032
0.8978
0.8824
Const
0.0014
2.3841
0.0006
1.2992
0.0004
0.7573
0.0004
0.6438
0.0004
0.7111
0.0003
0.4540
C$-i 0.6651
12.1792
0.8257
21.4695
0.8313
22.3918
0.8138
20.6120
0.8107
20.3787
0.8249 20.3688
REAL
0.0117
2.8748
0.0109
1.7985
0.0134
1.9827
0.0156
2.3954
0.0120
2.2785
0.0133
1.5467
INFL
0.0078
1.9672
0.0093
2.1251
0.0131
2.4450
0.0199
2.8117
0.0203
2.8145
0.0178 2.5655
FFR -0.0076
-1.1236
-0.0069
-1.2458
-0.0048
-0.8107
-0.0038
-0.5312
-0.0036
-0.4854
-0.0013
-0.1757
SLOPE
-0.0517
-3.2341
-0.0316
-2.5422
-0.0304
-2.2919
-0.0372
-2.3119
-0.0407
-2.4817
-0.0354
-2.1603
MKT 0.0013
1.0491
-0.0001
-0.1058
0.0000
-0.0371
0.0001
0.0820
-0.0003
-0.2444
-0.0007
-0.5518
VIX 0.0044
4.1365
0.0038
3.9214
0.0044
4.1716
0.0055
4.3548
0.0055
4.3248
0.0047 3.6345
Table A.9: OLS regression coefficients for AAA credit spreads on regressors
Maturity
lYr
2Yr
3Yr
5Yr
7Yr
lOYr
Adj. R2
0.8511
0.9099
0.9196
0.9183
0.9150
0.9021
Const
-0.0001
-0.1259
-0.0003
-0.5091
-0.0004
-0.6204
-0.0002
-0.2438
0.0001
0.1589
0.0002
0.3553
CSt-i 0.7561
16.7965
0.8340
23.6269
0.8258
23.6394
0.8161
22.5977
0.8056
22.0602
0.7979
21.0342
REAL
-0.0235
-2.3555
-0.0138
-1.6259
-0.0177
-1.9726
-0.0245
-2.3646
-0.0252
-2.4467
-0.0209
-2.1214
INFL
0.0227
1.9754
0.0229
2.0989
0.0277
2.4121
0.0336
2.6176
0.0345
2.7109
0.0336
2.6868
FFR
0.0038
0.4544
-0.0004
-0.0603
0.0002
0.0269
-0.0013
-0.1541
-0.0034
-0.4042
-0.0032
-0.3903
SLOPE
-0.0388
-2.0052
-0.0299
-1.8182
-0.0326
-1.9224
-0.0437
-2.2713
-0.0527
-2.7395
-0.0548
-2.9206
MKT
0.0007
0.4670
-0.0006
-0.4767
-0.0006
-0.4150
-0.0003
-0.1984
-0.0007
-0.4351
-0.0010
-0.6923
VIX
0.0066
4.7967
0.0061
4.8545
0.0070
5.2339
0.0078
5.3469
0.0079
5.4891
0.0073
5.1939
Table A. 10: OLS regression coefficients for AA credit spreads on regressors
Maturity lYr
2Yr
3Yr
5Yr
7Yr
lOYr
Adj. R2
0.9172
0.9442
0.9489
0.9471
0.9440
0.9337
Const
-0.0006 -0.7400
-0.0006 -0.8944
-0.0007
-0.9573
-0.0005 -0.6172
-0.0002 -0.2239
0.0001 0.1364
C$_! 0.8059
20.5299
0.8613 27.0646
0.8542
27.0828
0.8484 26.2869
0.8444 26.0583
0.8344 24.4374
REAL
-0.0380 -2.9959
-0.0255 -2.3485
-0.0291 -2.6000
-0.0343 -2.7695
-0.0339 -2.7674
-0.0312 -2.6294
INFL
0.0263 1.9783
0.0259 2.0429
0.0306 2.3709
0.0365 2.6423
0.0374 2.7393
0.0376 2.8047
FFR 0.0109 1.1151
0.0050 0.5794
0.0058
0.6588
0.0046 0.4962
0.0023 0.2557
0.0019 0.2140
SLOPE
-0.0289 -1.3693
-0.0284 -1.5214
-0.0316
-1.6860
-0.0423 -2.0765
-0.0512 -2.5214
-0.0561 -2.8227
MKT 0.0003 0.1881
-0.0008 -0.4988
-0.0006 -0.3977
-0.0002 -0.1052
-0.0004 -0.2238
-0.0006
-0.3709
VIX 0.0080
4.9879
0.0073 5.0122
0.0081 5.3877
0.0087 5.5214
0.0085 5.5743
0.0079
5.3160
Table A . l l : OLS regression coefficients for A credit spreads on regressors
Maturity
lYr
2Yr
3Yr
5Yr
7Yr
lOYr
Adj. R2
0.9584
0.9686
0.9677
0.9615
0.9541
0.9421
Const
-0.0004
-0.4045
-0.0005
-0.5427
-0.0006
-0.6580
-0.0006
-0.5477
-0.0003
-0.3151
-0.0002
-0.1387
CSt-!
0.8685 28.0947
0.8959
33.8695
0.8867
32.4830
0.8702
29.3963
0.8575
27.3248
0.8414
24.7578
REAL
-0.0484
-3.1536
-0.0386
-2.7803
-0.0434
-2.9137
-0.0535
-3.0759
-0.0584
-3.1382
-0.0617
-3.1608
INFL
0.0376
2.0558
0.0390
2.2114
0.0252
2.5095
0.0361
2.8813
0.0416
2.0139
0.0471
2.1400
FFR 0.0106 0.9021
0.0065
0.6022
0.0082
0.7117
0.0093
0.7138
0.0086
0.6145
0.0099
0.6615
SLOPE
-0.0374
-1.4983
-0.0402
-1.7397
-0.0426
-1.7506
-0.0527
-1.8973
-0.0634
-2.1226
-0.0710
-2.2540
MKT -0.0003
-0.1394
-0.0009
-0.4565
-0.0004
-0.2120
0.0005
0.2001
0.0006
0.2520
0.0008
0.3017
VIX 0.0093
4.7948
0.0090
4.9194
0.0101
5.1454
0.0117
5.2629
0.0124
5.2272
0.0125
5.0414
Table A. 12: OLS regression coefficients for BBB credit spreads on regressors
Maturity
lYr
2Yr
3Yr
5Yr
7Yr
lOYr
Adj. R2
0.8691
0.8778
0.8810
0.8839
0.8795
0.8670
Const
-0.0093
-2.0465
-0.0084
-1.9243
-0.0079
-1.8377
-0.0065
-1.5809
-0.0055
-1.3510
-0.0045
-1.1217
CSt-i
0.7005
13.7510
0.7322
15.1626
0.7445
15.8632
0.7637
17.0164
0.7758
17.6438
0.7842
17.8634
REAL
-0.2227
-3.3544
-0.1941
-3.0523
-0.1836
-2.9401
-0.1663
-2.7529
-0.1472
-2.5002
-0.1248
-2.1595
INFL
0.1893
2.2590
0.1905
2.3358
0.1952
2.4229
0.1955
2.5010
0.1898
2.4624
0.1825
2.3678
FFR 0.1365
2.2850
0.1124
1.9589
0.1009
1.7943
0.0790
1.4665
0.0632
1.1958
0.0501
0.9503
SLOPE
0.1443
1.1401
0.1036
0.8511
0.0771
0.6484
0.0278
0.2452
-0.0034
-0.0302
-0.0257
-0.2327
MKT 0.0061
0.5960
0.0027
0.2716
0.0007
0.0748
-0.0029
-0.3087
-0.0067
-0.7275
-0.0112
-1.2249
VIX 0.0486
5.0875
0.0460
4.9785
0.0452
4.9735
0.0427
4.9171
0.0403
4.7624
0.0374
4.4798
Table A. 13: OLS regression coefficients for BB credit spreads on regressors
110
Maturity
lYr
2Yr
3Yr
5Yr
7Yr
lOYr
Adj. R2
0.9273
0.9298
0.9313
0.9340
0.9351
0.9320
Const
-0.0302
-2.6025
-0.0279
-2.4778
-0.0264
-2.3989
-0.0233
-2.2378
-0.0207
-2.0749
-0.0181
-1.8651
CSt-t
0.8294
20.6372
0.8376
21.2238
0.8418
21.5378
0.8508
22.2116
0.8583
22.6387
0.8623
22.3690
REAL
-0.3472
-2.0907
-0.3357
-2.0702
-0.3325
-2.0861
-0.3153
-2.0596
-0.2897
-1.9683
-0.2615
-1.8268
INFL
0.3887
1.9763
0.3877
1.9876
0.3916
1.9773
0.3869
1.9919
0.3736
1.9925
0.3609
1.9629
FFR 0.3774
2.4663
0.3449
2.3175
0.3252
2.2319
0.2872
2.0663
0.2559
1.9085
0.2295
1.7384
SLOPE
0.2953
0.9727
0.2305
0.7818
0.1826
0.6344
0.1006
0.3691
0.0430
0.1650
-0.0008
-0.0031
MKT -0.0282
-1.1374
-0.0282
-1.1668
-0.0277
-1.1756
-0.0285
-1.2741
-0.0319
-1.4932
-0.0388
-1.8744
VIX 0.1010
4.0329
0.0969
3.9853
0.0942
3.9718
0.0875
3.9030
0.0806
3.7754
0.0728
3.5367
Table A. 14: OLS regression coefficients for B credit spreads on regressors
I l l
A.4 Markov Regime Switching Estimation
Maturity lYr
2Yr
3Yr
5Yr
7Yr
lOYr
Regime 1
2
1
2
1
2
1
2
1
2
1
2
Adj. R2
0.8894
0.9602
0.9671
0.9553
0.9566
0.9639
Const 0.0025
7.9982 -0.0029 -1.7836
0.0001 1.1150 0.0005 1.7195
0.0005 7.6962 0.0002 0.5196
0.0010 0.0003
-0.0010 0.0006
0.0012 6.5477 -0.0006 -0.6020
0.0019 15.3284
-0.0005 -1.6927
CSt-i 1.1198
3.9219 0.5761
11.3321
0.8498 21.1861
0.8157 12.8333
0.9234 11.6369
0.8237 26.1342
0.8556 5.9474 0.7819
22.6640
0.8771 8.1834 0.7538
20.0207
0.8887 14.0252
0.6329 16.7866
REAL -0.0225 -1.5718 0.0327
4.3360
0.0091 1.3386 0.0153
2.9762
-0.0093 -1.4038 0.0100
3.0688
0.0121 0.9272 0.0240
3.2327
-0.0098 -0.4154 0.0237
3.0762
-0.0088 -0.4896 0.0296 1.2958
INFL 0.0220 1.2940 0.0050
2.6135
0.0097 1.0805 0.0154
2.3403
-0.0070 -0.5616 0.0152
2.6446
0.0250 1.9572 0.0160
2.3349
0.0159 0.6398 0.0258
2.0594
0.0268 0.0190 0.0136
2.5739
FFR 0.0077 0.4461
-0.0109 -1.7967
-0.0028 -0.6710 -0.0169
-2.9459
-0.0023 -0.6120 -0.0201
-2.5313
-0.0015 -0.3529 -0.0223 -1.4887
-0.0229 -1.5089 -0.0022 -0.4846
-0.0042 -0.3646 0.0020 0.4498
SLOPE 0.0246 0.7986
-0.0684 -6.3532
-0.0060 -0.7138 -0.0593
-5.2681
-0.0228 -2.9895 -0.0477
-3.1520
-0.0226 -2.4924 -0.0792
-3.2184
-0.0929 -2.9754 -0.0253
-2.4727
-0.0688 -3.6838 -0.0190 -1.9332
MKT -0.0015 -1.3144 0.0106
2.1064
-0.0016 -1.4556 -0.0003 -0.1401
-0.0008 -0.8716 -0.0002 -0.1327
-0.0011 -0.8552 -0.0003 -0.0968
-0.0016 -1.3246 -0.0011 -0.4042
-0.0016 -1.6284 0.0006 0.2925
VIX 0.0026
2.2135 0.0065
2.2532
0.0005 0.5322 0.0094
7.6377
0.0016 1.9284 0.0097
5.8266
0.0022 2.0973 0.0154
6.2167
0.0025 2.3820 0.0150
4.7064
0.0022 2.3613 0.0138
7.3022
Table A. 15: Markov regime switching model coefficients for AAA credit spreads on regressors
Maturity lYr
2Yr
3Yr
5Yr
7Yr
lOYr
Regime 1
2
1
2
1
2
1
2
1
2
1
2
Adj. R2
0.9338
0.9722
0.9762
0.9732
0.9727
0.9734
Const 0.0012
2.8723 -0.0021 -0.7730
-0.0013 -5.4356
0.0006 0.7685
0.0003 3.0878 -0.0009 -1.8527
-0.0014 -3.1226
0.0008 8.4425
-0.0012 -2.8656
0.0013 10.4628
-0.0005 -1.7077 0.0022
23.1075
CS-i 0.9560
14.4920 0.3500
3.5754
0.9877 4.3576 0.8357
32.8941
0.9955 8.1220 0.8364
32.7081
0.8477 9.1639 0.8038
30.2014
0.8287 10.9921
0.7653 26.4742
0.7756 13.0908
0.7086 23.9550
REAL 0.0148 1.0409
-0.0860 -3.0903
0.0159 1.0273
-0.0086 -1.3449
0.0225 1.3792
-0.0224 -3.8872
0.0124 0.7992
-0.0214 -2.7477
0.0140 0.9497
-0.0253 -3.2789
0.0210 1.3621
-0.0346 -4.1309
INFL -0.0181
-2.1135 0.0961
2.7951
-0.0015 -0.1351 0.0141
2.9825
-0.0082 -0.4340 0.0209
3.5042
0.0110 0.6229 0.0246
3.4463
0.0122 0.6958 0.0260
3.6418
-0.0036 -0.2817 0.0141
1.9756
FFR -0.0181
-3.0849 0.0391
2.0731
-0.0126 -1.7947 -0.0044 -1.1682
-0.0285 -2.0894 -0.0015 -0.3930
-0.0085 -0.6612 -0.0065 -1.4782
-0.0087 -0.7245 -0.0100
-2.2690
-0.0037 -0.2909 -0.0081 -1.8508
SLOPE -0.0376
-2.6048 -0.0394 -0.7437
-0.0228 -1.3292 -0.0271
-3.1797
-0.0395 -1.6970 -0.0234
-3.0083
-0.0296 -1.1994 -0.0404
-4.2189
-0.0383 -1.6115 -0.0535
-5.5537
-0.0860 -3.9962 -0.0506
-4.9623
MKT 0.0005 0.1907 0.0028 0.9424
0.0043 1.1407 0.0000 0.0425
-0.0001 -0.1109 0.0004 0.1780
0.0002 0.0683
-0.0003 -0.2593
-0.0005 -0.2089 -0.0009 -0.8000
-0.0022 -1.1248 -0.0011 -1.0015
VIX 0.0010 0.7188 0.0174
4.7025
0.0018 0.3428 0.0128
5.3356
0.0026 2.9392 0.0151
6.2740
0.0181 8.5420 0.0036
3.4920
0.0180 9.0463 0.0044
4.1405
0.0167 9.5984 0.0034
3.1755
Table A. 16: Markov regime switching model coefficients for A A credit spreads on regressors
113
Maturity lYr
2Yr
3Yr
5Yr
7Yr
lOYr
Regime 1
2
1
2
1
2
1
2
1
2
1
2
Adj. _R2
0.9598
0.9783
0.9832
0.9840
0.9819
0.9708
Const 0.0008 1.4659
-0.0011 -0.1853
0.0000 0.8565
-0.0033 -6.5662
0.0007 1.5904
-0.0019 -3.8528
0.0002 0.2194
-0.0078 -0.8457
0.0003 0.6377
-0.0069 -2.7133
-0.0009 -1.3659 0.0000
-0.0036
CSt-i 0.9500
21.9782 0.3628
3.8695
0.8923 35.5516
0.7833 6.5453
0.8530 25.3093
0.7440 5.5315
0.9892 9.8172 0.3569
3.7984
0.9678 21.2988
0.3572 4.3204
1.0415 8.8436 0.4389
4.6397
REAL 0.0066 0.4103
-0.1931 -2.7541
-0.0108 -1.1287 -0.0424 3.1243
-0.0216 -2.1349 -0.0221
-2.0736
0.0128 0.3227
-0.1134 -2.8605
0.0064 0.4540
-0.1100 -3.2511
0.0169 0.6125
-0.1176 -3.5198
INFL -0.0116 -1.1512 0.1093
2.5765
0.0142 3.5285 0.0644
1.9755
0.0130 1.7369 0.0332
3.7207
-0.0079 -0.6706 0.0637
2.1150
-0.0025 -0.3853 0.2762
5.7875
0.0104 0.5797 0.0903
3.4604
FFR -0.0145
-2.1267 0.0488 0.7578
0.0034 0.9323 0.0245 0.8565
0.0012 0.2026 0.0120 0.4810
-0.0085 -1.4099 0.0830 1.5016
-0.0078 -1.8441 0.0722
3.3233
-0.0043 -0.3808 0.0475 1.1596
SLOPE -0.0430
-2.7907 -0.0885 -0.4723
-0.0153 -2.5890 -0.0049 -0.3403
-0.0273 -2.4790 -0.0205 -0.6321
-0.0220 -1.2697 -0.0039 -0.0831
-0.0238 -1.8675 -0.0577
-1.9741
-0.0133 -0.6654 -0.0935
-4.4543
MKT -0.0004 -0.2350 0.0008 0.1127
-0.0004 -0.3432 0.0039 1.2111
-0.0009 -0.6608 -0.0026 -0.9063
0.0004 0.1538
-0.0002 -0.0446
-0.0002 -0.1875 0.0023 0.6040
-0.0012 -0.5090 0.0009 0.2824
VIX 0.0026 1.7996 0.0222
3.9386
0.0017 1.6078 0.0197
6.0123
0.0021 1.9439 0.0203
9.6121
0.0016 1.3355 0.0347
3.6194
0.0020 1.8758 0.0349
7.4425
0.0023 1.3876 0.0205
3.7003
Table A. 17: Markov regime switching model coefficients for A credit spreads on regressors
114
Maturity lYr
2Yr
3Yr
5Yr
7Yr
lOYr
Regime 1
2
1
2
1
2
1
2
1
2
1
2
Adj. i?2
0.9876
0.9911
0.9898
0.9901
0.9830
0.9777
Const -0.0015 0.2752 0.0004 0.9857
-0.0021 0.3568
-0.0031 -0.2411
-0.0003 -1.8454 0.0033 0.9247
-0.0007 -2.5062
0.0078 4.9930
0.0012 0.7352 0.0005 1.0146
-0.0006 -1.0553 0.0055
3.9644
CSt-! 0.9538 1.0194 0.8579
33.9610
0.8972 12.3747
0.8947 43.5943
0.9931 43.2413
0.5439 19.6351
1.0122 43.7808
0.4776 7.9258
0.8822 12.4711
0.8455 31.4550
0.9305 27.6513
0.6311 23.1531
REAL 0.0018 0.9847
-0.0333 -18.1722
-0.0086 -1.1078 -0.0188
-3.5268
-0.0113 0.8726
-0.2128 4.5682
0.0138 0.9823
-0.4388 -12.2073
-0.0666 -1.0307 -0.0410
-2.7487
-0.0139 -0.9089 -0.3841
-7.2695
INFL 0.0051 0.3627 0.0154
3.1742
0.0107 0.6731 0.0195
2.4215
0.0050 0.7839 0.0365
4.0265
0.0002 0.0225 0.0548
2.6205
0.0128 0.3545 0.0352
3.1739
0.0065 0.2724 0.0405
3.7747
FFR -0.0193 -1.4338 0.0129 1.6753
-0.0011 -0.0966 0.0070 1.1910
0.0036 0.9945 0.0271 0.1348
-0.0059 -0.9087 -0.0113 -0.4658
-0.0592 -2.1824
0.0088 1.1333
0.0037 0.5831 0.0659
2.0650
SLOPE -0.0521 -1.7906 -0.0147 -0.9946
-0.0293 -1.3063 -0.0248
-1.9677
-0.0068 -0.6089 -0.2327 1.4233
-0.0226 -1.8616 -0.5926
-8.7880
-0.2035 -2.7955 -0.0443
-2.5577
-0.0424 -2.5815 -0.3530
-3.7149
MKT 0.0062
2.6384 -0.0031 -1.7597
0.0044 1.7566
-0.0012 -0.7740
-0.0006 -0.7564 -0.0042 0.8721
0.0020 1.0679
-0.0001 -0.0103
-0.0093 -1.7321 -0.0002 -0.0826
-0.0060 -3.0029
0.0342 7.3967
VIX 0.0188 1.1938 0.0177
3.5255
0.0214 11.0257
0.0263 2.1442
0.0018 0.9237 0.0305
5.7563
0.0037 2.8265 0.0390
10.6517
0.0055 3.5730 0.0305
6.5524
0.0051 3.2772 0.0245
5.0412
Table A. 18: Markov regime switching model coefficients for BBB credit spreads on regressors
Maturity lYr
2Yr
3Yr
5Yr
7Yr
lOYr
Regime 1
2
1
2
1
2
1
2
1
2
1
2
Adj. R2
0.9476
0.9512
0.9454
0.9524
0.9412
0.9328
Const -0.0401
-2.2612 0.0015 0.3599
0.0056 0.2414 0.0016 0.3734
-0.0371 -1.3756 0.0009 0.2127
-0.0245 -2.1327 0.0014 0.4979
-0.0297 -1.6550 -0.0005 -0.1137
-0.0260 -2.0754
0.0000 -0.0041
csw 0.8237 8.3246 0.7094
13.8365
0.8518 6.2826 0.7269
15.6094
1.0869 6.7507 0.7801
18.8823
0.8275 5.9326 0.7558
12.2385
1.0801 6.8815 0.7859
17.4791
1.0785 7.3088 0.7845
18.6815
REAL -0.7694
-3.4635 -0.1216
-2.3234
-1.4828 -3.3945 -0.1088
-2.2472
-0.2495 -0.8627 -0.0773
-3.5709
-0.4160 -3.1525 -0.0841
-4.3992
-0.2009 -1.1252 -0.0707
-3.3885
-0.2072 -1.2000 -0.0546
-3.0977
INFL 1.1970
2.9732 0.0220
3.3119
0.1729 0.4817 0.0264
4.3404
0.5794 1.3678 0.0513
2.6393
0.4566 3.2595 0.0777
3.6482
0.3657 1.3404 0.1143
2.3544
0.3016 1.4596 0.1210
2.5505
FFR 0.3479
3.4162 0.0410 0.7590
0.4804 2.9543 0.0312 0.5676
0.2812 1.1181 0.0115 0.2162
0.2120 0.6424 0.0165 1.5225
0.2359 0.9345 0.0201 0.3506
0.2232 1.1983 0.0103 0.1960
SLOPE -0.4858 0.0001 0.0332 0.3116
-0.8282 -1.5572 0.0219 0.2063
0.1824 0.2433
-0.0100 -0.0929
-0.0210 -0.0570 0.0007 0.0138
0.1801 0.3359
-0.0076 -0.0641
0.1331 0.3120
-0.0205 -0.1875
MKT -0.0083 -0.3423 0.0053 0.6972
-0.0147 -0.6124 0.0032 0.4049
-0.0534 -1.8196 0.0018 0.2357
-0.0529 -3.0180 -0.0011 -0.1427
-0.0391 -1.3977 -0.0017 -0.2158
-0.0382 -1.4085 -0.0066 -0.8390
VIX 0.1257
3.1378 0.0220
2.6260
0.0348 0.5910 0.0219
2.4211
0.0935 2.5380 0.0203
2.4378
0.1042 6.8452 0.0185
3.2424
0.0800 3.5152 0.0207
2.4057
0.0725 3.6144 0.0194
2.4031
Table A. 19: Markov regime switching model coefficients for BB credit spreads on regressors
116
Maturity lYr
2Yr
3Yr
5Yr
7Yr
lOYr
Regime 1
2
1
2
1
2
1
2
1
2
1
2
Adj. R2
0.9611
0.9618
0.9687
0.9709
0.9824
0.9698
Const -0.0253
-2.3286 -0.0843 -1.0675
-0.0212 -2.0132 -0.1067 -1.1466
-0.0124 -1.3282 -0.3438
-25.5459
-0.1954 -9.5151 -0.0100 -0.8703
-0.0096 -1.0593 -0.0092 -1.1265
-0.0089 -0.9056 -0.0650 -0.6691
CSi_i 0.8979
21.9418 0.3132
3.0986
0.9113 21.1380
0.2731 2.5599
0.9408 24.5209
0.4795 6.2758
0.9759 5.4522 0.9329
20.8274
0.9644 20.8050
0.6459 6.9774
0.9206 16.8215
0.6491 4.5444
REAL -0.0501 -0.3604 -1.8640 3.0567
-0.0591 -0.4096 -1.3088
-3.5667
-0.0822 -0.5936 -1.5633
-2.7628
-0.6050 -0.7562 -1.0260
-3.2106
0.0934 0.7560
-0.9915 -4.2432
-0.0250 -0.1718 -1.3424
-2.7937
INFL 0.3624 1.8715 0.9902
2.9415
0.3159 1.6842 1.3848
3.2254
0.2086 1.2413 0.6188
2.6402
3.5731 4.0761 1.1537 0.7742
0.1347 0.8245 0.6175
3.0476
0.1582 0.8673 0.6915
2.8570
FFR 0.2915
2.0751 1.3484
2.0404
0.2526 1.8128 1.2990 1.1634
0.1712 1.4262 5.2149
10.5672
1.2886 2.0295 0.1007 0.7984
0.0135 0.1131 2.3582
12.3867
0.0833 0.5957 1.4455 1.0891
SLOPE 0.1520 0.5732 2.6881 0.0023
0.0732 0.2830 3.2497 1.1338
-0.1116 -0.5060 10.8152 5.8239
1.0215 1.1842
-0.0838 -0.3556
-0.0291 -0.1565 -4.0918
-6.9775
-0.0263 -0.1187 0.7971 0.2578
MKT -0.0249 -1.1715 0.1615 0.0001
-0.0234 -1.1004 0.1393
2.1321
-0.0130 -0.7045 0.0003 0.0067
-0.3860 -4.4033
0.0138 0.7372
0.0199 1.4898
-0.0119 -0.2004
0.0189 1.1669
-0.3371 -6.9086
VIX 0.0541
2.5139 0.3309
2.9654
0.0471 2.1072 0.3811
3.5278
0.0322 1.6731 0.4379
6.2480
0.0346 1.2830 0.4443
5.7224
0.0338 1.3850 0.1923
3.0058
0.0332 1.2953 0.1941
2.8636
Table A.20: Markov regime switching model coefficients for B credit spreads on regressors
117
A.5 Robustness Check: Tests with Moody's Data
Full Sample May 1994 to Aug 1998 Sep 1998 to Mar 2003 Apr 2003 to Jun 2008
Aaa -1.73861 -0.62319 -2.10791
-2.74863
Baa -1.59253 -0.51291 -0.63282
-3.90358
Table A.21: Phillips-Perron test results for Moody's Corporate Credit Spread series
Ratings
Aaa
Baa
Adj. i?2
0.9443
0.9643
Const CSt-i
•0.0012 0.8657
-1.3030 22.7058
0.0014 0.8638
1.2205 23.9641
REAL INFL
-0.0211 0.0256
-1.5395 2.5044
-0.0513 0.0149
•3,0558 2.8452
FFR SLOPE
0.0227 0.0232
1.9442 0.9158
0.0110 -0.0221
0.9149 -0.8479
MKT VIX
-0.0049 0.0083
•2.3914 3.7454
-0.0049 0.0090
•2.2737 3.9728
Table A.22: OLS model coefficients for Moody's corporate credit spread series
Rating
Aaa
Baa
Regime
1
2
1
2
Adj. #
0.9732
0.9837
Const C$i-\
-0.0003 0.9574
-0.3461 17.9843
-0.0049 0.3773
-1.6940 2.3676
0.0011 0.9104
0.8123 37.6073
0.0002 0.8439
0.7645 2,1435
REAL INFL FFR SLOPE
-0.0232 -0.0029 0.0219 0.0227
-1.3041 -0.3056 2.8222 1.3478
-0.0290 0.1864 0.0625 0.0887
•2.5929 3,8772 1.2291 0 . 1 3
-0.0235 0.0034 0.0153 0.0115
-1.4174 0.2316 1.4196 0.4743
-0.1002 0.0544 0.0125 -0.0710
-7.4412 2,7083 0.4323 -1.3018
MKT VK
-0.0038 0.0008
-2,0678 0.3148
-0.0174 0.0343
•2.5619 6,8289
-0.0037 0.0005
-1.6102 0.4048
-0.0021 0.0214
-0.5384 7,5534
Table A.23: Markov regime-switching model coefficients for Moody's corporate credit spread series
118
A.6 Figures
Lehman 5-Year Credit Spread Data, 5/1994 - 6/2007
•AAA
-AA
A
BBB
-BB
•B
- Regime Shift
993-D1-31 1995-10-28 1998-07-24 2001-04-19 2004-01-14 2006-10-10 2009-07-06
Date
Figure A.l: 5-year credit spreads and possible break dates, May 1994 to June 2007
Volatility Factor
50.00% -J
45.00% -
40.00% -
35.00%
30.00%
0)
=j 25.00%
20.00% -
15.00% -
10.00%
5.00%
0.00%
•VIX
1993-01-31 1995-10-28 1998-07-24 2001-0419 2004-01-14 2006-10-10 2009-07-06
Date
Figure A.2: VIX, May 1994 to June 2007 to
o
Markov Smoothed Probabilities -5 Year A Credit Spreads
60% A
40<M
fl»N
1994-06 1995-10 1997-03 1998-07 1999-12 2001-04 2002-09 2004-01 2005-05 2006-10
Date
Figure A.3: Smoothed regime probabilities, 5 year A credit spreads
Appendix B
Chap. 2 Proofs, Tables, and
Figures
B.l Credit Spread Expression
As stated earlier, the price of a one-period default-risky bond can be expressed as
B = -^~ (B.l)
where B is the risky bond price, c is the face value it pays upon maturity, r is the risk-free rate, and s is the credit spread, representing the extra yield demanded by investors for a default-risky bond over the risk-free rate.
The default-risky bond can also be priced as a contingent claim, contingent on the default event. Along the path with no default, the bond simply returns the face value c, while, along the path with default, the bond returns rA where r again is the risk-free rate and A is the current recovery value of the bond upon default. The Arrow-Debreu price of the default event is qo and, accordingly, the A-D price of survival is (1 — g£>). The bond price is, therefore, the sum of the payoffs along different events paths discounted by the risk-free rate or
B = -(l-qD) + qDA (B.2)
Setting the two bond pricing expressions equal and performing some algebra, we find
122
123
that the expression of the spread in terms of the bond pricing parameters is
r W £ -A) , N
s = qDK\ >—- B.3 c(l - qD) + rqDA
We define the loss given default as percentage of the value loss upon default. From our definition of the payoffs under different states, the loss given default / is
- — A I = r—r- (B.4)
Substituting this expression into the expression for credit spreads B.3, we obtain
s = rT^ (B-5)
B.2 The Market Price of Risk and the Risk-Neutral
Measure
To price a contingent claim, which is how we evaluate a corporate bond to determine its spread, we must evaluate the expectation of its contingent payment at a future date under the risk-neutral measure, an approach underlying most of continuous-time finance. Since the corporate bond we are pricing depends upon the firm's cash flow, which we model as a primitive process, we must simply determine the expectation of future firm cash flows under the risk-neutral or Q measure. Pricing under the risk-neutral measure simplifies to this relationship:
y(0) = e-rTEq[V(T)}
To evaluate the price of a bond, we must simply therefore evaluate its payoff along different possible paths or scenario and discount back by the the risk-free rate of the appropriate maturity under the risk-neutral measure. We must then, therefore, determine how to model the price process V(t) under the risk-neutral measure to simulate its paths. The risk-neutral measure is an equivalent martingale measure under which every price process discounted by the price of a risk-free bond is a martingale.
B(0) L5(T)J
where B{T) is the payoff of a risk-free bond and B(0) = 1.
124
If we assume that the price process V(t) is
dV{t)
V(t) = fj,(t)dt + a(t)dW{t)
under the real measure, then, under some technical conditions, V(t) has the following representation under the risk-neutral measure by Girsanov's Theorem
^=IM(t)dt + *(t)dW*(t)
where dW(t) = dWq{t) - fj,(t)dt
The term //(£) that adjust the general price process to the risk-neutral measure is called the market price of risk.
Since we have an economy with explicit explicit preference specifications, we can directly specify the market price of risk, which adjust the price process to the risk-neutral measure. As derived by Lettau and Uhlig (2002), the market price of risk in the case of a power utility function is simply the relative risk aversion coefficient, 7. Therefore, in our model, we try to price the firm's cash flow, which under the real measure, is postulated to be
9K (t) = Q9t + 6 + P^Kef + aKy/l - p2e
where gt represents the growth in aggregate output, <?#•(£) is the growth of the marginal firm's cash flow, g is the sensitivity of the firm's cash flow growth to aggregate output growth, £f is the mean of firm-specific cash flow growth, p is the correlation of output growth and firm-specific cash flow growth, &K is volatility of firm cash flow growth, and ef, ef ~ iV(0,1) independent of each other.
Under the risk-neutral measure, the cash flow growth process is
9K{t) = Q9t + & - Ipo-c^K + PO-RC? + o-K^l-p2ef (B.6)
where ac is the volatility of consumption growth.
After simulating the firm's cash flow growth process under the risk-neutral measure, we can then price contingent claims on the firm's cash flow by taking an expectation along different simulated paths and discounting by the risk-free rate of the appropriate tenor.
B.3 Regression Test Coefficients
125
Ratings
A A A
AA
A
BBB
BB
B
Maturity
1 Yr
4 Yr
10 Yr
1 Yr
4 Yr
10 Yr
1 Yr
4 Yr
10 Yr
1 Yr
4 Yr
10 Yr
1 Yr
4 Yr
10 Yr
1 Yr
4 Yr
10 Yr
Adj. R2
0.017142
0.018305
0.090266
0.054131
0.050117
0.113329
0.084525
0.069319
0.158236
0.087802
0.156368
0.283167
0.275245
0.467394
0.617847
0.559428
0.712586
0.773477
Const
2.87E-06 2.196326 0.000295 2.215974
0.006769 7.610134
0.000315
5.251523 0.001887 4.811875 0.009657 9.214268 0.000946 7.499122
0.003163 6.140926 0.016463
12.31004 0.001032 7.7634
0.011618 12.0407
0.042143 21.51603 0.012779 21.52676
0.082723
39.71867 0.203372 69.58464
0.067078 54.96241
0.230885 95.81524 0.435189
157.4966
9t -0.00146 -13.3039 -0.1533 -13.7501
-2.33965
-31.4085 -0.12024
-23.9199 -0.75406 -22.9641
-3.12569 -35.6131 -0.32034 -30.3242
-1.17552
-27.2491 -4.83134
-43.1397 -0.34462
-30.9539 -3.46299 -42.8569 -10.2466 -62.4687 -3.04663
-61.2861
-16.2265 -93.0344
-30.8633 -126.1 -11.433 -111.866 -31.5019 -156.108 -42.3299
-182.933
7T*
3.47E-05 1.166342 0.003752
1.240606
0.026477
1.310343
0.002017
1.479549 0.011683
1.311676 0.029843 1.253482 0.004366 1.523528 0.016349
1.39712
0.036307
1.195129 0.004484
1.484781 0.030112 1.373804 0.060309 1.355437 0.022057 1.635694
0.073162
1.546388 0.04901
0.738193 0.042788 1.543376 0.032217 0.588549 -0.07928
-1.263
Table B . l : Regression of forward-looking default probabilities on contemporaneous output growth and inflation.
Ratings AAA
AA
A
BBB
BB
B
Maturity 1 Yr
4 Yr
10 Yr
1 Yr
4 Yr
10 Yr
1 Yr
4 Yr
10 Yr
1 Yr
4 Yr
10 Yr
1 Yr
4 Yr
10 Yr
1 Yr
4 Yr
10 Yr
Adj. R2
0.042197
0.031394
0.223458
0.112457
0.120064
0.256394
0.16354
0.164427
0.300232
0.166849
0.289734
0.383985
0.346449
0.507206
0.639084
0.533304
0.69762
0.821846
Const 0.00000 1.44194 0.00001 1.37358 0.00005
2.98259 0.00003
2.78614 0.00005
2.16652 0.00007
3.57753 0.00009
3.53017 0.00008
2.48007 0.00013
4.91325 0.00010
3.63704 0.00024
4.27772 0.00038
9.56764 0.00106
7.75085 0.00220
16.95086 0.00264
35.36535 0.00631
18.72564 0.00802
44.01298 0.00851
87.37827
S t - l
0.16272 16.54662
0.12945 13.09460
0.39391 43.67327
0.26189 27.42157
0.28919 30.47670
0.41306 46.49036
0.31540 33.78856
0.33998 36.60994
0.42304 48.46917
0.31735 34.03920
0.42759 48.89528
0.39948 47.15600
0.41283 48.08396
0.38381 48.17375
0.28667 38.93279
0.40109 51.32028
0.27398 39.50623
0.14947 25.92743
9t -0.00024
-11.42325 -0.00703
-11.24655 -0.03296
-21.44806 -0.01817
-18.10075 -0.03241
-16.27113 -0.04298
-23.98976 -0.04640
-21.45246 -0.04879
-18.47166 -0.06616
-28.84779 -0.04984
-21.78441 -0.13251
-26.98801 -0.14639
-41.53342 -0.42470
-35.88039 -0.65933
-56.88010 -0.55032
-82.89121 -1.73796
-58.20048 -1.50395
-94.12992 -1.10617
-141.36248
V"i
0.00001 1.45641 0.00025 1.47748 0.00116
2.85037 0.00059
2.20844 0.00111
2.06982 0.00148
3.12276 0.00151
2.61880 0.00172
2.43838 0.00214
3.56899 0.00160
2.62878 0.00457
3.55034 0.00397
4.44414 0.01359
4.46775 0.01650
5.88127 0.00961
6.39214 0.04699
6.54332 0.02429
6.88394 0.00853
5.27901
Table B.2: Regression of credit spreads on one-quarter lagged credit spreads and contemporaneous output growth and inflation.
Ratings AAA
AA
A
BBB
BB
B
Maturity 1 Yr
4 Yr
10 Yr
1 Yr
4 Yr
10 Yr
1 Yr
4 Yr
10 Yr
1 Yr
4 Yr
10 Yr
1 Yr
4 Yr
10 Yr
1 Yr
4 Yr
10 Yr
Adj. R2
0.00205
0.005215
0.018033
0.115046
0.022935
0.033709
0.151231
0.037002
0.083682
0.169208
0.115569
0.23907
0.348856
0.410843
0.555764
0.542525
0.649196
0.821513
Const
0.00000 1.30401 0.00000 0.58801 0.00000 1.08265 0.00000
3.07643 0.00000 1.26535 0.00000 1.37305 0.00001
4.01071 0.00000 1.52154 0.00001 1.84012 0.00001
4.18434 0.00002
2.50528 0.00005
3.34738 0.00026
9.22445 0.00058
9.10471 0.00119
22.22070 0.00298
22.00356 0.00444
33.27960 0.00712
84.34375
St-l
-0.00231 -0.23130 -0.00064 -0.06400 0.08833
8.88632 0.25518
26.70615 0.10209
10.28920 0.13756
13.93095 0.28985
30.80124 0.14300
14.49859 0.23815
24.67370 0.30882
33.08964 0.27378
28.74265 0.40094
44.75989 0.39719
46.12140 0.43575
52.53087 0.35417
45.59808 0.38599
49.50554 0.33064
45.74079 0.16553
28.73276
9t 0.00000
-4 .83856 -0.00001
-7 .49212 -0.00130
-9.72376 -0.00083
-19.38710 -0.00052
-10.62438 -0.00265
-11.28071 -0.00345
-22.24494 -0.00142
-11.87845 -0.00701
-14.36614 -0.00378
-23.14955 -0.01308
-17.32220 -0.02972
-23.05116 -0.09320
-37.75482 -0.22774
-40.88427 -0.31207
-65.41077 -0.72261
-60.73231 -0.94893
-80.39608 -0.95196
-139.31380
7T*
0.00000 -0.40240 0.00000 1.40593 0.00005 1.38410 0.00003
2.60731 0.00002 1.46330 0.00009 1.47950 0.00011
2.69304 0.00005 1.52520 0.00024 1.84547 0.00012
2.83554 0.00044
2.15005 0.00104
3.06087 0.00284
4.49945 0.00718
5.09333 0.00711
6.30167 0.01896
6.66997 0.01927
7.18517 0.00846
5.98888
Table B.3: Correlation of firm cash flow growth with output growth, rho, is 0.4. Regression of credit spreads on one-quarter lagged credit spreads and contemporaneous output growth and inflation.
Ratings AAA
AA
A
BBB
BB
B
Maturity 1 Yr
4 Yr
10 Yr
1 Yr
4 Yr
10 Yr
1 Yr
4 Yr
10 Yr
1 Yr
4 Yr
10 Yr
1 Yr
4 Yr
10 Yr
1 Yr
4 Yr
10 Yr
Adj. R2
0.03706
0.178215
0.328705
0.162892
0.281831
0.349687
0.218963
0.309826
0.389495
0.224123
0.383778
0.478969
0.407213
0.576666
0.684716
0.57112
0.727916
0.820649
Const 0.00001 1.50606 0.00011 2.44078 0.00027
7.28379 0.00024
3.11894 0.00027
4.07312 0.00034
8.48324 0.00047
3.91118 0.00039
4.98127 0.00050
10.87913 0.00049
3.97566 0.00097
8.91332 0.00104
17.59516 0.00284
9.05460 0.00451
25.99404 0.00395
45.58344 0.01175
22.18200 0.01108
53.12692 0.00943
89.12216
St-l
0.14023 14.21548
0.35791 38.84084
0.39793 45.84481
0.32303 34.64891
0.42539 48.46188
0.39196 45.50186
0.36877 40.60484
0.43022 49.57514
0.38028 44.82841
0.37262 41.13031
0.41232 48.80464
0.34596 42.38748
0.43388 52.18846
0.32681 42.49994
0.24286 34.28013
0.37913 49.57889
0.23543 35.03147
0.14109 24.39737
9t -0.00783
-12.19978 -0.06866
-18.59650 -0.11300
-35.26578 -0.13160
-20.42194 -0.15099
-26.11425 -0.13487
-38.36801 -0.24658
-24.06060 -0.19556
-29.28937 -0.17859
-44.04745 -0.25897
-24.37895 -0.38363
-40.20744 -0.29972
-57.13953 -1.11616
-40.66535 -1.09401
-70.57957 -0.71689
-94.52897 -3.06588
-64.92492 -1.87109
-104.03788 -1.19648
-141.90552
TTi
0.00029 1.67373 0.00243 2.46751 0.00314
3.80118 0.00442
2.57208 0.00514
3.38523 0.00359
3.99809 0.00830
3.06130 0.00639
3.66922 0.00451
4.41329 0.00875
3.11448 0.01084
4.47569 0.00688
5.40310 0.03536
5.09020 0.02259
6.23805 0.01016
6.06564 0.07498
6.72432 0.02485
6.36905 0.00844
4.84481
Table B.4: Correlation of firm cash flow growth with output growth, rho, is 0.8. Regression of credit spreads on one-quarter lagged credit spreads and contemporaneous output growth and inflation.
Ratings AAA
AA
A
BBB
BB
B
Maturity 1 Yr
4 Yr
10 Yr
1 Yr
4 Yr
10 Yr
1 Yr
4 Yr
10 Yr
1 Yr
4 Yr
10 Yr
1 Yr
4 Yr
10 Yr
1 Yr
4 Yr
10 Yr
Adj. i?2
0.041555
0.082425
0.301174
0.156296
0.209678
0.32404
0.215053
0.254157
0.367097
0.220571
0.366856
0.477391
0.414962
0.605007
0.741343
0.605567
0.789841
0.889473
Const 0.00000 1.69631 0.00003
1.85038 0.00013
4.99933 0.00008
3.44792 0.00011
2.93524 0.00018
6.14901 0.00021
4.42583 0.00017
3.59985 0.00032
8.62122 0.00022
4.54415 0.00057
7.03907 0.00085
16.50101 0.00210
10.53445 0.00441
28.25220 0.00469
58.27485 0.01156
26.98781 0.01349
70.59951 0.01225
134.57722
St-l
0.14842 15.07128
0.23611 24.44701
0.42217 48.37649
0.30802 32.88615
0.38083 41.92696
0.41711 48.16728
0.35447 38.85938
0.41009 46.11571
0.40349 47.30912
0.35838 39.38913
0.43513 51.39920
0.36131 44.36626
0.42130 50.69514
0.33002 43.90691
0.21094 31.84955
0.36719 49.18374
0.19851 32.42671
0.08116 17.14338
9t -0.00092
-12.85009 -0.02020
-14.30154 -0.06582
-29.08230 -0.04262
-21.07209 -0.06985
-21.01349 -0.08487
-32.59950 -0.09993
-25.15819 -0.09999
-24.15418 -0.12593
-39.20007 -0.10638
-25.52579 -0.25091
-35.82302 -0.25359
-55.58539 -0.75046
-42.84129 -1.03156
-74.06814 -0.75406
-109.47683 -2.67383
-70.53909 -1.97651
-124.26737 -1.25083
-190.59571
7T*
0.00003 1.67477 0.00069
1.82211 0.00211
3.57256 0.00139
2.57360 0.00247
2.79765 0.00257
3.82024 0.00327
3.11777 0.00350
3.20251 0.00349
4.26610 0.00345
3.13532 0.00791
4.40072 0.00605
5.44081 0.02327
5.27708 0.02119
6.58140 0.00888
5.99604 0.06303
7.15286 0.02125
6.35095 0.00491
3.75175
Table B.5: Mean of idiosyncratic firm cash flow growth is -2% per annum. Regression of credit spreads on one-quarter lagged credit spreads and contemporaneous output growth and inflation.
Ratings AAA
AA
A
BBB
BB
B
Maturity 1 Yr
4 Yr
10 Yr
1 Yr
4 Yr
10 Yr
1 Yr
4 Yr
10 Yr
1 Yr
4 Yr
10 Yr
1 Yr
4 Yr
10 Yr
1 Yr
4 Yr
10 Yr
Adj. i?2
0.018654
0.013163
0.1416
0.085417
0.061109
0.173962
0.125152
0.092612
0.233257
0.12767
0.211995
0.320636
0.293772
0.429421
0.544614
0.471549
0.610856
0.743263
Const 0.00000 1.83249 0.00000
0.99151 0.00003
2.04210 0.00001
2.39705 0.00002
1.67098 0.00004
2.37455 0.00004
2.90217 0.00004
1.93542 0.00006
3.11309 0.00005
3.03247 0.00012
2.99401 0.00018
5.85804 0.00056
6.10317 0.00113
10.68685 0.00150
22.62192 0.00359
13.73548 0.00472
28.67569 0.00564
57.75592
St-l
0.07496 7.53689 0.06719
6.74536 0.32203
34.34316 0.22824
23.61995 0.19865
20.36621 0.35377
38.31167 0.27719
29.19013 0.25144
26.15718 0.40066
44.62824 0.27907
29.41557 0.38208
42.10649 0.42022
48.49730 0.39817
45.27415 0.41811
50.64848 0.33656
42.85196 0.41842
51.79978 0.33146
44.32775 0.21803
33.05142
9t -0.00007
-11.00451 -0.00213
-9.09225 -0.01760
-16.39968 -0.00789
-16.01982 -0.01431
-13.14509 -0.02330
-18.37644 -0.02210
-18.89270 -0.02356
-14.89360 -0.03614
-22.11405 -0.02425
-19.16329 -0.07305
-21.23237 -0.08482
-31.82255 -0.24485
-31.02896 -0.41640
-44.77182 -0.38863
-65.64723 -1.14118
-49.45279 -1.09896
-74.74829 -0.91379
-109.25395
Kt
0.00000 0.86339 0.00008
1.31240 0.00063
2.18608 0.00025
1.90935 0.00049
1.68388 0.00084
2.46587 0.00074
2.37603 0.00081
1.90978 0.00128
2.95013 0.00078
2.30785 0.00257
2.80818 0.00262
3.79275 0.00794
3.86547 0.01192
5.11799 0.00846
6.03434 0.03371
5.91873 0.02268
6.68659 0.01123
6.24319
Table B.6: Mean of idiosyncratic firm cash flow growth is 2% per annum. Regression of credit spreads on one-quarter lagged credit spreads and contemporaneous output growth and inflation.
131
Ratings AAA
AA
A
BBB
BB
B
Maturity 1 Yr
4 Yr
10 Yr
1 Yr
4 Yr
10 Yr
1 Yr
4 Yr
10 Yr
1 Yr
4 Yr
10 Yr
1 Yr
4 Yr
10 Yr
1 Yr
4 Yr
10 Yr
Adj. R2
0
0
0
0
0
0
0
0
0.001837
-0.00028
0.00286
0.014154
0.021283
0.152485
0.38981
0.180878
0.458793
0.73678
Const 0.00000 0.06459 0.00000
-0.52709 0.00000 1.54456 0.00000 0.06459 0.00000
-0.52709 0.00000 1.54456 0.00000 0.06459 0.00000
-0.52709 0.00000
-0.10960 0.00000 0.06459 0.00000 0.20189 0.00000 0.98028 0.00000 1.20921 0.00007
2.38173 0.00041
9.93457 0.00011
3.73934 0.00147
12.85370 0.00496
56.58320
St-l
-0.00846 -0.84555 -0.01557 -1.55721 0.00523 0.52244
-0.00846 -0.84555 -0.01557 -1.55721 0.00523 0.52244
-0.00846 -0.84555 -0.01557 -1.55721 -0.00169 -0.16910 -0.00846 -0.84555 0.00952 0.95218 0.07587
7.62187 0.09200
9.26453 0.32783
35.09913 0.39748
47.02724 0.33151
35.79998 0.40584
49.78090 0.21755
32.66451
9t 0.00000 0.66683 0.00000
-0.84454 0.00000 1.20864 0.00000 0.66683 0.00000
-0.84454 0.00000 1.20864 0.00000 0.66683 0.00000
-0.84454 -0.00001
-4.65627 0.00000 0.66683
-0.00001 -5.56618
-0.00259 -8.91175
-0.00025 -10.82439
-0.04365 -17.82018
-0.15290 -42.40106
-0.05705 -22.47793
-0.49840 -49.28117
-0.81050 -107.73689
TTi
0.00000 -0.09152 0.00000 0.78720 0.00000
-0.90215 0.00000
-0.09152 0.00000 0.78720 0.00000
-0.90215 0.00000
-0.09152 0.00000 0.78720 0.00000 1.46822 0.00000
-0.09152 0.00000 1.30273 0.00010 1.24774 0.00001 1.65889 0.00153
2.33230 0.00411
4.51298 0.00188
2.78540 0.01353
5.41124 0.00992
6.11942
Table B.7: Standard deviation of idiosyncratic firm cash flow growth is 10% per annum. Regression of credit spreads on one-quarter lagged credit spreads and contemporaneous output growth and inflation.
Ratings AAA
AA
A
BBB
BB
B
Maturity 1 Yr
.4 Yr
10 Yr
1 Yr
4 Yr
10 Yr
1 Yr
4 Yr
10 Yr
1 Yr
4 Yr
10 Yr
1 Yr
4 Yr
10 Yr
1 Yr
4 Yr
10 Yr
Adj. R2
0.348073
0.422699
0.516314
0.482626
0.489582
0.536023
0.523152
0.513814
0.569234
0.526489
0.584488
0.635125
0.644904
0.725423
0.770465
0.750356
0.824682
0.874056
Const 0.00105
7.74355 0.00104
10.26658 0.00117
19.91324 0.00398
14.48589 0.00191
15.37876 0.00134
21.90000 0.00576
17.78214 0.00233
17.59787 0.00168
25.59159 0.00593
18.07008 0.00390
25.38290 0.00258
34.72173 0.01576
32.92125 0.00948
50.43988 0.00614
66.89640 0.03369
56.23516 0.01672
83.85174 0.01168
117.69324
St-l
0.41458 48.33843
0.42010 50.73904
0.34661 43.42116
0.41630 51.83482
0.39196 48.76824
0.33794 42.79496
0.40465 51.48360
0.38003 47.85739
0.32320 41.75364
0.40395 51.49714
0.34331 45.05496
0.28901 39.12642
0.34449 47.55924
0.25419 37.82228
0.19079 30.06758
0.26759 41.32909
0.17923 31.41637
0.11431 22.87314
9t -0.42364
-35.88283 -0.39156
-43.80790 -0.32157
-61.37375 -1.23515
-50.90975 -0.59895
-54.09577 -0.35356
-64.48437 -1.62756
-56.67173 -0.68445
-58.01202 -0.41059
-69.90363 -1.66240
-57.13202 -0.96238
-70.09128 -0.54161
-82.08076 -3.29695
-78.35083 -1.65623
-101.67278 -0.91989
-119.16066 -5.34419
-105.94939 -2.24021
-139.17448 -1.29049
-173.95372
7T<
0.01358 4.47569 0.01129
5.03185 0.00728
5.79590 0.03599
6.03739 0.01546
5.72259 0.00776
5.95455 0.04479
6.45656 0.01698
5.96411 0.00845
6.14832 0.04545
6.47561 0.02084
6.49765 0.00956
6.38605 0.07087
7.38659 0.02420
6.82404 0.00956
5.84455 0.08224
7.55939 0.02103
6.31661 0.00709
4.75041
Table B.8: Standard deviation of idiosyncratic firm cash flow growth is 40%. Regression of credit spreads on one-quarter lagged credit spreads and contemporaneous output growth and inflation.
Ratings AAA
AA
A
BBB
BB
B
Maturity 1 Yr
4 Yr
10 Yr
1 Yr
4 Yr
10 Yr
1 Yr
4 Yr
10 Yr
1 Yr
4 Yr
10 Yr
1 Yr
4 Yr
10 Yr
1 Yr
4 Yr
10 Yr
Adj. R2
0.04082
0.030041
0.258087
0.109594
0.117622
0.290317
0.159822
0.161451
0.331832
0.163044
0.284537
0.412879
0.338857
0.494887
0.66363
0.520911
0.682627
0.835956
Const 0.00000 1.93416 0.00001
1.84743 0.00007
3.97320 0.00004
3.55139 0.00006
2.84411 0.00010
4.77133 0.00011
4.47410 0.00010
3.28674 0.00017
6.51569 0.00012
4.59833 0.00030
5.59189 0.00049
12.29130 0.00123
9.51106 0.00241
19.45447 0.00315
43.14372 0.00691
21.48139 0.00831
47.36447 0.00947
102.00524
St-l
0.16324 16.59210
0.12995 13.13896
0.43744 49.56118
0.26311 27.51942
0.29025 30.56029
0.45369 52.19500
0.31704 33.91441
0.34132 36.71132
0.45892 53.69133
0.31904 34.16918
0.43011 49.06537
0.42621 51.24858
0.41630 48.29875
0.38952 48.45847
0.29929 41.68548
0.40664 51.55205
0.28029 39.70693
0.15539 28.04139
9t -0.00023
-10.80949 -0.00664
-10.64041 -0.03169
-20.54264 -0.01724
-17.18257 -0.03065
-15.39770 -0.04205
-23.12716 -0.04405
-20.36310 -0.04613
-17.46725 -0.06633
-28.13484 -0.04734
-20.68125 -0.12553
-25.51270 -0.15263
-41.44285 -0.40484
-34.03017 -0.63418
-53.97214 -0.58006
-84.32254 -1.67119
-55.15725 -1.46791
-89.30982 -1.12385
-145.57633
n 0.00001 1.00212 0.00017
1.04969 0.00078
1.93390 0.00040
1.52901 0.00077
1.46058 0.00096
2.03675 0.00100
1.76177 0.00117
1.69403 0.00132
2.19889 0.00105
1.75301 0.00285
2.24900 0.00248
2.73385 0.00831
2.77328 0.00968
3.48647 0.00483
3.23740 0.02740
3.85266 0.01369
3.88780 0.00144
0.94689
Table B.9: Coefficient of relative risk aversion is 10. Regression of credit spreads on one-quarter lagged credit spreads and contemporaneous output growth and inflation.
Ratings AAA
AA
A
BBB
BB
B
Maturity 1 Yr
4 Yr
10 Yr
1 Yr
4 Yr
10 Yr
1 Yr
4 Yr
10 Yr
1 Yr
4 Yr
10 Yr
1 Yr
4 Yr
10 Yr
1 Yr
4 Yr
10 Yr
Adj. R2
0.050654
0.066978
0.290813
0.176079
0.184416
0.317123
0.2378
0.232991
0.353546
0.239485
0.353737
0.445028
0.419083
0.568108
0.701435
0.590364
0.751557
0.861449
Const 0.00000 2.26172 0.00002
2.10770 0.00010
4.77161 0.00005
4.11692 0.00008
3.20022 0.00013
5.77575 0.00014
5.17776 0.00013
3.80689 0.00023
7.95154 0.00015
5.23310 0.00042
6.89298 0.00065
14.97173 0.00166
11.41434 0.00351
25.94451 0.00390
51.63829 0.00948
26.91699 0.01139
63.83647 0.01084
119.04969
St-l
0.18215 18.58946
0.21845 22.47525
0.45415 52.26369
0.34720 37.53992
0.37282 40.64128
0.45978 53.50097
0.39774 44.31802
0.41497 46.37385
0.45124 53.15288
0.39786 44.35807
0.47029 55.71749
0.41088 50.02530
0.45913 55.82789
0.37796 49.24574
0.27102 39.18422
0.41034 54.72055
0.24393 37.62282
0.12884 24.93113
9t -0.00033
-11.84106 -0.00856
-12.16389 -0.04179
-23.13698 -0.02241
-19.41962 -0.03936
-17.32304 -0.05528
-26.06877 -0.05699
-22.96021 -0.05954
-19.78898 -0.08633
-31.81687 -0.06153
-23.20281 -0.16591
-29.56496 -0.19088
-46.53301 -0.52075
-38.59808 -0.82028
-63.62671 -0.65730
-93.65968 -2.09670
-63.10480 -1.73787
-107.19658 -1.18050
-162.52765
VTi
0.00001 0.97215 0.00023
1.22064 0.00095
2.03961 0.00052
1.70730 0.00102
1.70700 0.00115
2.11950 0.00127
1.96916 0.00150
1.91431 0.00159
2.32285 0.00140
2.03427 0.00361
2.53074 0.00296
2.97533 0.01051
3.14794 0.01099
3.71943 0.00444
2.97875 0.03069
4.04240 0.01088
3.24395 0.00025
0.17445
Table B.10: Coefficient of relative risk aversion is 25. Regression of credit spreads on one-quarter lagged credit spreads and contemporaneous output growth and inflation.
135
Ratings AAA
AA
A
BBB
BB
B
Maturity 1 Yr
4 Yr
10 Yr
1 Yr
4 Yr
10 Yr
1 Yr
4 Yr
10 Yr
1 Yr
4 Yr
10 Yr
1 Yr
4 Yr
10 Yr
1 Yr
4Yr
10 Yr
Adj. i?2
0.005629
0.010807
0.080781
0.032977
0.03693
0.101378
0.056161
0.051961
0.1447
0.057537
0.120477
0.260458
0.180275
0.360645
0.543398
0.332238
0.539713
0.575723
Const 0.00000 1.10086 0.00003
1.40878 0.00024
6.84047 0.00010
3.74530 0.00017
3.58998 0.00035
8.13129 0.00030
5.60783 0.00029
4.75523 0.00061
10.87625 0.00033
5.73514 0.00118
9.46569 0.00175
19.09981 0.00424
14.62709 0.00983
27.13682 0.00916
46.91798 0.02404
26.49322 0.02866
46.88648 0.02622
53.62860
S t - l
0.00006 0.00585
-0.00016 -0.01652 0.01098 1.14512 0.00041 0.04176 0.00037 0.03753 0.01451 1.53168 0.00164 0.16878 0.00187 0.19251 0.01636 1.77064 0.00188 0.19351 0.01087 1.15886 0.01540 1.78637 0.00725 0.79962 0.01266 1.57624 0.00312 0.45732 0.00937 1.14293 0.00446 0.65089 0.00573 0.87201
9t -0.00024
-7.70920 -0.00518
-10.56861 -0.02523
-29.21094 -0.01233
-18.34578 -0.02217
-19.47344 -0.03408
-33.05715 -0.03149
-24.12512 -0.03473
-23.19860 -0.05438
-40.41228 -0.03380
-24.42821 -0.10954
-36.40920 -0.12523
-57.98687 -0.31835
-45.99658 -0.61633
-73.25055 -0.45951
-105.97595 -1.45442
-68.93054 -1.42887
-105.31179 -1.20943
-113.30136
7T<
0.00002 0.65732 0.00053 1.13786 0.00102 1.24637 0.00080 1.25160 0.00166 1.53826 0.00128 1.31229 0.00168 1.35531 0.00217 1.53007 0.00174 1.35916 0.00175 1.33449 0.00392 1.37471 0.00119 0.58082 0.00776 1.18154 0.00224 0.27996
-0.00371 -0.89472 0.01714 0.85395
-0.00575 -0.44375 -0.00404 -0.39583
Table B . l l : Coefficient of monetary policy smoothing is 0.9. Regression of credit spreads on one-quarter lagged credit spreads and contemporaneous output growth and inflation.
8 ^ CD P rt cT
Co I CL tO CO ' '
g H P P & • %
O ST o >-i
5. " St O 2 CD P 2-so a
O c+
c i— e-t- CO
era o
P CD
CO
EC P
§• 8 , o
CD
CD P a co o
o cp
P
P cm era
CD
O O
,0£Z
~J CD
O 4^ Cn i—1
00 -J
I
M O • 39
1
#». to
o b to -a oo o
I—1
o
i-i
p Cn cn m oo 4^ oo
o .023
Cn O
o b o oo o 00
i H .225
4^ 00
O
b o o 1—' CD
on oo 7361
o CO
o 4^ bO ai CD 4 -
HJ
O Or • 892.
* » •
-i
o b 4^ C35 00 -J
4^
i-i
p Cn CO Cn to I—'
CO
o .026
co C5
o b o to CD OO
i h—» .496
Oi 03
p b o o 4^ o
CO h-»
• 185;
CO -J
p Cn CO to o o
1
CJ CD • 33
1
14 ^ i
O Cn i—1
CD oo o
tt)
1—'
tf p CO
to CD -a 4^ -a
o .024
K Cn
p b o 4^ oo Ci
i i—> .590
oi H
I
O b o -a i—* CO
at CO 978!
o h-»
O I—1
4^ 4 > oo oo
M
o -J 553;
CO to
o b to 00 CD to
h-»
o
p cn 4^ to CD Cn co
o
800'
4^ oo
p b o o co CD
i
p 4^ oo
o OS
O
b o o o oo
00 © 764
#>. o
o bo 4^ oo co 4^
i -J to • 799
to ^
p i—1
co to -J Oi
4^
i-S
O
co Cn to 00 -a Oi
o .009
05 o
p b o Oi 00 Cn
i
o b -a 4^ CO to
p b o o -Q 4^
H-» 00 181
CO -1
p to CD Cn Cn to
, ^ P
£86
CO co
1
p b ~a co to to
cd td
I—1
tf p 1—»
oo co oo o Cn
O .004
Cn O
O
b o to 05
-a
i
o .348
CO - i
p b o co o 4^
to to • 023
-* to
p Cn 4^ CO CO 4^
, CN 00 •
9S0
~ j
cn
o to oo -a -a - j
H
o
tf p to Cn 05 CO 00 I—1
O b o i—» - j OS
o b o 4^ Oi CD
i
O h- 1
OO oo oo CO
p b o o 4 to
M to • 351i
OS 00
p Cn 00 O 4^ -a
i CO ^1 139
00
o 1
p Cn
co -a CD O
4^
p t—'
to CO to CO co
p b o CO Cn
O
b O Cn 4^ to
i
p I— 1
to to CO I—1
1
p b o i—1 o -J
00 613
co 00
i
o b to Cn 00 O
I
to Cn 8
74
00
o i
i — i
b co h — i
o OS
Cd Cd Cd
i—1
tf o b C35 CO to -<l •f^
o
000'
^ o 1
o b o o to Ol
1
o b co H-1
co
1
p b o o oo -J
M 00 838:
to <i
K^
b 05 to I—1
Cn
! ^ M •
908
00 CO
1
p h-'
oo M Cn i—1
i—» O
tf O I—1
•1^ oo Cn CD -~a
o
000'
ai oo
p b o CD -a oo
i
p b 1 — '
a> K
o b o o i—1 C5
<I 2871
~* CO
i
p b H CO O Cn
I
to oo 821;
CO hi
1
o bo Cn oo oo o
rf^ < I-i
o b Cn 4^ I—» 4^ 4^
O
000'
oo oo t
o b o o h-' 00
1
o b oo Cn co
i
p b o o oo H
00
002
00 M
O
b 1—J
Cn Cn -a
, to ^ 879
on 00
i
o CO C75 to co oo
>
H
O
b Cn 00 -J Oi 00
o
ooo-
co -CI
o b o o H Cn
i
o '.033
- j
oo
p b o o -a CO
M © 888
i—1 CO
i—i
b CD to o oo
• 00
^ 003
CO 05
I
p Cn -J GO 00 oo
I—1
O
i-i
O i—1
O Cn co i—1
CD
O
000'
4^ I—1
p b w O to co
i
o b CO oo to CD
O
b o o 00 co
on 965
00 M
i
O I—1
o -J as co
i M CO 9
22
00 05
O CD 4^ to Oi Cn
4^
*
O
b CO oo to oo oo
o
000'
to co i
o b o H o 03
1
o .023
co Oi
o b o o OJ Oi
p 108:
to CO
1
o b Cn OJ Cn co
i M co 1
58
to CO
i
I—1
O oo Cn I—>
> >
h-'
^
O
b co Cn 4^
O
000'
h-'
#=-1
p b o o Cn Oi
i
O (.012
oo as
i
p b o o 4^ Cn
CO 376
H^ 00
p ^ CD Cn h-»
to
to co
026
to -1
1
p ^ Cn -Q Cn CO
i—1
o
i-i1
O
b oo CO to Cn CO
O
000'
to co
p b o - j Cn CO
i
o b to oo i—»
o
1
p b o o 4^ co
00 024
00 l-»
1
p b Oi oo Cn Cn
! M O
9S2
to 00
1 I— 1
b CO Cn 4^ i—1
4^
i-i
O
b o co CD oo 00
o .000
o Cn
i
o b o o a> oo
i
o b o Cn O Cn
©
b o o co CO
to bo M I— 1
o oo i
o b oo OJ o -a
i KI 3
20
cn CO
i
H
t—»
co 4^ -a o
> > >
i—1
tf o b o 4^ CO oo Cn
o
000'
o o
o b o o co OJ
1
o b o o to 4^
i
O b o o o to
Rati
p TO cn
vlatur
>
S3 to
Cons
c-t-
C/i I
£
co 0 5
137
B.4 Impulse Response Functions
138
Cons, growth (quarterly)
10 11 12
Figure B.l: Impulse response of macroeconomic conditions to adverse technology/productivity shock
139
AAA1 Yr M l Yr A1 Yr
2 4 6 8 10 12 2 4 6 8 10 12 2 4 6 8 10 12
BBB 1 Yr BB1 Yr B1 Yr
2 4 6 8 10 1 2 4 6 8 10 1: 2 4 6 8 10 12
Figure B.2: Impulse response of 1 year credit spreads to adverse technology/productivity shock. The y-axis represents the credit spread, while the x-axis represents the number of quarters
140
AAA 4 Yr AA4 Yr A4 Yr
0.1
2 4 6 8 10 12 2 4 6 8 10 12 0.05
2 4 6 8 10 12
BBB 4 Yr BB4 Yr B4 Yr
0.1
0.05 0.05 0.05 2 4 6 8 10 12 2 4 6 8 10 12 2 4 6 8 10 12
Figure B.3: Impulse response of 4 year credit spreads to adverse technology/productivity shock. The y-axis represents the credit spread, while the x-axis represents the number of quarters
141
AAA 10 Yr AA10 Yr A 10 Yr 0.07
0.06
0.05
0.04 0.04 0.04 2 4 6 8 10 12 2 4 6 8 10 12 2 4 6 8 10 12
BBB 10 Yr BB 10 Yr 0.08 r-—•—• 1 0.08
B10Yr
0.07
0.06
0.05
0.07
0.06
0.05
0.08
0.07
0.06
0.05 2 4 6 8 10 12 2 4 6 8 10 12 2 4 6 8 10 12
Figure B.4: Impulse response of 10 year credit spreads to adverse technology/productivity shock. The y-axis represents the credit spread, while the x-axis represents the number of quarters
142
-0.005 h
Cons, growth (quarterly)
6 7
Inflation
5 6 7 8
Interest Rate
12
10 11 12
Figure B.5: Impulse response of macroeconomic conditions to positive monetary policy shock
143
1 0 i \AA1 Yr x10"4AA1 Yr x K f A I Y r
2 4 6 8 10 12 2 4 6 8 10 12 2 4 6 8 10 12
x 10-£BB 1 Yr BB 1 Yr 0.03, , 0.06
0.02
B1 Yr
0.01
2 4 6 8 10 12 2 4 6 8 10 12 2 4 6 8 10 12
Figure B.6: Impulse response of 1 year credit spreads to positive monetary policy shock. The y-axis represents the credit spread, while the x-axis represents the number of quarters
144
<I0"&AA4 Yr AA4 Yr A4 Yr
2 4 6 8 10 12 2 4 6 8 10 12 2 4 6 8 10 12
BBB4 Yr BB4 Yr B4 Yr
0.02
0.01
0.04
0.02
2 4 6 8 10 12 2 4 6 8 10 12 2 4 6 8 10 12
Figure B.7: Impulse response of 4 year credit spreads to positive monetary policy shock. The y-axis represents the credit spread, while the x-axis represents the number of quarters
145
A M 10 Yr AA10 Yr 0.02
0.015
0.01
0.005
0.02
0.015
0.01
0.005
2 4 6 8 10 12
0.02
0.015
0.01
0.005
n
A1 0 Yr
•
•
2 4 6 8 10 12 2 4 6 8 10 12
BBB 10 Yr 0.02
0.015
0.01
0.005
0.02
0.015
0.01
0.005
n
BB 10 Yr
\
\
\
B10Yr 0.02
0.015
0.01
0.005
2 4 6 8 10 12 2 4 6 8 10 12 2 4 6 8 10 12
Figure B.8: Impulse response of 10 year credit spreads to positive monetary policy shock. The y-axis represents the credit spread, while the x-axis represents the number of quarters
Appendix C
Chap. 3 Proofs and Tables
C.l Market Price of Risk
To price a contingent claim, which is how we evaluate a corporate bond to determine its spread, we must evaluate the expectation of its contingent payment at a future date under the risk-neutral measure, an approach underlying most of continuous-time finance. Since the corporate bond we are pricing depends upon the firm's cash flow, which we model as a primitive process, we must simply determine the expectation of future firm cash flows under the risk-neutral or Q measure. Pricing under the risk-neutral measure simplifies to this relationship:
V(0) = e-rTEq[V(T)]
To evaluate the price of a bond, we must simply therefore evaluate its payoff along different possible paths or scenario and discount back by the the risk-free rate of the appropriate maturity under the risk-neutral measure. We must then, therefore, determine how to model the price process V(t) under the risk-neutral measure to simulate its paths. The risk-neutral measure is an equivalent martingale measure under which every price process discounted by the price of a risk-free bond is a martingale.
v(o) -YM- E®\viT)] V{0) ~ B(0) ~ E [B(T)1
where B{T) is the payoff of a risk-free bond and B(Q) = 1.
146
147
If we assume that the price process V(t) is
dV(t)
V(t) fi(t)dt + a(t)dW(t)
under the real measure, then, under some technical conditions, V(t) has the following representation under the risk-neutral measure by Girsanov's Theorem
^ | = »{t)dt + a(t)dW®(t)
where dW(t) = dWQ(t) - n(t)dt
The term p,(t) that adjust the general price process to the risk-neutral measure is called the market price of risk.
Since we have an economy with explicit explicit preference specifications, we can directly specify the market price of risk, which adjust the price process to the risk-neutral measure. As derived by Lettau and Uhlig (2002), the market price of risk in the case of a power utility function is simply the relative risk aversion coefficient, 7. However, under internal habit persistence, the market price of risk actually equals the elasticity of the pricing kernel with respect to consumption innovations. Lettau and Uhlig (2002) derive this to be, under the assumption loglinearity of the pricing kernel and lognormality of all relevant random variables,
7 1 + f3b2e~9(l+^
where g is the steady-state growth rate of consumption, x is the steady-state ratio of habit to consumption, and b is the coefficient of the previous period consumption in the habit specification in the utility function.
In our model, we try to price the firm's cash flow, which under the real measure, is postulated to be
9K (t) = Q9t + it + P°K<^ + VKV1 - P 2 e f
where gt represents the growth in aggregate output, gx(t) is the growth of the marginal firm's cash flow, g is the sensitivity of the firm's cash flow growth to aggregate output growth, £t is the mean of firm-specific cash flow growth, p is the correlation of output growth and firm-specific cash flow growth, GK is volatility of firm cash flow growth, and ef, ef ~ iV(0,1) independent of each other.
Under the risk-neutral measure, the cash flow growth process is
9K(t) = ggt + 6 - r,™cpacaK + paKef + aKyJ\ - p2ef (C.l)
148
where oc is the volatility of consumption growth and r\^'ic reflects the elasticity of the pricing kernel to innovations in consumption growth.
After simulating the firm's cash flow growth process under the risk-neutral measure, we can then price contingent claims on the firm's cash flow by taking an expectation along different simulated paths and discounting by the risk-free rate of the appropriate tenor.
C.2 Regression Test Coefficients
Ratings AAA
AA
A
BBB
BB
B
Maturity 1 Yr
4 Yr
10 Yr
1 Yr
4 Yr
10 Yr
1 Yr
4 Yr
10 Yr
1 Yr
4 Yr
10 Yr
1 Yr
4 Yr
10 Yr
1 Yr
4 Yr
10 Yr
Adj. R2
0.017909
0.015756
0.076977
0.056171
0.043155
0.098818
0.084729
0.060301
0.142128
0.087829
0.142407
0.265857
0.270307
0.452774
0.607937
0.549468
0.703733
0.770879
Const 0.00000 2.92947 0.00020 2.52844 0.00589
7.68930 0.00027
6.41854 0.00152
5.03862 0.00860
9.40121 0.00083
8.71071 0.00267
6.41985 0.01514
12.66543 0.00090
8.93561 0.01063
12.58538 0.04057
22.39260 0.01226
23.71492 0.08210
41.90582 0.20593
73.58938 0.06670
58.88270 0.23354
101.13211 0.44026
167.27936
9t -0.00081
-13.61381 -0.09058
-12.76875 -1.95698
-28.92104 -0.08987
-24.45237 -0.56715
-21.30579 -2.67709
-33.14090 -0.25573
-30.46624 -0.93345
-25.38658 -4.29882
-40.70675 -0.27678
-31.06896 -3.04031
-40.76029 -9.62422
-60.14299 -2.77638
-60.83382 -15.72249
-90.86594 -30.72044
-124.30211 -11.03344
-110.29335 -31.37281
-153.82886 -42.52863
-182.96928
V£
0.00001 0.79860 0.00180 0.95240 0.02010 1.11469 0.00109 1.10834 0.00802 1.13037 0.02157 1.00204 0.00273 1.22104 0.01143 1.16644 0.02622 0.93171 0.00293 1.23392 0.02229 1.12126 0.04806 1.12701 0.01545 1.27008 0.05617 1.21810 0.01305 0.19822 0.02905 1.08968 0.00172 0.03165
-0.09658 -1.55922
Table C . l : Regression of forward-looking default probabilities on contemporaneous and inflation.
Ratings AAA
AA
A
BBB
BB
B
Maturity 1 Yr
4 Yr
10 Yr
1 Yr
4 Yr
10 Yr
1 Yr
4 Yr
10 Yr
1 Yr
4 Yr
10 Yr
1 Yr
4 Yr
10 Yr
1 Yr
4 Yr
10 Yr
Adj. R2
0.052106
0.058085
0.276128
0.170283
0.170067
0.305571
0.218936
0.215351
0.341609
0.228244
0.339787
0.426486
0.402389
0.546754
0.677164
0.573513
0.729194
0.843842
Const 0.00000 2.19369 0.00001 2.03647 0.00008
4.40955 0.00005
3.84921 0.00007
3.01850 0.00012
5.29124 0.00013
4.85192 0.00011
3.57511 0.00020
7.23153 0.00014
4.91535 0.00037
6.31281 0.00056
13.47681 0.00147
10.59223 0.00305
23.29991 0.00343
46.10380 0.00834
24.63871 0.01012
57.16553 0.00992
107.00139
St-l
0.18739 19.14339
0.20174 20.67070
0.44695 51.07106
0.34539 37.27266
0.35906 38.86341
0.45841 53.09303
0.38263 42.22967
0.40039 44.33780
0.45415 53.27235
0.39195 43.47920
0.46879 55.20774
0.41838 50.54996
0.45779 55.20953
0.38963 50.13558
0.28857 40.69552
0.41922 55.30654
0.26290 39.36740
0.14660 26.99181
9t -0.00028
-11.65271 -0.00759
-11.67716 -0.03749
-21.96590 -0.01993
-18.60756 -0.03534
-16 .57117 -0.04931
-24.65577 -0.05168
-21.99283 -0.05364
-18.90440 -0.07692
-29.98419 -0.05509
-22.23368 -0.14849
-27.98201 -0.17104
-43.75068 -0.47234
-36.86107 -0.74620
-59.95466 -0.61174
-87.59751 -1.92260
-60.14085 -1.63056
-100.12294 -1.14648
-150.49763
n 0.00001 0.94375 0.00019 1.13147 0.00088
1.98859 0.00047
1.66194 0.00092
1.64286 0.00106
2.06793 0.00119
1.94560 0.00137
1.84723 0.00146
2.24778 0.00126
1.96034 0.00331
2.44134 0.00271
2.84074 0.00956
2.99692 0.01054
3.65421 0.00471
3.13618 0.02975
4.03531 0.01177
3.44929 0.00107
3.71545
Table C.2: Regression of credit spreads on one-quarter lagged credit spreads and contemporaneous output growth and inflation.
151
Ratings Maturity AAA lYr
4Yr
lOYr
AA lYr
4Yr
lOYr
A lYr
4Yr
lOYr
Regime L
H
L
H
L
H
L
H
L
H
L
H
L
H
L
H
L
H
Adj. R2
0.00027
0.04671
0.00012
0.04527
0.00003
0.14728
-0.00027
0.10216
0.00012
0.10108
0.00003
0.16086
0.00192
0.12524
0.00012
0.12108
0.00003
0.18138
Const 0.00000 0.99873 0.00000
-0.98186 0.00000
-1.53393 -0.00002 -1.30283 0.00000 0.30677
-0.00013 -2.44587
0.00000 1.36392
-0.00005 -1.67418 0.00000
-1.53393 -0.00013
-2.01004 0.00000 0.30677
-0.00014 -2.35655
0.00000 5.33145
-0.00011 -1.54820 0.00000
-1.53393 -0.00018
-2.18717 0.00000 0.30677
-0.00014 -1.80489
St-l
0.00000 1.34868 0.17874
12.27943 0.00000
-1.47666 4.61410
11.83308 0.00000
-1.43791 19.44407
18.11847 -0.00129 -0.07219 10.60683
16.20919 0.00000
-1.47666 22.21483
16.96320 0.00000
-1.43791 22.00478
17.44482 -0.01015 -0.21111 24.31079
16.81542 0.00000
-1.47666 31.28069
17.77467 0.00000
-1.43791 24.91307
15.48597
9t 0.00000
-0.81007 -0.00044
-6.46414 0.00000 1.50965
-0.01219 -6.74934
0.00000 -1.39023 -0.08934
-17.97782 0.00000
-1.40274 -0.03944
-13.01610 0.00000 1.50965
-0.07222 -11.90935
0.00000 -1.39023 -0.11968
-20.49019 -0.00001
-3.67655 -0.10756
-16.06608 0.00000 1.50965
-0.11707 -14.36590
0.00000 -1.39023 -0.18381
-24.67431
7T*
0.00000 -0.08745 0.00000
-1.24933 0.00000
-0.00346 0.00004 0.11633 0.00000
-0.47214 0.00009 0.08786 0.00000 0.85517 0.00010 0.16828 0.00000
-0.00346 0.00007 0.05839 0.00000
-0.47214 0.00017 0.14362 0.00000
-0.60373 0.00032 0.23402 0.00000
-0.00346 0.00014 0.08395 0.00000
-0.47214 0.00047 0.31097
Table C.3: Regression of AAA-A credit spreads with regime switching on one-quarter lagged credit spreads and contemporaneous output growth and inflation.
152
Ratings Maturity BBB lYr
4Yr
lOYr
BB lYr
4Yr
lOYr
B lYr
4Yr
lOYr
Regime L
H
L
H
L
H
L
H
L
H
L
H
L
H
L
H
L
H
Adj. R2
0.00147
0.12802
0.00012
0.17955
0.00003
0.23011
0.10526
0.21008
0.07805
0.27889
0.06888
0.30247
0.17336
0.27655
0.12617
0.30513
0.13992
0.31605
Const 0.00000
5.55824 -0.00011 -1.50533 0.00000
-1.53393 -0.00030 -1.83553 0.00000 0.30677 0.00018 1.57091 0.00002
41.30838 0.00074 1.93776 0.00001
31.47698 0.00361
10.08167 0.00001
28.87154 0.00607
32.24364 0.00073
58.37372 0.01317
14.12400 0.00057
44.09497 0.01906
44.16364 0.00193
47.31271 0.01954
126.88663
S t_l
-0.01113 -0.20056 25.88789
16.92799 0.00000
-1.47666 58.40557
17.42646 0.00000
-1.43791 25.24195
10.55921 1.39768 0.89913
132.36116 16.55827
0.57350 1.06485
57.72695 7.74413 0.26828 0.22812
12.72014 3.24371 34.25463 0.86412
238.21676 12.26141
34.73198 0.85715
27.26243 3.03125
127.16300 0.99215
12.05100 3.75519
9t -0.00001
-3.22549 -0.11590
-16.36686 0.00000 1.50965
-0.35596 -22.93621
0.00000 -1.39023 -0.36685
-33.14091 -0.00118
-25.01402 -1.01158
-27.32880 -0.00035
-21.20930 -1.36461
-39.53406 -0.00071
-19.89198 -0.78914
-43.45833 -0.04007
-33.38108 -3.35693
-37.31459 -0.03400
-27.70629 -1.82386
-43.79424 -0.11410
-29.39506 -0.66480
-44.73724
VTi
0.00000 -0.97650 0.00027 0.18778 0.00000
-0.00346 0.00110 0.34905 0.00000
-0.47214 0.00246 1.09309 0.00001 1.01728 0.00542 0.72128 0.00000 0.48806 0.01376
1.96325 0.00000 0.46611 0.00854
2.31718 0.00045
2.68753 0.02789
4.52740 0.00031 1.14986 0.02181
4.58012 0.00106 1.23075 0.00453
3.50176
Table C.4: Regression of BBB-B credit spreads with regime switching on one-quarter lagged credit spreads and contemporaneous output growth and inflation.