Credit Risk Premia and a Link to the Equity Premium

Tobias Berg, Christoph Kaserer

Draft, this version: October 08, 2007

Abstract

While the equity premium is - both from a conceptual and empirical perspective - awidely researched topic in finance, the analysis of credit risk premia has only recentlyattracted wider attention. Credit risk premia are usually measured as the multiplebetween risk neutral (derived from bonds or CDS spreads) and actual default proba-bilities (derived from ratings of rating agencies or market implied rating as MoodysKMV or Altmans Z-Score), which we will call Q-to-P-ratio. Empirical studies havethough shown, that these Q-to-P-ratios are not simply a measure of risk aversion, butalso depend on factors such as credit quality and industry sector.

In this paper, we propose a different measure for extracting risk premia out of creditvaluations which is based on structural models. This approach is able to - qualitatively- explain the observed variations in the Q-to-P-ratio from empirical studies and hasseveral advantages: First, it is only based on observable parameters; second, it is con-sistent with classical portfolio theory; third, it is robust with respect to model changes(besides the classical Merton model we examine the Duffie/Lando (2001) model withunobservable asset values and deviations from the log-normal assumption) and fourth- and most importantly - it directly yields the market sharpe ratio and therefore allowsfor a direct comparison with the equity premium.

Based on an CDS spreads of the 125 most liquid CDS in the U.S. from 2003 to 2007,we show that appr. 80% of the CDS spreads can be explained by credit risk basedon structural models with unobservable asset values. We derive an average implicitmarket sharpe ratio of appr. 40%, adjusting for taxes yields an average market sharperatio of appr. 30%. This confirms research on the equity premium, which indicates,that the historically observed sharpe ratio of 40-50% (corresponding to an equity pre-mium of 7-8% and a volatility of 15-20%) was partly due to one-time effects.

Keywords: credit risk premium, equity premium, credit risk, structural models of default

Tobias Berg, Department of Financial Management and Capital Markets, Munich University of Tech-nology

Prof. Christoph Kaserer, Head of the Department of Financial Management and Capital Markets,Munich University of Technology

I

1 INTRODUCTION 1

1 Introduction

Risk premia in equity markets are a widely researched topic. The risk premium in equitymarkets is usually defined as the equity premium, e.g. the excess return of equities over riskfree bonds. Adjusting for different volatilities yields the sharpe ratio, i.e. the excess returndivided by the overall market volatility. Generally, the measurement of equity premia canbe dividend into three main approches: Models based on historical averages, dividend orcash-flow-discount models and models based on utility functions. While historical averageshave long dominated theory and practical applications, current research suggest an upwardbias, e.g. the ex post realized equity returns do not correctly mirror the ex ante priced equitypremium.1 Dividend-/Cash-flow-discount models have become more popular, but are alsosubject to debate, especially for their rather high sensitivity to growth/dividends/earningsforecasts. While approaches based on utility functions have been subject to intensive debatein the academic literature2, its use in practical applications is currently of minor importance.

Risk premia are though not limited to equity markets, the same logic does also apply tocredit markets in theory and it has been observed in empirical studies. For example, theaverage 5-year CDS spread for a A-rated obligor in the CDS.NA.IG-index over the past 3years has been 36 bp, whereas the average annual expected loss over 5 years is less than 10bp, e.g. 5 year CDS investments (approximately) yield an average return of appr 26 bp overthe risk free rate, see figure 1. In absolute terms, this premium increases with decreasingcredit quality (i.e. the expected net returns increase with increasing riskyness). Measuredrelative to the expected loss (or the actual default probability) it decreases with decreasingcredit quality. Over the last year, research about this default risk premium has developed,but there has not yet emerged consensus on the methodology for measuring this defaultpremia.

Duffie and Singleton (1999) model the default event via default intensity processes asan inaccesible stopping time. This approach has gained popularity especially for hedgedbased pricing applications restricted to debt markets. Assuming a recovery of 0%, a riskyzero coupon bond can simply be priced via a simple additional discount factor to the riskfree interest rate, which represents the loss in survival probability over the next instance oftime, i.e.

B(T ) = EQ[e T0 r(s)+(s)ds],

where r captures the risk free rate dynamics whereas captures the risk neutral defaultintensity dynamics.Fixing the risk attitude of investors, it is though not clear if - when examinign differ-ent obligors - the absolute default risk premium should be a fixed percentage of the actual

1Reasons are survivorship bias, risk premium volatility, interest rate level and state of the economy, seefor example Claus/Thomas (2001), Illmanen (2003), Fama/French (2002).

2This debate is mainly based on the so called Equity Premium Puzzle by Mehra/Prescott (1985),see Mehra (2003) for an overview about different utility based approaches including alternative preferencestructures, disaster states and survivorship bias and borrowing constraints.

1 INTRODUCTION 2

Figure 1: Relationship between 5-year CDS spreads (mid) and 5-year annualized expectedloss per rating grade for investment grade ratings.

default probability (i.e. Q = c P ), a fixed amount (in bp) independent of the ratinggrade/actual default probability (i.e. Q = P + c) or if other (e.g. logarithmic) relation-ships are to be preferred. Many researches opt for a fixed multiple3, i.e. they examinethe relationship between risk neutral and real world default probability or default intensity,e.g.

Q

P, which we will denote throughout this paper as Q-to-P-ratio4. Berndt et.al. (2005)

come to the conclusion, that the Q-to-P-ratio is higher for high quality firms and lower forlow quality firms. They use a linear and a log/log relationship between CDS spreads anddefault probabilities, resulting - especially for the linear relationship - in an intercept ofroughly 50 bp, which is more 30 times the standard error and does not seem to be plau-sible even accounting for liquidity risk. Amato and Remolona (2005) principallly confirmthese findings. In addition, Amato (2005) finds a significantly smaller default premium forfinancial services companies. Both papers find a positive correlation between risk premium

3Among others Berndt et.al. (2005), Amato (2005), Driessen (2003), Liu et.al. (2000).4Dependent on the specific rating methods, credit risk methodology and the overall context this ratio

is also referred to as risk neutral to actual default intensity (see for example Berndt et.al. (2005)) or - ifadjusted for recovery rates - CDS-to-PD or CDS-to-EDF-ratio (if KMVs expected default frequencies areuse) by other authors.

1 INTRODUCTION 3

and average default probability, i.e. an increasing average default probability in the marketis accompanied by an increasing risk premium. Driessen (2003) uses constant Q-to-P-ratiosby modelling the risk neutral default intensity as a constant multiple of the actual defaultintensity (over time and for different obligors). On average, he reports a Q-to-P-ratio of 2.31(S&P) and 2.15 (Moodys) respectively, although the standard errors are quite large (> 1%).

The main contribution of this paper is threefold. First, we explain the variations in theQ-to-P-ratio for different rating grades and different industry sectors observed in the litera-ture based on a simple Merton style model. Second, this paper is - to our best knowledge -the first to theoretically analyze the Q-to-P-ratio in an incomplete information setting (basedon the model of Duffie/Lando (2001)) - which enables us to model the effect of uncertainty onthe Q-to-P-ratio - and the effect in deviations from the lognormal distribution assumption.We show, that higher Q-to-P-ratios may - besides higher risk aversion - also be explained byhigher uncertainty of the current asset value, due to a concave relationship between defaultprobability and Q-to-P-ratio. Third - and most importantly - we directly extract the marketsharpe ratios out of CDS spreads, which gives us a direct link to the equity premium. Wemeasure an upper bound for the average sharpe ratio of 29 - 42%, which is consistent withresearch of equity markets.

We use a simple relationship between risk neutral and real world default probability inthe Merton framework, i.e.

Qdef (t, T ) = [1(P def (t, T )) + SRAssets

T t

](1)

The resulting relationship between risk neutral and actual default probability5 (Q-to-P-ratio) is neither linear nor log-linear, but involves the cumulative normal distribution. Pleasenote, that we do not try to estimate the actual and risk neutral default probability seper-ately; we simply assume that we know the actual default probability from rating information;calibrate the model accordingly and from there derive risk neutral default probabilities; e.g.we do not have to calibrate the asset value, default barrier or volatility separately. With thismodel, we are able to include the correlation and volatility of a firms assets and thereforederive a result which is consistent with classical portfolio theory (e.g. the dependence ofreturns on the systematic, rather than total, risk inherent in a claim). We will then beable to explain the high intercept from Berndt et.al. (2005) (which is due to a non-linearrelationship between real world and risk neutral default probability), the observed high riskpremia for high quality firms (which is a combined effect of the non-linearity of real worldand risk neutral PD and a larger part of ideosyncratic risk incorporated in high quality firms).

5Please note, that we there is minor a difference between the ratio of cumulative default probabilities, i.e.Qdef (t,T )Pdef (t,T )

and the ratio of the default intensities QP =ln(1Qdefdef(t,T )ln(1Pdefdef(t,T ) defined by Q := ln(1Q

defdef(t,T ))T

and P := ln(1Pdefdef(t,T ))T respectively, which is in first order equal to the ratio of cumulative default

probabilities. Throughout this paper, we will always analyze the ratio of the default intensity for reasons oftheoretical consistency.

1 INTRODUCTION 4

The model also allows for an extraction of the risk premium out of CDS spreads, whichis - in theory - not dependent on the quality of the firm or the sector the firm operates,but merely mirrors the risk attitude of investors and directly yields the sharpe ratio of themarket portfolio:

SRM :=M rM

1(Qdef (t, T )) 1(P def (t, T ))

T t 1

E,M=: Merton, (2)

where E,M denotes the correlation between the market portfolio and the equity value of thefirm.6 Since credit spreads are very sensitive to the sharpe ratio, this is a very convenientway: for example, a BBB-rated obligor would have a 5-year credit spread of 73 bp with asharpe ratio of 10% compared to a 5-year credit spread of 280 bp with a sharpe ratio of 40%,see subsection 2.1 for details. This difference shows, that the common noise in the data willnot significantly reduce the possibility to extract the sharpe ratio out of market prices.

Extending the simple Merton model to more advanced models (we examine a classical firstpassage time model with observable asset values, a model with unobservable asset valuesproposed by Duffie et.al. (2001) and deviations from the lognormal assumption.) showsan astonishing robustness of the results (1) and (2) derived in the Merton framework forinvestment grade companies as long as the asset volatility is larger than approximately 10%,which can reasonably be assumed for all companies outside the financial services sector.7

For smaller asset volatilities, the results of the first passage time models deviate from thestandard Merton model. With these models we are able to theoretically explain the lowerobserved risk premia for financial service companies (which is due to a significantly smallerasset volatility of financial services companies)8 More generally speaking, by introducing a(model- and parameter-dependent) adjustment factor by

SRM = Merton AF, (3)we show that

The adjustment factor is close to one for all models analyzed as long as the asset volatil-ity is below 10% and the resulting actual default probability belongs to an investmentgrade rating.

The adjustment factor can be accurately determined simply based on knowledge of thematurity and the actual default probability (i.e. parameters that can be observed in

6The correlation between equity value and market portfolio was used as a proxy for the correlationbetween asset value and the market portfolio. In the Merton framework as well as in more general structualmodels, it can be shown that for reasonable parameter choices (e.g. resulting in a credit quality / ratinggrade of B or higher) these two correlations do not differ by more than 1%, see section 2.

7In our sample of 125 companies of the CDS index CDX.NA.IG appr. 90% of all non-financial companieshad an asset volatility of 10% or larger based on data from Moodys KMV. In contrast, for fincial servicescompanies, the volatility is 10% or smaller in appr. 75% of all cases.

8Due to a lack of data of financial services companies with asset volatilities above 10% and/or non-financial services companies with an asset volatility below 10%, it is though hard to empirically verifiy thesetheoretical results, see section 3 for details.

2 MODEL SETUP 5

the market) as long as > 10%, i.e. for a given combination of default probabilityand maturity, parameters that can not be observed easily (e.g. asset volatility, defaultbarrier, asset value or accounting noise) do not significantly affect the adjustmentfactor.

Furthermore, the model directly yields the sharpe ratio of the market portfolio and thereforeallows a direct comparison with the equity premium. Applied to all NYSE-listed companiesin the investment grade CDS index CDX-NA.IG from 2003 to 2007 and using EDFs fromKMV as a proxy for the actual default probability results in an average implied marketsharpe ratio of 42% and an average company sharpe ratio of 20%. Adjusting for tax effectsyields a market sharpe ratio of 32% and an average company sharpe ratio of 16%. UsingMoodys ratings instead of EDFs shows similar results (market sharpe ratio of 39% and 29%after tax adjustments). We used CDS spreads as they are unfunded exposures, so that therates should be less sensitive to liquidity effects.9 Based on the theoretical models, appr.80% of the CDS spreads can be explained by credit risk.

The remainder of the paper is structured as follows. Section 2 describes the theoreticalframework for credit risk premia based on asset value models, including a discussion ofthe impact of different asset models and deviations from the lognormal distribution on thewidely used Q-to-P-ratios. We examine a classical Merton model, a first passage time model,a model based on unobservable asset values as proposed by Duffie/Lando (2001) and assetdistributions based on actual S&P returns. Using a model with unobservable asset values isfundamental in our point of view, since only these models are able to explain credit spreadsobserved in the markets and yield a default intensity, which constitutes the basis of moderncredit pricing models. Section 3 describes our data sources and shows our empirical resultsfor the risk premium. Section 4 concludes.

2 Model setup

This section discusses the theoretical framework for extracting risk premia out of CDS prices.The basic idea is to use structural asset models to derive a relationship between risk neutraland actual default probability. Most structural models show a quite poor performance inempirical studies10. One of the main reasons is the calibration process usually needed tospecifiy structural models, e.g. determination of leverage, asset volatilities, etc. In contrastto most parts of the literature11, we do though not aim to derive actual and risk neutraldefault probabilities from structural models, we are simply interested in the relation between

9Compare Berndt et.al. (2005) for similar arguments.10See Schonbucher (2003) for an overview of empirical studies.11Huang/Huang (2003), Bohn (2000) and Delianedis/Geske (1998) use a similar approach, our approach

differs though in at last three ways: First, we work in an incomplete information setting, whereas we explicitlyfocus on models with information uncertainty; second, we use CDS spreads, which should be less sensitiveto liquidity distortions; third, we are - to our best knowledge - the first who directly aim to extract the riskattitude out of credit prices.

2 MODEL SETUP 6

risk neutral and actual default probabilities. We simply assume, that there exists a struc-tural model yielding the correct actual default probability and from there derive the riskneutral default probability. We can therefore omit (most of) the calibration process whichresults in stable calculation.

Subsection (2.1) discusses a simple Merton model and shows, that most empirical resultsabout the Q-to-P-ratio can already be explained - at least qualitatively - in this setting.Subsection (2.2) expands the framework to a simple first passage time model. The results donot materially differ compared to the simple Merton framework as long as the asset volatilityis above 10 Percent. Asset volatilities smaller than 10% (which are usually only observedfor financial services companies) lead to a quite large deviation from the standard Mertonmodel. We are then able to explain the impact of volatility on the Q-to-P-ratio, especiallylower observed Q-to-P-ratios for financial services companies.Subsection (2.3) examines the model of Duffie/Lando (2001) and again shows the robustnessof the simple Merton model for our purpose. Subsetion (2.4) analyzes the impact of devia-tions from the assumption of lognormally-distributed asset returns.

The following subsections will always be structurd in the same way: first, the model setupincluding the main assumptions is given and formulaes for calculating actual and risk neu-tral default probabilities are given; second, the Q-to-P-ratio is analyzed within the respectivemodel; third, the implications for estimating the sharpe ratio are analyzed.

2.1 Merton

Model setup:

Structural models for the valuation of debt and the determination of default probabili-ties were already mentioned in the well-known Black/Scholes (1973)12 paper. The Mertonframework presented in the subsection is based on the famous paper of Merton (1974)13,which explicitely focusses on the pricing of corporate debt.

In this framework, a companys debt simply consists of one zero-bond. Default occurs,if the asset value of the company falls below the nominal value of the zero bond at thematurity of the bond. A company can therefore only default at one point in time, whichobviously poses a simplification of the real world.

Before we will derive an estimator for the market price of risk () based on the Mertonframework, the main assumptions will be presented:

Assumption 2.1 (Assumptions Merton framework)

12See Black/Scholes (1973).13See Merton (1974).

2 MODEL SETUP 7

1. The Assets Vt follow a geometric Brownian motion with constant drift = V (underP) and r (under Q) and constant volatility = V > 0

14, i.e.

under P : dVt = Vtdt + VtdBt VT = Vt e( 122)(Tt)+(BTBt)

under Q : dVt = rVtdt + VtdBt VT = Vt e(r 122)(Tt)+(BTBt).2. The companys debt consist of one single zero-bond with nominal N and maturity T.

3. Default occurs if VT < N (a default can therefore only occur at t=T).

Assumptions 1 is a standard assumption in financial economics, assumption 2 and 3 aresimplifications of the real world and will be relaxed in the next subsections.

Under these assumptions, the real world default probability P def (t, T ) between t and T15

can be calculated as follows:

P def (t, T ) = P [ VT < N ] = P [ Vt e ( 122)(Tt)+(WTWt) < N ]= P

[ (WT Wt) < ln

(N

Vt

) ( 1

22) (T t)

]=

[lnN

Vt ( 1

22) (T t)

T t

]. (4)

The default probability under the risk neutral measure Q can be calculated accordingly as

Qdef (t, T ) = Q[ VT < N ] =

[lnN

Vt (r 1

22) (T t)

T t

]. (5)

Combining (4) and (5) yields16

Qdef (t, T ) =

[1(P def (t, T )) +

r

T t]

(6)

andV rV

=1(Qdef (t, T )) 1(P def (t, T ))

T t (7)

respectively, see figure 2 for an illustration. Please note a main advantage of this formula:it directly yields the sharpe ratio of the assets; i.e. neither V and V nor Vt, N or r have tobe estimated separately (and - on the other hand - can not be inferred from (7) separately).

14For practical reasons, the index V will usually be omitted in the following formulas. To avoid potentialconfusion, the index V will though be used in the main formulas and results of this section.

15Of course, in the Merton framework, a default is only possible at time T, for reasons of consistency withthe following subsections, we will always talk about default probabilities between t and T in this context,too.

16See for example Duffie/Singleton (2003) for the first equation.

2 MODEL SETUP 8

Figure 2: Illustration of the relationship between actual and risk neutral default probabilitiesin the Merton framework. PDcum: actual cumulative default probability, Qcum: risk neutralcumulative default probability, SRV : sharpe ratio of the assets, T: maturity.

Q-to-P-ratio:

The resulting relationship between CDS spreads / risk neutral default probabilites and ac-tual default probabilities is shown in figure 4. This relationship is increasing and concave,as can be seen by the first two derivatives of (6) with respect to P def :

Q

P=

12 pi e

0.5(SRT+1(P def )2 1

P def

= e0.5(SR2T+2SRT1(P def ) > 0

and

2Q

P 2= Q

P SR

T2pie0.5(

1(P def ))2 < 0,

If one tries to make a linear regression between the risk neutral and the actual defaultprobability, the intercept will be significantly above zero, which is a result of the concavity of

2 MODEL SETUP 9

Figure 3: Relationship between CDS spread and actual default probability in the Mertonframework and results of a linear interpolation (i.e. assumption of a credit quality indepen-dent Q-to-P-ratio. Parameters: r=5%, SRA = 30%, RR=50%.

the relationship between the risk neutral and the actual default probability implied by (10)and confirms the empirical research, see for example Berndt et.al. (2005). Please note thatthe specific parameters of the regression (slope and intercept) are rarely arbitrarely based onthe choosen interval of actual default probabilities and the choosen data points (which areusually clustered at regions of investment grade default probabilities, e.g. one-year-defaultprobabilities up to appr. 0.3% and 5-year cumulative default probabilities up to appr. 3.5%).Plotting the Q-to-P-ratio and the actual default probabilities (or, similarly, the rating) basedon (10) shows, that the Q-to-P-ratio declines with declining credit quality (see figure 4),which underlines empirical research on this topic (See Berndt et.al. (2005) and Amato(2005)). Again, this is a direct result of the concave funcion between the risk neutral andthe actual default probabilities implied by (10).

Market sharpe ratio:

If we try to extract the market sharpe ratio out of (6), we are faced with an additionalproblem: the sharpe ratio of the assets V r

Vwill usually differ from the market sharpe ratio,

since the assets Vt will usually not be on the efficient frontier. In other words, the sharperatio of the assets does not only capture the risk attitude of investors, but also depends on

2 MODEL SETUP 10

Figure 4: Relationship between the Q-to-P-ratio and the rating/actual default probabilityin the Merton framework. Parameters: r=5%, SRA = 20%.

the correlation of the assets with the market portfolio. The market price of risk can thoughbe calculated via a straight forward application of the CAPM:

V = r +M rM

V,M V 17 M rM

=V rV

1V,M

, (8)

where V,M denotes the correlation coefficient between the asset return and the market re-turn.18

17Without loss in generality we assume V,M 6= 0.18An application of the CAPM does require the respective assets to be traded or - equally - it requires

a self-financing trading strategy resulting in a contingent claim equal to the asset payoff. Asset values areusually not traded on financial markets, we do though believe that this application is justified for two reasons.First, the asset value can - in theory - be duplicated by the equity value and a risk free bond in the Mertonframework, so the asset value lies in the asset span of the market. Second, most of the claims on the assetsof large corporations are either directly traded (e.g. equity and bonds) or market values can be inferred fromdirectly traded instruments with a certain accuracy (e.g. bank loans via bonds). In addition, non-tradableparts like insolvency costs in more advanced models do have an impact on the choice of the optimal capitalstructure, given the actual default probabilities observed in the markets (most large companies are ratedinvestment grade), these do not have a significant effect on the asset value and are also hedgeable in thesemodel-setups.

2 MODEL SETUP 11

Therefore, in addition to the sharpe ratio of the assets, we will need an estimate of thecorrelation between the asset value and the market portfolio. At first, this correlation (V,M)seems to be a problem for practical applications, since it can neither be directly measurednor implicitly inferred e.g. from option prices. In the following applications, we will ap-proximate V,M by the correlation between the corresponding equity return and the marketreturn (denotet by E,M), i.e. we will assume that

V,M E,MThe error of this approximation is almost negligible, since - within the Merton framework -the equity value of a company equals a deep-in-the-money call option on the assets. For allreasonable parameter choices, the approximation error is less than 1%, see appendix A fordetails. Hence the following approximation holds:

M rM

1(Qdef (t, T )) 1(P def (t, T ))

T t 1

E,M

and we can formulate a Merton estimator for the market price of risk = MrM

:

Merton :=1(Qdef (t, T )) 1(P def (t, T ))

T t1

E,M; (9)

or, solved for the risk neutral default probability:

Qdef (t, T ) =

[1(P def (t, T )) + SRM

1

E,M

](10)

Please note, that we will need a sufficient sensitivity of the risk neutral default probabilityQdef (t, T ) with respect to the sharpe ratio in order for an empirical application. Otherwisenoise in the data (e.g. bid-ask-spreads, inaccuracies in determining correlations and actualdefault probailities) will result in a very inaccurate estimation. That this sensitivity is indeedgiven can be seen by calculating the first derivative of (10) with respect to the sharpe ratio,i.e.

Qdef (T )

SRV=

12pi

e 12(SRV T+1(P def (T )))2

T. (11)

see figure 5 for an illustration. If we look for example at a BBB-rated obligor with a 5-yearcumulative actual default probability of appr 2.17%, the resulting risk neutral default prob-ability should be either 3.6% (for an asset sharpe ratio of 10%) or 13% (for an asset sharperatio of 40%) respectively. Assuming a recovery rate of 50% transforms this into a CDSspread of either 73 bp or 279 bp19, a difference that is certainly above any noise induced byliquidity effects, bid/ask-spread or inaccurate measurement of actual default probability.

19Using the approximation CDS spread = Q LGD and by deriving the risk neutral default intensity fromthe risk neutral cumulative default probability by Qdef (t, T ) = 1 eQ(Tt).

2 MODEL SETUP 12

Figure 5: Influence of the sharpe ratio on risk neutral default probabilities in the Mertonframework for different sharpe ratios (10%-40%). Other Parameters: T=5.

Please note again, that this relationship is independent of the asset volatility, the assetvalue Vt and the default barrier L (which is a direct implication of (10)). A calibration of theasset value model will therefore not be necessary to specify the relationship between actualand risk neutral default probabilities.

Although we found a compelling result for an estimation of the market price of risk, theassumptions made in the Merton framework do not fully reflect the true world. In the fol-lowing subsections we will relax the assumptions about the default timing (see subsection2.2) and the assumption about complete information (see subsection 2.3) and look at moregeneral first passage time models. The subsections will always be structured as follows: first,the model setup is explained and formulas for actual and risk neutral default probabilities arederived; second, the resulting Q-to-P-ratio is analyzed; third, the impact on the estimationof the sharpe ratio is discussed.

Before moving on, please note that our estimator for the market price of risk contains acertain kind of robustness against changes in the underlying assumptions: Since both de-fault probabilities (P def and Qdef ) are measured within the same model and substracted

2 MODEL SETUP 13

from each other, the effect of changes in the default modelling (e.g. first passage time) orin the assumptions about distribution of returns (e.g. deviation from the assumption of alog-normal distribution of the asset returns) is - qualitatively spoken - reduced significantly.

2.2 First passage time modell with certain default barrier

Model setup:

In first passage time models, a default occurs as soon as the asset value20 falls below acertain barrier. The asset value and the default barrier can both be either observable orunobservable. The model with a certain default barrier and a certain asset value based onis treated in this section. A model with an uncertain default barrier and unobservable assetvalue based on Duffie/Lando (2001) is analyzed in the following subsection.

For this subsection, the following assumptions apply:

Assumption 2.2 (Assumptions first passage time model with certain default barrier)

1. The Assets Vt follow a geometric Brownian motion with constant drift = V (underP) and r (under Q) and constant volatility = V > 0, i.e.

under P : dVt = Vtdt + VtdBt VT = Vt e( 122)(Tt)+(BTBt)

and

under Q : dVt = rVtdt + VtdBt VT = Vt e(r 122)(Tt)+(BTBt)

2. Default occurs as soon as the asset value Vt falls below a predefined and certain barrierL (i.e. L R)(a default can therefore occur at any time in (t,T].

Under these assumptions, the real world default probability P def (t, T ) between t and Tcan be calculated as

P def (t, T ) = P [Vs < L for any t s T] = 1 P[Vs L, t s T]= 1 P [ min

tsTVs L] = 1 P

[mintsT

[Vt e( 122)(st)+(WsWt)

] L;

]= 1 P

[mintsT

[ 1

22)(s t) + (Ws Wt)

] ln( L

Vt)

]=

(bm(T t)T t

) e 2mb2

(b+m(T t)T t

)(12)

20Some authors use an even more general version of an ability-to-pay process, see Bluhm/Overbeck/Wagner(2003) for example.

2 MODEL SETUP 14

with

b = ln(L

Vt); m = 1

22; = V ,

see especially Musiala/Rutkowski (1997) for the last step. The default probability under therisk neutral measure Q can be calculated accordingly as

Qdef (t, T ) =

(b m(T t)T t

) e 2mb2

(b+ m(T t)T t

)(13)

with

b = b = ln(L

Vt); m = r 1

22; = V .

There is - in contrast to the Merton framework - no closed form solution for the sharpe ratioV rV

in this model.

Unfortunately, we have two equations ((12) and (13) with three unknown variables ( LVt, , ,

assuming we can derive r, T t, P def , Qdef from bond/CDS market data), so we can notdirectly derive and in order to determine the sharpe ratio r

Q-to-P-ratio. We will

though be able to determine as a function of and therefore also the sharpe ratio of theassets (r

) as a function of . This function is nearly constant for > 10%, see figure 6, so

apart from assets with very low asset volatility we will again be able to estimate the sharperatio without calibrating the asset volatility , the asset value Vt or the default barrier L.Please note, that the sharpe ratios in figure 6 do not imply, that assets with a lower volatilityhave a higher sharpe ratio, rather they implicitly derive sharpe ratios out of CDS spreadsbased on empirical data and a first passage time model. A company with a low asset volatil-ity will therefore (all other parameters being equal, especially the actual default probabilityand the asset correlation) trade at lower spreads than a company with a high asset volatility.

Q-to-P-ratio:

The resulting slope of the Q-to-P-ratio as a function of the asset volatility is shown infigure 7 for different rating classes (identified by specific cumulative default probabilities).Again, one can see that the Q-to-P-ratio is higher for higher rating classes, (almost) inde-pendent of the asset volatility for < 10% and sharply declining for asset volatilites smallerthan 10%. It converges to 1 for all rating classes, thus, the slope is more pronounced forhigher rating classes. The dependency on the asset volatility can be explained by the defaulttiming: for lower asset volatilities, the expected default time conditional on a default untilT is lower for lower asset volatilities conditional on a fixed cumulative default probability upto T.21 In other words, fixing the cumulative default probability until time T, defaults willoccur with a higher probability at the beginning of the period if the volatility is low. Since

21Please note that - all other parameters being equal - the default probability declines with declining assetvolatility. Therefore, the expected value of the default time will also decrease. In this case, the decliningasset volatility is always balanced by a lower t0-asset-value to result in order to yield the same rating grade.

2 MODEL SETUP 15

Figure 6: Relationship between the asset volatility and the implicit sharpe ratio derivedfrom CDS spreads for rating class A. P5, mean and P95 denote observed CDS spreads forthe CDX-NA-IG-index from 2004-2006 for all obligor rated A1/A2/A3 by Moodys at the5% (8 bp), 50% (19 bp) and 90% percentile (42 bp). Risk neutral default probabilities werederived with a recovery rate assumption of 50%. The slope of the graph shows quite similarpatterns for other rating classes or recovery rate assumptions.

the Q-to-P-ratio decreases with decreasing maturity, see figure 4 and the related discussionin subsection 2.1, it also decreases with decreasing asset volatility in the first passage timeframework. In addition, one can easily see from (6), that the Q-to-P-ratio converges to oneif the maturity (T-t) converges to zero.

Estimation of the sharpe ratio:

We now want to test the robustness of the simple Merton estimator for the sharpe ratio. Wetherefore introduce an adjustement factor AFFP by

SRM :=M rM

=:1(Qdef (T ) 1P def (T )

T 1M,E

AFFP = Merton AFFP , (14)

i.e. the adjustment factor shows, how far the estimate of the market sharpe ratio via thestandard Merton model deviates from the true market sharpe ratio if a first passage modelapplies. Again, we have assumed that V,M = E,M , i.e. that the correlation between marketand asset returns equals the correlation between market and equity returns. This equation

2 MODEL SETUP 16

Figure 7: Relationship between the Q-to-P-ratio and the asset volatility for different ratinggrades the first passage time model. Parameters: r=5%, SRA = 15%.

holds true for reasonable parameter choices in the first passage time framework as well, aswe will show in the next subsection.The adjustment factor is dependent on the volatility, the sharpe ratio and the credit quality(interpreted as actual default probability) of the company. The relationship is shown infigure 8 for a sharpe ratio of 20%. It shows, that the adjustment factor increases withdecreasing credit quality and with increasing volatility, but for investment-grade titles anda volatility smaller than 10% the adjustment factor is very close to 1.

2.3 A model with unobservable asset values

Model setup:

Although an improvement compared to the Merton model, credit spreads predicted by sim-ple first passage time models are still not able to predict the credit spreads that can beobserved on the markets.22 Especially for short term maturities, market credit spreads (orrisk neutral default probabilities respectively) are higher than a simple first passage timemodel would suggest.

22See Duffie/Lando (2001), Duffie/Singleton (2003) and Schonbucher (2003) for a detailed discussion.

2 MODEL SETUP 17

Figure 8: Credit quality smile: Adjustment factor in the first passage time framework.Parameters: r=5%, SRA = 20%. Please note that the majority of traded bonds and CDS(by volume) has an investment grade rating.

These higher credit spreads for short term maturitites seem to be mainly attributable tocredit risk and are not due to liquidity effects, other risk categories or market imperfections(See for example Schonbucher (2003).). Several methodological reasons are discussed in theliterature:

The asset value may be uncertain due to imperfect information structures, or - tobe more precise - the current asset value can not be observed exactly. Therefore,the current asset value becomes a random variable, which in turn has the effect ofincreasing short term default probabilities.

The default barrier may be uncertain/unobservable.This is consistent with the fact that the recovery rate is usually assumed to be a randomvariable rather than a fixed value.23 In the simple first passage time model presentedin the last subsection, the barrier was certain, therefore the recovery rate should alsobe non-random. An unobservable default barrier leads to a significant increase in the

23See for example Moodys (2007). A random recovery rate could though also be induced by introducingrandom insolvency costs, i.e. costs incurred at default due to direct insolvency expenses, losses in asset valuedue to a forced sale in an insolvency process and revaluation of assets serving a specific purpose for therespective company.

2 MODEL SETUP 18

short term default probability, long term default probabilities are less effected, sincethe asset volatility dominates uncertainty for longer time periods.

Asset values are not lognormally distributed and may include jumps. This increasesthe short term probability, that the asset value will fall below the default barrier.24

The arguments all have an economic foundation and seem to be reasonable. It is not withinthe scope of this thesis to evaluate, which one (or which combination) best produces thetrue economic world. Incorporating jumps in the asset value process has been analyzed byZhou (1997). Duffie/Lando (2001) have developed a first passage time model with uncertainasset value. The assumption of an uncertain default barrier has been implemented in thecommercial model CreditGrades by Finger et.al. (2002).

In this subsection, we will focus on the model of Duffie/Lando (2001) and show that -although the default probabilities implied by this model differ substantially from the clas-sical Merton model - the difference between risk neutral and actual default probabilities isalmost the same than in the Merton model as long as the asset volatility is above 10%. Tobe more precise, we will show that the simple Merton estimator for the market sharpe ratio(see (9)) is - to a certain degree of accuracy - still valid in the Duffie/Lando-framework forasset volatilities above 10%.

We choose the Duffie/Lando model for this analysis as it is the most general frameworkfor credit risk modeling we know of. It incorporates a sophisticated structural model ofdefault (i.e. a strategic setting of the default barrier based on the asset value process, taxshield and insolvency costs) and - even more important for this setting - an unobservableasset process resulting in realistic default term strucures with a default intensity.25

The main assumptions in the Duffie/Lando model are as follows:26

Assumption 2.3 (Assumptions Duffie/Lando model)

1. The Assets Vt follow a geometric Brownian motion with constant drift m = (under P) and r (under Q) - where denotes the asset payoff rate - and constantvolatility = V > 0, i.e.

under P : dVt = mVtdt + VtdBt VT = Vt e(m 122)(Tt)+(BTBt)24Note that for long term maturities, a higher volatility has the same effect as adding jumps to the process,

therefore long term default probabilities will not be effected in the same manner than short term defaultprobabilities.

25Defaults in a Merton framework can not be described via default intensity processes, since the probabilityof a default from t (today) until t+ t is always zero or one for a sufficient small t. A default intensity doesalso not exist in the the Zhou (1997) framework, since the default time cannot be represented by an totallyinaccessible stopping time (which is a consequence of the fact, that the default barrier may be hit/crossedvia the normal diffusion process with positive probability), see Duffie/Lando (2001) for details.

26See Duffie/Lando (2001) for details.

2 MODEL SETUP 19

and

under Q : dVt = r Vtdt + VtdBt VT = Vt e(r 122)(Tt)+(BTBt)

2. Default occurs as soon as the asset value Vt falls below a predefined and certain barrierVB (i.e. L R), which is determined by the equity holders in t=0 as to optimize thetotal initial firm value (equity + proceeds of the sale of debt).

3. The bond holders / CDS investors are not able to observe the asset process directly.

Instead they receive imperfect information Y (ti) := ln(Vti

)= ln(Vti) + Uti, where

U(ti) is normally distributed and independent of B(t) and is a parameter specifyingthe degree of noise in the information received by the bond/CDS investors. Therefore,the information filtration given to the bond/CDS investors is27

Ht = ({Y (ti), ..., Y (Tn), 1s : 0 s t}) (15)

We will examine the case n=1. Under these assumptions, the conditional density of Zt :=ln(Vt), > t

28 conditional on the noisy observation Yt and with a fixed starting pointln(V0) =: z0 can be calculated as

29

b(x, t|Yt, z0) := P [ > t Zt dx|Yt) = P [ > t Zt = x Yt = Yt]P [Yt = Yt]

=P [ > t Zt = x Ut = Yt x]

P [Yt = Yt]=P [ > t|Zt = x] P [Zt = x] P [Ut = Yt x]

P [Yt = Yt],

where we used the equivalance of Yt() = Yt and Ut() = Yt x under the condition thatZt() = x in the second step and Bayes rule in the first and third step. Conditioning on > t30, this yields the conditional density g(x|Yt, z0, t) of the log asset value Yt conditionalon the information Yt = Yt and > t:

g(x|Yt, z0, t) = b(x|y, z0, t)P [ > t|Yt, z0]

This conditional density can be explicitly calculated, if it is assumed, that asset values followa geometric Brownian motion and Ut has a normal distribution, see Duffie/Lando (2001) fordetails.To calculate cumulative default probabilities requires a weighted application of (12) over all

27Of course, the bond/CDS investors can also obvserve, if a default has already occured.28 denotes the stopping time representing the default point.29In the following, we will use the conventional informal notations P [X = x] or P [X dx].30Please note that the investors are of course able to observe, that no default has taken place up to time

t.

2 MODEL SETUP 20

possible asset values Vt, where the weight is - roughly speaking - the probability of the assetvalue Vt

31, i.e.

P def (t, T ) =

VB

P defFP (t, T, x) PD(first passage time), if Vt=x

g(x|Yt, z0, t) Prob., that Vt=x

dx (16)

where P defFP (t, T, x) denotes the probability, that an asset value process starting in t=t atVt = x will fall below the default barrier up time t=T, see (12), and g is the conditionaldensity of the asset value at t=t given the filtration Ht.

Q-to-P-ratio:

Formula (16) can now be used to calculate the risk neutral as well as the real world de-fault probability by either using the risk neutral drift of the asset (e.g. the risk free rate lessthe payout rate) or the actual drift of the assets.32

Unfortunately, there is no closed form solution for (16), we though have to draw back onnumerical solutions. Figure 9 shows the Q-to-P-ratio for = 10% and varying actual defaultprobabilities (identified as ratings) and asset volatilities.It can be seen, that - for a given rating grade - the Q-to-P-ratio is again almost independentof the asset volatility as long as > 10% and sharply declines for < 10%.Uncertainty (measured by and T1) results in slightly higher Q-to-P-ratios. This effect israther small but increasing with increasing credit quality, see figure 10 for = 15% andvarying values for ). The explanation is rather simple:

To calculate default probabilities in the Duffie/Lando model, we simply have to calculatethem in the first passage time framework with observable asset values and average the resultfor different possible levels of PDcum (representing different levels of the actual asset level Vt)with respect to the conditional density of Vt. Since the Q-to-P-ratio is a convex function of the

default probability in the Merton framework (see figure 4), i.e. E[Qdef (P def )P def

] Qdef (E[P def ])E[P def ]

as a result of Jensens inequality, an increasing leads to an increase in the Q-to-P-ratio,although the effect is rather small for the parameters analyzed. The effect is though stillquite interesting: An increase in the Q-to-P-ratio does not necessarily mean, that agentshave become more risk averse, it can also reflect a higher (actual or perceived) noise in assetvalues. This effect is larger for high quality companies, so that in states of higher uncertainty,spreads on good quality companies should increase most. This finding can also have effectson trading strategies which try to use risk aversion measures depicted from credit marketsfor other market segments (e.g. equities)33

31Of course the probability of a single value Vt will be zero for non-degenerated parameter choices, since weoperate in a continious setting. We will still use this informal notation to allow for a better understanding.

32Please note, that the drift changes P defFP (t, T, x) as well as the conditional density g(x|Yt, z0, t). Theinfluence on the conditional density decreases with decreasing .

33There are several tactical asset allocation strategies which use risk aversion derived from credit markets,equity markets and money markets (e.g. TED spreads) to allocate funds to different asset classes. Using

2 MODEL SETUP 21

Since asset volatilites and accounting noise is usually hard to estimate, this a very convenientresult for our purpose. If we want to extract the risk premium out of CDS spreads, assetvolatilites and accounting noise does merely play a role given a specific actual default prob-ability. The basic logic behind this is quite simple: the accounting noise does not pose anysystematic risk, therefore the difference between risk neutral and real world default proba-bility is merely unaffected; the same logic applies to the payout rate, the asset value leveland the time between the start of the asset value process and the first noise observation, too.

Figure 9: Relationship between the Q-to-P-ratio and the asset volatility for different ratinggrades in the Duffie/Lando model. Parameters: = 10%, m=2.5%, SRA = 20%, T=5, s=1.

Estimation of the sharpe ratio:

simple measures as the development of average CDS spread or Q-to-P-ratios may be misleading in somemarket environments, where a high uncertainty about true values exists. An example is probably thesubprime crisis in 2007, which lead market participants highly unsecure about where risks actually turnedabout and what dimension potential losses may have. It is not within the scope of this paper, to empiricallyanalyze the market reactions on credit and equity markets, but this could provide an area for further researchon this topic.

2 MODEL SETUP 22

Figure 10: Relationship between the Q-to-P-ratio and the rating/actual default probabilityin the first passage time model for different levels of . Parameters: r=5%, SRA = 15%,=15%.

As in subsection 2.2, we now want to test the robustness of the Merton model estimator,e.g. we again define an adjustment factor AFDL by

M rM

= Merton AFDL (17)

This adjustment factor may depend on all parameters included in the model, which we willgroup into two different classes: Class 1 captures all parameters that can easily be observedin the market, i.e. the actual default probability (which is actually a combined parameter ofall other input parameters) and the maturity. Class 2 captures all parameters that can notbe easily observed in the markets, i.e. the asset volatility , the payout rate or the riskneutral net asset growth rate m := r , the starting point of the asset value process in t=0(Z0), the default barrier VB, the noise asset value observed at t=t (Vt) and the accountingnoise . If the adjustment factor depends on any class-2 parameter, this will affect ourability to correctly measure the market sharpe ratio, since these parameters will be subjectto possibly significant calibration errors.

2 MODEL SETUP 23

We have evaluated (17) for different combinations of input parameters34, the main resultscan be summarized as follows:

The adjustment factor is close to 1 for all parameter combinations as long as theasset volatility is below 10% and the resulting actual default probability belongs to aninvestment grade rating, see figure 11. This can be explained by looking at the impactof the parameters introduced in the Duffie/Lando framework: all of them do effect theactual default probability as well as the risk neutral default probability in the samedirection, e.g. increasing the information uncertainty increases the actual as well asthe risk neutral default probability. The share ratio is the only parameter that simplyhas an effect on the actual default probability only. This explains qualitatively, whythe adjustment factor is close to one in many cases.

The adjustment factor can be accurately determined simply based on knowledge ofthe class-1 parameters and the actual default probability as long as > 10%, i.e. fora given combination of default probability and maturity, parameters that can not beobserved easily (e.g. asset volatility, default barrier, asset value or accounting noise)do not significantly affect the adjustment factor.

Please note the special role of the actual default probability: For example, an adjustmentfactor of appr. 1.7 (i.e. significantly above 1) occurs for an asset value of 108, =10%,T=5, SR=40% and =0%. If this were due to any class-2 parameter, empirical applicationswould be hardly possible due to calibration errors of class-2 parameters. But as soon aswe change any of these parameters so that the resulting actual default probability belongsto an investment grade rating (e.g. increasing Vt, decreasing , decreasing (up to a levelof 10%)) the adjustment factor will be close to 1. In other words, any combination ofthese parameters, that yields a given actual default probability also yields (almost) the sameadjustment factor. All in all, class-2 parameters do actually have a significant influence onthe adjustment factor; but this influence can (almost fully) be captured simply by the ratingsmile.If we take a closer look at the parameter combinations leading to the minimal and maximaladjustment factor in figure 11, we see, that these actually belong to unlikely parametercombinations. Table 6 shows the dependeny of the adjusment factor for a maturity of 5years and a Baa-rating35 for = 0% and s = 0, e.g. for the extreme of observable asset

34Input parameters used were: : 3%30% (The 5% and 95% quantile for the asset volatility from KMVwas 6% and 25% respectively), sharpe ratio of the ability-to-pay process: 10% to 40% (The market sharperatio is usually assumed to be anywhere between 20% and 50%, due to a correlation of lower than 1, theasset sharpe ratio should be smaller), m : 0% 5% (m < 0 would imply, that the payout rate is larger thanthe risk free rate, m=5% was choosen as an upper limit to reflect (almost) zero payout at a risk free interestrate of 5%.), : 0% 30% ( = 0% reflects the classical first passage model with observable asset values,Duffie/Lando use 10% as a standard value, the upper limit of 30% is also based on Duffie/Lando(2001)),Vt = Z0 and VB for all combination that resulted in rating grades from AA to B. The case Vt > Z0 andVt < Z0 was also analyzed, the results merely differ from the case Vt = Z0 and are available upon request.

35We choose this as an example, since 5-year CDS are the most liquid ones usually used in empiricalstudies, see for example Berndt et.al. (2005) and Amato (2005) and Baa is the most common rating amongnon-financial companies.

2 MODEL SETUP 24

values, table 7 for = 30% and s = 3 representing large uncertainty about the asset value.The bold numbers depict the minimum and the maximum values for the adjustment factor.The absolut maximum is taken for small asset volatilites, no uncertainty and a high riskneutral asset growth rate (i.e. a low payout rate). Even ignoring the unrealistic default termstructure implied by a certain asset value, one would usually expect small asset volatilitiesto belong to a value firm whereas low payout rates usually apply to growth companies.The absolut minimum is taken for small asset volatilities and high payout rates (i.e. a lowrisk neutral asset growth rate m), which would suit the usual assumptions about value firms,but for a high uncertainty about the current asset value, which one would usually assume forgrowth companies. Depicting values one would usually assume for value firms and growthfirms, we can see that the adjustment factor will lie even closer around the mean value.

Figure 11: Adjustment factor in the Duffie/Lando model for different rating grades. Theminimum and maximum is taken over the parameters 10% 30%, 0 30%,0 m 5%, 10% SRV 40%, 0 s 3. Other parameter: T=5.

2.4 Deviations from the lognormal assumption

It is a widely discussed topic in empirical finance, that log asset returns are not normallydistributed (as a geometric Brownian motion suggests) but rather fat tailed or leptokurtic,i.e. their standardized third moment is larger than the third moment of a standard normaldistribution (which has a fourth moment of 3). In first passage time structural models, a

2 MODEL SETUP 25

default occurs, as soon as the asset value has fallen below a predefined default barrier. Sincethese default probabilities are usually small (e.g. the 5-year default probability of a A-ratedobligor is 0.60%, the 5-year default probability of a BBB-rated obligor is appr. 2.2%36),leptokurtis of the asset value returns has a significant influence on default probabilities, and- as we will show in this subsection - on the difference between risk neutral and real worlddefault probabilities as well.

To analyze the impact of leptokurtic asset returns, we will return to a Merton setting,where default can only occur at maturity of the bond/CDS. The reason for this approachis mainly its simplicity and tractability. Using fat-tailed distributions in first-passage-timesetting usually involves quite sophisticated mathematical framework based on discontiniousmartingales.37 In addition, the results usually loose the simplicity and intuition.

We will assume in the following, that the log asset return RT := ln(VTVt) at time T (the

maturity) has a distribution identified by its cumulative distribution function F (x). Thedistribution is supposed to have first and second order moments, so that we can standardizeit by RT =

RTTT

. The expected one-year return is supposed to equal under the realworld probability measure and r under the risk neutral probability measure. Furthermore,we assume stationary and independent increments of the 1-year returns, so that T = Tand T =

T , which yields RT =

RTTT . Therefore, the real world default probability

can be calculated as

P def (t, T ) = P [ VT < N ] = P

[RT < ln(

N

Vt)

]= P

[RT