Managing Corporate Bond Risk: New Evidence in the Light of the Sub-prime Crisis

Giovanni Barone-AdesiNicola CarcanoHakim Dall’O

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Agenda

• Motivation• Model• Data• Results• Conclusions

3

Motivation (I)• Most commonly used techniques to hedge bond

portfolios against interest rate risk (PCA, Duration Vector, Key Rate Duration) assume the following dynamics of the zero-coupon risk-free rate R(t,Dk) of maturity Dk:

where:– Fl

t represents the change in the l-th factor between time t and t+1

– clk represents the sensitivity of the zero-coupon rate of maturity Dk to this change

f

l

ltlkk FcDtdR

1

,

4

Motivation (II)• When the number of hedging assets is not smaller

than the number of risk factors, these techniques set the sensitivity of the hedged portfolio to each individual risk factor equal to zero

• For high-quality hedging relying on more than one risk factor, these techniques have been found both by academicians and practitioners to be unstable and to lead to contradictory results (see, for example, Falkenstein and Hanweck (1997))

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Motivation (III)• For default risk-free bonds, Carcano (2009) and Carcano

and Dall’O (2011) show that accounting for the modeling errors in the development of the hedging equations can make these techniques more robust, thus reducing hedging errors as well as transaction costs

• In this framework, the dynamics of the zero-coupon risk-free rate are described by the following equation:

where e represents the model error.

kf

l

ltlkk DtFcDtdR ,,

1

6

Motivation (IV)• The main goals of our research were:

– To assess if the inclusion of modeling errors in the hedging equations can lead to better hedging also of corporate bond portfolios

– To test the performance of alternative hedging instruments before and during the financial crisis of 2007-2009. In particular, we intended to test the following hedging instruments:• US T-Note / T-Bond Futures• S&P500 Futures• CDX Contracts

– To test the effectiveness of liquid credit derivatives, like CDS, to hedge credit spreads in times of financial distress

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Model (I) – Dynamics of BBB-rated Bonds

• where:– ⍵i(t,Dk) indicates the present value of bond i cash-flow with maturity Dk as

a percentage of its total present value at time t.– P indicates the bond dirty price– RBBB(t,Dk) indicates the zero-coupon market yield for BBB-rated bonds with

duration Dk at time t

– and ei(t) indicates the idiosyncratic return provided by bond i from time t to time t+Dt

–

tDtDDtdRP

PE

P

Pi

n

kkikkBBB

ti

ttit

ti

tti

1,

,

,

, ,,

kBBB

L

l

ltkBBBlkBBB DtFcDtdR ,,,

1,,

8

Model (II) – Dynamics of T-Bond Futures

• where:– FP indicates the future price and s is the maturity of the future contract– RRF(t,Dk) is the zero-coupon yield for risk-free bonds with duration Dk – CTD(y) indicates the cheapest-to-deliver bond of the futures contract y

– for k > s, where cf stands for cash-flow

– for k = s

n

ktkyCTDkkRF

ty

ttyt

ty

tty DDtdRFP

FPE

FP

FP

1,),(

,

,

,

, ,

tkCTDkDDtR

kCTD

CTD

DDtR

t

k

k

t

t

De

cf

CF

e

FP

D

DtR

FP

FP

kk

ss

,,,

,,

,

tsCTDs

n

sktkCTDs

n

skDDtR

kCTD

CTD

DDtR

t

s

s

t

t

DDe

cf

CF

e

FP

D

DtR

FP

FP

kk

ss

,,1

,,1

,

,,

,

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Model (III) – Equations for Hedging only through T-Bond Futures

• If we represent the dynamics of risk-free rates as:

the unexpected return of the hedged portfolio is:

where Ai represents the amount invested in bond i and fy represents the optimal weight to be invested in the y-th futures contract.

kRF

L

l

ltkRFlkRF DtFcDtdR ,,,

1,,

,

,,

4

1 1 1,,,),(,

8

1 1 1,,,

y

n

kkRF

L

l

ltkRFltkyCTDkty

ii

n

kkBBB

L

l

ltkBBBlkiktit

DtFcD

tDtFcDtDA

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Model (IV) – Equations for Hedging only through T-Bond Futures

• In the context of PCA, if we assume that errors eBBB and eRF as well as idiosyncratic returns ei are independent from each other and from risk factors F, the partial derivatives of the variance of unexpected return relatively to the weight fy is approximated by:

• The optimal hedging strategy is obtained when the last equation is set equal to zero for each of the four considered T-bond futures

tkyCTDkkRF

n

k jtkjCTDtjk

n

ktkyCTDkkRFl

n

k j

n

ktkjCTDkkRFltj

ikitikkBBBl

L

l

lt

ty

t

DDtED

DcDcDtADcFEE

,),(2

1

4

1,),(,

1,),(,,

1

4

1 1,),(,,,

8

1,,,

1

2

,

2

,2

, 2

11

• We define the dynamics of the S&P500 futures as:

• We perform the same steps presented above including the S&P500 futures in the estimation of the PCA parameters and adding one additional weight fS&P in the set of hedging equations

tFcPFutS

PFutSE

PFutS

PFutSPS

L

l

ltPSl

t

ttt

t

tt&

1&,&

&

&

&

tFcDtFcD

tDtFcDtDA

PS

L

l

ltPSltPS

y

n

kkRF

L

l

ltkRFltkyCTDkty

ii

n

kkBBB

L

l

ltkBBBlkiktit

&1

&,,&

4

1 1 1,,,),(,

8

1 1 1,,,

,

,,

Model (V) – Equations for Hedging through T-Bond and S&P500 Futures

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• Because of lack of observations to estimate one integrated PCA including CDX, we followed a two-stage approach:– In the first stage, the optimal portfolio of T-bond futures has been identified following the

procedure presented above– In the second stage, we determined the optimal weighting for the CDX contract by following

a traditional approach based on linear regression:

where:– eFUT represents the hedging error produced by the strategy including only the T-bond futures

– RCDX represents the return to the protection buyer of the CDX contract

– eCDX represents the hedging error remaining after the inclusion of the CDX in the hedging

portfolio.

Model (VI) – Equations for Hedging through T-Bond Futures and CDX

tCDXtCDXtFUT eRe ,,,

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• In order to explain the residual error obtained by using the CDX contracts, we estimated the hedging error which would have been obtained by using CSF free of liquidity as well as counterparty risk

• The market rate of these CSF has been calculated as:

where YBBB(t,Dk) is the yearly-compounded par market yield.

The hedging regression now is:

where CSDBBB is the Credit Spread Duration of the considered spread

Model (VII) – Equations for Hedging through T-Bond Futures and Credit Spread Forwards

sDkD

sDsRF

kDkRF

sDkD

sDsBBB

kDkBBB

kBBBDtY

DtY

DtY

DtYDsttCSF

11

,1

,1

,1

,1,,

tCSFtkRFtkBBBkBBBkBBBtFUT eDttYDttYDtttCSFDtttCSDe ,, ,,,,,,

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• In order to perform realistic tests, we intended to build corporate bond portfolios displaying both a good correlation with market yields as well as a certain level of idiosyncratic risk: bond portfolios consisting of 8 bonds of different issuer and maturity displayed these characteristics

• Following Eom et al. (2004), we restricted our bond selection to BBB-rated industrial issuers

• We defined 8 time buckets with maturity equal to – respectively – 2, 4, 6, 8, 10, 16, 20, and 26 years. At the beginning of each year, we selected new, fixed-coupon bonds based on three conditions: a publicly held face value outstanding of at least 100 million US$, an already-paid first coupon and a maturity as close as possible to the one of the corresponding time bucket

• Relying on results reported by Carcano (2009) and Carcano and Dall’O (2011), we performed our tests on equally-weighted portfolios

• Bond prices and reference data have been kindly provided by Wilshire Associates

Data (I) – Corporate Bond Portfolios

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• Zero and par yield curves for BBB-rated bonds issued by industrial companies as well as the continuous S&P500 future series have been provided by Bloomberg

• We referred to the next expiring contract of US T-bond and T-note futures with denomination of – respectively – 2, 5, 10, and 30 years. For each contract and period, we identified CTD bonds following the net basis method and relying on the monthly baskets of deliverable bonds and conversion factors (CF) kindly provided by the CME. Future closing prices have been provided by Datastream.

• We extracted all data related to US Treasury bonds needed for estimating the sensitivity of each T-bond futures to the risk factors from the CRSP database. We calculated the present value of each individual cash-flow using US Treasury zero-coupon rates estimated through the Unsmoothed Fama-Bliss methodology described in Bliss (1997).

• 5-year CDX.NA.IG series from 2004 to 2010 (no. 3 to 13) have been downloaded from Bloomberg. Markit supported us on the unexpected return calculation.

Data (II) – Other Data

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• Sub-periods include – respectively – 119, 191, and 48 by-weekly observations

• Since our hedging strategies assume that idiosyncratic bond returns are independent from market returns, they should not be expected to hedge the residual idiosyncratic return provided by the bond portfolio.

• Accordingly, the fourth column of the table provides us with a rough estimate of the maximum variance reduction which can be expected from the tested strategies.

Results (I) – Summary Statistics

AverageCorrelatio

n

Average Correlatio

nBond Portfolio:

Explanatory

BBB Systematic Risk:

Explanatory

Average Ratio Spread

Variance /

PeriodBBB Rates

BBB Spreads

Power of BBBSystematic Risk

Power of Risk-Free Rates

Total Variance

1995-Sep. 2000 75.5% 30.2%  N.A. 87.8% 27.2%Oct. 2000 - 2007 65.0% 37.0% 83.8% 83.5% 26.8%2008-2009 76.0% 94.9% 75.5% 43.4% 57.6%

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• Sub-periods include – respectively – 100, 72, and 47 by-weekly observations

• Results for the period 2000-2007 compare well to the maximum variance reduction of 50% reported by previous studies hedging corporate bonds through T-bond and S&P500 futures. We attribute this improvement to the use of four futures contracts and to the consideration of modeling errors.

Results (II) – Variance Reduction Obtained by Alternative Hedging Strategies

Hedging Instruments T-Bond Futures

T-bond Futures

and S&P500 Futures

T-bond Futuresand CDX

T-bond Futures and

Credit Spread Forward

 Bond

Portfolio Variance R2 of Hedging Error Variance Variance Variance

Period Variance Reduction on Spread Changes Reduction Reduction Reduction

2000-2004 0.015% 82.3% 0.6% 76.8% N.A.  83.1%

2005-2007 0.008% 84.7% 0.0% 84.0% 86.7%  84.9%

2008-2009 0.035% 23.5% 49.8% (***) 26.1% 30.3% 64.3%(**)

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Results (III) – Level of corporate bond spread and CDX spread from 2004 to 2009

-2

-1

0

1

2

3

4

CDX Basis

CDX Spread

Spread to Swap

The CDX Spread has been obtained by Bloomberg for the 5-year Investment Grade CDX contract. The Spread to Swap has been calculated as the difference between the average yield of the CDX basket and the 5-year swap rate. The CDX Basis has been calculated as the difference between the CDX Spread and the Spread to Swap.

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Conclusions (I) – From 2000 to 2007

• A hedging strategy based only on T-bond futures would have reduced the variance of the bond portfolio by circa 83.5%.

• This reduction is significantly higher than the 50% reported by previous studies attempting to hedge corporate bonds through T-bond futures.

• Hedging errors tend to be independent from the dynamics of the average credit spread paid on BBB-rated bonds. This suggests that they are mainly due to the idiosyncratic returns provided by the bond portfolio to be hedged.

• Accordingly, it is plausible to think that increasing the number of bonds in the portfolio might lead to even better hedging.

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Conclusions (II) – 2008 and 2009• During this period – which was characterized by credit spreads of

BBB-rated bonds in excess of 2% - properly hedging the dynamics of these spreads was of paramount importance

• The S&P500 future would have not helped a lot, since the weights assigned to it by our strategies are quite low. This is due to the low correlation of this future with changes in BBB yields (even though we followed Marcus and Ors (1996) in considering the S&P500 future only in periods of low consumer confidence)

• Surprisingly, also the CDX would have not helped a lot; the good performance of hedging through CSF indicates that the problem is not the methodology, but rather the specific dynamics of the CDX basis due to liquidity issues and/or counterparty risk

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Conclusions (III) – Summing up…• In periods of financial stability, corporate bond portfolios can be

hedged very effectively by simply using T-bond futures• However, when credit spreads exceed ordinary levels indicating

financial distress, this is no longer the case and hedging instruments capturing the dynamics of credit spreads should be used

• Unfortunately, all currently liquid derivatives which could plausibly hedge these spreads do not seem to perform well

• A part of the problem is that - to use a non-financial analogy - credit derivatives subject to counterparty risk are like fire-extinguishers stored within inflammable cases!

• This motivates launching fully-collateralized CDS or credit spread forward contracts which should move more in-line with actual bond spreads

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References• Bliss, Robert R., 1997. Testing term structure estimation methods. Advances in

Futures and Options Research 9, 197-231.• Carcano, N., 2009. Yield curve risk management: adjusting principal

component analysis for model errors. Journal of Risk 12, n.1, 3-16.• Carcano, Nicola, and Hakim Dall’O, 2011, Alternative models for hedging yield

curve risk: An empirical comparison, Journal of Banking and Finance, 35, November, 2991-3000.

• Eom, Young, Jean Helwege, and Jing-zhi Huang. 2004. Structural Models of Corporate Bond Pricing: An Empirical Analysis. Review of Financial Studies 17:499–544.

• Falkenstein, E., and Hanweck, J. , 1997. Minimizing basis risk from non-parallel shifts in the yield curve. Part II: Principal components. Journal of Fixed Income (June), 85–90.

• Marcus, Alan. and Evren. Ors, 1996, Hedging Corporate Bond Portfolios Across the Business Cycle, Journal of Fixed Income, Vol. 4, No. 4 (March), pp. 56-60.