Fixed Income

Fixed Income

1. Credit Risk Modeling

   • Structural Models
     • Merton 1974
       • 1.Value of asset follows a log diffusion process 2. Total value of firm is made by a single zero-coupon bond plus equity 3. considers zero-coupon debt only 4. Default only at T; never prior to maturity Main contradiction: The short end of the yield spread curve in calibrated versions of the Merton model typically remains essentially zero for months, in strong contradiction with observations.
     • Black & Cox 1976
       • B&C considers infinite life debt with endogenous default  B&C allow early default before the maturity date of a risky debt. Default happens when the firm value process reaches a constant safety threshold level from above. Default would occur at the first time that assets fall below a boundary D (which may or may not be the face value of debt), thereby turning the pricing problem into one of computing first-passage" probabilities.
       • Link
         • The Merton model gives a closed form solution for defaultable zero-coupon bond prices. Longstaf and Schwartz provided an approximate numerical solution for B(t; T ) in their setting. Subsequently, Collin- Dufresne and Goldstein [2001b] provided an ecient numerical method for computing the B(t; T ).
     • Black and Scholes 1973
       • An explicit equilibrium model for valuing options. 
     • Geske 1977, 1979
       • extended the Merton model to the case of bonds of different maturities. 1.Application of compound option pricing techniques on Corporate bond valuation2. Generalize B&C model to value options on options or any corporate liabilities3. Simplify the numerical evaluation4. New formula for valuation of subordiniated bond is introduced
       • Geman et al. 1995
         • Change of Numeraire and its application on GESKE compound option pricing
     • Leland 1994
       • -Introduced taxes, default costs, and endogenous default -Derived closed-form solutions to debt and equity values, the default boundary, and optimal leverage. Paper results:-Credit spreads fall as the riskless rate rose-For bonds near default (“junk”), prices could rise (and credit spreads fall) as asset value volatility increases.-Optimal debt for firms with higher default costs may have a lower credit spread. (Lower leverage)
       • Leland Assumptions: 1) Underlying asset value follows an exogenous process: Underlying asset is the value of operational cash flows and  coincides with the value of an un-levered firm 2) Process is a diffusion with constant volatility and total payout rates 3) The risk-less rate is constant 4) Debt and equity are contingent claims on underlying asset value 5) Firms cannot sell assets to meet debt servicing payments 6) Debt and equity have no issuance costs or (il)liquidity premia 7) Implicit assumption that underlying value is a traded asset 8) No info. asymmetry: the value process is perfectly observed 9) Default endogenously determined (given fixed coupon) 10) Static capital structure (constant amount of debt or coupon) 11) Default costs a constant fraction of value at default 12) Infinite-life debt (can’t examine term structure of yield spreads) 13) Single type/priority of debt 14) Managers make decisions in shareholders’ interests(but possible agency costs between stock and bond holders) 15) No personal taxes 
       • Longstaff & Shwartz 1995
         • (Relaxes Assumption 3) 1.Consider stochastic risk-free interest rate 2.Interest rate follows Vasicek diffusion model 3.Provides simple closed-form valuation expressions for fixed and floating rate debt. Findings: 1.correlation between default risk and the interest rate has a significant effect on the properties of the credit spread. 2. credit spreads are negatively related to interest rates and that duration of risky bonds depend on the correlation with interest rates.
         • Dynamic Capital Structure, Collin-Dufresne, Goldstein (2001, JF)
           • (relaxes Assumption 8) Introduce: Dynamic Capital structuring Introduce stationary leverage ratio model(mean-reverting leverage ratio) means: firms tend to issue more debt when leverage ratio reduces Consider stochastic term structure
       • Anderson & Sundaresan (1996, RFS)
         • Strategic Default (relaxes Assumption 9) This article models the violation of absolute priority cases: In practice we see cases where debt payment is reduced while equity is still alive and positive after restructuring.
         • Link
           • Anderson and Sundaresan [1996] and Mella-Barral and Perraudin [1997] solve simplified bargaining games to obtain close-form expressions for their default boundaries.
       • LT- Leland & Toft 1996
         • (Relaxes assumption 10) Maturity T, straight line amortization rate P/T. New bonds are issued and old bonds retired at a rate of p=P/T per year, where P is the total principal value of all outstanding bonds.  This condition ensures that as long as the firm remains solvent, at any time during the life of the firm, the total outstanding debt principal will be invariant and equal to P. It also insures total coupon paid for all of the outstanding bonds is stationary. Roll-over of principal preserves time independence—but complex In the Leland and Toft [1996] model, a firm continuously issues new debt with coupons that are paid from the firm's payout  A. The default boundary is endogenous, because equity-holders can decide whether or not to issue new equity to service the debt in the event that the payout is not large enough to cover the dividends. Their model gives a closed-form expression for B(t; T) under the assumption of constant r.
         • He & Xiong (2011, JF)
           • While previous studies typically interpret this finding as a liquidity-premium effect, this model highlights that through the rollover-risk channel, an increase in liquidity premium also feeds back to firms’ endogenous default thresholds and causes higher default probabilities. This implication cautions against the common practice of decomposing credit spreads as simple sums of default premia and liquidity premia, and justi fies the large liquidity-premium effect documented in the empirical studies. Moreover, it suggests time-varying market liquidity as an explanation for the common latent factors discovered by Duffie et al (2009) in firms' ’default probabilities.
       • Mella-Barral & Perraudin (1997, JF)
         • (Relaxes assumption 9) Strategic Default Formulate the model so as to create a trade-off between renegotiation costs and tax advantages of debt.
       • Dynamic Capital Structure, Leland 1998
         • (Relaxes assumption 10) A model to examine optimal firm decisions. It provides quantitative guidance on the amount and maturity of debt, on financial restructuring, and on the firm's optimal risk strategy. Infinite life debt, BUT retired proportionately (at par) at rate m  Debt of each vintage declines exponentially, replaced with new debt that with same principal (and declines exponentially)Total debt principal, coupon remains time independent.
       • Duffie & Lando (2001, Econometrica)
         • As shown by Duffie and Lando [2001], more plausible levels of short-term spreads are obtained in structural models by making the assumption that bondholders measure firm's assets with error. Once measurement errors are introduced, this basic structural model becomes mathematically equivalent to an intensity-based, reduced-form model. Imperfect Information (relaxes Assumption 8)Contrary to the perfect-information case, there exists a default-arrival intensity process. That intensity is calculated in terms of the conditional distribution of assets. Credit yield spreads are characterized in terms of accounting information. Generalizations are provided.
         • Fan Yu (2005, JFE)
           • Uses cross-sectional regression and Nelson-Siegel yield curve Find that firms with higher AIMR disclosure rankings tend to have lower credit spreads. Firms with more accurate information disclosure have lower short-term credit spreads.
       • Ericsson & Reneby (1998, AMF)
         • (Relaxes assumption 9) Strategic Default Consider a corporate coupon bond with discrete payments and debt subject to strategic debt service.
         • Link
           • Equally importantly, many structural models also imply that credit spreads tend to asymptote to zero with increasing maturity. This is a consequence of the fact that (riskneutrally) A (asset value) drifts away from the default boundary at the rate r so the forward probability of default, HT (At=F; rt; T −t) converges to zero as T goes to inifinity. To address this counter-factual implication, Collin-Dufresne and Goldstein [2001b] and Tauren [1999] attribute a target debt/equity ratio to issuers, and Ericsson and Reneby [2001] assume a positive growth rate of total nominal debt.
       • Fan & Sundaresan (2000, RFS)
         • Endogenous Cash holding/Dividend Policy (relaxes Assumption 2) Strategic Default (relaxes Assumption 9)
         • Ericsson & Renault (2006, JF)
           • Develop a model to illustrate the impact of liquidity risk on the yield spreads of corporate bonds. Debt (bonds) are less liquid than equity, investors demand extra return. Levels of liquidity spreads are likely to be positively correlated with credit risk and that they should be decreasing functions of time to maturity.
       • Bhamra Kuehn Strebulaev (2011, RFS)
         • Our structural-equilibrium framework embeds a contingent-claim corporate financing model within a consumption-based asset-pricing model. We investigate the effect of macroeconomic conditions on asset valuation and optimal corporate policies, and of preferences on capital structure. While capital structure is pro-cyclical at dates when firms re-lever, it is counter-cyclical in aggregate dynamics, consistent with empirical evidence. We also find that financially constrained firms choose more pro-cyclical policies and that leverage accounts for most of the macroeconomic risk relevant for predicting defaults, but is a poor measure of how preferences impact capital structure.
     • Optimal Dynamic Capital Structuring, Structural equilibrium framework
       • Bhamra Kuehn Strebulaev (2011, RFS)
         • Macroeconomic Equilibrium Setting (relaxes Assumption 2) Strebulaev (this conference paper 2009): Epstein-Zinn aggregate investor embeds a contingent-claim corporate financing model within a consumption-based asset pricing model.
   • Reduced form Models
     • Duffie & Singleton 1997
     • Lando 1998
       • Assume recovery of a fraction of face value: 1-LGD
     • Duffie & Singleton 1999
       • Assume recovery of market value RMV

2. Credit Spread Puzzle, Empirical Studies of Credit Spreads

   • *Dai Singleton (2003, RFS)
     • Dai Singleton (2000) introduced multifactor ATSMs model of short rate 2003 is a Survey
   • Jump-Diffusion, Liquidity is Important, Huang and Huang (2003, WP)
     • Jump-diffusion models can explain short end of the default, spread curve Problem with diffusion models: Any model assuming a pure diffusion process will be incapable of explaining shorter term default probabilities, spreads, because default risk rate must go to zero as horizon goes to zero Have closed form solutions Results suggest illiquidity is important in spreads They find that for high-grade debt, only a small fraction of the total spread can be explained by credit risk. For lower quality debt a larger part of the spread can be attributed to default risk.
   • CDS Related studies
     • Extract default and non-default component of spread from CDS,Longsta ff, Mithal & Neis (2005, JF)
       • non-default component is time varying and mean reverts rapidly. The non-default component of spreads is strongly related to measures of bond-specific il-liquidity such as the bid–ask spread and the outstanding principal amount. There is only weak support for the hypothesis that the non-default component is due to taxes. Using the Treasury curve, the nondefault componentsrange from 18.8 to 104.5 basis points, with a mean of 65.0. Using the Refcorpcurve, the nondefault components range from −8.1 to 89.8 basis points witha mean of 47.7. Similarly, using the swap curve, the nondefault componentsrange from −57.8 to 50.5 basis points with a mean of 8.6. These nondefaultcomponents are significantly different from 0 for virtually every firm in thesample
   • Chen Collin-Dufresne Goldstein (2009, RFS)
     • Macroeconomic Equilibrium Setting (relaxes Assumption 2) Combined with model generating countercyclical default rates, can explain Baa-Aaa spreads (not Baa-Treasury or Aaa-Treasury spreads)Leland says: I suggest countercyclical liquidity spreads also could do this“credit spread puzzle” can be resolved if one accounts for the fact that default rates and Sharpe ratios strongly covary
   • *Davydenko (2011)
     • Links the default boundary to firm's asset market value just before default. Find that default's boundary is 66% of asset face value for medium firms.
   • *Elkamhi Ericsson & Parsons (2012, JFE)
     • Trade-off default cost an Tax shield