Structural Models of Credit Risk are Useful: Evidence from Hedge Ratios on Corporate Bonds

1. Structural Models of Credit Risk are Useful:Evidence from Hedge Ratios on Corporate Bonds Stephen M. Schaefer London Business School International Financial Research Forum Financial Risks New Developments in Structured Products and Credit Derivatives Paris, 27-28 March 2008

2. Joint work with: Ilya A. Strebulaev Stanford University Structural Models are Useful

3. Structural Models • Structural models of credit risk represent default in terms of the value of the firm’s assets (that collateralise the debt): • falling short of the face value of the debt at maturity (Merton model); or • hitting a lower threshold representing (e.g.) the point at which lenders will intervene and liquidate the firm (“second generation” models – e.g., Leland) • Structural models represent the best framework currently available to analyse “fundamental” value in credit setting BUT ….. Structural Models are Useful

4. Introduction • Structural models fail to explain size of yield spreads on corporate bonds • e.g. Huang and Huang (2003) - 5 models: Structural Models are Useful

5. What do credit spreads in structural models represent? • The credit spread in a structural model is approximately: • So possible reasons for underestimating spreads: • underestimating default probability (or LGD) • underestimating hedge ratio (beta) of debt to equity (or equity risk premium) • or .. impact of other variables (liquidity etc.) Structural Models are Useful

6. Yield Spread, Default Loss Rate and Calculated Credit Spread (10-year bonds) Source: Huang & Huang Structural Models are Useful

7. So, is it all bad news? Structural Models are Useful

8. Structural Models and Default Probabilities • Actually structural models appear to provide reasonably good (or at least not bad) estimates of default probabilities • Leland (2002), Huang and Huang (2003) • Moody’s KMV Structural Models are Useful

9. Leland’s Estimates of Default Probabilities • Leland uses default boundary model with realistic input parameters to calculate default probabilities • B-rated bonds; asset volatility is 32% • A-rated bonds; asset volatility is 23% (Base case). Dotted line is actual. Dotted line is actual. Source: Leland, H. “Predictions of Expected Default Frequencies in Structural Models of Debt”, Working Paper, Univ. of California, Berkeley, September 2002. Structural Models are Useful

10. Short-Term vs. Long-Term Default Probabilities • Long-term (7-8 years and longer) default frequencies fit quite well • Short-term (1-6 years and below) default frequencies are too low Structural Models are Useful

11. What do we do in this paper? • Existing research: • default probabilities – (partial) success • spreads – failure (i.e., so far – no success for variable that depends on prices) • This paper: • perhaps the risk premium component is underestimated • do structural models predict hedge ratios of corporate debt to equity? Structural Models are Useful

12. Why are hedge ratios important - I? • Determine risk premia • In all structural models the bond value is determined as the price of the replicating portfolio • in theory portfolio and equity and riskless debtreplicates payoff on bond • composition of the replicating portfolio is determined by the hedge ratios • so bond price is determined by the hedge ratios • If observed hedge ratios consistent with those predicted by models, then that failure of models to predict spreads likely to be due to non credit risk factors Structural Models are Useful

13. low hedge ratio (slope) = low credit exposure high hedge ratio (slope) = credit exposure Why are hedge ratios important - II? • Hedge ratio: • measures exposure of debt value to value of collateralising assets • hedge ratio High => high credit risk • hedge ratio low => low credit risk Structural Models are Useful

14. Focus of Paper • Estimate hedge ratio regressions: • In a world governed by structural models, hedge ratio regressions would produce • coefficientsbj,E close to one • high explanatory power (R2 close to 1) • ... but not exactly as a result of (a) non-linearity; (b) discreteness • We show that (a) and (b) are not important and test hypothesis that bj,E = 1 • Consider other systematic factors (a la Collin-Dufresne) and examine their relation to underlying credit risk Structural Models are Useful

15. Main Findings - 1 • Simple structural model (Merton, 1974) provides reasonably good estimates of hedge ratios of corporate debt to equity • BUT returns on corporate bonds also strongly related to: • SMB and HML (Fama-French factors) .. But NOT in a way that is related to exposure to underlying equity and NOT in a way that appears linked to credit exposure • S&P (or VIX) .. but NOT in way that is linked to credit risk • Thus these factors seem to have significant effects on prices / returns but not via credit risk channel • Another puzzle: • structural (Merton) model fails to explain LOW empirical hedge ratios of debt to riskless bonds (duration) Structural Models are Useful

16. Main Findings - 2 • While it is true that structural models underestimate corporate yield spreads • .. if spreads reflected credit risk alone then the sensitivity of bond returns to equity would be higher to be consistent with reasonable estimates of the equity risk premium • in fact .. empirical sensitivitiescorrespond quite well to predictions of simple structural model. Structural Models are Useful

17. Data • Merrill Lynch: Corporate Master Index and Corporate High Yield Index • covers nearly all corporate bond issues in the U.S. (2114 issuers; 10370 issues) • Monthly price data from 12.1996–12.2003 (388,000 bond-month observations) • final sample satisfies additional standard criteria (only US bonds, matching with CRSP/COMPUSTAT, no financials, only straight bonds) • Entire and final sample: Structural Models are Useful

18. Descriptive Statistics: Final Dataset Structural Models are Useful

19. A Simple Time-Series Hedging Regression • We run the following regression: • Results: : • estimated hedge ratios – small (0.006 – 0.04 for IG) but highly (statistically) significant • R2 much less than 100% • sensitivity to Treasury returns (“duration”) low Structural Models are Useful

20. Hedge Ratios • Are these hedge ratios reasonable? • compare with hedge ratios implied by Merton model • In one-factor structural models the hedge ratio, bE, is: where DE is the “delta” of equity to the firm’s asset value and L is the debt-to-asset value ratio Structural Models are Useful

21. Hedge Ratios from the Merton Model Structural Models are Useful

22. The Volatility of Corporate Assets Structural Models are Useful

23. Testing the Merton Model’s Hedge Ratio Predictions • Estimates of individual hedge ratios are very noisy: calculate as mean hedge ratio within sub-rating : Structural Models are Useful

24. Using the Merton Model to Predict Hedge Ratios Source: Schaefer / Strebulaev Structural Models are Useful

25. Explaining Structural Model Spreads with Empirical Betas • Implies default-boundary structural models produce very similar hedge ratios to Merton Structural Models are Useful

26. The story so far • Merton model produces hedge ratios in line with empirical estimates • so, structural models appear to capture credit exposure quite well • But the R2 are lower than the model predicts • equity and risk-free debt should account for large fraction (80% plus) of debt return variation • BUT we find R2 ~ 50% – 70% for investment grade bonds and R2 ~ 30 – 40% for non-investment grade • What other factors influence corporate bond returns? • “Inside” the model: stochastic interest rates • “Outside” the model: other systematic factors Structural Models are Useful

27. Stochastic Interest Rates • In Merton model ( effectively, Black-Scholes) riskless interest rates are fixed • formally, need model that allows for uncertainty in riskless rate • Also, puzzle of low interest rate sensitivity of corporate debt Structural Models are Useful

28. Low Duration Puzzle: Regressions on riskless bonds only Structural Models are Useful

29. Including Stochastic Interest Rates • Merton (1974) with affine interest rates (Shimko et. al. (1993), Lando (2004)) • For simplicity, consider one-factor Vasicek model • Results on hedge ratios unchanged Structural Models are Useful

30. Other Factors: Running a “kitchen sink” regression • Hedge ratios for equity and riskless debt are not much changed Structural Models are Useful

31. Sensitivity to SMB • Sensitivity to corporate debt returns to SMB: • not result of sensitivity of underlying assets to SMB • not strongly connected to credit exposure (!!) Structural Models are Useful

32. Conclusion 1 • Hedge ratios provide agood measure of credit exposureand, in this sense, structural models seem to capture credit exposure better than commonly supposed: • But do NOT explain level of credit spreads Structural Models are Useful

33. Conclusion 2 • Understanding identity and role of non credit risk related factors: still incomplete • liquidity • taxes: ?? • imperfect substitution between equity, riskless bonds and corporate bonds • fluctuations in capital allocated to credit risky instruments Structural Models are Useful