Structural Models IV

Transcription

1 Structural Models IV Implementation and Empirical Performance Stephen M Schaefer London Business School Credit Risk Elective Summer 2012 Outline Implementing structural models firm assets: estimating value and volatility of return Empirical performance of structural models in predicting: default frequency credit spreads hedge ratios Other risk factors in corporate bond returns Structural Models IV 2

2 Structural Model Inputs Structural models depend on value and volatility of firm assets neither is directly observable Asset value Value of Value of Value of Value of other liabs = + + Assets Equity Debt (e.g., preferred stock) Value of equity = stock price x number of shares Value of debt may be problematic Structural Models IV 3 Potential problems Valuing the Debt which debt to include e.g., should we include shortterm as well as long-term debt? should we net all or some short-term debt against shortterm assets? what is value of debt? Only part of total debt will be traded. borrowings observed only in periodic accounts (quarterly, annual etc.) No simple solution using book value rather than market value of debt is usually adequate Structural Models IV 4

3 The Volatility of Corporate Assets All AAA AA A BBB BB B Quasi-Market Leverage Mean Std.Dev Equity Volatility Mean Std.Dev Estimated Asset Volatility Mean Std.Dev Quasi-market leverage ratio Book Value of Debt (Compustat items 9 and 34) Book Value of Debt + Market Value of Equity Estimated asset volatility σ Ajt = (1 Ljt ) σ Ejt + Ljtσ Djt + 2 Ljt (1 Ljt ) σ ED, jt Structural Models IV 5 Empirical Predictions of Structural Models: Default Probability Structural Models IV 6

4 Leland (2003) Predictions of Default Probability Uses Leland-Toft model to estimate default probabilities default probability is first hitting time given previously except that m, the mean of the distribution of continuously compounded returns on V, allows for possibility of non-zero payout ratio, d, from firm (dividends + debt payments): ( 2 m / σ 2 V ) ln( B / V ) mt B ln( B / V ) + mt 1 2 N N, where m µ d σ σ T + V = T 2 V V σ V Leland s Parameters Riskless rate (r) 8% Payout rate (d) 6% Asset risk premium (λ) 4% Expected ret. on asset (µ) 12% Structural Models IV 7 More Evidence on Consistency of Structural Models with Actual Default Frequency A-rated bonds asset volatility is 23% (Base case). Dotted line is actual. B-rated bonds uses asset volatility of 23% but actual asset vol. for firms issuing riskier debt is higher. Dotted line is actual. Source: Leland, H. Predictions of Expected Default Frequencies in Structural Models of Debt, Working Paper, Univ. of California, Berkeley, September Structural Models IV 8

5 Using a higher asset volatility Model (roughly in line with Schaefer/Strebulaev) fits default Probabilities for B-Rated Debt better Dotted line is actual. Source: Leland Structural Models IV 9 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 14.0% Leland - Default Prob w. Asset Vol =23% 2.5% Leland - Default Prob w. Asset Vol =23% 12.0% Moody's Default Prob. BBB % Moody's Default Prob. BBB Default Probabiity 10.0% 8.0% 6.0% 4.0% 2.0% Default Probabiity 1.5% 1.0% 0.5% 0.0% Horizon 0.0% Horizon Structural Models IV 10

6 Empirical Predictions of Structural Models: Credit Spreads Structural Models IV 11 Further Empirical Evidence on Structural Models and Credit Spreads Simple application of Merton model appears to produce yield spreads that are too small recent paper by Huang & Huang provides evidence on the performance of a number of second generation structural models By implication these results suggest that credit spreads are partly determined by factors other than credit risk recent paper by Dufresne et. al. is consistent with this view and finds risk factor in residuals which is: common across issues not connected with obvious market factors such as interest rates and asset prices liquidity? Structural Models IV 12

7 Huang and Huang Calibrate a number of structural models to fit average data for each credit rating on: leverage ratio (S&P) loss given default (Altman & Kishore) default probability (Moody s) equity risk premium (Bhandari(1988) dependence on leverage) Investigate a number of structural models: Longstaff & Schwartz (stochastic interest rates) Leland-Toft (extension of Leland) Collin-Dufresne & Goldstein (mean reverting leverage ratio) Anderson-Sundaresan (strategic default) Structural Models IV 13 Credit Rating Huang-Huang: Base Case Parameters Lev. Ratio Equity Prem. Target Cum Def. Prob 4 yrs 10 yrs Recov. Rate Av. Yield Spread 4 yrs 10 yrs Aaa Aa A Baa Ba B Source: Huang, J. & Huang, M., How Much of the Corporate-Treasury Yield Spread is Due to Credit Risk, Working Paper, Smeal College of Business, Penn State Univ. Structural Models IV 14

8 Yield Spread and Default Loss Rate (10-year bonds) 5% 4% Yield Spread Default Loss Rate 3% 2% 1% 0% AAA AA A BBB BB B Credit Rating Source: Huang & Huang Structural Models IV 15 Huang and Huang: Results using Longstaff-Schwartz Model Maturity = 10 years Target Implied Credit Rating Lev. Ratio (%) Equity Prem. (%) Cum Def. Prob. (%) Asset Vol. (%) Asset Risk Prem (%) Calc Credit Spread (bps) Av Yield Sprd (bps) % of Sprd due to Def. Aaa Aa A Baa Ba B Source: Huang & Huang Structural Models IV 16

9 Yield Spread, Default Loss Rate and Calculated Credit Spread (10-year bonds) Longstaff-Schwartz 5% 4% Yield Spread Default Loss Rate Calculated Credit Spread 3% 2% 1% 0% AAA AA A BBB BB B Credit Rating Source: Huang & Huang Structural Models IV 17 Credit Rating Huang and Huang: Calculated Credit Spreads for Different Models Maturity = 10 years Longstaff-Schwartz CDG Count Base Case Stochastic Interest Rates Mean Rev. Lev. Cycle. Risk prem. With Jumps In asset Value Aaa Aa A Baa Ba B Source: Huang & Huang Structural Models IV 18

10 Leland / Huang and Huang: Conclusions Structural models DO appear to be able to capture realistic levels of default frequency using empirically reasonable parameters (consistent with Leland (2003)) but Using the same structural models, credit risk appears to account for only a fraction of yield spread (e.g., 16% (AAA) - 83% (B) using Longstaff & Schwartz model) Different methods of calibrating model may give somewhat different results.. but unlikely to change very much. Structural Models IV 19 Empirical Predictions of Structural Models: Hedge Ratios Structural Models IV 20

11 Structural Models and Hedge Ratios Even though structural models provide poor predictions of level of spreads perhaps they do better in predicting sensitivity of bond prices to equity ( hedge ratios ) Table shows results of a regression of corporate bond returns on (a) returns on the 10-year Treasury and (b) returns on the equity of the issuing firm. All AAA AA A BBB BB B Intercept year bond (x 100) Return on Equity (x 100) R-squared Sample size The coefficients on: equity returns are betas of corporate debt against the underlying equity 10-year (Treasury) bond returns are duration ratios of corporate debt to 10-year Treasuries Source: Schaefer / Strebulaev (2005) Structural Models IV 21 Sensitivity of Bond Price (Return) to Equity Low level of spreads predicted by structural models implies that sensitivity of bond prices to changes in asset values is also low (i.e., default put is typically a long way out of the money) low sensitivity 4.0 Bond Value and Bond Spread 25% % million Bond Value (LHS) Bond Spread (RHS) 15% 10% % - 0% value of assets of firm ( million) low spread Structural Models IV 22

12 Using the Merton Model to Predict Hedge Ratios 1.4 Average Ratio of Emprical to Theoretical Hedge Ratio Source: Sc chaefer / Strebulaev 0.0 All AAA AA A BBB BB B And, in fact, the actual sensitivity of bond returns to equity values turns out to be in line with the predictions of the model Structural Models IV 23 Why are hedge ratios important? A bond s sensitivity to equity (or firm assets) is a measure of its credit exposure. Results on hedge ratios suggest that, even though structural models do a poor job predicting spreads, even the Merton model does quite a good job in in predicting credit exposure. However, while in theory, interest rates and equity should explain 100% of the returns on credit risky, in practice the R 2 is much lower: about ~ 20% - 40%. it therefore appears that other factors influence returns on corporate debt Structural Models IV 24

13 Understanding Huang and Huang s Predicted Credit Spreads The expected return on a corporate bond is approximately equal to the yield (y) minus the expected loss rate (default probability, p, times LGD). E( R) y p LGD The risk premium on the debt can also be expressed as the beta of the debt versus equity multiplied by the risk premium on equity: RPDebt = E( R) rf = βd, ERPEquity y rf p LGD Spread βd, ERPEquity + p LGD = Spread - p LGD Structural Models IV 25 Calculating HH s Spreads from Debt Betas Source: Schaefer & Strebulaev Structural Models IV 26

14 Other Sources of Risk in Corporate Debt Returns Structural Models IV 27 Collin-Dufresne, Goldstein & Martin Estimate regression of changes in credit spreads against number of plausible explanatory variables consistent with structural approach together these account for ~ 20% of variability in spreads Then analyse properties of residuals Structural Models IV 28

15 Collin-Dufresne et. al.: Explanatory Variables Changes in: government bond rate (yield on 10 year benchmark Treasury) slope of yield curve (10 year minus 2 year) leverage (Book value Debt / (Book Debt + Mkt. Equity) like equity, leverage essentially picks up changes in V volatility (ISD on near-the money S&P Options) probability of jump (Slope of smirk in ISD for S&P Options) business climate (Returns on S&P) Structural Models IV 29 Collin-Dufresne et. al.: Results for 25-35% Leverage, Long Maturity Variable Estimated Coeff. T-statistic leverage r_ ( r_10) slope Volatility return_s&p jump smirk R Structural Models IV 30

16 Is the sensitivity of Corporate Debt to SMB & HML related to credit risk? Table below shows sensitivity of bond returns to SMB and HML in regression that also includes equity returns, Treasury returns, S&P and VIX Curious result: All AAA AA A BBB BB B SMB T-statistic HML T-statistic sensitivity of returns on credit risky bonds to SMB & HML highly significant, especially for investment grade but for SMB, sensitivity is very similar for all risk categories from AAA BB suggests effect of SMB is not related to credit risk results for HML similar (though less marked) Source: Schaefer / Strebulaev Structural Models IV 31 Collin-Dufresne et. al.: Analysis of Residuals main results confirm those of others: structural models account for only part of the behaviour of spreads but C-D et. al. also analyse residuals: Assign each month s residuals to one of 18 bins according to: three maturity groups (<12 years, years, >18 years) six leverage groups Then carry out principal components analysis on residuals: 71% of variation accounted for by first PC eigenvector associated with first PC shows that this factor places roughly equal weight on each of the eighteen classes conclusion: some other persistent factor influencing yield spreads but C-D et. al. fail to identify good proxy Structural Models IV 32

17 Characteristics of Common Factor Chart below shows weights on each maturity/leverage buckets in first principal component Since weights are very similar across maturity/leverage buckets, this implies a factor that affects corporate spreads equally across credit ratings short medium long weight in first principal component Low High Low High Low High Maturity / Leverage Structural Models IV 33 Summary Structural models, despite limited success so far, remain best framework for understanding the fundamental determinants of credit risk Most attention so far paid to performance in pricing: poor Recent evidence suggests that models may perform quite well in estimating: default probabilities hedge ratios BUT: clear that prices of credit risky bonds are also sensitive to variables that have little connection with credit risk WHY? Liquidity? Supply/Demand imbalances?.. the usual suspects Structural Models IV 34