From default probabilities to credit spreads: Credit risk models do explain market prices

Transcription

1 From default probabilities to credit spreads: Credit risk models do explain market prices Presented by Michel M Dacorogna (Joint work with Stefan Denzler, Alexander McNeil and Ulrich A. Müller) The 2007 IACPM Spring General Meeting, Rüschlikon,

2 Page 2 Outline of the Talk Definition of credit risk and credit spread From default probability to credit spread Cash flow valuation and the Brownian Motion Model The Power Law Model Testing the model: forecasting power and validity across industries Conclusions

3 Page 3 Credit Risk and Credit Spreads Any financial security is subject to credit risk the uncertainty regarding the ability of the firm issuing this security to meet its financial obligations. A priori it is impossible to discriminate between firms that will default and firms that will not. The only realistic way is to derive a probabilistic statement on the likelihood of default of a company. Default-risky bonds thus pay a spread over the risk-free government bonds. This spread is called credit spread. The corporate bond market is the source for yield data coming from the debts of companies put on the market.

4 Page 4 Assessing the Credit Risk Since more than ten years structural credit risk models like Moody s KMV provide expected default frequency (EDF) for publicly traded firms on a monthly basis. Rating agencies provide credit ratings to firms. These can be easily related to the default probability of the firm through statistical studies. The difference between the corporate bond yield and the government bond yield is called the credit spread of a bond. From the name credit spread it is obvious that it should be related to the credit risk of the company issuing the bond. The question is how?

5 Page 5 Objectives of the Study To develop a quantitative model linking default probabilities obtained from a credit risk model to the credit spreads observed in the market. The model should: be tractable, provide a closed form solution, allow for empirical testing, be useful on a whole industry level as well as for corporates.

6 Page 6 Parameters Influencing the Credit Spread The credit spread of a corporate bond will depend on the following parameters: The default probability of the company, The time to maturity of the bond, The recovery rate in case of default, The coupon payments of the bond. For ease of notation, we shall only deal with zero coupon bonds.

7 Page 7 The Data Monthly yields of various maturities determined at month end for the North American bond market for industries as well as government. Bloomberg provides yield data only for certain industries, rating classes, and for maturities going from 1 to 20 years. Moody s KMV monthly EDFs for the appropriate industries. Those are probabilities of default during the forthcoming year. The sample period is constrained to the availability of EDF data: from November 1995 until December 2004 (i.e. 110 independent observations in the time domain). From Moody s KMV CreditMonitor we construct the industry EDFs from individual firms.

8 Page 8 Cash Flow Valuation of Bonds If we assume risk-neutrality, the risk-neutral probability of default, q(t) for a maturity T, can be computed from the risk-free yield, Y G (T), and the yield of the risky bond, Y(T): where R is the universal recovery rate (percentage of the final cash flow paid in case of default). We assume it here to be 40% independent of the maturity or the industry sector. The annual default probability is simply obtained by the relation (assuming independence from year to year): YT ( ) qt ( ) = 1 1 R 1 + YG ( T) % 1/ qt ( ) = 1 (1 qt ( )) T T

9 Page 9 The Brownian Motion Model To derive a probability of default for a certain maturity based on an annual probability, we need to assume a process for the dynamics of the default probability over time. Let us first assume that a distance to default D, a monotonic function of the default probability, follows a Brownian Motion over time: D = D +σ W t 0 p t where D 0 > 0 is the initial distance to default, σ p > 0 represents the volatility of the rating process and W t is a standard Brownian Motion.

10 Page 10 The Default Probability The derivation of the risk-neutral default probability relies on Brownian motions hitting a lower boundary and first passage times (see J. Michael Harrison, 1985): qt ( ) 2 T p T = Φ Φ where the function Φ is the Gaussian distribution, T 0 is the lowest maturity available, p is the one-year EDF value and q is not annualized. This is done using the equation seen above.

11 Page 11 Results of the BM Model Annualized Default Probability (in %), Finance A, Maturity 5 years Nov 95 Nov 96 Nov 97 Nov 98 Nov 99 Nov 00 Nov 01 Nov 02 Nov 03 Nov 04 one-year EDF value risk-neutral def. probability BM model, G =

12 Page 12 How to Improve on the BM Model? We want to remedy the weaknesses of the model. We note that the Brownian Motion assumes a power law between the maturity of exponent ½ consistent with the Gaussian distribution. In many financial asset studies, it was shown that the power law deviates from the Gaussian exponent of ½ because of the fat tails of the distribution. It is reasonable to assume that the process governing the drift of the default probabilities is of similar nature as for the financial returns.

13 Page 13 The Power Law Model Heuristically, we introduce two parameters to the risk-neutral default probability measure: qt ( ) 2 p c T 2 α T0 1 = Φ Φ If α=1/2 and c=1, then this is equivalent to the BM model. The parameter α represents heavy tails and long range dependence in the time series of q s. All the quantities here are annualized.

14 Page 14 The Power Law Model cont. d It plays the role of the Hurst exponent in the modelling of Fractional Brownian Motion. The exponent α mainly captures the overall movements and accounts for the scaling law of the risk-neutral default probability, q(t), with respect to the time to maturity. A possible economic interpretation of α would be a sort of market mood indicator. While c would describe the market price of credit risk. Both parameters are estimated by linear regression across all times to maturity.

15 Page 15 Comparison of Model Results Credit Spread / EDF Implied Spread (in bp), Global Index, Maturity 5 years Nov 95 Nov 96 Nov 97 Nov 98 Nov 99 Nov 00 Nov 01 Nov 02 Nov 03 Nov 04 credit spread BM model, G = PL model, G = 0.97

16 Page 16 Time to Maturity Behaviour Annualized Default Probability (in %), Global Index, Average M onth (Nov Dec. 2004) Time to Maturity in yrs risk-neutral def. prob. BM model PL model

17 Page 17 Out-of-Sample Testing Procedure

18 Page 18 Out-of-Sample Test of the Model Forecasts Credit Spread / EDF Implied Spread (in bp), Global Index, Maturity 20 years Nov 98 Nov 99 Nov 00 Nov 01 Nov 02 Nov 03 Nov 04 year credit spread PL forecast 1 month, G = 0.94 Global Index / 5 yrs / rec. = 40 % PL forecast 3 months, G = 0.70 PL forecast 6 months, G = 0.50

19 Page 19 Model Testing European Sector In practice yield data are only available for a limited set of companies, industry sectors, credit rating classes or countries. We want to test if we can use some universal parameters to study a particular market. Collect EDF s on the European industry sectors and calculate the annualized risk-neutral default probabilities. Compute this quantity according to our PL model with universal parameters α and c. We also compare the results for the credit spreads themselves.

20 Page 20 Testing on the European Bond Market Credit Spread / EDF Implied Spread (in bp), European Utility A, Maturity 10 years Aug 01 Feb 02 Aug 02 Feb 03 Aug 03 Feb 04 Aug 04 year credit spread Global BM model, Index / G 5 yrs = / rec. = 40 % PL model, G = 0.94

21 Page 21 Testing on the Corporate Level Credit Spread / EDF Implied Spread (in bp), Boeing Company Apr 96 Apr 97 Apr 98 Apr 99 Apr 00 Apr 01 Apr 02 year credit spread BM model, G = Global Index / 5 yrs / rec. = 40 % PL model; company params., G = 0.41 PL model; global params., G = 0.21

22 Page 22 Evidence for the PL Model Stochastic Simulations The PL model is based on heuristic considerations. We want to provide evidence for it through Monte Carlo simulations. The goal is to describe the behaviour of the annualized default probability as a function of different times to maturity. We define the creditworthiness of a firm as its distance to default. Solvent companies have a strictly positive distance to default. We start with a universe of N companies with the same initial credit rating and we let the default probabilities evolve in time according to a certain model. We shall observe a dying off of companies as the time to maturity increases.

23 Page 23 Modelling the Dynamics of the Credit Rating Express the credit rating of company i by its distance to default D i for a time to maturity T j max( Di( Tj 1) Xi( Tj),0) if Di( Tj 1) > 0 Di( Tj) = 0 if Di( Tj 1) 0 Where the relative changes of the firm s creditworthiness, X i, are coming from a loggamma distribution: Z ( T ) i j X ( T ) = ae b with Z Γ ( α, β) and a> 0, b 0 i j We evaluate the distance to default recursively.

24 Page 24 Results Simulation Study Annualized Default Probability (in %), Global Index Time to Maturity in yrs risk-neutral def. prob. PL model log-gamma sim. model (discr. = 0.25 yrs)

25 Page 25 Conclusion We have been able to explain most of the credit spread seen in the market by the probability of default given by structural credit risk models. This was possible by assuming a non-gaussian credit migration rate for the default probability. Simulation results show that a log-gamma type of distribution for the migration rate describes the process reasonably well. The model is powerful enough to explain credit spreads from general parameters obtained from the market. Thus the model can be used to compute the price of credit risk for a new bond.