1. Introduction to Credit Risk

2. Credit Risk - Definitions Credit risk - the risk of an economic loss from the failure of a counterparty to fulfill its contractual obligations. Credit Exposure (CE) or Exposure at Default (EAD) – the economic value of the claim on the counter party at time of default. Recovery Rate (RR) – the payment ratio given default Loss Given Default (LGD) – the fractional loss to default, which is equal to 1 - RR

3. Measuring Credit Risk-Distribution of loss Definitions b i - a “bernoulli” random variable that take the value of 1 if default occurs and 0 otherwise, with probability of p i. CE i - the credit exposure at the time of default. f i - the recovery rate (RR) (1-f i ) – the loss given default (LDG) N – number of instruments

4. Measuring Credit Risk-Distribution of loss The distribution of losses due to credit risk can be described as: Assuming the only random variable is b i :

5. Joint Events The CL distribution depends on the correlation between the default events. When the defaults events are uncorrelated: When the defaults events are perfectly correlated

6. Joint Events For Instance, p(A)=p(B)=1% In the uncorrelated case: In the perfectly correlated case:

7. Joint Events When  <1:

8. Joint Events Consider the pervious example and assume the  =0.5:

9. Credit VaR Consider a portfolio of $100M composed of 3 bonds A, B and C with the following default probabilities and CE: Default Prob. CE ($M)Bond 0.0525A 0.1030B 0.2045C For simplicity, assume: 1. Exposures are constant; 2. The recovery rates are zero; 3. The default events are independent

10. Credit Var  2 =(L-(EL)) 2 p(L) E(L)=Lp(L)C. Prob.P(L)L ($M)Default 120.800.68400.6840None 4.970.9000.72000.036025A 21.322.2800.79600.076030B 172.387.6950.96700.171045C 6.970.2200.97100.004055A&B 28.990.6300.98000.009070A&C 72.451.4250.99000.019075B&C 7.530.1001.00000.0010100A&B&C 434.713.25Sum

11. Credit VaR

12. With a confidence level of 95% the VaR is $45M The unexpected loss is:

13. Credit Diversification A portfolio of loan is less risky than single loans Consider different alternatives for $100M loan portfolio: One loan of $100M 10 loans each for $10M 100 loans each for $1M 1,000 loan each for $0.1M Assume a fixed default probability of 1% for all loans and are independence across loans

14. Credit Diversification In the first case:

15. Credit Diversification In the second case:

16. Credit Diversification In the third case:

17. Credit Diversification In the last case: This reflects the Central Limit Theory by which the distribution of the sum of independent variables tends to normal distribution.

18. Credit Diversification The loans diversification does not effect the expected loss but decreases the variance. With N independent defaults events with the same probability of p, we have:

19. Credit Diversification In reality, there is some correlation between the defaults events, which are all affected by the general state of the economy: many more defaults occur in a recession than in expansion. In this case the distribution will lose its asymmetry more slowly. The solution for this is to limit the exposure to a particular sectors – defaults are more correlated among sectors than across sectors.

20. Historical Default Rates Cumulative default rate measure the total frequency of default at any time between the starting date and year T. According to the S&P experience - from 10,000 BBB rated firms, there where 36 defaults over one year, and 96 defaults over 2 years. Based on the Cumulative default rate one can derives the marginal default rate, which is the frequency of default during year T.

21. Historical Default Rates Definitions M T – The number of issuers rated R that default in year T N T – The number of issuers rated R that have no default by the beginning in year T. d T – The marginal default rate during year T – the proportion of issuers, relative to the number at the beginning of year T. S T – The survival rate - The number of issuers rated R that will not have default by T. P T – The probability of defaulting in year 2. C T – The cumulative default rate at the end of year T

22. Historical Default Rates The marginal default rate during year T: The survival rate: The probability of defaulting in year 1: In order to default in year 2, the firm must have survived the first year and default in the second

23. Cumulative Default Rates Thus, the cumulative default rate at end of year 2: In order to default in year 3, the firm must have survived the first and the second years and default in year 3.

24. Default Process Default No default d1d1 1-d 1 d2d2 d3d3 1-d 3 1-d 2

25. Historical Default Rates Numerical Example Consider a BBB rated firm that has default rates of d 1 =4%, d 2 =6% and d 3 =8% What are the survival rates at the end of years 1,2 and 3? What is the probability of defaulting in years 1,2 and 3? What is the cumulative default rates at the end of years 1,2 and 3?

26. Historical Default Rates Numerical Example

27. Recovery Rates Credit rating agencies measure recovery rates using the historical observations of the value of the debt right after default. The historical observations reveal that the RR depend on: The state of the economy The seniority of debtor – the proceeds from liquidation should be divided according to the absolute priority rule

28. Recovery Rates Credit rating agencies measure recovery rates using the historical observations of the value of the debt right after default. The historical observations reveal that the RR depend on: The state of the economy The seniority of debtor – the proceeds from liquidation should be divided according to the absolute priority rule

29. Recovery Rates Priority rule Secured creditors – up to the extent of secured collateral Priority creditors – post-bankruptcy creditors and taxes. General creditors – unsecured creditors before bankruptcy Shareholders

30. Recovery Rates S&P’s Historical RR for Corporate Debt Weighted AverageSeniority Ranking 49.32Senior secured 47.09Senior unsecured 32.46Subordinated 35.51Junior subordinated 40.23Total