Portfolio Credit Risk II

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1 University of Toronto Department of Mathematics Department of Mathematical Finance July 31, 2011

2 Table of Contents 1 A Worked-Out Example Expected Loss Unexpected Loss Credit Reserve 2 Examples Problem 3 Methods for Calculating Credit Exposure Interest Rates and Spreads Algorithm

3 A Worked-Out Example

4 A Simple Bond A Worked-Out Example Expected Loss Unexpected Loss Credit Reserve Example (A Simple Bond) Consider a bond issued from a default-prone party, paying two $5 coupons after the end of the second and fourth years. We assume throughout the duration of the bond the interest rates are 0% (this assumption simplifies discounting). The default-prone party has a yearly defalut probability of 7% and when it defaults no money can be recovered (recovery rate= 1 severity= 0). We assume that the deafult-free party maintians a risk-capital to cover the standard deviation of losses that is is adjusted annually and that it demands a certain return on this risk-capital.

5 Expected Loss Unexpected Loss Credit Reserve Survival and Default Probabilities 1 p (1 p) 2 (1 p) 3 (1 p) 4 ½ ¼º ¼º ¼º ¼ ¼º ½ Æ Æ Æ Æ Æ ¼º¼ ¼º¼ ½ ¼º¼ ¼ ¼º¼ Survival and Default Probabilities.. where D =Default. ND =Not Default. Nodes are one year apart.

6 Expected Loss Calculation Expected Loss Unexpected Loss Credit Reserve Expected loss calculation: There are two, equivalent in this case, ways to compute the expected loss. Since the value of the contract is always non-negative to the default-free party we do not need to discard any future events (as already explained this not a limitation, as every contract can be decomposed into contracts that have always non-negative or non-positive value). One way to compute the expecetd loss is to compute the expected cashflows. Recall that there are two such cashflows: 1 $5 at t = 2, 2 $5 at t = 4. but we also need to factor in the probabilities of default within these periods.

7 Expected Loss Unexpected Loss Credit Reserve Expected Loss Calculation Continued... Continuing... There are two cashflows of $5 each, and the expected cashflow is: EC = 5 p ND (0,2] +5 p ND (0,4] = $8.065 where p ND (i,j] is the probability that the default-prone party does not default in the time interval between years i and j (i < j). The expected loss is: EL = 10 EC = $1.935

8 Expected Loss Unexpected Loss Credit Reserve An Equivalent Way to Calculate Expected Loss The second way is to calculate loss: Is based on the yearly exposure: Exposure(year 1 ) = $10 Exposure(year 2 ) = $10 Exposure(year 3 ) = $5 Exposure(year 4 ) = $5 where no correction is due to discounting was included, since interst rates are flat at %0 and Exposure(year 1 ), the value of the contract just before year 1.

9 Expected Loss Unexpected Loss Credit Reserve An Equivalent Way to Calculate Expected Loss Continued... Continuing... The expected losses are: EL = Exposure(year 1 ) p D (0,1] +Exposure(year 1 ) p D (1,2] +Exposure(year 3 ) p D (2,3] +Exposure(year 4 ) p D (3,4] = = $1.935 where p D (2,3] is the probability that the default-prione party defaults in the time interval between years 2 and 3.

10 The Unexpected Loss Expected Loss Unexpected Loss Credit Reserve Recall that the unexpected loss is the variance of the losses, so: V(L [0,1) ) = Exposure(year 1 ) 2 p D (0,1] ( ) Exposure(year 1 2 ) p D (0,1] ( ) = (EL(1)) = 6.51 p D [0,1) V(L [1,2) ) = Exposure(year 2 ) 2 p D (1,2] ( ) Exposure(year 2 2 ) p D (1,2] ( ) 1 = (EL(2)) 2 1 = 6.08 p D [1,2) ( ) V(L [2,3) ) = (EL(3)) = 1.42 p D [2,3) ( ) V(L [3,4) ) = (EL(4)) = 1.33 p D [3,4) where V(X) = Var(X) = E [ X 2] (E[X]) 2

11 Credit Reserve A Worked-Out Example Expected Loss Unexpected Loss Credit Reserve Credit Reserve: If, for any example, a risk-capital of two standard deviations is required, the default-free party anticipates to use risk-capital equal to: 1 $5.10 at year 0, 2 $4.93 at year 1, 3 $2.38 at year 2 and 4 $2.31 at year 3. A yearly return of 10% on such capital leads to an additional surcharge of $1.47. Remark Notice that a high enough return rate would lead to the possibility of arbitrage (in this case arbitrage corresponds to an intial credit-risk premium of more than $10).

12 Examples Problem

13 A Worked-Out Example Examples Problem : is the unexpected credit loss, at some confidence level, over a certain time horizon. If we denote by f(x) the distribution of credit losses over the prescribed time horizon (typically one year), and we denote by c the confidence level (i.e. 95%), then the Worst-Credit-Loss (WCL) is defined to be: f(x)dx = 1 c WCL and = (Worst-Credit Loss) (Expected Credit Loss) }{{} Leads to Reserve Capital

14 Examples Problem Example 23-5: FRM Exam 1998 Example (Example 23-5: FRM Exam 1998) A risk analyst is trying to estimate the for a risky bond. The is defined as the maximum unexpected loss at a confidence level of 99.9% over a one month horizon. Assume that the bond is valued at $1M one month forward, and the one year cumulative default probability is 2% for this bond What is your estimate of the for this bond assuming no recovery? a) $20,000 c) $998,318 b) $1,682 d) $0

15 Solution 23-5: FRM Exam 1998 Examples Problem Solution (Solution 23-5: FRM Exam 1998) What is your estimate of the for this bond assuming no recovery? a) $20,000 b) $1,682 c) $998,318 d) $0 Why? If d is the monthly probability of default then: (1 d)12 = (0.98), so d = , ECL =$ 1,682, WCL(0.999)=WCL( )=$1,000,000, CVaR=$1,000,000 1, 682 =$998,318.

16 Example 23-6: FRM exam 1998 Examples Problem Example (Example 23-6: FRM exam 1998) A risk analyst is trying to estimate the for a portfolios of two risky bonds. The is defined as the maximum unexpected loss at a confidence level of 99.9% over a one month horizon. Assume that both bonds are valued at $500,000 one month forward, and the one year cumulative default probability is 2% for each of these bonds. What is your best estimate of the for this portfolio assuming no default correlation and no recovery? a) $841 c) $10,000 b) $1,682 d) $249,159

17 Examples Problem Solution: Example 23-6: FRM exam 1998 Solution (Solution: Example 23-6: FRM exam 1998) What is your best estimate of the for this portfolio assuming no default correlation and no recovery? a) $841 b) $1,682 c) $10,000 d) $249,159 Why? If d is the monthly probability of default then: (1 d)12 = (0.98), so d = , ECL =$ , WCL(0.999)=WCL( )=$250,000, CVaR= $250,000 $840 =$249,159.

18 Examples Problem Solution: Example 23-6 Continued... Credit Loss Distribution As before, the monthly discount is d = The 99.9% loss quantile is about $500,000 Also we have that: EL =$ , WCL(0.999)=WCL( )=$250,000, CVaR= $250, 000 $840 =$249,159. Default Probability Loss P L 2 Bonds d 2 = $500,000 $1.4 1 Bond 2 d (1 d) = $250,000 $ Bonds (1 d) 2 = $0 $0

19 Problem A Worked-Out Example Examples Problem Example Consider a stock S valued at $1 today, which after one period can be worth S T : $2 or $0.50. Consider also a convertible bond B, which after one period will be worth max(1,s T ). Assume the stock can default (p = 0.05), after which event S T = 0 (no recovery). Determine which of the following three portfolios has the lowest 95%-Credit-VaR: 1 B 2 B S 3 B +S

20 Methods for Calculating Credit Exposure Interest Rates and Spreads Calculating Credit Exposure

21 A Worked-Out Example Methods for Calculating Credit Exposure Interest Rates and Spreads Credit exposure: How much can one lose due to counterparty default? max(swap Value t,0)

22 Continued Methods for Calculating Credit Exposure Interest Rates and Spreads What is the 99% credit Var? Sort losses and take the 99 th percentile.

23 Expected Shortfall A Worked-Out Example Methods for Calculating Credit Exposure Interest Rates and Spreads Expected loss given 99% VaR: Take the Average of the exposure greater than the 99% percentile

24 Simulation A Worked-Out Example Methods for Calculating Credit Exposure Interest Rates and Spreads Run a Monte Carlo Simulation: 10,000 Simulations Simulate: 1 Interest Rates. 2 Credit Spreads.

25 Interest Rates A Worked-Out Example Methods for Calculating Credit Exposure Interest Rates and Spreads Black-Karsinski Model: dln(r) = a ( ) θ a ln(r) dt +σ r dw We estimate the following from the bonds: Tenor Initial IR Mean Volatility 0.5 Years 8.18% 7.99% 5.98% 10 Years 10.56% 8.93% 5.64%

26 Spreads A Worked-Out Example Methods for Calculating Credit Exposure Interest Rates and Spreads Vasicek Model ( ) θ ds = a a s dt +σ s dw Estimated values: Tenor Initial IR Mean Volatility 0.5 Years 2.4% 2.546% 0.535%

27 Algorithm A Worked-Out Example Methods for Calculating Credit Exposure Interest Rates and Spreads Interest rate spread: Choleski Decomposition. Sample from the normal distribution. Interest rate-spread correlation: The correlation matrix above is estimated from bond rates and new car sales from Each column represents the correlation 6 months, 5 years and 10 years of spreads to 5 year interest rates.

28 Algorithm Continued... Methods for Calculating Credit Exposure Interest Rates and Spreads Continuing: 1 Iterate the Black-Karasinski. 2 Calculate the Value of the Swap as the difference of the values of Non-Defaultable Fixed and Floating Bonds. 3 After 10,000 calculate the credit VaR and the expected shortfall.

29 Simulation: Credit Exposure Methods for Calculating Credit Exposure Interest Rates and Spreads Where: 99% Exposure 95% Exposure Credit Exposure

30 Simulation: Expected Shortfall Methods for Calculating Credit Exposure Interest Rates and Spreads

Risk Management. Exercises

Risk Management. Exercises Risk Management Exercises Exercise Value at Risk calculations Problem Consider a stock S valued at $1 today, which after one period can be worth S T : $2 or $0.50. Consider also a convertible bond B, which