P2.T6. Credit Risk Measurement & Management Malz, Financial Risk Management: Models, History & Institutions Portfolio Credit Risk Bionic Turtle FRM Video Tutorials By David Harper, CFA FRM 1
Portfolio Credit Risk Define default correlation for credit portfolios. Identify drawbacks in using the correlation-based credit portfolio framework. Assess the effects of correlation on a credit portfolio and its Credit VaR. Describe the use of a single factor model to measure portfolio credit risk, including the impact of correlation. Describe how Credit VaR can be calculated using a simulation of joint defaults with a copula. 2
Define default correlation for credit portfolios. Default correlation drives the likelihood of having multiple defaults in a portfolio of debt issued by several obligors. The simplest framework for understanding default correlation is the case of only two firms (or obligors or credits). The assumptions are: Two firms (or countries, if we have positions in sovereign debt). With respective probabilities of default = and Over some time horizon ( ) And a joint default probability (probability that both default over ) equal to Then the default correlation, =, is given by: (1 ) (1 ) 3
Identify drawbacks in using the correlation-based credit portfolio framework. Curse of dimensionality: If there are N credits in the portfolio, we need to define N default probabilities and N recovery rates. In addition, we require N(N 1) pairwise correlations. In modeling credit risk, we often set all of the pairwise correlations equal to a single parameter. But that parameter must then be non-negative, in order to avoid correlation matrices that are not positive-definite and results that make no sense: Not all the firms event of default can be negatively correlated with one another. For most companies that issue debt, most of the time, default is a relatively rare event. This has two implications: Default correlation is hard to measure or estimate using historical default data. Most studies have arrived at one-year correlations on the order of 0.05. However, estimated correlations vary widely for different time periods, industry groups, and domiciles, and are often negative. Default correlations are small in magnitude. However, an optically small correlation can have a large impact. 4
Assess the effects of correlation on a credit portfolio and its Credit VaR. Portfolio credit VaR is a quantile of the credit loss minus the expected loss of the portfolio. Default correlation has a tremendous impact on portfolio risk. Default correlation affects the volatility and extreme quantiles of loss but does not impact the expected loss (EL). If default correlation in a portfolio of credits is equal to 1.0, then the portfolio behaves as if it consisted of just one credit. No credit diversification is achieved. If default correlation is equal to 0, then the number of defaults in the portfolio is a binomially distributed random variable. Significant credit diversification may be achieved. Credit VaR = Quantile of - Credit Loss Expected Loss 5
Assess the effects of correlation on a credit portfolio and its Credit VaR (continued) Portfolio value (millions), A $1,000.0 $1,000.0 $1,000.0 Default probability, PD 0.50% 2.00% 5.00% EL (millions) = A*PD $5.0 $20.0 $50.0 Number of positions, n= 1 Value per position (mm), V $1,000.0 $1,000.0 $1,000.0 Confidence 95.0% Prop of defaults, d/n (d = defaults) 0.00% (0.0) 0.00% (0.0) 100.00% (1.0) loss quantile, q = d*v $0.0 $0.0 $1,000.0 95.0% CVaR = UL = q - EL -$5.0 -$20.0 $950.0 Confidence 99.0% Prop of defaults, d/n (d = defaults) 0.00% (0.0) 100.00% (1.0) 100.00% (1.0) loss quantile, q = d*v $0.0 $1,000.0 $1,000.0 99.0% CVaR = UL = q - EL -$5.0 $980.0 $950.0 Number of positions, n= 50 Value per position, V $20.0 $20.0 $20.0 Confidence 95.0% Prop of defaults, d/n (d = defaults) 2.00% (1.0) 6.00% (3.0) 10.00% (5.0) loss quantile, q = d*v $20.0 $60.0 $100.0 95.0% CVaR = UL = q - EL $15.0 $40.0 $50.0 Confidence 99.0% Prop of defaults, d/n (d = defaults) 4.00% (2.0) 8.00% (4.0) 14.00% (7.0) loss quantile, q = d*v $40.0 $80.0 $140.0 99.0% CVaR = UL = q - EL $35.0 $60.0 $90.0 The table here replicates Malz Table 8-1 which summarizes the results of credit VaR for n = 1, 50, 1,000, for default probabilities π =0.5%, 2%, 5%, and at confidence levels of 95 and 99 percent. Number of positions, n= 1,000 Value per position, V $1.0 $1.0 $1.0 Confidence 95.0% Prop of defaults, d/n (d = defaults) 0.90% (9.0) 2.80% (28.0) 6.20% (62.0) loss quantile, q = d*v $9.0 $28.0 $62.0 95.0% CVaR = UL = q - EL $4.0 $8.0 $12.0 Confidence 99.0% Prop of defaults, d/n (d = defaults) 1.10% (11.0) 3.10% (31.0) 6.70% (67.0) loss quantile, q = d*v $11.0 $31.0 $67.0 99.0% CVaR = UL = q - EL $6.0 $11.0 $17.0 6
Assess the effects of correlation on a credit portfolio and its Credit VaR (continued) 7
Assess the effects of correlation on a credit portfolio and its Credit VaR (continued) 8
Assess the effects of correlation on a credit portfolio and its Credit VaR (continued) 9
Assess the effects of correlation on a credit portfolio and its Credit VaR (continued) For a default probability of 2%, we illustrate how credit Var varies for each value of n: For a single credit portfolio(n=1), the extreme loss given default is equal to $1,000,000,000 as we assume recovery to be zero. Since the expected loss is subtracted from the extreme loss to get VaR, if default probability is π = 2%, the credit VaR is $20,000,000. For n= 50, each position has a future value, if it doesn t default, of $20,000,000. The expected loss is π 1,000,000,000 which is the same as for the single-credit portfolio. If π = 2 %, the 95th percentile of the number of defaults is 3 and the credit loss is $60,000,000. Subtracting the expected loss of $20,000,000, we get a credit VaR of $40,000,000. For the same default probability (π = 2 %), if n =1,000 the 95th percentile of defaults is 28, and the credit loss is $28,000,000, so the credit VaR is $8,000,000. This shows that as we continue to increase the number of positions(n) and decrease their size, keeping the total value of the portfolio constant, the variance of portfolio values decreases. 10
Describe the use of a single factor model to measure portfolio credit risk, including the impact of correlation. To use the single-factor model to measure portfolio credit risk, we start with a number of firms =1,2,..., where each firm has: Its own correlation to the market factor, Its own standard deviation of idiosyncratic risk, 1, and Its own idiosyncratic shock Firm s return on assets is given by: = + 1 = 1,2, 11
Describe the use of a single factor model to measure portfolio credit risk, including the impact of correlation (continued) We assume and are standard normal variates, and further not correlated with one another. In addition, assume the are not correlated with one another: ~ 0,1 ~ 0,1 = 1,2,, = 0 = 1,2,, = 0, = 1,2, Under these assumptions, each is a standard normal variate. Since both the market factor and the idiosyncratic shocks are assumed to have unit variance, the beta of each credit to the market factor is equal to. The correlation between the asset returns of any pair of firms and is : = 0 = 1,2, = + 1 = 1 = 1,2,, = + 1 + 1 =, = 1,2, 12
Describe the use of a single factor model to measure portfolio credit risk, including the impact of correlation (continued) The single-factor model has a feature that makes it an especially handy way to estimate portfolio credit risk: Conditional independence: It is the property that once a particular value of the market factor is realized, the asset returns and hence default risks are independent of one another. Conditional independence is a result of the model assumption that the firms returns are correlated only via their relationship to the market factor. To see this, we let take on a value such that three things happen: 1. The conditional probability of default is greater or smaller than the unconditional probability of default, unless either = 0 or = 0. 2. Given a realization of less than or equal to triggers default. As we let vary from high to low values, a smaller idiosyncratic shock will suffice to trigger default. 3. The conditional variance of the default distribution is 1, so the conditional variance is reduced from the unconditional variance of 1. It makes the asset returns of different firms independent. The are independent, so the conditional returns 1 and 1 and thus the default outcomes for two different firms i and j are independent. Thus, the unconditional default distribution is a standard normal while conditional default is a normal with a mean of and standard deviation of 1 13
Describe the use of a single factor model to measure portfolio credit risk, including the impact of correlation (continued) The conditional cumulative default probability function can be represented as a function of as: = Φ = 1,2,. 1 For a given realization of the market factor, the asset returns of the various credits are independent standard normals. Extending this single factor model to number of firms in the portfolio, by applying the law of large numbers, we can estimate the credit VaR of a granular, homogeneous portfolio. When is very large, for each level of the market factor, the loss level, ( ) that is, the fraction of the portfolio that defaults, converges to the conditional probability that a single credit defaults, given for any credit by: = = Φ So, = This solves for the that gives a specific quantile of the standard normal distribution. 14
Describe the use of a single factor model to measure portfolio credit risk, including the impact of correlation (continued) As shown in the table below, say a firm has default as 1% so that = 2.33, then: Example (Malz 8.4) = 0.4 and unconditional probability of If we enter a modest economic downturn, with = 1.0, a 1.926 will trigger default, the conditional asset return distribution is ( 0.4, 0.9165), and the conditional default probability is the probability that this distribution takes on the value 2.33 which is found to be 1.78%. If we have a more severe economic downturn, with = 2.33, a 1.394 will now trigger default, the firm s conditional asset return distribution is ( 0.932, 0.9165) and the conditional default probability is 6.41 %. 15
Describe the use of a single factor model to measure portfolio credit risk, including the impact of correlation (continued) Example (Malz 8.4) 16
Describe how Credit VaR can be calculated using a simulation of joint defaults with a copula. We start with a set of univariate default time distributions that we assume for each of the credits in the portfolio. Then we simulate joint defaults. We need a multivariate default time distribution. However, The default time distributions that we have are univariate, and we do not know how to connect them with one another to derive the behavior of portfolios of credits. Copulas may be used as it permits us to separate the issue of the default-time distribution of a single credit from the issue of the dependence of default times for a portfolio of credits. We can combine the default-time distributions we assume with a multivariate normal distribution. Suppose we have a portfolio with securities of issuers. We have estimated singleissuer default- time distributions 1 1,..., ( ). We do not know the joint distribution ( 1,..., ). So, we specify a copula function 1,..., specifying that: 1,..., = ( 1,..., ) 17
Describe how Credit VaR can be calculated using a simulation of joint defaults with a copula (continued) There are four steps in computing a credit VaR : 1. Specify the copula function that we ll use. 2. Simulate the default times. 3. Apply the default times to the portfolio to get the market values and P&Ls in each scenario. 4. Add results to get portfolio distribution statistics. 18
The End P2.T6. Credit Risk Measurement & Management Malz, Financial Risk Management: Models, History & Institutions Portfolio Credit Risk 19