# Modeling Default Correlation

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Modeling Default Correlation
University of Chicago Masters in Financial Mathematics Week 3 Jon Frye Federal Reserve Bank of Chicago The views expressed are the author’s and do not necessarily represent the views of the management of the Federal Reserve Bank of Chicago or the Federal Reserve System.

Modeling Default Correlation Outline of presentation
A. Setting the stage B. Correlation in variants of CreditManager C. Summary and conclusion

Modeling Default Correlation Outline of presentation
A. Setting the stage 1. Default correlation and portfolio loss Two state (default-only) model Default correlation Multi-state (marked-to-market) model 2. Latent variable probability model

Default correlation and portfolio loss Two state (default-only) model
The bank has exposure of \$1 to each of two firms. The credit capital analysis has a horizon (one year). Either the maturity of the exposures equals the horizon, or the bank does not care about the marked-to-market value of its exposure portfolio. Therefore, the bank experiences a loss only if a firm defaults. Loss given default is equal to 100%.

Default correlation and portfolio loss
Let D1 = 1 if firm 1 defaults, D1 = 0 otherwise Let D2 = 1 if firm 2 defaults, D2 = 0 otherwise Correlation is defined as follows:

Default correlation and portfolio loss
Var(D1) = E [ (D1 - E [ D1 ])2 ] = E [ D D1 E [ D1 ] + (E [ D1 ])2 ] = E [ D1 ] - 2 (E [ D1 ])2 + (E [ D1 ])2 = p1 - p12 = p1(1-p1) where p1 = Prob [ D1 = 1 ]

Default correlation and portfolio loss
Cov(D1, D2) = E [ (D1 - E [ D1 ]) (D2 - E [ D2 ]) ] = E [ D1D2 ] - E [ D1 ] E [ D2 ] = p12 - p1p2 where p12 = Prob [ D1 = 1 and D2 = 1 ]

Default correlation and portfolio loss
For any two events (such as D1 and D2), if we have their probabilities p1 and p2, then p12 implies correlation and correlation implies p12 For example, if p12 = p1 p2, then Corr = 0 Next: Calculating default correlation

Default correlation and portfolio loss
Multi-state (marked-to-market) model

Default correlation and portfolio loss
Multi-state (marked-to-market) model With 3 states, there are 4 free probabilities With 4 states, there are 9 free probabilities... How can we estimate, for example, the joint probability that Firm 1 becomes junk and Firm 2 defaults? We do not have the data we want. If we had a long historical series of a large number of firms similar to Firm 1 and Firm 2, we could hope to estimate the joint probabilities as observed frequencies.

Default correlation and portfolio loss
Multi-state (marked-to-market) model A common way (and the way used by CreditManager) around the data problem is to assume the joint probabilities reflect a simpler underlying relationship between “latent variables” having fewer parameters. Most commonly (and in CreditManager), the latent variables are modeled as bivariate normal.

2. Latent variable probability model
Properties of bivariate normal distribution Latent variables, joint probabilities, and event correlations

Latent variable probability model Normal latent variable
For X ~ U[0, 1], default if X < p0 = 0.07% For Z ~ N[0, 1], default if Z < -1(p0) = -3.19 X = <=> Z = 1.56 <=> transition to A rating

Latent variable probability model Normal latent variable
The choice of latent variable distribution does not matter for modeling the marginal probabilities (probability that a single firm transitions to default or another state) Modelers specify bivariate normal latent variables, in part, because the entire distribution (and therefore the probabilities of joint events, such as Firm 1 becomes junk and Firm 2 defaults) is controlled by the single parameter, .

Latent variable probability model
Properties of bivariate normal distribution The fully elaborated bivariate normal density has five parameters, 1, 2, 1, 2, and . When the variables are standardized, only  is free.

Latent variable probability model
Properties of bivariate normal distribution Excel: the bivariate normal (2D)

Latent variable probability model
Properties of bivariate normal distribution Excel: two firm portfolio

Latent variable probability model
Properties of bivariate normal distribution Given , the density of the latent variables is known. Given the density of the latent variables and both firms’ transition probabilities, the probabilities of joint events are known. Given the probabilities of joint events (such as p12), the default (and other event) correlations are known. Thus,  determines all joint probabilities and all event correlations.

Latent variable probability model
Properties of bivariate normal distribution Generalizing to a portfolio of many firms, multivariate standard normal latent variables capture the probability distribution of all joint events, given: the transition probabilities for each firm the correlation matrix of the latent variables With a portfolio of thousands of firms, the correlation matrix itself becomes too big to estimate consistently. CreditMetrics assumes that the latent variables arise from a simpler structure of underlying factors.

Correlation in variants of CreditManager

Correlation in variants of CreditManager
B. Correlation in variants of CreditManager 3. Single factor model of latent variables Model Latent variable correlation matrix Variance of default rate Fitting the model to default data 4. A neglected correlation: recovery 5. Multi- factor model of latent variables 6. Correlation in multi-state CreditManager

Correlation in variants of CreditManager
So far, we have modeled joint probabilities by using latent variables. Now we model the latent variables as rising from underlying factors. The factors are normal, so the latent variables are normal as before. We start with the simplest case: one factor, in a two-state (default mode) model. Then we move to the multi-factor, multi-state framework of conventional CreditManager.

Single factor model of latent variables
Let Aj be the latent variable for firm j: Therefore, Aj is standard normal

Single factor model of latent variables
All firms are affected by X X is called the systematic risk factor wj is called firm j’s loading on the systematic risk factor Each firm is affected by its own independent Zj The Zj are called the idiosyncratic risk factors

Single factor model of latent variables
The loading wj controls how much of the variance of Aj comes from X. The remainder comes from Zj. A more “cyclical” firm will have a higher loading on the systematic risk factor than will a less cyclical firm. Cyclical firms are affected by the same underlying factor that affects other (cyclical and non-cyclical ) firms

Single factor model of latent variables
Latent variable correlation matrix The correlations between latent variables are controlled by the loadings {wj} on the systematic risk factor. If two firms have the same loading, w, the correlation between their latent variables is w2.

Single factor model of latent variables
Latent variable correlation matrix Suppose all firms have w = 0.5. Aj = 0.5 X Zj Corr(Aj, Ak) = 0.52 = 25%, for all j, k The correlation matrix has 1’s along the diagonal and 25% in other elements.

Single factor model of latent variables
Latent variable correlation matrix Suppose there are two kinds of firms Cyclical firms have w = 0.5. Non-cyclical firms have w = 0.3. Acyclical, j = 0.5 X Zcyclical, j Anon-cyclical, k = 0.3 X Znon-cyclical, k

Single factor model of latent variables
Latent variable correlation matrix Cyclical Non-cyclical 1 25% 25% 15% 15% 15% Cyclical 25% 1 25% 15% 15% 15% 25% 25% 1 15% 15% 15% 15% 15% 15% 1 9% 9% Non-cyclical 15% 15% 15% 9% 1 9% 15% 15% 15% 9% 9% 1

Single factor model of latent variables Variance of default rate
We can simulate this by drawing the systematic risk factor and the idiosyncratic risk factors: Acyclical, j = 0.5 X Zcyclical, j Anon-cyclical, k = 0.3 X Znon-cyclical, k We can use the simulated latent variables in either a two-state (default only) model or in a multistate (marked-to-market) model.

Single factor model of latent variables Variance of default rate
For simplicity, we will use a two-state model. A loss is experienced only if a firm defaults. We will fit this single factor, two-state latent variable model to Moody’s default data. The key feature of the Moody’s data is the variance (from year to year) of the default rate. First consider the variance of the default rate in general terms, and then perform the actual estimation.

Single factor model of latent variables
Variance of default rate Variance of the default rate depends on default correlations: We know that higher loadings {wj, wk} produce higher latent variable correlations {j,k}, which produce higher default correlations {Corr(Dj, Dk)}, which produce a higher variance of the default rate. Next we look at two examples of that chain of effect.

Single factor model of latent variables Variance of default rate
Example 1: Suppose wj = 0 for all firms. Aj = 0 X + Zj Each Aj depends only on its own independent Zj. The {Zj} are independent, so the {Aj} are independent, therefore the {Dj} are independent. On each run, if there are many firms, the simulated default rate will equal the expected default rate (+/- minor sampling variation). There will be little variation in the default rate from year to year.

Single factor model of latent variables Variance of default rate
Example 2: Suppose wj = 0.9 for all firms. Each Aj depends mostly on X, and hardly on Zj. A high value of X causes most of the Aj to be high, so most of the firms do well. Only firms that get a very bad Zj will default. A very low value of X causes most of the Aj to be low. Even firms with average values of Zj may default The default rate varies significantly from year to year.

Single factor model of latent variables Variance of default rate
We will maintain for estimation purposes the (grossly simplified) assumption that all firms have the same loading on the systematic risk factor. Then we can use the year-to-year variation in the default rate to estimate this loading. To stay consistent with “Collateral Damage,” denote the common loading as p. Since a higher p causes greater variability in the default rate, there is only one level of p consistent with the observed variation of default.

Single factor model of latent variables
Fitting the model to default data Default data universe: US Domicile Industry Group not Financial, Banking, Insurance, … Moody’s Rated Aaa, Aa1, Aa2, Aa3, A1, A2, A3, Baa1, Baa2, Baa3, Ba1, Ba2, Ba3, B1, B2, B3, Caa, Ca, C

Single factor model of latent variables
Fitting the model to default data

Single factor model of latent variables
Fitting the model to default data

Single factor model of latent variables
Fitting the model to default data A key assumption: the law of large numbers applies. There are enough firms that the observed default rate equals its expectation (conditioned on X):

Single factor model of latent variables
Fitting the model to default data Adapting this key assumption to the data set, with T years: t = 1,…,T ( ) and R rating grades: r = 1,…,R (Aaa, Aa1,…) In year t, the default rate of a firm rated r is Assume that PDr is the long-term average default rate in grade r

Single factor model of latent variables
Fitting the model to default data Let hr,t be the fraction of firms rated r in year t, so: Then the default rate for the portfolio in year t is: Note that the function g is monotonic in x.

Single factor model of latent variables
Fitting the model to default data The default rate is a monotonic function of the systematic risk factor, X, which has a known probability density: We can find the density of DRt by change of variable, and, assuming independence from year to year, we can find the density of {DRt}. The density of {DRt} depends on the parameter p. Given the {DRt} actually observed, a certain value of p provides the maximum likelihood estimate.

Single factor model of latent variables
Fitting the model to default data The density of DRt equals By taking the derivative, and with independence year-to-year, we have the joint distribution of the set of observed default rates:

Single factor model of latent variables
Fitting the model to default data The density of {DRt} is p enters this expression both directly and through g-1. Maximizing the density with respect to p provides the maximum likelihood estimate.

Single factor model of latent variables
Fitting the model to default data The default rate of bonds has a certain variation from year to year. This variation stems from the randomness of X, which has a standard normal distribution. The loading p connects the variation of X to the variation of the default rate: the greater is p, the more the default rate will vary from year to year. There is only one level of p consistent with the default rate variation that is actually observed.

Single factor model of latent variables
Fitting the model to default data The maximum likelihood estimate of p is 0.23 for this data set. The implied level of correlation between the latent variables of two firms is (0.23)2 = 5.3%. (Note, this is below the average correlation between stocks, thought to be in the range of 20-30%.) Given p, the monotonic function g is determined and we can “back out” the level of X consistent with each year’s default rate:

Single factor model of latent variables
Default Rate and { }

Single factor model of latent variables
Fitting the model to default data The loading of 0.23 provides the best fit for the variance of the default rate for this data set. A higher loading, such as suggested by stock price correlation, would produce too much default rate variance. To fit a default-only model, we used only default data. A multi-state model could be fit using transition data, and would probably have a different average loading. In practice, a risk manager wants to know the loading of each firm, not just an overall average, to detect cyclically sensitive credits.

4. A neglected correlation: recovery
Evidence of systematic recovery risk Model Fitting the model to recovery data

A neglected correlation: recovery Evidence of systematic recovery risk
Recovery universe: US Domicile, Non-Financial Bonds, Not Loans Not backed by second entity Where Moody’s has a Default Price (observed one month after default, measures recovery)

A neglected correlation: recovery Evidence of systematic recovery risk

A neglected correlation: recovery Evidence of systematic recovery risk

A neglected correlation: recovery
Model We model each recovery as depending on the systematic risk factor (with loading q), and on an independent factor idiosyncratic to that recovery. We need two more parameters to model the mean and the variance of recovery. Mean recovery is assumed to depend on the seniority of the bond. The variance of recovery is assumed uniform for all bonds.

A neglected correlation: recovery
Model There are T years: t = 1, …, T There are J seniority classes: j = 1, …, J Senior Secured (SS) Senior Subordinated (SR) Senior Unsecured (SU) Subordinated (SB) There are Nj,t bonds of class j in year t: i = 1, …, Nj,t Bond recovery is specified as follows:

A neglected correlation: recovery Fitting the model to recovery data
The {Xt} were estimated from the default data. (This is sequential MLE, rather than simultaneous MLE.) Treating Xt as known, each recovery has a normal distribution that depends on μj, σ, and q. Therefore, average recovery in a year has a normal distribution that depends on {μj}, σ, and q. Maximizing the joint density with respect to the parameters produces the estimates σ = 0.32, q = 0.17.

A neglected correlation: recovery Fitting the model to recovery data
σ = 0.32: Recovery is risky. The difference between zero recovery and 96% recovery is only 3 SD. q = 0.17: Recovery varies with the systematic risk factor. A low realization of X causes both a high level of default and a low level of recovery. To obtain the relation between the default rate and the average recovery rate, we can substitute a range of X into the relevant equations. The following chart assumes the average portfolio distribution among rating grades and among seniority classes.

A neglected correlation: recovery Fitting the model to recovery data

A neglected correlation: recovery Fitting the model to recovery data
Bond recovery might fall by 25% or more in a bad year, e.g., from 45% to 20%. If in a bad year, loan recovery falls by 25%–say from an average of 75% to 50%–LGD doubles. Conventional credit models, which ignore systematic recovery risk, might miss half the risk.

A neglected correlation: recovery Fitting the model to recovery data
Existing portfolio credit models do not allow a role for systematic recovery risk. Therefore, they understate credit risk, perhaps by 50% in aggregate (see Depressing Recoveries, November 2000 Risk Magazine). The understatement is greater for deals having high expected recovery. The performance of a default-only credit model might be improved by a work-around called the “EL-equivalent substitution” (see Collateral Damage, April 2000 Risk Magazine).

5. Multi-factor model of latent variables
Structure of correlation matrix

Multi-factor model of latent variables
This model has two systematic risk factors, X1 and X2. These risk factors are not usually uncorrelated The correlation between two firms depends on the correlation between the risk factors. Each firm also depends on an idiosyncratic factor.

Multi-factor model of latent variables Structure of correlation matrix
Suppose there are two kinds of firms: Suppose Corr(X1, X2) = 0

Multi-factor model of latent variables Structure of correlation matrix
Industry 1 Industry 2 1 25% 25% 15% 15% 15% Industry 1 25% 1 25% 15% 15% 15% 25% 25% 1 15% 15% 15% 15% 15% 15% 1 25% 25% Industry 2 15% 15% 15% 25% 1 25% 15% 15% 15% 25% 25% 1

Multi-factor model of latent variables Structure of correlation matrix
A multi-factor model can: capture “pockets of correlation,” where the correlation is higher within a group than between groups. capture differences in the timing of the default (or other transition) cycle between industries or countries. These features are important, if the credit portfolio can become concentrated in a particular industry or country or other pocket of correlation. As before, the latent variables can drive a default-only model or a marked-to-market model.

6. Correlation in multi-state CreditManager
Orientation A few modest assumptions

Correlation in multi-state CreditManager
Orientation Multi-factor model: Allows richer structure to latent variable correlation matrix, as seen before. Multi-state model: The latent variables determine upgrades and downgrades as well as defaults. For example, there might be 8 states: AAA, AA, A, BBB, BB, B, C, and D

Correlation in multi-state CreditManager
Orientation There are many ways to calibrate a multi-state model. The first generalizes the approach of Collateral Damage, which used variation of the annual default rate. The generalization would fit loadings (or correlations) to data involving all rating transitions, not just transition to default. The model would fit the variation of transition frequencies from year to year. This approach might still miss pockets of correlation that interest the practitioner.

Correlation in multi-state CreditManager
Orientation A second approach would look directly to the objects of interest: the prices of credit-risky instruments such as bonds. The behavior of bond spreads might imply the parameters of the credit model. This approach is limited by the quality of corporate bonds data. Bonds have embedded options of uncertain value, which confounds the value of the default option. Further, when a credit becomes weak, the bond often becomes illiquid, and pricing becomes uncertain.

Correlation in multi-state CreditManager
Orientation The third approach, which is the one used in CreditManager, has several advantages. It uses data that is more available, consistent, and readily interpreted than corporate bond data. It can identify pockets of correlation more readily than methods that rely on transitions data. However, this approach requires a few modest assumptions.

Correlation in multi-state CreditManager
The latent variables are asset returns!!!! Merton model of default If firm’s asset value < liability value, the stockholders give the firm to the debt holders and walk away. The firm’s asset value has analogous thresholds for rating transitions to non-default rating grades. There are corresponding thresholds for asset returns.

Correlation in multi-state CreditManager
Asset returns can be inferred from stock returns!!!! KMV applies option pricing theory to the stockholders’ option to default. The level and volatility of a firm’s asset value should imply the level and volatility of its stock price. Working backward, KMV observes the level and volatility of the stock price and implies the firm’s asset value each period. These in turn imply the asset return. Another approach simply assumes a linear relationship between assets and stocks, so that the correlation between latent variables equals the correlation between stocks.

Correlation in multi-state CreditManager
Stock returns can be proxied in a factor model!!!! Stock returns are generally not available for all firms in the portfolio. If only some of the correlations are estimated from stock returns, and other correlations come from other sources, it is likely that the correlation matrix is not positive definite. In practice, stock prices are not used; a factor representation of stock prices is used instead. The correlation matrix of a factor model is always positive definite.

Correlation in multi-state CreditManager
The factors are country and industry indices!!!! The factors used to explain stock prices are the set of available stock indices. Any source of correlation beyond the movements of these indices is ignored.

Correlation in multi-state CreditManager
The loadings can be estimated by intuition!!!! For firms that have stock prices, the factor representation is the regression of the firm on the indices. This regression throws away information. The rationale for the approach comes from extending it to firms that do not have (enough history of) a stock price. Then the loadings are estimated intuitively, based on a knowledge of the firm’s domicile, industry, and so forth.

Correlation in multi-state CreditManager A few modest assumptions
The latent variables are asset returns!!!! Asset returns can be inferred from stock returns!!!! Stock returns can be proxied in a factor model!!!! The factors are country and industry indices!!!! The loadings can be estimated by intuition!!!! What can be said about this chain of assumptions?

Correlation in multi-state CreditManager A few modest assumptions
The assumptions made in CreditManager establish a chain of inference that begins with the multi-variate distribution of stock index returns and ends with the probability distribution of the ratings of all firms in the portfolio.

Correlation in multi-state CreditManager
A few modest assumptions-disadvantages Though many of the assumptions made in CreditManager have intuitive plausibility, few of them have been subject to explicit tests, and none of them can be considered “true.” In the aggregate, there is no assurance that the correlations that arise within the stock index factor framework are consistent with the severity observed in the default cycle or the rating transition cycle.

Correlation in multi-state CreditManager
A few modest assumptions-advantages To make inferences about specific firms, one needs data on those firms. Of several sources of data on specific firms (bond prices, balance sheets, earnings forecasts), stock data probably is the most reliable.

C. Summary and conclusion

C. Summary and conclusion
What matters in variants of CreditManager are the probabilities of joint events, such as firm A defaults while firm B gets a double downgrade. Many credit models assume the joint probabilities depend on correlated latent variables. The key question then becomes the level of correlation among the latent variables.

Summary and conclusion
One way to calibrate latent variable correlation is to assume it takes the level that matches historical variation in the default rate (e.g., Collateral Damage). (By the way, when recovery is also allowed to correlate, an economically significant risk is revealed.) This method could be extended to multi-factor models (which have a richer correlation structure) or to multi-state models (which match a richer set of transitions data than simply the transition to default).

Summary and conclusion
CreditManager finds latent variable correlation at the firm level, using stock index data. This approach has the potential to discover pockets of correlation, but it depends on a long chain of strong assumptions. The resulting correlation matrix generally will not reproduce the severity of the default cycle.

Modeling Default Correlation
University of Chicago Masters in Financial Mathematics Week 3 Jon Frye Federal Reserve Bank of Chicago The views expressed are the author’s and do not necessarily represent the views of the management of the Federal Reserve Bank of Chicago or the Federal Reserve System.

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