In the preceding chapters, we created a framework for measuring and quantifying the risk of isolated credit facilities such as loans. We now need to do the same for a portfolio of credit risks. However, this effort requires a different set of tools because we need to include the effects of correlation. Correlation describes the extent to which loans tend to default at the same time. Intuitively, we would expect that companies would have some tendency to default together. This could happen because the whole economy is in recession, forcing many companies into bankruptcy at the same time, or it could be that the default of one company triggers the default of another company. For example, the collapse of a car factory would tend to push suppliers and businesses in the local town closer to default. These correlation effects produce loss distributions that are highly skewed: there are many years of low losses, then a few years of “surprisingly” high loss. The description of this correlation and skew is one of the most difficult problems in credit-risk measurement. In this chapter and the next, we describe the five most common approaches used to measure the credit risk of a portfolio.
Each approach estimates the probability distribution for the credit losses and specifically estimates the portfolio’s expected loss, unexpected loss, and economic capital. This allows us to determine how much capital the bank should hold to maintain its credit rating. Also, in Chapter 22, we use the economic capital to calculate the risk-adjusted profitability of each loan in the portfolio.
Correlation effects for credit defaults are difficult to observe because defaults are relatively infrequent. For example, in any given year we would only expect 1 in 1000 single-A-rated companies to default. This contrasts with market-risk analysis in which we collect new price data on all traded securities every day. Given the scarcity of data, credit-risk models use assumptions and financial theories to estimate loss statistics based on the small amounts of observable data. There are five common approaches:
• The covariance model
• The actuarial model
• The Merton-based simulation model
• The macroeconomic default model
• The macroeconomic cash-flow model
The fundamental difference between each model is the approach taken to characterize the correlation. We discuss each model in detail, but for reference, the main features of each model are shown in Table 20-1. In this table, the correlation mechanism is the approach used to model the tendency for defaults to occur at the same time. The parameterization column shows the most important variables that must be quantified to capture the portfolio effects. The capital column shows the method used to evaluate the portfolio’s economic capital.
Each of these model approaches has been integrated into software packages created by different companies. In this discussion, we do not describe the specific details of each package, but describe the fundamental concepts behind each model and possible ways in which each model can be developed.
In this chapter, we describe the covariance approach to modeling credit risk. We spend a lot of time on this approach for two reasons: it lays the foundation for the other approaches, and it is the easiest to use if you need to build a credit-portfolio model from scratch. In the next chapter, we discuss the underlying mechanisms for the other types of models.
The covariance portfolio model is also sometimes referred to as the Markowitz model. It is largely the same as the parametric approach for VaR except that the correlations used are default correlations and the probability distribution is assumed to be a Beta distribution rather than a Normal distribution. Another difference is that for credit risk, one of the results we are concerned about is the mean of the distribution (the expected loss), whereas for market risks, we assume the mean is zero. In this discussion, we use the term “loan” to refer to any general credit exposure. This discussion applies equally to other types of credit exposures, such as bonds, lines, and derivatives exposures.
TABLE 20-1 Summary of Credit-Portfolio Models
There are four steps required in the covariance model:
1. Defining the expected loss (EL) and unexpected loss (UL) of the portfolio in terms of the EL and UL of the individual loans.
2. Estimating the default correlation, which is the longest step.
3. Estimating the portfolio’s overall probability distribution based on its EL and UL.
4. Allocating the capital of the whole portfolio to the individual loans using the concept of unexpected loss contribution.
The expected loss for the portfolio (ELP) is simply the sum of the expected losses for the individual loans (ELi) within the portfolio:
As with parametric VaR, the standard deviation for the portfolio (ULP) is obtained from the sum of the variances for the individual loans. If there are only two loans, ULP would be as follows:
Where ρ1,2 is the loss correlation between loan 1 and loan 2. For N loans, we can use summation notation:
This can also be expressed in matrix notation:
The next step is to estimate values for the correlations.
Loss correlations can be estimated based on the historical data of losses or on asset correlations. The approach taken in any given situation depends mostly on the data available. Ideally, correlations should be calculated using both methods, and then compared.
Let us begin by considering the simple case of a portfolio of two loans. The UL for this portfolio is given by the variance equation:
If we knew the values for ULP, UL1, and UL2 we could solve the equation to get ρ1,2:
If we have a large number of loans, we said that ULP is given by the following:
We can get an estimate for the correlation if we assume that the correlation between each loan is identical:
By doing this, we have neglected the fact that the correlation between a loan and itself must equal one. This becomes unimportant if there is a large number of loans, but the proof is quite tedious, and is therefore relegated to Appendix A. Given the assumption of a fixed correlation, we can separate the two summations because they no longer depend on each other:
If we now assume that each loan has the same UL, we can estimate the correlation as follows:
Here, N is the total number of loans in the portfolio. You would use this approach (giving ULP) if the only information available was a time series of losses from the portfolio and (giving PD and therefore ULi) an estimate of the average percentage loss given default. As an example, consider the loss information in Table 20-2. This is the type of information that you could get from the bank’s annual reports. The loss is written in percentage terms to reduce the effect of the changing size of the portfolio. The historical expected loss of the portfolio (ELP,H) is simply the average of the losses and the historical percentage UL (ULP,H) is the standard deviation.
TABLE 20-2 Example of Historical Losses Used to Estimate the Unexpected Loss of the Portfolio
In dollar terms, the UL for the portfolio is the UL as a percentage, multiplied by the total size of the portfolio:
Here, 
is the average loan size, and N is the number of loans.
We now wish to estimate UL for the individual loans. If we assume that the loss given default (LGD) is 75%, then the average probability of default for loans in this portfolio is 1.9%:
The UL for an individual loan is estimated as follows:
We can now calculate the correlation as follows:
The calculation of 
using this method is useful for getting estimates of to use in the calculation of the unexpected loss contribution, as discussed later. It is also a useful cross-check on results obtained using other methods.
A better estimate of can be made if we have an idea of the distribution of the creditworthiness of the loans. This extension to the methodology is demonstrated in Appendix B.
The calculation of loss correlations directly from historical data can produce a quick estimate for the correlation, but it has a couple of problems. First, it only gives us one correlation for the whole portfolio so it is not possible to differentiate between loans that have different correlations. The second problem is that the reported historical loss data is often corrupted because the amount of losses reported each year is manipulated by the bank’s managers.
The manipulation of reported losses is done to maintain the appearance of stable earnings. Banks tend to underreport the losses in bad times and overreport them in good times. Banks can postpone the declaration of losses by restructuring loans that are about to default. For example, if a company is unable to pay back a loan, the bank has two choices: it can force the company into bankruptcy and take over its remaining assets, or the bank could give it a new loan that it can use to repay the old loan and thereby delay the official default. The bank will probably never recover its money, but by making such deals, banks have had some leeway in manipulating the reported losses, thereby reducing ULP and the accuracy of this approach.
This corruption of the data exacerbates the problems of only having a few data points, and can produce results that significantly undercount the risk.
The analysis above used portfolio-level data. We now discuss the asset-correlation approach, which calculates the loss correlation between two individual companies. The loss correlation is estimated based on the correlation between their net asset values or equity prices.
The approach calculates the loss correlation from the joint default probability. The joint default probability between two loans, 1 and 2 (JDP1,2), is the probability that both loans will default at the same time. The relationship between the joint default probability and the loss correlation is detailed in Appendix C to this chapter. The result is that the loss correlation between loans 1 and 2 (ρ1,2) can be expressed in terms of the joint default probability and the individual probabilities of default for the loans (P1, P2) as follows:
We can estimate P1 and P2 from the calculation of default probabilities as in Chapter 19. The problem now is to calculate JDP1,2.
The calculation of the joint default probability can be based on the Merton approach that was discussed in Chapter 19. In Chapter 19, we estimated the probability of default for a single company as the probability that its asset value would fall below its debt value. For this calculation, we assumed that the equity (E) was a good measure of the difference between the assets and debts. The probability of default was then calculated as the probability that the equity price would fall below zero:
We transformed this equation to say that the default probability was equal to the probability that a Standard Normal variable would fall below a critical value (C), which depends on the equity price and its volatility:
/σE is the distance to default or the critical value, and φ is the Standard Normal probability-density function. This gave us the probability that one company will default. We now want to know the probability of two companies’ defaulting at the same time.
Let us define z1 and z2 as follows:
Here, E1 is the equity value for company 1, and E2 is the value for company 2. If we assume that the equity values are Normally distributed, then z1 and z2 will be Normally distributed, each with a mean of 0 and standard deviation of 1. The probability of each company’s defaulting is as follows:
The correlation between z1 and z2 equals the correlation between the equity prices, ρE. Figure 20-1 shows a scatter plot for 1000 possible changes in z1 and z2 for 2 companies with an equity correlation equal to 0.7. Notice that when one company has a low value, the other also tends to have a low value. The probability of both companies’ having a particular value is described by the joint-probability density function. The formula for the joint probability density for 2 correlated Standard Normal variables is given below:
This function is plotted in Figure 20-2 for a correlation of 0.7. Notice that the plot is flattened and twisted along a 45-degree line. If the correlation was lower, the twist along the 45-degree line would be less pronounced. Figure 20-3 shows the probability-density function for a correlation of 0.1. Notice that the height of the function is now much less in the area where both companies have low values.
FIGURE 20-1 Illustration of the Correlation Between the Standard Asset Values of Two Companies
This figure depicts a scatter plot for 1000 possible changes in asset values z1 and z2 for two companies with an equity correlation equal to 0.7.
FIGURE 20-2 Illustration of the Joint Normal Probability-Density Function with a Correlation of 0.7
This figure plots the joint probability density for two Standard Normal variables with a correlation of 0.7. The height of the plot corresponds to the probability of z1 and z2 both falling in the given cell.
The probability of z1 and z2 falling in a small area is given by the probability-density function, multiplied by the size of the area:
The joint probability that both companies will default is equal to the probability that z1 falls in the range below –C1, and z2 falls in the range below –C2. This probability is evaluated by integrating the probability-density function over the area in which both companies default:
Let us summarize the results so far. We calculated the critical distance to default from the equity-price information:
FIGURE 20-3 Illustration of the Joint Normal Probability-Density Function with a Correlation of 0.1
This figure plots the joint probability density for two Standard Normal variables with a correlation of 0.1.
We calculated the probability of default for a single company using the Merton model and the cumulative Standard Normal probability-density function, Φ:
We calculated the joint probability of default from the critical values, the two-dimensional Standard Normal probability-density function, and the correlation between the equity prices:
Finally, the loss correlation was calculated from the probabilities and the JDP:
The process just described is a reasonable way of estimating the default correlation; however, it requires that we know all the equity information. Alternatively, if we already know the individual probabilities of default (e.g., if we know the credit grades) we can take a short-cut and calculate the critical values from Table 19-4 or by using the inverse of the cumulative probability function:
Φ-1 is available in most spreadsheet programs (e.g., “norminv(P,0,1)” in Excel). Knowing the critical values and the correlation between equity prices, we can step directly to evaluating the JDP.
The evaluation of the integral cannot be done analytically, and instead, numerical integration is required. Numerical integration takes the sum of the probability over many small areas:
In this scheme, we evaluate φ at a total of N2 points. Δ1 is the distance between each point in the z1 direction, and Δ2 in the z2 direction. The size of the steps is chosen so that the difference between evaluation points is not too large. N is chosen so that φ((−C1 – NΔ1), (−C2 – NΔ2), ρ) is very small.
Table 20-3 and Table 20-4 show the results for companies of differing credit quality and asset correlation. In each table, the credit quality of the companies is increased from a default rate of 0.01% per year (1 basis point) to 10% per year (1000 basis points). As the default rate increases, the default correlation also increases. Comparison of Table 20-3 and Table 20-4 also shows that an increase in the equity correlation causes an increase in the default correlation.
TABLE 20-3 Default Correlation for Companies with Equity Correlation of 20%
TABLE 20-4 Default Correlation for Companies with Equity Correlation of 40%
From the discussions above, we can calculate the portfolio’s EL and UL. We now wish to use those results to estimate the portfolio’s economic capital. To do this, we need to estimate the probability distribution of the portfolio’s losses.
In the covariance model, the losses are typically assumed to have a Beta distribution. The Beta distribution is used for three reasons:
1. It can have highly skewed shapes similar to the distributions that have been observed for historical credit losses.
2. It only requires two parameters to determine the shape (ELP and ULP).
3. The possible losses are limited to being between 0 and 100%.
The formula for the Beta probability-density function for % losses (L) is as follows:
The Beta function can be integrated numerically and is available in most spreadsheet programs. To use it, the variables a and b can be expressed in terms of the required mean (ELP) and standard deviation (ULP):
Figure 20-4 shows three Beta distributions for which (ULP) is held constant at 1%, and ELP equals 1%, 2%, and 3%. Notice that as EL becomes larger relative to UL, the distribution becomes more like a Normal distribution. Portfolios of lower-quality loans (e.g., credit cards) have a higher ratio of EL to UL, and therefore, have Beta distributions that tend to look like Normal distributions.
FIGURE 20-4 Illustration of Beta Distribution for Credit Losses
From the tail of the Beta distributions, we can obtain estimates of the economic capital required for the portfolio. In Chapter 2, we discussed risk measurement at the corporate level and said that the common definition of economic capital for credit losses is the maximum probable loss minus the expected loss:
ECP = MPLP − ELP
The maximum probable loss is the point in the tail where there is a “very low” probability that losses will exceed that point. The “very low” probability is chosen to match the bank’s desired credit rating. For example, a single-A-rated bank would require that there should be only 10 basis points of probability in the tail, whereas AAA banks require around 1 basis point. For the 3 distributions shown in Figure 20-4, the 10-basis-point confidence level is close to 7%, and the 1-basis-point confidence level is between 8% and 9%.
Figure 20-5 shows the process for calculating the economic capital using the covariance model with default correlations estimated from asset correlations.
We have now calculated the UL and economic capital of the portfolio as a whole. Using the unexpected loss contribution, we can allocate the capital to the individual loans in the portfolio. The unexpected loss contribution (ULC) allocates the total UL of the portfolio to the individual loans in the portfolio in such as way that the ULCs sum back up to the ULP of the portfolio. In Chapter 22, we use this capital allocation to calculate the risk-adjusted profitability for each loan.
FIGURE 20-5 Illustration of the Process for the Covariance Approach
The unexpected loss contribution takes the same approach as the Value at Risk contribution (VaRC). One approach is to differentiate ULP with respect to ULi, as shown in Appendix D. An alternative derivation is simply to rearrange the terms in the summation notation:
We now divide both sides by ULP:
By analogy, with the case in which all the correlations were assumed to be equal, the second term is called the square root of the average correlation between loan i and the rest of the portfolio:
With this notation, the portfolio UL can now be written as a sum of the individual ULs weighted by their average correlations:
The term inside the summation is called the unexpected loss contribution (ULC). By construction, the sum of the ULCs equals the portfolio UL:
Let us summarize the ULC approach. By using the variance formula, we can calculate the ULP for the portfolio, based on the UL of the individual loans:
Knowing the ULP, we can calculate the unexpected loss contribution in such a way as to ensure that the sum gives us the ULP:
ULC is useful because we can use it to allocate the economic capital of the whole portfolio (ECP) the individual loans. The economic capital contribution for loan i (ECCi) can be allocated according to its unexpected loss contribution:
This has the property that the sum of the individual ECCs equals the total portfolio capital. The ratio of the economic capital to the unexpected loss is sometimes called the capital multiplier, M:
This allows the economic capital contribution to be written simply as follows:
ECCi = M × ULCi
Knowing the economic capital contribution for an individual loan, we can calculate the risk-adjusted return on capital (RAROC) for that loan and set the required price if it is a new loan. Pricing is discussed in Chapter 22 once we have explored the other approaches for quantifying the risk of credit portfolios.
In this chapter, we described the covariance approach to measuring the credit risk of a portfolio. In the next chapter, we discuss four alternatives to the covariance approach.
For a large portfolio (large N), the correlation (or systematic risk) becomes the dominant term in determining ULP, and the correlation of each loan with itself becomes unimportant. To see this, consider a portfolio of N identical loans. Assume that all the off-diagonal correlations are equal, and the UL for each loan is equal:
The summation can now be written as:
If N is much larger than 
(e.g., if N is greater than 100), the second term dominates, and we can approximate ULP as follows:
This equation shows that for a large portfolio, the systematic risk dominates the idiosyncratic risk of the individual loans.
In the section on estimating the loss correlation from historical data, we assumed that all the loans were identical. As a next step of complication, assume that we know the historical loss on the portfolio as before in Table 20-2, and in addition, we have an idea of the distribution of the creditworthiness of the loans in the portfolio as in the first two columns of Table 20B-1.
With the information in Table 20B-1, we no longer need to assume that all the loans are the same, and we can go back to the correlation equation and develop it more carefully. Assuming that the correlation is the same for all loans, the correlation equation was as follows:
Now we can sum up the ULs of the individual loans according to the loan’s credit grade:
TABLE 20-B1 Example of a Portfolio’s Credit-Grade Allocation Used to Estimate the Unexpected Loss of the Individual Grades
In this equation, ULi can be estimated as follows:
Here, Ei is the exposure at default for the loan, and Pi is the probability of default for the given credit grade. If we assume that LGD is fixed for the whole portfolio, we can calculate the sum of UL for each grade from the sum of the total exposure to the credit grade:
This can be expressed as a percentage by dividing by the total exposure of the portfolio:
Here, RG is just the proportion of the portfolio’s exposure that is in each grade:
The result is shown in the fifth column of Table 20B-1.
From Table 20-2, we know that UL%P is 1.2%, and from Table 20B-1, we know that the sum of UL for the individual loans is 5.88%. From this we can calculate :
Recall that correlation between two variables, x and y, can be defined from the variances and covariance of the two variables:
Where the covariance is calculated as follows:
The covariance between two losses L1 and L2 is therefore:
In Chapter 18, our formula for loss was made up of an indicator function (I), the severity of loss in the event of default (S), and the exposure at default (E).
L = I × S × E
Substituting this into the covariance equation gives a fairly complex expression:
The usual approach for estimating the correlation is to bypass the complexity and assume that the severities and exposures are constants. This allows us to simplify the equation to the following:
We can now simplify the summation term. If I1 or I2 equals 0, the term inside the summation equals 0. Therefore, we are only concerned with the case where both I1 and I2 equal 1, i.e., that both companies default. P(I1 = 1, I2 = 1) is called the joint default probability of I1 and I2. The JDP is the probability that both companies default at the same time. We can now write the loss covariance in terms of the joint default probability:
We can find similar equations for the variances:
Note that this is the familiar expression for UL for an individual facility. If we put these expressions into the equation for the correlation, the severities and exposures cancel out, allowing us to write the correlation in terms of the independent probabilities and the joint probability:
Notice that if the defaults were independent, then the joint probability would simply be the product of their independent probabilities (P1 P2) and therefore the correlation would equal zero.
Strictly speaking, ρ1,2 is the default correlation; if we assume that the severity and exposure amounts are constant, it is also equal to the loss correlation. Be aware that in many credit-risk discussions “default correlation” and “loss correlation” are used interchangeably.
The unexpected loss contribution can be thought of as the sensitivity of the portfolio UL to a change in the UL of an individual transaction. We can obtain ULC by differentiating ULP with respect to ULi:
Where:
