1. Risk Measurement for a Credit Portfolio: Part One
CHAPTER 20
Risk Measurement for a Credit Portfolio: Part One

2. INTRODUCTION
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

3. 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

4. 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

5. 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

6. MODELING TECHNIQUES TO MEASURE AND QUANTIFYA PORTFOLIO OF CREDIT RISKS
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 in1000 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

7. There are five common approaches:
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

8. THE COVARIANCE CREDIT-PORTFOLIO MODEL
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

9. THE COVARIANCE CREDIT-PORTFOLIO MODEL
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

10. THE COVARIANCE CREDIT-PORTFOLIO MODEL
This discussion applies equally to other types of credit exposures, such as bonds, lines, and derivatives exposures
There are four steps required in the covariance model:

11. THE COVARIANCE CREDIT-PORTFOLIO 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

12. Defining the Expected Loss and Unexpected Loss
The expected loss for the portfolio (ELp) is simply the sum of the expected losses for the individual loans (ELi) within the portfolio:

13. Defining the Expected Loss and Unexpected Loss
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:

14. Defining the Expected Loss and Unexpected Loss

15. Estimation of Loss Correlation
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

16. Using Historical Observations to Estimate Loss Correlation
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:

17. Using Historical Observations to Estimate Loss Correlation

18. Using Historical Observations to Estimate Loss Correlation

19. Using Historical Observations to Estimate Loss Correlation

20. Using Historical Observations to Estimate Loss Correlation

21. Using Historical Observations to Estimate Loss Correlation

22. Using Asset Correlations to Estimate Default Correlations
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

23. Using Asset Correlations to Estimate Default Correlations
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 can be expressed in terms of the joint default probability and the individual probabilities of default for the loans (P1, P2) as follows:

24. Using Asset Correlations to Estimate Default Correlations
We can estimate P1 and P2 from the calculation of default probabilities as in Chapter 19
The problem now is to calculate JDP1,2

25. Using Asset Correlations to Estimate Default Correlations
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

26. Using Asset Correlations to Estimate Default Correlations
The probability of default was then calculated as the probability that the equity price would fall below zero:

27. Using Asset Correlations to Estimate Default Correlations
E/σE is the distance to default or the critical value
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:

28. Using Asset Correlations to Estimate Default Correlations

29. Using Asset Correlations to Estimate Default Correlations
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:

30. Using Asset Correlations to Estimate Default Correlations

31. Using Asset Correlations to Estimate Default Correlations
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:

32. Using Asset Correlations to Estimate Default Correlations

33. Using Asset Correlations to Estimate Default Correlations

34. Using Asset Correlations to Estimate Default Correlations
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

35. Using Asset Correlations to Estimate Default Correlations

36. Using Asset Correlations to Estimate Default Correlations

37. Using Asset Correlations to Estimate Default Correlations

38. Using Asset Correlations to Estimate Default Correlations

39. Using Asset Correlations to Estimate Default Correlations

40. Using Asset Correlations to Estimate Default Correlations
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 (1000basis 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

41. Using Asset Correlations to Estimate Default Correlations

42. Using Asset Correlations to Estimate Default Correlations

43. Estimation of the Economic Capital for Credit Losses Using the Covariance Portfolio Model
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

44. Estimation of the Economic Capital for Credit Losses Using the Covariance Portfolio Model
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:

45. Estimation of the Economic Capital for Credit Losses Using the Covariance Portfolio Model

46. Estimation of the Economic Capital for Credit Losses Using the Covariance Portfolio Model
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

47. Estimation of the Economic Capital for Credit Losses Using the Covariance Portfolio Model

48. Estimation of the Economic Capital for Credit Losses Using the Covariance Portfolio Model
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:

49. Estimation of the Economic Capital for Credit Losses Using the Covariance Portfolio Model
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

50. Estimation of the Economic Capital for Credit Losses Using the Covariance Portfolio Model
For the 3 distributions shown in Figure20-4, 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.

51. Estimation of the Economic Capital for Credit Losses Using the Covariance Portfolio Model

52. Calculation of the Unexpected Loss Contribution
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

53. Calculation of the Unexpected Loss Contribution
In Chapter 22, we use this capital allocation to calculate the risk-adjusted profitability for each loan
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:

54. Calculation of the Unexpected Loss Contribution

55. Calculation of the Unexpected Loss Contribution

56. Calculation of the Unexpected Loss Contribution
This allows the economic capital contribution to be written simply as follows:
ECCi = M x 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