instead of EL to denote the mean loss:To quantify credit risk, we wish to obtain the probability distribution of the losses from a credit portfolio and also to have a measure of the contribution of each loan to the portfolio loss. In this chapter, we lay the foundation for quantifying the credit risk of a portfolio by quantifying the risk for individual facilities.
We start by considering the simple case of modeling the losses that could occur over one year due to a default. We then add the complication of additional losses due to the possibility of downgrades. Finally, we look at how the risk can be calculated for a credit exposure that lasts for multiple years (e.g., a five-year loan).
For a single facility, we describe the credit-loss distribution by the mean and standard deviation of the loss over a year. The mean is commonly called the expected loss (EL). The expected loss can be interpreted as being the amount that a lender could expect to lose on average over a number of years, from transactions with similar levels of creditworthiness, exposure, and security. The EL can be viewed as a cost of doing business because over the long run, the bank should expect to lose an amount equal to EL each year. The standard deviation of the loss is typically called the unexpected loss (UL).
In this section, we consider the potential losses due to default. The actual loss (L) can be described as the exposure at default (E), multiplied by the loss in the event of default or severity (S), multiplied by an indicator of default (I). The indicator is a discrete variable that equals one if a default occurs, and zero otherwise:
L = I × S × E
L = 0 if I = 0
L = S × E if I = 1
In deriving expressions for EL and UL, we first get results quickly, using the simplifying assumptions that S and E are fixed. We then repeat their derivation in more detail without the simplifying assumptions.
If we assume that LIED and EAD are fixed, the only remaining uncertainty is whether a default occurs. The two possible conditions at the end of the year are: there is a default or there is no default, with probabilities of P and (1 − P) respectively. The expected loss is then the amount that is lost in each condition, multiplied by the probability of being in each condition:
EL = P[1 × E × S] + (1 − P)[0 × E × S]
= P × E × S
This is a fairly intuitive result, as it says that the average is the probability of default times the amount that is lost if default occurs. If we loaned $100 to a BBB-rated company, then P would be around 22 basis points. Assuming that S is 30%, the EL is 0.066 :
EL = 0.0022 × $100 × 0.3 = $0.066
The next step is to calculate the standard deviation of losses (UL). Let us again assume that LIED and EAD are fixed. For a discrete variable, such as our indicator function, the variance is the probability of each state times the difference between the result and the mean squared. For clarity, we use instead of EL to denote the mean loss:
By taking the square root, we obtain the simple expression for the unexpected loss:
For the loan of $100 to a BBB company, the UL would be $1.41:
This result can also be obtained by realizing that the indicator, I, is a Bernoulli variable. A Bernoulli variable can only equal one or zero and the variance of a Bernoulli variable is P(1 − P).
These simple expressions for the EL and UL are the ones most commonly used. However, if we also want to include the possibility of uncertainty in LIED and EAD, we must use more a complex derivation. The math gets quite detailed, so you may want to review the derivations quickly, focus on the results, then come back to the math if you need it later.
Now we repeat the derivations of EL and UL but without assuming fixed values for the exposure at default (E) and the loss in the event of default (S). In this general case, the expected loss is the mean loss in each state, multiplied by the probability of being in that state:
Here, p(I) is the probability of event I taking a particular value (0 or 1). pr(S, E|I) is the joint probability-density function for S and E given that I equals a particular value. In practice, we only care about pr(S, E|I = 1), which is the probability-density function for S and E in the case when there is a default. Notice that the term inside the double integral is similar to the definition of the covariance of S and E:
Using this definition, the equation for EL can be simplified to the following result:
This result is the same as the equation we had before for EL, but with an additional term for the covariance between S and E. If the S and E are uncertain, but the correlation between them is zero, then the expected loss is simply the product of the expected value of each component:
Let us now turn our attention to UL. In the general case where E and S are uncertain, the variance of the loss (UL2) is the square of the deviation of the loss in each state, multiplied by the probability of being in that state. Again, we use to denote the expected loss:
has already been found and is constant over the summation and integration, so it can be pulled out and evaluated separately, as in the equations below:
We can now carry out the summation over the two possible values for I. When I equals zero, the term inside the integrals equals zero:
This double integral is difficult to evaluate unless we assume that the values of S and E are independent, meaning that the severity of the loss is independent of the exposure at the time of default. In that case, the probability distribution can be separated into the product of two distributions, each of which can be integrated separately:
Now we can recognize that the integrals are almost the same as the equations for variance, and use this to produce the following result:
This result is the same equation as before for UL, but with additional terms for the variances of S and E. For a loan, the variance of the exposure, E, is close to zero.1 As we see in the next chapter, the standard deviation of the loss in the event of default (LIED or S) is typically around 25%. For our example of the BBB loan, this gives a UL of $1.83, which is a 30% increase from the UL that was calculated assuming that the LIED was fixed:
In summary, if the loss given default and exposure amount are fixed, the EL and UL for the losses due to default are given by the following:
If E and S can vary, EL is given as follows:
If S and E are uncorrelated, UL is given as follows:
In the section above, we calculated the statistics of loss due to the possibility of default. It is also possible for bonds and loans to lose value because the issuing company is downgraded. When a company is downgraded, it means that the rating agency believes that the probability of default has risen. A promise by this downgraded company to make a future payment is no longer as valuable as it was because there is an increased probability that the company will not be able to fulfill its promise. Consequently, there is a fall in the value of the bond or loan.
As an example, consider a case in which the one-year (risk-free) government note is trading at a price of 95 cents per dollar of the nominal final payment (implying a risk-free discount rate of 5.25%). A promise by a single-A company to make a payment in one year would have a price around 94 cents per dollar, and a BBB company would trade around 93 cents on the dollar. This implies discount rates of 6.35% and 7.5%, or credit spreads of 1.1% and 2.25% relative to the risk-free rate. If you had bought the bond of the single-A company for 94 cents, and then it was downgraded to BBB, you would lose 1 cent.
To obtain the EL and UL for this risk, we require the probability of a grade change and the loss if such a change occurs. The probability of a grade change has been researched and published by the credit-rating agencies. Table 18-1 shows the probability of a company of one grade migrating to another grade over one year. (Appendix A describes how this table is derived.) To understand how to read this table, let us use it to find the grade migration probabilities for a company that is rated single-A at the start of the year. Looking down the third column, we see that the company has a 7-basis-point chance of becoming AAA rated by the end of the year. It has a 2.25% chance of being rated AA, a 91.76% chance of remaining single-A, and a 5.19% chance of being downgraded to BBB. Looking down to the bottom of the column, we see that it has a 4-basis-points chance of falling into default. Notice that the last row gives the probability of a company’s defaulting over the year; for example, a company rated CCC at the beginning of the year has a 21.94% chance of having defaulted by the end of the year.
From Table 18-1, we can get any company’s probability of moving to a different grade by the end of the year. Associated with each grade is a discount rate relative to the risk-free rate. This varies with the market’s appetite for risk, but Table 18-2 shows the spread between U.S. corporate bonds and U.S. Treasuries in October, 2001. These spreads are calculated for zero-coupon bonds.
With these spreads, we can calculate the value of a bond or loan, and thereby estimate the change in value if the grade changes. As an example, let us calculate the EL and UL for a BBB-rated bond with a single payment of $100 that is currently due in 3 years. At the end of the year the bond will have 2 years to maturity. If we assume a risk-free discount rate of 5%, and the bond is still rated BBB, the value will be $88.45:
TABLE 18-1 Probability of Grade Migration (bps)
TABLE 18-2 Corporate Bond Spreads Above the Risk-Free Rate (basis points)
However, if the bond rating falls to single B, the value of the bond will fall to $81.90 :
This is a loss of $6.55:
We can repeat this calculation for all possible grades to give the values shown in Table 18-3. The value in default corresponds to our assumption of a 30% LIED. Table 18-3 also shows the loss in value compared with the value if the bond retained its BBB-rating.
From Table 18-1, we have the probability of a change in credit grade, and from Table 18-3, we have the loss amount if the bond changes grades. We can now bring these together to calculate EL and UL. The expected loss is calculated from the probability of being in a given grade (PG), multiplied by the loss for that grade (LG):
The unexpected loss is given by the square root of the probability-weighted sum of the differences squared:
TABLE 18-3 Change in Values for a BBB Bond due to Credit Events
Table 18-4 shows the elements of the EL and UL calculation for the example BBB bond. The result is that EL is $0.18 and UL is $1.25.
It is also interesting to compare this with the result of considering only the default. If we only consider default, EL is the probability of default times the loss, given default:
EL = PDLD
= 0.0022 × $26.5 = $0.058
This is much less than the EL calculated including downgrades. To calculate UL for default only, we only consider the default and zero-loss cases:
UL for just default is a little less than UL including downgrades. The downgrades are relatively insignificant to UL because UL is largely determined by the extreme losses, and the extreme losses depend mostly on defaults and not on downgrades.
The next logical step would be to add complication to the analysis by considering the uncertainty to the loss in the event of default (LIED). This is normally done by simulation, in which we first randomly choose the grade at the end of the year. If the grade is CCC or higher, we value the bond as above. If the grade is a default, we then randomly choose the LIED. We discuss simulation for credit risk in great depth in the later chapter on portfolio analysis.
TABLE 18-4 Calculation of Expected and Unexpected Loss, Including Downgrades
In the discussion above, we dealt with the probability of the company’s defaulting or being downgraded at some point over the next year. We are now going to tackle the problem of quantifying the risk over multiple years.
One approach is to look at historical data. The approach groups the companies according to their ratings at one snapshot of time. Then calculates how many defaulted in the first year, how many in the second year, and so forth. If we want to look out over many years, this requires a large amount of data. An alternative is to derive the default rate over several years from the one-year grade-migration matrix in Table 18-1.
From our discussion on grade migrations, we know the probability of a company’s transitioning from one grade to another by the end of a year. And we know the probability of each grade’s defaulting. From these two pieces of information, we can estimate the probability of the company’s defaulting in the second year. The probability of default in the second year (PD,2) is given by the probability of transitioning to each grade (PG), multiplied by the probability of default for a company of that grade (PD|G):
For a single-A rated bond, the probability of default in the first year is 4 basis points. The probability of defaulting by the end of the second year is given by the following sum:
Here, PAAA is the probability of becoming AAA rated, and PD|AAA is the one-year probability of default for a company that is rated AAA at the start of a year. Notice that this calculation multiplies the elements of the third column (the single-A row) of the migration matrix in Table 18-1 by the bottom row of the matrix.
If we want to know the probability of default by the end of the third year, we need to calculate the probability of migrating to each grade over the first year, followed by the probability of migrating to another grade over the second year, followed by the probability of default over the third year. This is quite painful to do with normal algebra, but is easy to do with matrix multiplication if we treat the migration matrix as a state transition matrix.
Let us use the symbol M for the migration matrix in Table 18-1, and define G to be a vector giving the probability of being in each grade:
At the beginning of the first year, a single-A-rated bond has a 100% chance of being rated single A:
The probability distribution of ratings at the end of the year is given by M times G:
GT=1 = M GT=0
This equation can be used recursively to get the probability distribution of grades after N years:
GT=1 = M GT=0
GT=2 = M GT=1
= MM GT=0
GT=N = MN GT=0
For a company that is initially rated single A, we get the following results:
The most interesting element of this result is the bottom row, which gives the probability of having fallen into the default grade. From these results, the probability of default after 2 years is 0.11%, after 3 years it is 0.20%, and after 4 years it is 0.33%.
Figure 18-1 shows the probability of default over 10 years for companies with different initial grades. The results are also shown numerically in Table 18-5 for reference.
Notice that these are cumulative probabilities of default. Two other measures of default probability are also of interest to us: the marginal probability of default, and the conditional probability of default. The marginal probability is the probability that the company will default in any given year. It is calculated by taking the difference in the cumulative probability:
PD,Marginal,T = PD,Cumulative,T − PD,Cumulative,T−1
FIGURE 18-1 Probability of Default over 10 Years Depending on Initial Rating
TABLE 18-5 Cumulative Probability of Default over 10 Years Depending on Initial Rating (basis points)
The conditional probability is the probability that it will default in the given year, given that it did not default in any of the previous years. The conditional probability is the marginal probability divided by the probability that it has survived so far:
Figure 18-2 shows the conditional probability of default over 10 years. Notice that as time increases, the probabilities converge. One way of thinking about this is to say that if a company is still surviving many years from now, we cannot predict what will be its rating.
FIGURE 18-2 Conditional Probability of Default over 10 Years Depending on Initial Rating
In this chapter, we explored quantitative frameworks for calculating the credit losses that can occur due to defaults and downgrades. In the next chapter, we explain the process for calculating the values of the parameters used in these equations.
The migration matrix in Table 18-1 is based on the observation of historical data. Such studies are published by the rating agencies. The agencies take several years of historical data on bond ratings and observe the ratings at the beginning and end of each year. From this, they calculate the probability of a grade change. The results of such a study are shown in Table 18-A1.
TABLE 18-A1 Historical Grade-Migration Probabilities (basis points)
The category “NR” means that at the end of the year, the companies had not defaulted but were not rated. The rating may be withdrawn for a variety of reasons, including that the company had no outstanding rated debt at the end of the year, or the company was in sufficient financial trouble that it was no longer cooperating with the rating agencies. For simplicity, we can assume that the unrated companies have the same probability distribution as the other companies that did not default. By doing this, we can remove the NR row and rescale the other migration probabilities.
We also make one other change to the data to obtain the migration matrix. We assume that once a company goes into default, it stays in default. So, if its initial rating is D, there is a 100% chance that the final rating will be D. With these modifications, we produce the migration matrix in Table 18-1.
Notice that the probability of default for AAA companies is shown to be 0. In reality, AAA companies do not have a 1-year probability of default that is exactly 0. Other studies have found that AAA companies can default within a year, but because the true default rate is around 1 basis point, we would only expect to observe a default every 10,000 company-years, which means that there is typically insufficient data to estimate reliably the default rate for AAA companies. A common alternative assumption is that AAA companies have a default rate of 1 basis point, but for the purpose of the migration examples, we will leave the 1-year default rate for AAA companies equal to 0.
1 There may be some variance if the loan is amortizing and the time of default is unknown.