1. Risk Measurement for a Single Facility
CHAPTER 18
Risk Measurement for a Single Facility

2. INTRODUCTION
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).

3. DETERMINING LOSSES DUE TO DEFAULT
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 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)

4. DETERMINING LOSSES DUE TO DEFAULT
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 (LIED) or severity (S), multiplied by an indicator of default (I)
The indicator (I) is a discrete variable that equals one if a default occurs, and zero otherwise

5. DETERMINING LOSSES DUE TO DEFAULT
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

6. DETERMINING LOSSES DUE TO DEFAULT

7. DETERMINING LOSSES DUE TO DEFAULT
The next step is to calculate the standard deviation of losses (UL)

8. DETERMINING LOSSES DUE TO DEFAULT
An Example: if we loaned $100 to a BBB-rated company, then P (probability of default) would be 22 basis points. Assuming that S is 30%, then:

9. DETERMINING LOSSES DUE TO DEFAULT
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:

10. DETERMINING LOSSES DUE TO DEFAULT

11. DETERMINING LOSSES DUE TO DEFAULT
Now, let us turn our attention to UL
In the general case, E and S are uncertain (namely, random variable)
The variance of loss (UL2) is the square of the deviation of the loss in each state, multiplied by the probability of being at each state.

12. DETERMINING LOSSES DUE TO DEFAULT

13. DETERMINING LOSSES DUE TO DEFAULT

14. DETERMINING LOSSES DUE TO DEFAULT

15. DETERMINING LOSSES DUE TO DEFAULT
In summary, two situations are considered:
S and E are fixed and constants
S and E are random variables

16. DETERMINING LOSSES DUE TO BOTH DEFAULT AND DOWNGRADES
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.

17. DETERMINING LOSSES DUE TO BOTH DEFAULT AND DOWNGRADES
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.

18. DETERMINING LOSSES DUE TO BOTH DEFAULT AND DOWNGRADES
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.

19. DETERMINING LOSSES DUE TO BOTH DEFAULT AND DOWNGRADES

20. DETERMINING LOSSES DUE TO BOTH DEFAULT AND DOWNGRADES
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

21. DETERMINING LOSSES DUE TO BOTH DEFAULT AND DOWNGRADES
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.
Table 18-2 shows the spread between U.S. corporate bonds and U.S. Treasuries in October, These spreads are calculated for zero-coupon bonds.

23. DETERMINING LOSSES DUE TO BOTH DEFAULT AND DOWNGRADES
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:

24. DETERMINING LOSSES DUE TO BOTH DEFAULT AND DOWNGRADES
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

25. DETERMINING LOSSES DUE TO BOTH DEFAULT AND DOWNGRADES
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.

26. DETERMINING LOSSES DUE TO BOTH DEFAULT AND DOWNGRADES

27. DETERMINING LOSSES DUE TO BOTH DEFAULT AND DOWNGRADES
From Table 18-1, we have the probability of a change in credit grade
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

28. DETERMINING LOSSES DUE TO BOTH DEFAULT AND DOWNGRADES
The expected loss is calculated from the probability of being in a given grade (PG), multiplied by the loss for that grade (LG):

29. DETERMINING LOSSES DUE TO BOTH DEFAULT AND DOWNGRADES
>It 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

30. DETERMINING LOSSES DUE TO BOTH DEFAULT AND DOWNGRADES
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:
This is much less than the EL calculated including downgrades.

31. DETERMINING LOSSES DUE TO BOTH DEFAULT AND 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.

32. DETERMINING DEFAULT PROBABILITIES OVER MULTIPLE YEARS
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.

33. DETERMINING DEFAULT PROBABILITIES OVER MULTIPLE YEARS
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):

34. DETERMINING DEFAULT PROBABILITIES OVER MULTIPLE YEARS
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:

35. DETERMINING DEFAULT PROBABILITIES OVER MULTIPLE YEARS

36. DETERMINING DEFAULT PROBABILITIES OVER MULTIPLE YEARS
Let us use the symbol M for the migration matrix in Table 18-1

37. DETERMINING DEFAULT PROBABILITIES OVER MULTIPLE YEARS
Define G to be a vector giving the probability of being in each grade:

38. DETERMINING DEFAULT PROBABILITIES OVER MULTIPLE YEARS
At the beginning of the first year, a single-A-rated bond has a 100% chance of being rated single A:

39. DETERMINING DEFAULT PROBABILITIES OVER MULTIPLE YEARS
The probability distribution of ratings at the end of the year is given by M times G:

40. DETERMINING DEFAULT PROBABILITIES OVER MULTIPLE YEARS
This equation can be used recursively to get the probability distribution of grades after N years

41. DETERMINING DEFAULT PROBABILITIES OVER MULTIPLE YEARS
For a company that is initially rated single A, we get the following results:

42. DETERMINING DEFAULT PROBABILITIES OVER MULTIPLE YEARS
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

43. DETERMINING DEFAULT PROBABILITIES OVER MULTIPLE YEARS

44. DETERMINING DEFAULT PROBABILITIES OVER MULTIPLE YEARS

45. DETERMINING DEFAULT PROBABILITIES OVER MULTIPLE YEARS
Note that the probabilities above are cumulative probabilities of default. Two other measures of default probability are also of interest to us:
The marginal probability: the probability that the company will default in any given year
The conditional probability: the probability that it will default in the given year, given that it did not default in any of the previous years

46. DETERMINING DEFAULT PROBABILITIES OVER MULTIPLE YEARS
The marginal probability: the probability that the company will default in any given year
The marginal probability is calculated by taking the difference in the cumulative probability:

47. DETERMINING DEFAULT PROBABILITIES OVER MULTIPLE YEARS
The conditional probability: 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 calculated by the marginal probability divided by the probability that it has survived so far:

48. DETERMINING DEFAULT PROBABILITIES OVER MULTIPLE YEARS
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.

49. DETERMINING DEFAULT PROBABILITIES OVER MULTIPLE YEARS