CHAPTER 18 Risk Measurement for a Single Facility.

CHAPTER 18

Risk Measurement for a Single Facility

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).

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)


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


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


The next step is to calculate the standard deviation of losses (UL)


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:


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:


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.


In summary, two situations are considered: S and E are fixed and constants S and E are random variables

DETERMINING LOSSES DUE TO BOTH DEFAULT AND DOWNGRADES

In the section above, we calculated the statistics of loss due to the possibility of defaultIt is also possible for bonds and loans to lose value because the issuing company is downgradedWhen a company is downgraded, it means that the rating agency believes that the probability of default has risenA 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 promiseConsequently, 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 dollarThis implies discount rates of 6.35% and 7.5%, or credit spreads of 1.1 % and 2.25% relative to the risk-free rateIf 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.


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.

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:


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

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):


>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


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.


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.

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.


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:


Let us use the symbol M for the migration matrix in Table 18-1


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:


This equation can be used recursively to get the probability distribution of grades after N years


For a company that is initially rated single A, we get the following results:


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


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


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:


The conditional probability: the probability that it will default in the given year, given that it did not default in any of the previous yearsThe conditional probability is calculated by 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.

CHAPTER 18 Risk Measurement for a Single Facility.

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Transcript of CHAPTER 18 Risk Measurement for a Single Facility.