Transcript Chapter 18

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)
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
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
DETERMINING LOSSES DUE TO
DEFAULT
DETERMINING LOSSES DUE TO
DEFAULT
The next step is to calculate the standard deviation of
losses (UL)
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:
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:
DETERMINING LOSSES DUE TO
DEFAULT
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.
DETERMINING LOSSES DUE TO
DEFAULT
DETERMINING LOSSES DUE TO
DEFAULT
DETERMINING LOSSES DUE TO
DEFAULT
DETERMINING LOSSES DUE TO
DEFAULT
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 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.
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.
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.
DETERMINING LOSSES DUE TO BOTH
DEFAULT AND DOWNGRADES
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
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, 2001. These
spreads are calculated for zero-coupon bonds.
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 BBBrated 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:
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
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.
DETERMINING LOSSES DUE TO BOTH
DEFAULT AND DOWNGRADES
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
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):
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
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.
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.
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.
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):
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:
DETERMINING DEFAULT PROBABILITIES
OVER MULTIPLE YEARS
DETERMINING DEFAULT PROBABILITIES
OVER MULTIPLE YEARS
Let us use the symbol M for the migration matrix in Table
18-1
DETERMINING DEFAULT PROBABILITIES
OVER MULTIPLE YEARS
Define G to be a vector giving the probability of being in
each grade:
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:
DETERMINING DEFAULT PROBABILITIES
OVER MULTIPLE YEARS
The probability distribution of ratings at the end of the
year is given by M times G:
DETERMINING DEFAULT PROBABILITIES
OVER MULTIPLE YEARS
This equation can be used recursively to get the
probability distribution of grades after N years
DETERMINING DEFAULT PROBABILITIES
OVER MULTIPLE YEARS
For a company that is initially rated single A, we get the
following results:
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
DETERMINING DEFAULT PROBABILITIES
OVER MULTIPLE YEARS
DETERMINING DEFAULT PROBABILITIES
OVER MULTIPLE YEARS
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
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:
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:
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.
DETERMINING DEFAULT PROBABILITIES
OVER MULTIPLE YEARS