Credit Portfolio Management phần 10 potx

Now we can use the expression for the portfolio variance equationA.29 to calculateand so we have that A.32 To show that equation A.32 obeys equation A.31, simply sum over allthe assets i

Risk Contribution

Another important concept in modern portfolio theory is the idea of therisk contribution of an asset to a portfolio It is important to rememberthat it can be defined in more than one way, but the most common defin-ition is what we call marginal standard deviation—the amount of varia-

tion that a particular asset (call it “A”) adds to the portfolio If the asset has a weight or number of shares of w A, then its risk contribution is de-fined by:

(A.30)

The fraction on the right-hand side after “w A” contains the “partialderivative” symbol (“∂”) All this means is that (in the fraction) we are

calculating the change in the portfolio standard deviation with respect

to a very small change in the weight (i.e., dollar amount or number of

shares) of the particular asset Another way of saying this is that this

de-rivative tells us the sensitivity of the portfolio standard deviation with respect to asset A Calculating RC A in this way is equivalent to adding

up all the elements of the row (or column, since it is symmetric)

corre-sponding to asset “A” of the covariance matrix we introduced earlier

and dividing this sum by the portfolio standard deviation This confirms

that the sum of all the risk contributions (RC i) is equal to the portfoliostandard deviation—that is,

∂

Now we can use the expression for the portfolio variance equationA.29 to calculate

and so we have that

(A.32)

To show that equation A.32 obeys equation A.31, simply sum over allthe assets in the portfolio:

Derivation of the Default Event Correlation Formula

We now turn to the derivation of the equation for default event correlation

as a function of a joint probability equation A.19

Assume two binomial random variables X and Y (that can have values

of 0 or 1) have a joint probability distribution p(x,y) Assume a joint ability p(1,1) = J.

prob-Start with definition of correlation (see equation A.15):

(A.33)

We now try to calculate cov[X,Y] using its definition calculated using

the joint probability density for discrete random variables:

12

σσσ

j j

p(1,1) = J p(1,0) = µx – J p(0,1) = µy – J

and after canceling terms, we get

Inserting this into equation A.33 yields equation A.19 for the defaultevent correlation (ρxy).

IMPORTANT PROBABILITY DISTRIBUTIONS

We now turn our attention to the specifics of the probability distributionsthat we have encountered in this book and that are useful to understand incredit portfolio management

Normal

By far the most common of any distribution, the normal distribution is

also called the bell curve or the Gauss distribution, after the prominent

mathematician and physicist, Carl Friedrich Gauss, of the early eighteenthcentury Many distributions converge to the normal one when certain lim-its are taken For example, the normal distribution is the limiting distribu-tion of the binomial one when the number of trials tends to infinity (more

2 2

2

πσ

µ σ

EXHIBIT A.16 Comparison of the Normal Distribution to the Binomial

Distribution for the Same Parameters as in Exhibit A.4

8% default rate, 100 loans

Number of Defaults

Binomial Normal

Note that the normal distribution extends out to negative values and isperfectly symmetric about the mean (=8%) The binomial and normal areclose approximations to each other, but there are times when one is not ap-propriate and the other will be, for example, in the modeling of probabili-ties The fact that the normal permits negative values means that it will not

be used (typically) to describe the probability of default

Upper/Lower Bounds The normal distribution has no lower or upperbounds It has a non-zero value all the way up (down) to infinity (minusinfinity)

Parameters The bell curve has two parameters that describe it pletely—the mean (µ) and the standard deviation (σ) or variance (=σ2).Principal Applications The normal distribution has applications every-where—in every science, social science (e.g., economics and finance), engi-neering, and so on It is the most basic (nontrivial) distribution, and manyother distributions are compared to it

com-Lognormal

The lognormal distribution is a variation on the normal distribution, inwhich the natural logarithm of a random variable is normally distributed.When plotted against the random variable itself, the distribution is asym-metric and allows only positive (or zero) values

The lognormal distribution is given by the following density function:

nor-Upper/Lower Bounds The lower bound is zero, and the upper is infinity

Parameters The lognormal distribution has two parameters directly

re-lated to the corresponding normal distribution Suppose y is a normally

distributed random variable with mean µ and variance σ2 (that is, y ~

2 2

2

πσ

µ σ

N(µ, σ2)) Then the variable x = e yis log-normally distributed with a meanequal to exp(µ + σ2/2) and a variance equal to exp(2µ + σ2)[exp(σ2) – 1].8

Principal Applications In econometrics, the lognormal distribution hasbeen particularly useful in modeling size distributions, such as the distribu-tion of firm sizes in an industry or the distribution of income in a country

In financial applications it is widely used in modeling the behavior of stockprices, which are always positive (greater than zero) This implies that thelogarithm of the stock price is normally distributed This assumption isused in the Black–Scholes option pricing formula and theory

Binomial

The binomial distribution comes out of the answer to the question:

“What is the probability that I will get v number of aces after tossing a die n times?

The binomial distribution gives the probability that v events will occur

in n trials, given an event has a probability p of occurring, and is explicitly

where the exclamation mark “!” denotes the factorial symbol (e.g., 5! =

5⋅4⋅3⋅2⋅1 = 120, and by definition, 0! = 1) This distribution converges to

the normal distribution when n tends to infinity It converges more rapidly

if p is close to 1/2 Note that default probabilities are much smaller than 1/2,reflecting the fact that default probability distributions (and loss distribu-tions, for that matter) are far from normally distributed

Upper/Lower Bounds The lower bound is zero and the upper is n, the

total number of “trials” (e.g., rolls of the die, spins of the wheel, names

in the loan portfolio, and so on) See Exhibit A.18 Note the differentvertical scales

Parameters The binomial distribution has two parameters: the total

number of “trials,” n (e.g., the number of loans in a portfolio), and the probability of an event to occur, p (e.g., the probability of default) The ex- pected value of the distribution is given by np, and the variance is given by np(1 – p).

Principal Applications In credit risk modeling, the binomial distribution isused sometimes as a starting point for more complex models (which re-quire the incorporation of correlation effects between obligors) For exam-ple, the normal (Gauss) distribution can be derived from the binomial

when taking the limit that n goes to infinity.

EXHIBIT A.18 Binomial Distribution with Three Different Probabilities of

Default (Panel A: 1%, Panel B: 3%, Panel C: 8%.)

100 loans, 1% default rate

Number of Defaults

2 0

The Poisson distribution is the mathematical distribution governing a dom variable in which one counts “rare” events, but at a definite averagerate This is called a “Poisson process”—a process in which discrete eventsare observable in an area of opportunity—a continuous interval (of time,length, surface area, etc.)—in such a manner that if we shorten the area of

EXHIBIT A.18 (Continued)

100 loans, 8% default rate

opportunity enough, we obtain three conditions: (1) the probability of serving that exactly one success in the interval is stable; (2) the probability

ob-of observing that more than one success in the interval is 0; (3) the rence that a success in any one interval is statistically independent of that

occur-in any other occur-interval Examples occur-include findoccur-ing the probability of the ber of:

num-Radioactive decays per second Stoppages on a production line

Imperfections per square meter Customers arriving at a service

Telephone calls per hour Requests arriving at a server

Cashews per can of mixed nuts Accidents at a particular

Bacteria in a given culture intersection per month

Cases of a rare disease per year

The distribution was invented by the French mathematician Denise Poisson (1781–1840) and was first applied to describe the proba-bility of a particular number of Prussian soldiers being killed by beingkicked by horses Actuaries use a Poisson distribution to model events like

Simeon-a hurricSimeon-ane’s striking Simeon-a specific locSimeon-ation on the eSimeon-astern seSimeon-aboSimeon-ard of theUnited States

The Poisson distribution is related to a “rare” event (though rare is arelative term) in which the time of arrival is exponentially distributed—

that is, the probability of arrival time decreases exponentially (i.e., as e –rt,

where r is some average arrival rate and t is time) with increasing time The Poisson distribution describes the probability of having v “arrivals”

(e.g., defaults) in a fixed time interval, as already discussed Later on, the

gamma distribution is discussed, and it is also part of the same family The

gamma distribution describes the probability distribution of the time of

the kth“arrival” (if k = 1 then the gamma distribution becomes the

It turns out that this distribution, pµ(n), that is, the probability of n

events occurring in some time interval, is equal to:

bi-Parameters The Poisson distribution takes only one parameter: the

aver-age of the distribution, µ It turns out that the variance is equal to themean, so the standard deviation is

Principal Applications There is an incredibly large array of applicationsfor the Poisson distribution In physics it is well-known for being the distri-bution governing radioactive decay In credit portfolio management, it isused as the starting point to model default event probability distributions

in the Credit Risk+ model, described in Chapter 3.

EXHIBIT A.19 Comparison of the Poisson Distribution to the Binomial

100 loans, 8% default rate

Number of Defaults

Poisson Binomial

The beta distribution is described by two parameters (α and β) and applies

to a random variable constrained between 0 and c > 0.

The beta distribution density is given by

(A.38)

whereΓ( ) is the gamma function, not to be confused with the gamma

distribution The mean and variance of this distribution are given by

and

This functional form is extremely flexible in the shapes it will modate It is symmetric if α = β, asymmetric otherwise, and can be hump-shaped or U-shaped In credit risk analysis, these properties make it idealfor modeling the distribution of losses for a credit, given that default hasoccurred for that credit (called loss given default, or LGD) Exhibit A.20shows four examples of the beta distribution

accom-Upper/Lower Bounds The lower bound is zero, while the upper can be set

to any constant (c), but usually 1.

Parameters The beta distribution is described by two parameters (α andβ).The mean is equal to α/(α + β) and the variance is αβ/[(α + β)2(α + β + 1)]

Gamma

As mentioned earlier, the gamma distribution derives from the same family

as the exponential and the Poisson distributions The general form of the

Gamma distribution is given by

whereΓ( ) is the gamma function (see next section).9The mean is P/λ

and the variance is P/λ2 The gamma distribution has been used in a variety

of settings, including the study of income distribution and productionfunctions

Exhibit A.21 shows the gamma distribution as compared to the normal one for the same mean (8) and standard deviation (2.71) The twodistributions are quite similar, though the gamma distribution has a bitmore leptokurtosis

log-Upper/Lower Bounds The lower bound is zero and the upper is infinity

Parameters The gamma distribution has two parameters, P andλ, both

determining the mean (= P/ λ) and the variance (= P/λ2)

Principal Applications In econometrics, this function has been used in avariety of settings, including the study of income distribution and produc-

tion functions (W H Greene, Econometric Analysis, Macmillan

Publish-ing Company, 1980)

Gamma Function The gamma function is not a probability distribution but

is an important mathematical function used in the gamma and beta

distrib-EXHIBIT A.20 Four Examples of the Beta Distribution, Showing the Mean (µ) and Standard Deviation ( σ) of Each

utions It comes up in many areas of science and engineering, as well aseconomics and finance The function is generally written as an integral:

For an integer p,

which follows the recursion

For p greater than 1, the gamma function may be regarded as a alization of the factorial operation for non-integer values of p In addition,

gener-For integer values of p,

Γ 12

For values of p that are not multiples of 1/2, the gamma function must

be numerically approximated Exhibit A.22 shows a plot of the gamma

function up to p = 5.

NOTES

1 Note that a discrete random variable may take an infinite number ofvalues, but only because the interval considered is infinite, while a con-tinuous variable has an infinite number of values in a finite interval(e.g., 0 to 1) as well

2 This is written in mathematical shorthand as “[–1, ∞)”—the bracketmeans inclusive, and the parenthesis means exclusive

3 Notice that we have N – 1, not N, in the denominator of the

right-hand side of equation A.5 The reason for this is due to the fact thatfor a small number of measurements the best estimate for the standarddeviation is not clear (or rather it will take more space than is allowed

by this book to explain) It makes little difference when N is large (e.g.,

greater than 20)

4 Refer to pp 314–324 for details concerning each distribution function

5 We define the continuously compounded return as ln(Si/S i–1), where

“ln( )” means “natural logarithm of,” and S is today’s share price

12

12

EXHIBIT A.22 Gamma Function Plotted from x = 0 to x = 5

6 Anyone familiar with stock price modeling or option pricing knowsthat this statement should be taken with a grain of salt It is wellknown that stock returns are not exactly normally distributed and ex-hibit leptokurtosis over the normal distribution That is, real data ex-hibit returns that should occur much less frequently than are predicted

by a normal distribution with the same mean and standard deviation

7 The symbol “∩” denotes “intersection of,” and may be remembered asbeing similar to the letter capital “A,” standing for “and.”

8 To calculate the mean (i.e., E[x]) use equation A.2.b with equation A.34 for f(x) and make the substitutions x = e y and dx = e y dy Com-

plete the square in the exponential to simplify the integral For thevariance, use the result for the mean along with equations A.8 and

A.34 and make the same substitutions just mentioned for x and dx.

9 If one thinks of x as being time and λ as being an average “arrival”rate (per unit time), then equation A.39 describes the probability dis-

tribution in time of the Pthevent’s arrival Note that if P = 1 one

ob-tains the exponential distribution