BINOMIAL DISTRIBUTION OF SUM OF BERNOULLI VARIABLES

The binomial distribution is the probability of the sum Y of n Bernoulli variables X. that are independent. Let n be number of binomial trials, p the probability of success. The first two moments of the binomial distribution are:

HXl,Xy Xy ... are independent, identically distributed (i.i.d.) random variables, all Bernoulli distributed with "true" probability p, then:

The binomial distribution is characterized by the number n of Bernoulli variables and the probability p of a "true" value. Hence Fis binomial (n,p).

Using again default events as an example, each one is a Bernoulli variable. When default events are independent, and have the same probability, the distribution of the random number of defaults in a portfolio of loans is the binomial distribution. The binomial distribution applies to a sum of random independent losses of equal size since it is the distribution of a count of defaults.

Using a simple default probability of 1%, and 100 obligors, Figure 10.5 shows the distribution of the number of defaults. Although the distribution has the fat tail characterizing credit risk loss distributions, this distribution falls short from realistic correlated losses distribution. Nevertheless, the binomial distribution provides a good introduction to credit risk loss distributions.

Binomial distributions are distributions of the number of defaults within a portfolio when all defaults are independent. Using the uniform default probability d — 1%, uniform across the portfolio, and n by the number of firms, say 100, we find that the expected number of defaults

Binomial distribution of the number of independent defaults

FIGURE 10.5 Binomial distribution of the number of independent defaults

is 100 x 1% = 1 and the variance of the number of defaults is 100 x 1% x 99% = 0.99. The number of defaults is the sum of n variables taking values 1 for default and 0 under no default for the n obligors of the portfolio.

 
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