INDEPENDENT DEFAULT EVENTS:THE BINOMIAL DISTRIBUTION
When defaults events are independent, and have the same probability, the distribution of the number of defaults in a portfolio is the binomial distribution. Unfortunately, the binomial distribution cannot capture the effect of size discrepancies and of correlations. Nevertheless, the binomial distribution provides a good introduction to credit risk loss distribution. We use it here as a reference case and to show how diversification due to adding new facilities alters the distribution.
Simulating Increased Diversification with Loss Independence
Consider a portfolio with N loans of same maturity, each one with same exposure X net of recovery (or same LGD). Considering a unique horizon equal to the common maturity allows considering the loss distribution in default mode only. The default probability is identical for all obligors. This default probability is equal to the expected loss, if loss given defaults are also identical and equal to 1. The weights of each loan are identical.
Each random default is a Bernoulli variable with value 1 in the event of default, with probability a, and value 0 in the event of no default, with a probability 1 - a. It is the indicator function: 1 (default) — 1 under default and 0 if no default. The expected individual loss for obligor "i" is:
The individual borrower's loss variance is Xa(l - a). The number of defaults is the sum of all random individual defaults. The distribution of number of defaults is the binomial distribution.
The number of defaults has an expected value Na. The portfolio number of defaults has a variance equal to the sum of all individual default variances, since defaults are independent of №x(l - a) or N multiplied by individual default variance. The portfolio loss variance is Nx Xa(l - a) and the total portfolio loss volatility is the square root, ovX^Na(l - a).
With N exposures, of which unit value size is X— 2, and a — 10%, the unit variance in value is 10%(1 - 10%) x 22= 0.36 and the total variance is 0.367V. The loss distribution volatility in value is the square root, or ^(0.36^. Note that with a large number of obligors, the binomial distribution would tend to the normal distribution under independence. In a later section, we will see how the limit normal distribution for large uniform granular portfolios behaves when there is a positive uniform correlation.