THE BETA DISTRIBUTION
The beta distribution depends upon two parameters, n and r. The formula of the beta density is:
X is the random variable, with values between 0 and 1. The expectation and the variance of the beta distribution are:
The beta distribution models continuous variables between 0 and 1. It notably applies to random recovery rates. It can be bell-shaped, U-shaped, skewed, or even flat between 0 and 1. These are attractive properties for recoveries of which distributions can have all these shapes. If recovery data is available, it is possible to attempt fitting a beta distribution on the data. In addition, modeling recovery uncertainty serves for enhancing the modeling of the loss under default, which depends on exposures, default event and recoveries. Uncertain recoveries increase the volatility of such losses.
THE STUDENT DISTRIBUTION
The Student distribution is, as the normal distribution, characterized by g and [i, plus v, the degree of freedom. It is a symmetric curve that becomes identical to the normal curve when the degrees of freedom increase. The degree of freedom controls fat tails, or the degree of leptokurtosis. The smaller is v, the fatter is the tail. When the degree of freedom v gets larger, the Student distribution converges towards the normal distribution. Student distributions are usually considered as a better fit of actual distributions of stock returns. They are designated as tv where v is the degree of freedom.
APPENDIX: CALCULATION OF STANDARD DEVIATION FROM TIME SERIES
The formulas for calculating the mean and the standard deviations when using observed values of a random variable, the usual case when using times series of observations, are shown below. The random variable is X, the mean is E(X). Say it represents monthly earnings and we are interested in the volatility of these earnings. It is calculated as the arithmetic average of a time series of observations when all are considered of equal probability. For n observations:
The volatility, or standard deviation, is:
With time series, the probabilities are estimated by the frequencies of observing any given value. The probability of a single observed value is therefore lln. In other cases, probabilities have to be assigned to values, for instance by assuming that the distribution curve is given. The variance V(X) is identical to O2.
The example in Table 10.3 shows how to calculate a yearly volatility of earnings over a 12-month time series of earnings observations. The expectation is the mean of all observed values. The variance is the sum of squared deviations from the mean, and the standard deviation is the square root. The following is a sample of calculations using those definitions. Monthly observations of earnings are available for one year, or 12 observed values. Volatilities are in the same unit as the random variable. If, for instance, the earnings are in Euros, the standard deviation is also expressed in Euros, here €7.71.
TABLE 10.3 Example of a calculation of mean and volatility with a time series of observed data
*The mean is the sum of observed values divided by the number of observations (l2).The variance is the sum of squared deviations divided by 12.The volatility is the square root of variance.