A Bernoulli variable is a variable that can take only two values, such as 0 and 1 with probabilities (l-d) and d. Bernoulli variables serve notably for characterizing any discrete couples of "true" and "false" states. A Bernoulli variable represents for example a default event and could take value 1 if a default occurs and 0 if no default occurs. When considering a portfolio of loans, the number of defaults would be the number of default events, each one characterized by a Bernoulli (default, no default) variable.

Let us calculate the expectation and variance, when d is the probability that the Bernoulli variable takes value 1 (default) and (1 —d)v& the probability that the variable takes the value 0 (no default).

Using

These are, respectively, the expected loss and the loss volatility of a facility which has loss under default equal to 1. If the loss under default is LGD, simply, multiply expectation and volatility by LGD and obtain:

INDICATOR FUNCTION

The indicator function is not a random variable, nor a distribution but a mathematical function. A more general way of writing such a discrete variable with two possible values is as an indicator function which takes a value 1 if a certain event occurs and 0 otherwise. The indicator function is usually defined for a set of values of continuous or discrete variable. For example, the event "x belongs to a certain range of values" and the alternate event "x does belong to this set of values" can be the arguments of the function. In abbreviated notations we would write x g A and x <£. A, where A is a specified set of values. The indicator function is:

The indicator function can be seen as a Bernoulli variable if we assign probabilities to each of these two events that sum up to 1.

BERNOULLI DISTRIBUTION

The Bernoulli distribution is a discrete probability distribution, which can be seen as a sum of k Bernoulli variables. Considering again a portfolio, suppose that the default events are independent. We can make various trials, or economic scenarios, and see how many defaults would occur. The sum of defaults would be distributed as a Bernoulli distribution.

The common way of introducing this distribution is to say that we make a series of trials, each trial meaning taking a ball from a set of such balls, some being white and other black. A success is getting a white ball and a failure a black one. Obviously the chances of getting a white ball depend on the fraction p of white balls and the fraction 1 -p of black balls. The probability of success (white ball) isp and the probability of failure (black ball) is q — 1 —p. A single trial is a Bernoulli variable with probabilities p and q—1 -p. If we make k trials, the probability of A: successes is f(k,p) —-p)l~k- The expected value of a Bernoulli random variable Xis E(X) —p and its variance is V(X) —p(l -p).

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