A uniform function is a distribution function that can take any value between a lower bound "a" and an upper bound "Z>" with the same constant probability. It is a continuous function. The expectation of a standard uniform distribution between values a and b is (b - a)l2 and its variance is (b - a)2/2, and its standard deviation is (b - a)H 12.

Let w(x) be the uniform probability density function and U(X) the cumulative distribution function. The standard uniform distribution is U(0, 1) and is a distribution of a random variable talking values between 0 and 1 with equal probability. The expectation of the standard uniform function is Vi and the standard deviation is 1/Vl2. The standard uniform distribution is U(0, 1) is of special interest when simulating random variables with various distribution functions because each random value between 0 and 1 can be seen as the value of cumulative distribution function CDF of another variable, or a probability. Starting from any random uniform standard number, we can derive the value x of the other variable using x — F~u).

APPLICATION TO LOSS DISTRIBUTIONS AND LOSS PERCENTILES

If we assign a value to i^x), such as F(x) = w, a percentage value representing the probability that Xis lower or equal to x, the percentage is usually called a quintile or percentile. It is always between 0 and 1. Given the above equations, the value of x that matches a given percentile is the inverse function of that percentile, or x — F~l(u).

In finance, percentiles are used for VaR-based measures of risk. A percentile is often called a confidence level a. A loss percentile is an upper bound L(a) of the random loss L which is not exceeded with probability a, hence such that P(L < L(a)] = a. L(a) designates the threshold which is not exceeded with probability 1 - a, and, equivalently, which is exceeded with the low probability a. Cumulative distribution functions provide the probability that the random variable does not exceed the upper bound, and refer to 1 - a.

A common practice in risk management for designating loss percentiles is to refer to the "low" probability of exceeding an upper bound, or 1 %, while cumulative distribution functions provide the probability of being lower or equal than this upper bound. If the confidence level is a — 1%, then 1 - a — 99% and the corresponding probability of not exceeding the threshold value is F(99%). Hence, if we use a as a confidence level equal to the "low" probability of exceeding an upper bound, we should use 1 - a as argument of the CDF of the random loss L, or F(l - a). For avoiding any confusion, it is easier to refer to probabilities directly:

The threshold loss L(a) is such that: F[L(a)] — 1 - a. When using 1% as the "low" probability 1 - a that the loss exceeds the threshold L(l%), the probability that Lbe lower or equal to L(l%) is F(99%). Therefore, L(l%) — F~1(99%). For illustrating, we refer to the simple example of a random loss following a normal standard distribution and we use "low" probabilities of exceeding the threshold value. The standard normal variables thresholds matching various confidence levels are embedded in the normal distribution as multiples of its standard deviation o:

This is equivalent, using the normal distribution for F, to writing that:

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