VALUE AT RISK PROBABILITY DENSITY FUNCTION SERIES

VALUE AT RISK PROBABILITY DENSITY FUNCTION SERIES

However, the drawback is that this infinite series may not converge since the infinite integral of a positive exponential can indeed blow up. The key benefit of the MGF is that you can Taylor expand it asįrom which you can define the random variable’s moments. Well, this is very similar to the moment generating function (MGF) of, which is defined as Now all we need to do is consider taking the expectation of the exponential of the random variable, i.e. Which is very similar to the definition above involving the summation. Once we have this we can easily define the expectation of a random variable as Hence, our requirements to construct and evaluate such an integral (expectation) of random variables is aĬollectively known as a probability space. Therefore we need a probability measure to assign real numbers to each measurable event in order to evaluate an integral. The reason for this is that the random variable is an element of a sigma-algebra of events of some sample space and does not necessarily come to us as real numbers. However, now that we are discussing random variables and not numbers, this integral must be a Lebesgue integral. when then this summation becomes an integral. We’ve already discussed that when we have an infinite of these numbers, i.e. In our case, which is probability theory, the expected value of a random variable is defined, from first principles, as So let’s go back one step and take a look at the Expected Value a little more closely. Well, we could just define the Laplace transform to be precisely the expectation of like this: Where does the Laplace transform fit in to this? Īlternatively, we could define the probability distribution of losses as and then VaR is just the -quantile of the loss distribution. The -percentile VaR is defined as the smallest real value such that the probability of exceeding is less that. Since those values can each be considered as an infinite number of VaR numbers at some percentile (just separated by an infinitesimal percentile), we can say that the expected shortfall is the integral of over the range with respect to an infinitesimal change in – the standard definition of an integral: Hence, the expected shortfall of a range of values becomes an integral. See my previous article on this concept here. In the case of an infinity of numbers, the average becomes a (Riemann) integral. In most cases a computer is calculating these values, usually about ten thousand or so for VaR on any given day, and so we tend to have a finite number of values to either take a percentile or to take an average.īut what if we have an infinite number of values? This makes the risk measure more robust as it requires perfect calculation of every value in the tail of the distribution whereas VaR essentially ignores those values. Expected shortfall (or ES), in the context of financial risk, is similar to Value at Risk (or VaR) except that one is usually asked to consider the average of a range of values rather than a single percentile, as is done for VaR. Today at work we were discussing the concept of Expected Shortfall as a risk measure and how it is computed.

Value at risk probability density function