What is Value-at-Risk (VaR) and how to rekon it.
Value at risk is a measure of risk concerning to estimate limits for financial losses. Actuaries had been using this risk measure long before investment banking reinvented it, according to Mary R Hardy’s paper an introduction to risk measures for actuarial applications. Of course, for our purposes, it does not matter who did discover it or where.
In probabilistic terms, the VaR is a quantile of a probability distribution where that distribution models the returns of an asset or a portfolio. Let us suppose that the normal probability distribution explains the random behavior of stock returns, then the quantile we are about going to calculate is from a normal distribution.
Keeping it in mind, we can ask ourselves: what is the most financial loss we can suffer in a specific set time (such as a day) considering a probability P? And the answer takes this fashion: There is a probability (P = alpha) that we will lose no more than Y dollars in a particular time range. That is to say, the value at risk represents a loss limit that, with probability P, will not be exceeded. We always specified a VaR measure giving a probability and a time range.
We have two questions to solve. The first one is which probability we must choose, and the second one is how we can find the loss value? The first point is the easy one, although it involves a little guess and subjectivity. The portfolio manager, based on his experience of the business, chooses the probability in question, say 95%. Now, he wants to know how much is the maximum financial loss he will suffer given the time range he takes and a probability of 95%.
Now, supposing normal distribution for the data and the fact that VaR is a quantile, being L the loss amount and picking a for alpha denoting the probability chose we can mathematically define:
One can read this as the probability of a financial loss being at most Q is alpha. Now, assuming we can invert the function F, that is the cumulative distribution function, we need to isolate Q on the left side of the equality. Then we get the result below.
One can see that this is precisely the quantile relative to a normal distribution, considering its CDF, and, in our case, this CDF is of the loss random variable L, which we are considering normally distributed.
Supposing we have a normally distributed loss random variable of an investment portfolio with a mean equal 33 and a standard deviation of 109. Let us chose a probability alpha=0.95 and consider the time range as a month.
In other words, we will not suffer a loss bigger than 212.29 in a month with a probability of 95% — note that in the third step, we checked a standard normal distribution. That is it, as one can see, there is no mystery about VaR. One can face endeavoring work when it has to find the loss distribution of a big portfolio with a lot of highly correlated assets, but this is a matter for another moment. But if the reader desire something to keep in mind, think about our choice to model the loss variable, a nonfat-tailed normal probability distribution. It would have been wiser picking a fat-tailed probability distribution, for example, a type 2 Pareto distribution. One can check that this CDF is invertible and finding a close analytical form for VaR of a loss modeled by Type 2 Pareto distribution.
Further reading:
A easy to read book about loss models is: Klugman, Stuart A., Harry H. Panjer, and Gordon E. Willmot. Loss models: from data to decisions. Vol. 715. John Wiley & Sons, 2012.
A paper really interesting about this subject (and more): Hardy, Mary R. “An introduction to risk measures for actuarial applications.” SOA Syllabus Study Note 548 (2006).
