Value at risk metric:
The VaR metric is a real-valued function of the distribution of P1 conditional upon information available at time 0 and P0. An example of a risk metric that is not a VaR metric is standard deviation of cash flow because this generally cannot be expressed as a function of P0 and the conditional distribution of P1. VaR cannot anticipate changes in the composition of the portfolio during the day. Instead, it reflects the riskiness of the portfolio based on the portfolio's current composition.
Let us see sketchily how VaR is used. Let 0 be the current time. We know a portfolio's current market value, say P0 . We do not know its value at the end of one trading day, P1 . It is a random variable. The task is to assign P1 a probability distribution.
Sometimes a normal distribution is assigned, and its properties are used to study risk. Occasionally, time-series analysis is used to study historical patterns of data related to the securities.
Another approach is to model the portfolio's behavior, not in terms of individual assets, but in terms of specific risk factors. Depending upon the composition of the portfolio, risk factors might include exchange rates, interest rates, commodity prices, spreads, implied volatilities, etc. The modeled risk factors are called key factors. The idea is to list these factors into a vector, and then devise a valuation function that is a pricing formula which expresses prices of these assets as a function of the key factors. A linear function is occasionally used. Then the vector of prices (or the list of prices may be indexed into a scalar) of the assets is expressed as a function of vector of key factors. This function is called a portfolio mapping function. The vector (or scalar) of prices, however, is random. This involves mathematics that is beyond the scope of our discussion. There are three methods of calculating VAR: the historical method, the variance-covariance method, and monte-carlo simulation. We do not discuss these. You can look up your course on quantitative methods to understand these concepts in general terms. We may just mention that the historical method simply re-organizes actual historical returns, putting them in order from worst to best. It then assumes that history will repeat itself, from a risk perspective.