Value at Risk

Definition of Value at Risk (VaR) Value at risk is a statistical technique which measures the level of financial risk in a portfolio over a specific time frame. For example, if a firm states that it has a 1% one week value at risk of $5 million; this would mean that for any given week, the firm would have a 1% chance of losing $5 million. In order words, 1 out of every 100 weeks, the firm would expect to have a loss of $5 million. This can be viewed as the standard deviation of portfolio value during “normal” market movements. The reason we look at it in terms of losses even though VaR compromises of some risk metric like volatility is because VaR is a type of risk measurement and the calculation of VaR would determine the amount of risk one’s portfolio is exposed to at a given time frame. This is very important for institutions to manage risk. VaR has become popular because of its prospective nature, as compared to retrospective risk metrics like historical volatility. This VaR calculation is able to quantify market risk while trades are being taken. Despite its benefits, VaR is limited as it is unable to give any information about the severity of loss by which it is exceeded. Mathematical Definition Given a confidence level α in (0,1), the Value-at-Risk of a portfolio at α over the time period t is given by the smallest number l in R such that the probability of a loss L over a time interval t greater than l is α. VaRα = inf {l in R, P(L ≤ l) > α} History of Value at Risk The first VaR measure ever published was probably by Leavens (1945). He did not explicitly identify a VaR metric but he mentioned multiple times about the “spread between probable losses and gains”. Later on, Markowitz (1952) and Roy (1952) independently published VaR measures that were surprisingly similar. Their work basically encapsulates optimizing reward for a given level of risk. The 1970s and 1980s wrought sweeping changes for markets and technology. These