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# Value-at-Risk_(VaR) VALUE-AT-RISK (VAR) Value-at-Risk (VaR) The authors describe how to implement VaR, the risk measurement technique widely used in fi nancial risk management. by Simon Benninga and Zvi Wiener I n this article we discuss one of the modern risk- measuring techniques Value-at-Risk (VaR). Mathemat- ica is used to demonstrate the basic methods for cal- culation of VaR for a hypothetical portfolio of a stock and a foreign bond. VALUE-AT-RISK Value-at-Risk (VaR) measures the worst expected loss un- der normal market conditions over a specifi c time inter- val at a given confi dence level. As one of our references states: “VaR answers the question: how much can I lose with x% probability over a pre-set horizon” (J.P. Mor- gan, RiskMetrics–Technical Document). Another way of expressing this is that VaR is the lowest quantile of the potential losses that can occur within a given portfolio during a specifi ed time period. The basic time period T and the confi dence level (the quantile) q are the two ma- jor parameters that should be chosen in a way appropriate to the overall goal of risk measurement. The time horizon can differ from a few hours for an active trading desk to a year for a pension fund. When the primary goal is to satisfy external regulatory requirements, such as bank capital requirements, the quantile is typically very small (for example, 1% of worst outcomes). However for an internal risk management model used by a company to control the risk exposure the typical number is around 5% (visit the internet sites in references for more details). A general introduction to VaR can be found in Linsmeier, [Pearson 1996] and [Jorion 1997]. In the jargon of VaR, suppose that a portfolio manager has a daily VaR equal to \$1 million at 1%. This means that there is only one chance in 100 that a daily loss bigger than \$1 million occurs under normal market conditions. A REALLY SIMPLE EXAMPLE Suppose portfolio manager manages a portfolio which consists of a single asset. The return of the asset is nor- mally distributed with annual mean return 10% and annual standard deviation 30%. The value of the portfolio today is \$100 million. We want to answer various simple questions about the end-of-year distribution of portfolio value: 1. What is the distribution of the end-of-year portfolio value? 2. What is the probability of a loss of more than \$20 million dollars by year end (i.e., what is the probability that the end-of-year value is less than \$80 million)? 3. With 1% probability what is the maximum loss at the end of the year? This is the VaR at 1%. We start by loading Mathematica’s statistical package: Needs[“Statistics‘Master‘“] Needs[“Statistics‘MultiDescriptiveStatistics‘“] We fi rst want to know the distribution of the end-of- year portfolio value: Plot[PDF[NormalDistribution[110,30],x],{x,0,200}]; 50100150200 0.002 0.004 0.006 0.008 0.01 0.012 The probability that the end-of-year portfolio value is less than \$80 is about 15.9%. CDF[NormalDistribution[110.,30

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