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Unformatted text preview: V A L U EA TR I S K (V A R) ValueatRisk (VaR) The authors describe how to implement VaR, the risk measurement technique widely used in financial risk management. by Simon Benninga and Zvi Wiener I n this article we discuss one of the modern risk measuring techniques ValueatRisk (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. VALUEATRISK ValueatRisk (VaR) measures the worst expected loss un der normal market conditions over a specific time inter val at a given confidence level. As one of our references states: “VaR answers the question: how much can I lose with x % probability over a preset 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 specified time period. The basic time period T and the confidence 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 endofyear distribution of portfolio value: 1. What is the distribution of the endofyear 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 endofyear 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 first want to know the distribution of the endof year portfolio value: Plot [ PDF [ NormalDistribution [ 110,30 ] ,x ] ,{x,0,200} ]; 50 100 150 200 0.002 0.004 0.006 0.008 0.01 0.012 The probability that the endofyear portfolio value is less than $80 is about 15.9%.less than $80 is about 15....
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 Fall '09
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 Normal Distribution, Variance, Covariance matrix, Value at risk, Education and Research

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