Lecture 14

Lecture 14 - Value at Risk (VaR) Value at Risk is one of...

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Value at Risk (VaR) Value at Risk is one of the basic measures of the risk of a loss in a particular portfolio. It is typically stated in a form like, "there is a one percent chance that, over a particular day of trading, the position could lose" some dollar amount. We can state it more generally as VaR(X,N) = V, where V is the dollar amount of the possible loss, X is the percentage (managers might commonly be interested in the 1%, 5%, or 10%), and N is the time period (one day, one week, one month). To get from one-day VaR to N-day VaR, if the risks are independent and identically distributed, we multiply by . Or, in one irreverent definition, VaR is " a number invented by purveyors of panaceas for pecuniary peril intended to mislead senior management and regulators into false confidence that market risk is adequately understood and controlled " (Schachter, from gloriamundi.org). measure risk exposure; ensure correct capital allocation; provide information to counterparties, regulators, auditors, and other stakeholders; evaluate and provide incentives to profit centers within the firm; and protect against financial distress. Note that the final desired outcome (protection against loss) is not the only desired outcome. Being too safe costs money and loses business! VaR is an important component of bank regulation: the Basel Accord sets capital based on the 10-day 1% VaR (so, if risks are iid, then the 10-day Var is times larger than the one-day). The capital is set at least 3 times as high as the 10-day 1% VaR, and can be as much as four times higher if the bank's own VaR calculations have performed poorly in the past year. If a bank hits their own 1% VaR more than 3 times in a year (of 250 trading days) then their capital adequacy rating is increased for the next year. Poor modeling can carry a high cost! If we graph the distribution of possible returns of a portfolio, we can interpret VaR as a measure of a percentile. If we plot the cumulative distribution function (cdf) of the portfolio value, then the VaR is simply the inverse of the probability, X:
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Or if we graph the probability density function (pdf), then V is the value that gives a particular area under the curve: This is obviously similar to hypothesis testing in statistics. Typically we are concerned with the lower tail of losses; therefore some people might refer to either the 1% VaR or the 99% VaR. From the mathematical description above, the 99% VaR would seem to be the upper tail probability (therefore the loss in a short position) but this is rarely the intent the formulation typically refers to the idea that 99% of losses will be smaller (in absolute value) than V.
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There are three basic methods of computing the Value at Risk of some portfolio: parametric models that assume some error distribution,
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This note was uploaded on 04/04/2012 for the course FIN 420 taught by Professor Poniachek during the Spring '12 term at Rutgers.

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Lecture 14 - Value at Risk (VaR) Value at Risk is one of...

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