Lecture Notes 1

1. Value-at-Risk

Birger Nilsson, NEKN83/TEK180, Spring 2012

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1. Value-at-Risk (VaR) 1a. De…nition Value-at-Risk is the smallest loss ` such that the probability of a future portfolio loss L that is larger than the loss `, is less than or equal to 1 (V aR) mathematically: V aR (L) = min f` : Pr (L > `) 1 g. = 0:99 or = 0:95. . The following equation de…nes Value-at-Risk

From a probabilistic perspective, Value-at-Risk is essentially the quantile of the loss distribution (the distribution of the stochastic variable L). Typical con…dence levels are Typical time horizons (holding periods) over which future portfolio losses L are forecasted are h = 1 day or h = 10 days. To explicitly indicate the holding period we use the notation V aRh . Under the assumption of a continuous loss distribution, the equation de…ning Value-at-Risk may be rewritten as: Pr (L > V aR (L)) = 1 .

This alternative de…nition says that the probability of a future portfolio loss L that is larger than Value-at-Risk is equal to 1 valid de…nition of Value-at-Risk. Because gains (pro…ts) are simply equal to minus losses, G = de…ned in terms of gains: V aR (G) = max fg : Pr (G < g) 1 g . If the L, Value-at-Risk can also be . If the loss distribution is continuous, the two given de…nitions are equivalent. If the loss distribution is discrete, the alternative de…nition is not a

According to this de…nition (minus) Value-at-Risk is the largest gain g such that the probability of a future portfolio gain G that is smaller than the gain g, is less than or equal to 1 gain distribution is continuous the de…nition may be rewritten: Pr (G < V aR (G)) = 1 .

This alternative de…nition says that the probability of a future portfolio gain G that is less than (minus) Value-at-Risk is equal to 1 Example 1 (Discrete loss distribution). Assume that the distribution of future portfolio losses L are (probability in parentheses): 100 (1%), 50 (3%), 25 (5%), 10 (15%), etc. To determine Value-at-Risk we...