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# Maximum loss optimization search over the losses that

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- Maximum loss optimization: search over the losses that occur for intermediate as well as extreme values of the risk variables Chapter 9 Market Risk VaR: Historical Simulation Approach 1. Methodology Suppose that we calculate VaR for a portfolio using a one-day time horizon, a 99% confidence level, and 500 days of data. - Identify the market variables affecting the portfolio. They typically are exchange rates, equity prices, interest rates, and so on. - Collect data on the movements in these variables over the most recent 500 days. - The value of the market variable tomorrow under i th scenario is 1 i n i v v v (n – today, say n=500, i = 0, 1, … 499) - Get change in portfolio value under each scenario according to the calculated market variables. - VaR = 5 th worst number of the change in portfolio value. 2. The confidence interval: - Kendall and Stuart calculate a confidence interval for the quantile of a distribution when it is estimated from sample data. - The standard errors of the estimate is: 1 (1 ( ) q q f x n ) , where f(x) is the probability density function of the loss evaluated at x. 3. Weighting of observations : the basic historical simulation approach assumes that each day in the past is given equal weight. Boudoukh et alpha. Suggest that more recent observations should be given more weight – the weight given to the change between day n-i and n-i+1 is 1 (1 ) 1 i n λ λ λ . VaR is calculated by ranking the observations from the worst outcome to the best. Starting at the worst outcome, - 14 -

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