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# 3 incremental var is the incremental effect on var of

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3. Incremental VaR is the incremental effect on VaR of a new trade or the incremental effect of closing out an existing trade. It asks the question: “what is the difference between VaR with and without the trade.” If a component is small relative to the size of a portfolio, it may be reasonable to assume that the marginal VaR remains constant as xi is reduced to zero Æ ( ) i i VaR x x 4. Component VaR is the part of the VaR of the portfolio that can be attributed to this component: - The i th component VaR for a large portfolio should be approximately equal to the incremental VaR. - The sum of all the component VaRs should equal to the portfolio VaR. - 1 ( ) N i i i VaR VaR x x = = 5. Back testing : - The percentage of times the actual loss exceeds VaR . Let p = 1-X, where x is confidence level. m = the number of times that the VaR limits is exceeded, n the total number of days. Two hypotheses: a. The probability of an exception on any given day is p. b. The probability of an exception on any given day is greater than p. The probability (binominal distribution) of the VaR limit being exceeded on m or more days is: ! (1 ) !( )! n k k m n p p k n k n k = . We usually use confidence level as 5%. If the probability of the VaR limit being exceeded on m or more days is less than 5%, we reject the first hypothesis that the probability of an exception is p. The above test is one-tailed test. Kupiec has proposed a two-tailed test (Frequency-of-tail-losses or Kupiec test). If the probability of an exception under the VaR model is p and m exceptions are observed in n trials, then 2ln[(1 ) ] 2ln[(1 / ) ( / ) ] n m m n m m p p m n m + n should have a chi-square distribution with one degree of freedom. z The Kupiec test is a large sample test z Kupiec test focuses solely on the frequency of tail losses. It throws away potentially valuable information about the sizes of tail losses. This suggests that the Kupiec test should be relatively inefficient, compared to a suitable test that took account of the sizes as well as the frequency of tail losses. - 12 -

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Study Notes: Risk Management and Financial Institutions By Zhipeng Yan - Sizes-of-tail-losses test: compare the distribution of empirical tail losses against the tail-loss distribution predicted by model – Kolmogorov-Smirnov test (it is the maximum value of the absolute difference between the two distribution functions). Another backtest is Crnkovic and Drachman (CD) test. The test is to evaluate a market model by testing the difference between the empirical P/L distribution and the predicted P/L distribution, across their whole range of values. - The extent to which exceptions are bunched : in practice, exceptions are often bunched together, suggesting that losses on successive days are not independent.
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• Spring '10
• NanLi
• Normal Distribution, ........., Risk Management and Financial Institutions, Zhipeng Yan

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