actsc445unit9p3_-_handout

# Delta normal approximation cornish fisher expansion

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Delta-Normal Approximation Cornish-Fisher Expansion What if the distribution of a risk factor is positively skewed or has fat tails? Recall that if V n - V 0 ∼ N μ, ˆ σ 2 ) , then VaR α, n = ˆ σ z α - ˆ μ , where z α is the α percentile of a N ( 0 , 1 ) distribution. The idea of the Cornish-Fisher expansion is to replace z α in the above expression with a term z 0 α which takes into account the skewness (3rd moment) of the loss distribution. Actsc 445 (Spring 2011) Risk Measures 12/ 13 12 / 13

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Delta-Normal Approximation Cornish-Fisher Expansion (Cont’d) We calculate VaR CF α n = ˆ σ z 0 α - ˆ μ where z 0 α = z α - 1 6 z 2 α - 1 and the skewness of V n - V 0 is denoted by where = E n [( V n - V 0 ) - ˆ μ ] 3 o ˆ σ 3 In class examples. Actsc 445 (Spring 2011) Risk Measures 13/ 13 13 / 13

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