A perturbative approach is used to derive approximations of arbitrary order to estimate high percentiles of sums of positive independent random variables that exhibit heavy tails. Closed-form expressions for the successive approximations are obtained both when the number of terms in the sum is deterministic and when it is random. The zeroth order approximation is the percentile of the maximum term in the sum. Higher orders in the perturbative series involve the right-truncated moments of the individual random variables that appear in the sum. These censored moments are always finite. As a result, and in contrast to previous approximations proposed in the literature, the perturbative series has the same form regardless of whether these random variables have a finite mean or not. The accuracy of the approximations is illustrated for a variety of distributions and a wide range of parameters. The quality of the estimate improves as more terms are included in the perturbative series, specially for higher percentiles and heavier tails.
Using heuristic arguments, a correction to the single-loss approximation was proposed in @cite_38 for distributions with finite mean In the limit @math , the value @math is large, so that @math and the approximation given by ) becomes similar to ).
@cite_38
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Making the assumption that the distribution of operational-loss severity has finite mean, Klaus Bocker and Jacob Sprittulla suggest a refined version of the analytical operational VAR theorem derived in Bocker and Kluppelberg (2005), which significantly reduces the approximation error to operational VAR.