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Missing Contract-Related Time Periods

This means that 1 ← u 5 u 5 F S D 1 2 . (7) t t In our notation used before, ut 5 x . Whenever t 121/t is large—typically the case in the catastrophic, that is, rare, event situation—the following result due to Balkema, de Haan, Gnedenko, and Pickands (see EKM, Theorem 3.4.13(b)) is very useful. Theorem 4 Suppose X ,…, Xn are iid with df F. Equivalent are: 1 i) F [ MDA(Hj), j [ R, ii) for some function b : R → R , 1 1 lim sup u F (x) 2 G (x)u 5 0, (8) u j,b(u) u↑xF F 0,x,x 2u where Fu(x) 5 P(X 2 u # x u X . u), and the generalized Pareto df is given by 21/j x G (x) 5 1 2 S D 1 1 j , (9) j ,b b 1 for b . 0. M It is exactly the so-called excess df Fu that risk managers as well as reinsurers should be interested in. Theorem 4 states that for large u, Fu has a generalized Pareto df (9). Now, to estimate the tail ( F u 1 x) for a fixed large value of u and all x $ 0, consider the trivial identity F(u 1 x) 5 F(u) F (x), u, x $ 0. (10) u In order to estimate ( F u 1 x), one first estimates F(u) by the empirical estimator Nu ˆ (F(u)) 5 , n where Nu 5 # {1 # i # n : Xi . u}. In order to have a ‘‘good’’ estimator for ( F u), we need u not too large: Name /8042/03 04/21/99 09:19AM Plate # 0 pg 35 # 6 EXTREME VALUE THEORY AS A RISK MANAGEMENT TOOL 35 NAAJ (SOA) Figure 4 Log Histogram of the Fire Insurance Data Figure 5 Mean-Excess Plot of the Fire Insurance Data the level u has to be well within the data. Given such a u-value, we approximate u F (x) via (8) by ˆ (F (x)) 5 Gˆ ˆ , (x) u j b(u) for some estimators and (u), depending on u. For ˆ ˆ j b this to work well, we need u large (indeed, in Theorem (4ii), u ↑ x F , the latter being 1` in the Fre´chet case). A ‘‘good’’ estimator is obtained via a trade-off between these two conflicting requirements on u. The statistical theory developed to work out the above program runs under the name Peaks over Thresholds Method and is discussed in detail in Embrechts, Klu¨ppelberg, and Mikosch (1997, Section 6.5), McNeil and Saladin (1997), and references therein. Software (S-plus) implementation can be found at,mcneil/software. This maximum-likelihood-based approach also allows for modeling of the excess intensity Nu , as well as the modeling of time (or other co-variable) dependence in the relevant model parameters. As such, a highly versatile modeling methodology for extremal events is available. Related approaches with application to insurance are to be found in Beirlant, Teugels, and Vynckier (1996), Reiss and Thomas (1997), and the references therein. Interesting case studies using upto-date EVT methodology are McNeil (1997), Resnick (1997), and Rootze´n and Tajvidi (1997). The various steps needed to perform a quantile estimation within the above EVT context are nicely reviewed in McNeil and Saladin (1997), where a simulation study is also found. In the next section, we illustrate the methodology on real and simulated data relevant for insurance and finance