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Mediation Vs. Arbitration

Value at Risk: A Word of Warning We have already pointed out the similarity in estimating attachment points or retentions in reinsurance and VaR calculations in finance. Both are statistically based methods, where the basic underlying risk measure corresponds to a quantile estimate xˆ p of an unknown df. Through the work of Artzner et al. (1996, 1998) we know that a quantile-based risk measure for general (nonnormal) data fails to be coherent—such a measure is not subadditive, creating inconsistencies in the construction of risk capital based upon it. This Name /8042/03 04/21/99 09:19AM Plate # 0 pg 40 # 11 40 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 3, NUMBER 2 NAAJ (SOA) Table 3 Quantile Estimates for the ARCH Example y 15 20 25 30 P(max{j i,…, j n} # y) 0.28229 0.65862 0.83794 0.91613 p 0.05 0.01 0.005 0.0005 x p 34.47153 52.63015 63.04769 114.692 Figure 16 Time Series and Mean-Excess Plots of BMW Return Data situation typically occurs in portfolios containing nonlinear derivatives. Further critical statements concerning VaR are to be found in Danielsson, Hartmann, and de Vries (1992), Garman (1997), Longin (1997a, b), and Ca´rdenas et al. (1997). A nice discussion on calculating VaR in stochastic volatility models using EVT is given by Frey and McNeil (1998). A much better, and indeed (almost) coherent, risk measure is the conditional VaR (or mean excess) E(X u X . xˆ ). (15) p For the precise formulation of coherency see Artzner et al. (1996). We want to point out that the latter measure is well known in insurance but is only gradually being recognized as fundamental to risk management. It is one of the many instances where an exchange of ideas between actuaries and finance experts may lead to improved risk measurement. Note that in the equation above, E(X u X . p) 5 e( p x ˆ ˆ x ) 1 p. One could use the mean-excess plot {(u, e n xˆ (u)), u $ 0} to visualize the tail behavior of the underlying data and hence get some insight on the coherent risk measure (15) above. As a final example, we have plotted in Figure 16 the daily log-returns of BMW over the period January 23, 1973–July 12, 1996, together with the mean-excess plot of the absolute values of the negative returns (hence corresponding to down-side risk). In the latter plot, the heavy-tailed nature of the returns is very clear. Also clear is the change in curvature of the plot around 0.03. This phenomenon is regularly observed in all kinds of data. One can look at it as a pictorial view of Theorem 4; indeed, Smith (1990) indicates how to base on this observation a graphical tool for determining an initial threshold value for an extreme value analysis. See also Embrechts, Klu¨ppelberg, and Mikosch (1997, p. 356), for a discussion. However, we would like to warn the reader that because of the intricate dependencies in finance data, one should be careful in using these plots beyond the mere descriptive level.