# antitrust Risk Management Tool for Real Estate Brokerages

The basic assumption yields that X ,…, X are iid with df P(X # x) 1 100 1 5 2x/10 1 2 e , x $ 0. Therefore, for Mn 5 max(X1,…, Xn), 100 P(M . x) 5 1 2 (P(X # x)) 100 1 5 2x/10 100 1 2 (1 2 e ) . From this, we immediately obtain P(M $ 50) 5 0.4914, 100 P(M $ 100) 5 0.00453. 100 Name /8042/03 04/21/99 09:19AM Plate # 0 pg 33 # 4 EXTREME VALUE THEORY AS A RISK MANAGEMENT TOOL 33 NAAJ (SOA) Figure 3 Some Examples of Extreme Value Distributions Hj ;0,1 for j 5 3/4 (Fre´chet), j 5 0 (Gumbel), and j 5 23/4 (Weibull) However, rather than doing the (easy) exact calculations above, consider the following asymptotic argument. First, for all n $ 1 and x [ R, Mn P S D 2 log n # x 5 P(M # 10(x 1 log n)) n 10 2 n x e 5 S D 1 2 , n so that Mn 2 2x e lim P S D 2 log n # x 5 e [ L(x). n→` 10 Therefore, use the approximation x P(Mn # x) < L S D 2 log n 10 to obtain P(M $ 50) < 0.4902, 100 P(M $ 100) < 0.00453, 100 very much in agreement with the exact calculations above. Suppose we were asked the same question but had much less specific information on F(x) 5 P(X1 # x); could we still proceed? This is exactly the point where classical EVT enters. In the above exercise, we have proved the following. Proposition 1 Suppose X1,…, Xn are iid with df F , EXP(l), then for x [ R: lim P(lM 2 log n # x) 5 L(x). n n→` M Here are the key questions: Q1: What is special about L? Can we get other limits, possibly for other df’s F? Q2: How do we find the norming constants l and log n in general—that is, find an and bn so that Mn n 2 b lim P S D # x n→` an exists? Q3: Given a limit coming out of Q1, for which df’s F and norming constants from Q2, do we have convergence to that limit? Can one say something about second order behavior, that is, speed of convergence? The solution to Q1 forms part of the famous Gnedenko, Fisher-Tippett theorem. Theorem 2 (EKM Theorem 3.2.3) Suppose X1,…, Xn are iid with df F and (an), (bn) are constants so that for some nondegenerate limit distribution G, Mn n 2 b lim P S D # x 5 G(x), x [ R. n→` an Then G is of one of the following types: —Type I (Fre´chet): 0, x # 0 Fa(x) 5 H 2a a . 0 exp{2x }, x . 0 —Type II (Weibull): a exp{2(2x) }, x # 0 Ca(x) 5 H a . 0 1, x . 0 —Type III (Gumbel): L(x) 5 exp{2e }, x [ R. 2x M G is of the type H means that for some a . 0, b [ R, G(x) 5 H((x 2 b)/a), x [ R, and the distributions of one of the above three types are called extreme value distributions. Alternatively, any extreme value distribution can be represented as 21/j x 2 m H (x) 5 exp HS D J 2 1 1 j , x [ R. j ;m,s s 1 Here j [ R, m [ R, and s . 0. The case j . 0 (j , 0) corresponds to the Fre´chet (Weibull)-type df with j 5 1/a (j 5 21/a), whereas by continuity j 5 0 .