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STABILITY OF THE MAXIMUM OF THE EXTREME VALUE DISTRIBUTION 213 Example 7.8 (Example 7.7 continued) Suppose, in addition, that the indi- vidual losses are exponentially distributed with Then the distribution of the maximum loss for a k-year period has df Example 7.9 (Example 7.8 continued) Suppose instead that the individual losses are Pareto distributed with df Then the distribution of the maximum loss for a k-year period has df FMN (X) = 11 + P{ 1 - Fx (.)}I -T = [ 1+p (‘:;”-a]-T,X>o. - 7.4 STABILITY OF THE MAXIMUM OF THE EXTREME VALUE DISTRIBUTION The Gumbel, Frkchet, and Weibull distributions have another property, called “stability of the maximum” or “max-stabilty” that is very useful in extreme value theory. This is already hinted at in Examples 7.1, 7.2, and 7.3. First, for the standardized Gumbel distribution, we note that [Go(z + 1nn)ln = exp[-nexp(-z - Inn)] = exp [- exp (-x)] = Go (x) . Equivalently, [Go (x)]~ = Go (x - Inn). This shows that the distribution of the maximum of n observations from the standardized Gumbel distribution has itself a Gumbel distribution, after a shift of location of Inn. Including location and scale parameters yields 214 EXTREME VALUE THEORY: THE STUDY OF JUMBO LOSSES x - p - Blnn =Go( where p* = p + 0 Inn. Similarly, for the standardized Frkchet distribution [Gl,a(n'/"x)]n = exp (-n (TL'/~Z)-~) Equivalently, X iGl,a(X)l" = G1,a (m) . This shows that the distribution of the maximum of n observations from the standardized Frkchet distribution, after a scale change, has itself a Frkhet distribution. Including location and scale parameters yields = Gi,a,p,e+ (XI where 6' = en'/". The key idea of this section is that the distribution of the maximum, after a location or scale normalization, for each of the extreme value (EV) distri- butions also has the same EV distribution. Section 7.5 shows that these EV distributions are also approximate distributions of the maximum for (almost) any distribution. 7.5 THE FISHER-TIPPETT THEOREM We now examine the distribution of the maximum value of a sample of fixed size n (as n becomes very large) when the sample is drawn from any distribu- tion. As n + 00, the distribution of the maximum is degenerate. Therefore, in THE FISHER-TIPPETT THEOREM 215 order to understand the shape of the distribution for large values of n, it will be necessary to normalize the random variable representing the maximum. We require linear transformations such that x - b, 12-00 lim F, ( T) = G (x) for all values of x, where G (x) is a nondegenerate distribution. If such a linear transformation exists, Theorem [?] gives a very powerful result that forms a foundational element of extreme value theory. Theorem 7.10 Fisher-Tippett Theorem n If [. (*)I has a nondegenerate limiting distribution as n + cm, for some constants a, and b, that depend on n, then [.(%)In -+ G(x) as n + cm, for all values of x, for some extreme value distribution G, which is one of Go, GI?~ or G2,a for some location and scale parameters. The original theorem was given in a paper by Fisher and Tippett[37]. A detailed proof can be found in Resnick [99]. The Fisher-Tippett theorem proves that the appropriately normed maximum for any distribution (subject to the limiting nondegeneracy condition) converges in distribution to exactly one of the three extreme value distributions: Gumbel, Frkchet, and Weibull. This is an extremely important result. If we are interested in understanding how jumbo losses behave, we only need to look at three (actually two, because the Weibull has an upper limit) choices for a model for the extreme right-hand tail. The Fisher-Tippett theorem requires normalization using appropriate norm- ing constants a, and b, that depend on n. For specific distributions, these norming constants can be identified. We have already seen some of these for the distributions considered in the examples in Section 7.3. The Fisher-Tippett theorem is a limiting result that can be applied any distribution F(z). Because of this, it can be used as a general approximation to the true distribution of a maximum without having to completely specify the form of the underlying distribution F(x). This is particularly useful when we only have data on extreme losses as a starting point, without specific knowledge of the form of the underlying distribution. It now remains to describe which distributions have maxima converging to each of the three limiting distributions and to determine the norming con- stants a, and b,. Example 7.11 (Maximum of exponentials) Without any loss of generality, for notational convenience, we use the standardized version of the exponen- tial distribution. Using the norming constants a, = 1 and b, = - Inn, the 216 EXTREME VALUE THEORY: THE STUDY OF JUMBO LOSSES distribution of the maximum is given by Pr(Mn-bn 5~ =Pr(Mn5aa,x+bn) an = [Pr (X 5 a,x + b,)]" = [Pr (X 5 x - Inn)]" = [I - exp(-x - Inn)]" + exp(-exp(-x)) as n -+ 00. Having chosen [somehow) the right norming constants, we see that the limiting distribution of the maximum of exponential random variables is the Gumbel distribution. 0 Example 7.12 (Maximum of Paretos) Using the Pareto df and the norming constants a, = 0n'/"/a and b, = en1/" - 0, = [Pr (X 5 a,x + bn)ln n = Pr XI- x + 0n'la - 0)] [( This shows that the maximum of Pareto random variables has a Frkchet distribution with p = -a and t9 = a. MAXIMUM DOMAIN OF ATTRACTION 21 7 7.6 MAXIMUM DOMAIN OF ATTRACTION Definition 7.13 The maximum domain of attraction (MDA) for any distribution G, is the set of all distributions that has G as the limiting distrib- ution as n + m of the normalized maximum (M, - b,) la, for some norming constants a, and b,. Essentially, distributions (with nondegenerate limits) can be divided into three classes according to their limiting distribution: Gumbel, Frkchet and Weibull. If we can identify the limiting distribution, and if we are only inter- ested in modeling the extreme value, we no longer need to worry about trying to identify the exact form of the underlying distribution. We can simply treat the limiting distribution as an approximate representation of the distribution of the extreme value. Because we are interested in the distribution of the maximum, it is natural that we only need to worry about the extreme right-hand tail of the underlying distribution. Furthermore, the MDA should depend on the shape of only the tail and not on the rest of the distribution. This is confirmed in Theorem 7.14. Theorem 7.14 MDA characterization by tails A distribution F belongs to the maximam domain of attraction of an ex- treme value distribution Gi with norming constants a, and b, if and only lim nF (a,x + b,) = - In Gi(z). if n-+m This result is illustrated in Examples 7.15 and 7.16. Example 7.15 (Maximum of exponentials) As in Example ?',ll, we use the standardized version of the exponential distribution. Using the norming con- stants a, = 1 and b, = - Inn, the distribution of the maximum is given by nF(x+b,)=nPr(X>z+Inn) = nPr (X > IC +Inn) = n exp( -x - Inn) -n = exp (-x) = - lnGo(z). - exp(-x) n Having chosen the right norming constants, we see that the limiting dis- tribution of the maximum of exponential random variables is the Gumbel distribution. 0 218 EXTREME VALUE THEORY: THE STUDY OF JUMBO LOSSES It is also convenient, for mathematical purposes, to be able to treat distri- butions that have the same asymptotic tail shape in the same way. The above example suggest that if any distribution has a tail that is exponential, or close to exponential, or exponential asymptotically, then the limiting distribution of the maximum should be Gumbel. Therefore, we define two distributions FX and Fy as being tail-equivalent if where c is a constant. (Here the notation x -+ 00 should be interpreted as the x increasing to the right-hand endpoint if the distribution has a finite right-hand endpoint.) Clearly, if two distributions are tail-equivalent, they will be in the same maximum domain of attraction, because the constant c can be absorbed by the norming constants. Then in order to determine the MDA for a distribution, it is only neces- sary to study any tail-equivalent distribution. this is illustrated through the Example 7.16. Example 7.16 (Maximum of Paretos) Using the Pareto df and the norming constants a, = On-’/“ and b, = 0, and the tail-equivalence for large x, we obtain lim nF(a,x+ b,) N n-CX = x-0 =-lnG~(z). This shows that the maximum of Pareto random variables has a Frkhet distribution. 0 Because tail-equivalent distributions have the same MDA, all distributions with tails of the asymptotic form czPa are in the Frkchet MDA and all dis- tributions with tails of the asymptotic form Ice-”/* are in the Gunibel MDA. Then, all other distributions (subject to the riondegenerate condition) with infinite right-hand limit of support must be in one of these classes; that is, some have tails that are closer, in some sense, to exponential tails. Similarly, some are closer to Pareto tails. There is a body of theory that deals with the GENERAL /ZED PA RE TO DIS TRlBU TlONS 21 9 issue of “closeness” for the Frechet MDA. In fact, the constant c above can be replaced by a slowly varying function (see Definition 6.19). Slowly varying functions include positive functions converging to a constant and logarithms. Theorem 7.17 If a distribution has its right-tail characterized by F(x) - x-”C(x), where C(x) is a slowly varying function, then it is in the Frkchet maximum domain of attraction. Example 7.16 illustrates this concept for the Pareto distribution that has C(x) = 1. Distributions that are in the Frechet MDA of heavier-tailed distri- butions include all members of the transformed beta family and the inverse transformed gamma family that appear in Figures 4.1 and 4.2. The distributions that are in the Gumbel MDA are not as easy to charac- terize. The Gumbel MDA includes distributions that are lighter-tailed than any power function. Distributions in the Gumbel MDA have moments of all orders. These include the exponential, gamma, Weibull, and lognormal dis- tributions. In fact, all members of the transformed gamma family appearing in Figure 4.2 are in the Gumbel MDA, as is the inverse Gaussian distribu- tion. The tails of the distributions in the Gumbel MDA are very different from each other, from the very light-tailed normal distribution to the much heavier-tailed inverse Gaussian distribution. 7.7 GENERALIZED PARETO DISTRIBUTIONS In this section, we introduce some distributions known as generalized Pareto (GP) distributions’ that are closely related to extreme value distributions. They are used in connection with the study of excesses over a threshold. In operational risk, this means losses that exceed some threshold in size. For these distribution functions, we use the general notation W(x). Generalized Pareto distributions are related to the extreme value distributions by the simple relation with the added restriction that W(x) must be nonnegative, that is, requiring that G(x) 2 exp(-1). Paralleling the development of extreme value distributions, there are three related distributions in the family known as generalized Pareto distributions. W(x) = 1 + lnG(x) (7.4) *The ”generalized Pareto distribution” used in this chapter differs from the distribution with the same name used in Section 4.2. It is unfortunate that the term ”generalized” is often used by different authors in connection with different generalizations of the same distribution. Since the usage in each chapter is standard usage (but in different fields), we leave it to the reader to be cautious about which definition is being used. The same comment applies to the used of the terms ”beta distribution’’ and ”Weibull distribution.” 220 EXTREME VALUE THEORY: THE STUDY Of JUMBO LOSSES Exponential distribution The standardized exponential distribution has df of the form F(x) = Wo(x) = 1 - exp (-x) , x > 0. With location and scale parameters p and 6 included, it has df Note that the exponential distribution has support only for values of x greater than p. In the applications considered in this book, p will generally be set to zero, making the distribution a one-parameter distribution with a left-hand endpoint of zero. The df of that one-parameter distribution will be denoted by F(x) = Wo,e(x) = 1 - exp (-i), x>0. Pareto distribution The standardized Pareto distribution has df of the form With location and scale parameters p and 6 included, it has df F(x) = 1 - Note that the Pareto distribution has support only for values of 2 greater than p + 6. In the applications considered in this book, p will generally be set to -6, making the distribution a two-parameter distribution with a zero left-hand endpoint. The df of the two-parameter Pareto distribution will be denoted by Beta distribution The standardized beta distribution has df of the form With location and scale parameters p and 6 included, it has df Note that the beta distribution has support only for values of x on the interval [p - 6, p]. As with the Weibull distribution, it will not be considered THE FREQUENCY OF EXCEEDENCES 221 further in this book. It is included for completeness of exposition of extreme value theory. It should also be noted that the beta distribution is a (shifted) subclass of the usual beta distribution on the interval (0,l.) interval which has an additional shape parameter, and where the shape parameters are positive. Generalized Pareto distribution The generalized Pareto distribution is the family of distributions incorpo- rating, in a single expression, the above three distributions as special cases. The general expression for the df of the generalized Pareto distribution is F(z)=l- (1+3)-*. For notational convenience, it is often written as Because the limiting value of (1 + 7$)-’” is exp(-$) as y -+ 0, it is clear that Wo(x) is the exponential distribution function. When y (or equivalently a) is positive, the df W,,Q(X) has the form of a Pareto distribution. 7.8 THE FREQUENCY OF EXCEEDENCES 7.8.1 An important component in analyzing excesses (losses in excess of a threshold) is the change in the frequency distribution of the number of observations that exceed the threshold as the threshold is changed. When the threshold is increased, there will be fewer exceedences per time period; whereas if the threshold is lowered, there will be more exceedences. Let Xj denote the severity random variable representing the “gr~und-up”~ loss on the jth loss with common df F(x). Let NL denote the number of ground-up losses. We make the usual assumptions that the Xjs are mutually independent and independent of NL. Now consider a threshold d such that F(d) = 1 - F (d) = Pr(X > d), the survival function, is the probability that a loss will exceed the threshold. Next, define the indicator random variable Ij by Ij = 1 if the jth loss results in an exceedence and Ij = 0 otherwise. Then Ij has a Bernoulli distribution with parameter F(d) and the pgf of Ij is PI^ (2) = 1 - F(d) + F(d)z. From a fixed number of losses “The term “ground-up’’ is a term that comes from insurance. Often there is a deductible amount so that the insurer pays less than the full loss to the insured. A ground-up loss is the full loss to the insured, not the (smaller) loss to the insurer. In the operational risk context, ground-up losses are measured from zero and are not the losses measured from the threshold. 222 EXTREME VALUE THEORY: THE STUDY OF JUMBO LOSSES If there are a fixed number n of ground-up losses, NE = I1 + . . . + I, represents the number of exceedences. If I1,12,. . . are mutually independent, then NE has a binomial distribution with pgf PNE(Z) = [PI2 (2))" = [l + F(d)(z - l)]" Thus the binomial distribution with parameters n and F(d) represents the number of exceedences. This concept is similarly extended to the number of exceedences above some threshold dz, when the number of exceedences above a lower threshold dl is known and denoted by 7~1. In this case, the number of exceedences NE has a binomial distribution with parameters n1 and F(d2) /F(dl). It is often argued that the number of very rare events in a fixed time period follows a Poisson distribution. When the threshold is very high the probability of exceeding that threshold is very small. When also the number of ground- up losses is large the Poisson distribution serves as an approximation to the binomial distribution of the number of exceedences. This can be argued as follows: PNE(Z) = [PIj(z)]" = [1 +'S(d)(z- I)]" f exp (A (z - 1)) where X = nF (d) as n -+ a. Thus, asymptotically, the number of exceedences follows a Poisson distribution. 7.8.2 From a random number of losses In practice, the number of losses is unknown in advance. In this case, the number of exceedences over the threshold d is random. If there is a random number of exceedences, NE = I1 + . . . + INL represents the number of excee- dences. If Il,IZ,. . . are mutually independent and are also independent of NL, then NE has a compound distribution with NL as the primary distribution and a Bernoulli secondary distribution. Thus PNE(z) = PNL PI^ (211 = PNL [I + F(d)(z - I)]. In the important special case in which the distribution of NL depends on a parameter 6 such that PNL (z) = PNL (2; 6) = B[O. (2 - I)], where B(z) is functionally independent of 0 (as in Theorem 5.11), then PNE (2) = BIB. (1 - F(d) + F(d)z - l)] = B[F(d) .6. (2 - l)] = PNL (2; F(d)6). [...]... upper right corner, there is no evidence of dependence The corresponding Clayton copula pdf is shown in Figure 8.2 ARCHIMEDEAN COPULAS , > 0 02 04 06 U 08 1 U Fig 8.1 Clayton copula density (6 = 3) Fig 8.2 Clayton copula pdf (6 = 3) 243 244 MULTIVARIATE MODELS 1 1 08 06 06 06 , > 04 04 02 02 0 0 02 04 U 06 08 1 0 02 04 06 08 1 U fig 8.3 Gumbel copula density (0 = 3) Gumbel-Hougaard copula The Gumbel-Hougaard... - u) U , -1 c 02 ' 0 02 04 02 J U 06 04 OB 1 '0 02 04 U 06 08 1 fig 8.7 Ali-Mikhail-Haq copula density (6 = 0.8) Hence, the AMH copula is the form The AMH copula is also tuned through a single parameter 0 The measure of association,... focused on the modeling of specific risk types using univariate distributions This chapter will focus on addressing the issue of possible dependencies between risks The concern in building capital models for operational risk is that it may not be appropriate t o assume that risks are mutually independent In the case of independence the univariate probability (density) functions for each risk can be multiplied... threshold d is the Value-at -Risk xP = Va%(X), then we can write the Tail-Value-at -Risk as 8 Y + -+ 1-7 1-7 TVaR,(X) = zP XP 230 EXTREME VALUE THEORY: THE STUDY OF JUMBO LOSSES If the threshold d is less than the Value-at -Risk, xp = VaRp(X), then from formula (7 .6) , we can write the tail probability a s From this the quantile, xp =VaEtp(X), can be obtained as and the Tail-Value-at -Risk follows as TVaR,(X)... It should also be noted that there is no upper tail dependence when 6 = 1, and that tail dependence approaches 1 as 6 becomes large Frank copula The Frank copula [39] has generator ARCHlMEDEAN COPULAS 245 Fig 8.4 Gumbel copula pdf (6 = 3) Hence, the Frank copula is the form The Frank copula is also tuned through a single parameter 6 The measure of association, Kendall’s tau, for the Frank copula is... t-io 6 t? The lack of tail dependence can be seen from both Figures 8.5 and 8 .6, where there is no concentration of points in the upper right and lower left corners Frees, Carriere, and Valdez [40] use Frank’s copula for a study of joint lifetimes Ali-Mikhail-Haq copula The Ali-Mikhail-Haq (AMH) copula [4] has generator d ( u ) = In 1 - 6( 1 - u) U , -1 . /3 = 1.953125(1.5 36) = 3. Also, 0.4595 - 2.536C2 + 4-2 - 0.4595(4)-2 pE* = = 0.4 1 - 2.5 36- 2 as expected. For the second case, M* - 0.002 - 2.5 36- 2 + 4C2 - 0.002(4)-2. distribution given, the new parameters are r* = 2, p* : 3(0.512) = 1.5 36, and M* - 0.4 - 4-2 + 2.5 36- 2 - 0.4(2.5 36) -2 = 0.4595. 1 - 4-2 Po - 0 If we have values of the amounts. the Value-at -Risk, xp = VaRp(X), then from formula (7 .6) , we can write the tail probability as From this the quantile, xp =VaEtp(X), can be obtained as and the Tail-Value-at -Risk follows