Math 576: Quantitative Risk Management

Math 576: Quantitative Risk Management Haijun Li lih@math.wsu.edu Department of Mathematics Washington State University Week 11 Haijun Li Math 576: Quantitative Risk Management Week 11 1 / 21

Outline 1 Distributions Beyond High Thresholds 2 Tails of Specific Models Haijun Li Math 576: Quantitative Risk Management Week 11 2 / 21

Analyzing Extremes in Terms of Tail Property Let X, X 1,..., X n be iid from a distribution F, and M n = max{x 1,..., X n }. X F MDA(H) lim n P(M n c n + d n x) = lim n F n (c n + d n x) = H(x). Haijun Li Math 576: Quantitative Risk Management Week 11 3 / 21

Analyzing Extremes in Terms of Tail Property Let X, X 1,..., X n be iid from a distribution F, and M n = max{x 1,..., X n }. X F MDA(H) Rewrite: lim n P(M n c n + d n x) = lim n F n (c n + d n x) = H(x). log H(x) {}}{ ( n[1 F(c n + d n x)] ) n lim 1 = e log H(x) = H(x) n n is equivalent to n[1 F(c n + d n x)] log H(x), as n or for high linear threshold u n (x) = c n + d n x, ( ) np X > u n (x) log H(x), as n. Haijun Li Math 576: Quantitative Risk Management Week 11 3 / 21

Remark Let X, X 1, X 2,..., X n,... are stationary (not necessarily independent) with marginal distribution X F, and M n = max{x 1,..., X n }, n 1. Then, under some mild conditions (Leadbetter, 1983), ( ) np X > c n + d n x log H(x), as n implies that lim P(M n c n + d n x) = H θ (x), n where 0 θ 1 is known as the extremal index that measures series dependence among X, X 1, X 2,..., X n. Haijun Li Math 576: Quantitative Risk Management Week 11 4 / 21

Distributions Beyond High Thresholds The distribution of extremes = the tail distribution of random samples X i s exceeding over a sufficiently large threshold. Consider a random variable X with df F MDA(H( ; γ)) where γ > 0. High threshold u n := c n + d n x for x > 0, as n. We are interested in the tail probability: P(X t > x X > t) = P(X > x + t) P(X > t) = 1 F(x + t) 1 F (t) where t > c n + d n x for sufficiently large n. The random variable [X t X > t] is called the remaining life in reliability theory and survival analysis. The distribution F(x + t) F(t) F t (x) := P(X t x X > t) =, x, t R + 1 F(t) is called the tail conditional distribution. Haijun Li Math 576: Quantitative Risk Management Week 11 5 / 21

Generalized Pareto Distribution (GPD) For sufficiently large n, 1 F(u n ) 1 H 1/n ((u n c n )/d n ; γ) log H 1/n ((u n c n )/d n ; γ) = n 1 (1 + γ(u n c n )/d n ) 1/γ +. For a large threshold t, let n be sufficiently large, then P(X t > x X > t) (1 + γ(x + t c n)/d n ) 1/γ + (1 + γ(t c n )/d n ) 1/γ + = (1 + γx/dn) 1/γ + where d n = d n + γ(t c n ). That is, P(X t x X > t) 1 (1 + γx/d n) 1/γ +, n = sufficiently large Haijun Li Math 576: Quantitative Risk Management Week 11 6 / 21

A Deeper Result Pickands-Balkema-de Haan Theorem Let F t (x) := (F(x + t) F(t))/(1 F(t)) be the tail conditional distribution of a df F. Then F MDA(H( ; γ)) if and only if there is a positive, measurable function δ(t) such that lim t x F sup F t (x) G(x; γ, δ(t)) = 0 0 x<x F t where G( ; γ, δ(t)) is the GPD defined by { 1 (1 + γx/δ) 1/γ G(x; γ, δ) := 1 + log H(x; γ/δ) = + if γ 0 1 exp{ x/δ} if γ = 0. Haijun Li Math 576: Quantitative Risk Management Week 11 7 / 21

Example: Asymptotic Analysis of Tail Risk Let X > 0 be a loss variable having df F MDA(H( ; γ)). For sufficiently large t, F t (x) = P(X t > x X > t) (1 + γx/δ(t)) 1/γ + uniformly for all 0 < x <. Consider, for all x t and sufficiently large t, P(X > x) = P(X > t)p(x t > x t X > t) = F(t)F t (x t) F(t)(1 + γ(x t)/δ(t)) 1/γ + uniformly. Set P(X > x) = 1 α, we have x = VaR α (X). Thus for sufficiently large t, ( (1 VaR α (X) t + δ(t) ) α γ 1), γ F(t) uniformly for all α in a left neighborhood of 1. Haijun Li Math 576: Quantitative Risk Management Week 11 8 / 21

Example (cont d): Asymptotic Analysis of Tail Risk If 0 < γ < 1, then ES (expected shortfall) of loss X can be estimated as follows, for sufficiently large t, ES α (X) = 1 1 α 1 α VaR u (X)du VaR α(x) 1 γ which is asymptotically increasing in VaR α (X) as α 1. Consider the difference of two tail risk measures: + δ(t) γt 1 γ, mean excess loss = ES α (X) VaR α (X) ( = E X VaR α (X) ) X > VaR α (X) } {{ } mean remaining lifetime Note that the mean excess loss ES α (X) VaR α (X) is also increasing in VaR α (X), as α 1, with asymptotic slope γ/(1 γ). Haijun Li Math 576: Quantitative Risk Management Week 11 9 / 21

Domain of Attraction of Fréchet Distribution Fréchet distribution: H + (x; θ) = exp{ x 1/γ }, x > 0, γ > 0. Rewrite the tail probability: H + (x; γ) := 1 H + (x; γ) = 1 exp{ x 1/γ } } x 1/γ x 1/γ, x > 0. {{} slowly varying Fréchet survival distribution H + (x; γ) = L(x)x 1/γ is regularly varying. Haijun Li Math 576: Quantitative Risk Management Week 11 10 / 21

Domain of Attraction of Fréchet Distribution Fréchet distribution: H + (x; θ) = exp{ x 1/γ }, x > 0, γ > 0. Rewrite the tail probability: H + (x; γ) := 1 H + (x; γ) = 1 exp{ x 1/γ } } x 1/γ x 1/γ, x > 0. {{} slowly varying Fréchet survival distribution H + (x; γ) = L(x)x 1/γ is regularly varying. Gnedenko-de Haan Theorem: Fréchet MDA Case F MDA(H( ; γ)) with γ > 0 if and only if F(x) = 1 F(x) is regularly varying with heavy tail index 1/γ. Haijun Li Math 576: Quantitative Risk Management Week 11 10 / 21

Example: Pareto Distribution Consider random samples X 1,..., X n from the Pareto distribution F(x) = 1 x 1/γ, x > 1, γ > 0, which is in the domain of attraction of Fréchet distribution. Set normalizing constants c n = 0 and d n = F 1 (1 n 1 ) = n γ, and clearly, for x > 0, ( ) lim P Mn c n x = lim F n (c n + d n x) n n d n = lim n (1 x 1/γ /n) n = exp{ x 1/γ }. Haijun Li Math 576: Quantitative Risk Management Week 11 11 / 21

Example: t-distribution The t-pdf: f (x) = of ν 1 ν 3 Γ(ν/2) (ν 1)πΓ((ν 1)/2) 1 [x 2 /(ν 1)+1] ν/2, with a variance for ν > 3. The t distribution has heavier tails than the normal distribution. http://upload.wikimedia.org/wikipedia/commons/f/f4/t_distribution_1df.png The t-distribution is in the domain of attraction of Fréchet distribution with heavy tail index ν 1. T_distribution_1df.png (PNG Image, 480x480 pixels) Haijun Li Math 576: Quantitative Risk Management Week 11 12 / 21

Example: t-distributions With Various d.f. As ν goes to, the t-distribution becomes close to the standard normal distribution. Figure : Left = 2 d.f. Right = 10 d.f. Haijun Li Math 576: Quantitative Risk Management Week 11 13 / 21

Gamma Distribution A random variable X Ga(k, λ) is said to have a gamma distribution if its pdf f (x) = λk x k 1 e λx, x > 0, where Γ(k) = Γ(k) 0 x k 1 e x dx, If k is a positive integer, then the gamma function Γ(k) = (k 1)!. Widely used in reliability engineering and telecommunication. E(X) = k λ, V (X) = k λ 2. k = shape parameter, λ = rate parameter. Haijun Li Math 576: Quantitative Risk Management Week 11 14 / 21

k = shape parameter, λ = rate parameter, θ = λ 1 = scale parameter. A special case: When k = n (integer), the distribution is called an Erlang distribution (widely used in telecommunication). Haijun Li Math 576: Quantitative Risk Management Week 11 15 / 21

Example: Inverse Gamma Distribution If Y Ga(k, λ), then X := 1/Y Ig(k, λ) is said to have an inverse gamma distribution. The inverse gamma density: f (x) = λk x (k+1) e λ/x, x > 0, where Γ(k) = Γ(k) 0 x k 1 e x dx, The inverse gamma distribution is in the domain of attraction of Fréchet distribution with heavy tail index k. Haijun Li Math 576: Quantitative Risk Management Week 11 16 / 21

Domain of Attraction of Gumbel Distribution Gumbel or double-exponential distribution: H 0 (x) = exp{ e x }, x R. For sufficiently large x, the tail probability: H 0 (x) := 1 H 0 (x) = 1 exp{ e x } e x, decays exponentially. Gnedenko-de Haan Theorem: Gumbel Case F MDA(H 0 ( )) if and only if x F t 0 (1 F (x))dx < for some t 0 < x F and 1 F(t + xr(t)) lim = exp{ x}, x R t x F 1 F(t) where R(t) := x F t (1 F(x))dx/(1 F(t)), t < x F and c n = F (1 n 1 ) and d n = R(c n ). Haijun Li Math 576: Quantitative Risk Management Week 11 17 / 21

von Mises Distribution Let a(t) denote a positive and differentiable function with lim t a (t) = 0. Then the survival distribution { x } 1 H(x) = 1 H(x) = c exp z a(t) dt, z < x <, is called a von Mises distribution. The standard normal distribution Φ(z) is a von Mises distribution with a(t) = a t 1. Let F be twice differentiable on (z, ) with density f (x) = F (x) and F (x) < 0 on (z, ). Then F is a von Mises distribution if and only if F(x)F (x) lim x f 2 = 1. (x) Haijun Li Math 576: Quantitative Risk Management Week 11 18 / 21

Domain of Attraction of Gumbel Distribution, Again Gumbel or double-exponential distribution: H 0 (x) = exp{ e x }, x R. Haijun Li Math 576: Quantitative Risk Management Week 11 19 / 21

Domain of Attraction of Gumbel Distribution, Again Gumbel or double-exponential distribution: H 0 (x) = exp{ e x }, x R. Characterization Theorem F MDA(H 0 ( )) if and only if 1 F is a von Mises distribution, or 2 F is tail equivalent to a von Mises distribution G; that is, 1 F (x) lim = c, c > 0. x 1 G(x) Haijun Li Math 576: Quantitative Risk Management Week 11 19 / 21

Example: Gamma Distribution A random variable X Ga(k, λ) is said to have a gamma distribution if its pdf f (x) = λk x k 1 e λx, x > 0, where Γ(k) = Γ(k) Verify that F (x) = f (x) = f (x)(λ + (1 k)/x) < 0. Verify that 0 x k 1 e x dx. F (x) F(x) lim = λ, lim x f (x) x f (x) = lim f (x) x f (x) = 1/λ. The gamma distribution is in the domain of attraction of Gumbel distribution. Haijun Li Math 576: Quantitative Risk Management Week 11 20 / 21

Tails of Mixture Distributions Let R > 0 and X be independent. The scale mixture Y := RX has the survival function: P(Y > y) = 0 P(X > y/r)g(r)dr, where g(r) is the density of mixing variable R. The t distribution is the distribution of the following scale mixture: Y = Z, where Z N(0, 1), W χ 2 ν. W ν Breiman s Theorem (Leo Breiman, 1965) Let Y = RX where R > 0 and X are independent and E(R α+ɛ ) < for some ɛ > 0. If X is regularly varying with heavy tail index α > 0, then Y is also regularly varying with heavy tail index α, and P(Y > y) E(R α )P(X > y), y. Haijun Li Math 576: Quantitative Risk Management Week 11 21 / 21