Quant Risks - Loss Tail Analysis II - EVT

1 Extreme Value Thoery (EVT)

A beanch in pobability and statistic that examines the tail distribution of a distribution
This allows us to study the likelihood and magnitude of rare but catastrophic event (Black Swan event)

Two major methods in EVT

• Block maxima/minima
• Exceedance

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2 Block maxima/minima

Create blocks of loss (usually by time) and study the maximum / minimum of the Loss for the blocks

2.1 Maxima / minima

let $X_1, X_2, …, X_n$ be i.i.d RV and PDF is given by $f_X(x)$ and CDF is given by $F_X(x)$

Sample Maxima:

\[M_n = \max(\{X_i, i\in[1,n]\})\]

Sample Minima:

\[m_n = \min(\{X_i, i\in[1,n]\})\]

Then

CDF of $M_n$

\[\begin{align} F_{M_n} &= \mathbb{P}(M_n <= x)=\prod_{i=1}^n{\mathbb{P}(X_i <= x)}\\ &= F_X(x)^n \quad \square\\ \end{align}\]

CDF of $m_n$

\[\begin{align} F_{m_n} &= \mathbb{P}(m_n <= x)= 1- \mathbb{P}(m_n > x) \\ &=1-\prod_{i=1}^n{\mathbb{P}(X_i > x)} \\ &= 1-S_X(x)^n \quad \square\\ \end{align}\]

2.2 Limiting Distribution

If we simply take limit $\lim_{n\to\infty} M_n$

\[\lim_{n\to\infty} F_{M_n}(X) = \lim_{n\to\infty} F_X(x)^n\begin{cases} 0 & \text{if }F_X(x)<1\\ 1 & \text{if }F_X(x)=1\\ \end{cases}\]

This is not very meaningful.

• We will find some sequence $c_n, d_n$ such that

  \[\lim_{n\to\infty} \mathbb{P}(\frac{M_n-d_n}{c_n}<=x)=\lim_{n\to\infty} F_X(c_nx+d_n)^n=H(x)\]

  where $H(x)$ is non-degenerate probability

• $F_X(x) \in MDA(H)$ if $c_n, d_n$ exists

2.2.1 Examples

2.2.1.1 Expotential Distribution

2.2.1.2 Uniform Distribution

2.2.1.3 Pareto Distribution

2.2.2 GEV Distribution

Fisher-Tippett Theorem:

If $F_X \in MDA(H)$ then the limit distribution $H(x)$ convergence to 3 types

Type I: Frechet Distribution $\Phi_\alpha(x) = \exp(-x^{-\alpha})$ for $x > 0$

Type II: Weibull Distribution $\Psi_\alpha(x)= \exp(-(-x)^\alpha)$ for $x<=0$

Type III: Gumbel Distribution $\Lambda_\alpha(x)=\exp(e^{-x})$ for $x\in\mathbb{R}$

Choosing different $c_n, d_n$ will not change the type limit distribution, but only an affline transform of the function

\[V(x) = U(Ax+B)\]

where

\[A=\lim_{n\to\infty}{\frac{c_v}{c_u}}\]

and

\[B=\lim_{n\to\infty}{\frac{d_v-d_u}{c_u}}\]

The 3 types can be generalized into Generalized Extreme Value Distribution (GEV)

\[H_\xi(x) = \begin{cases} \exp(-(1+\xi x)^{-1/\xi}) & \text{if } \xi \ne 0 \\ \exp(-e^{-x}) & \text{if } \xi = 0 \\ \end{cases}\]

noted: the limit of is $\xi\to 0$ converges to the 2nd case, so this is continuous

2.2.2.1 Model fitting of GEV distribution

• To fit the GEV distribution, we need to fit 3 params
• $H_{\mu,\sigma,\xi}(x) = H_\xi(\frac{x-\mu}{\sigma})$
• This can be done by Maximum Log-Likelihood Estimateion

2.2.2.2 Using GEV to find VaR and ES

• If we set a fixed Block size, after fitting the GEV $H_{\mu, \sigma, \xi}(x)$

• This is an approximation of CDF of $M_n$

  \[F_{M_n} \approx H_{\mu, \sigma, \xi}\]

• VaR of $M_n$

  \[\begin{align} \text{Let }v &=\operatorname{VaR}_\alpha(M_n) \\ F_{M_n}(v) &\approx H_{\mu, \sigma, \xi}(v)=\alpha \\ H_\xi(\frac{v-\mu}{\sigma}) &= \alpha \\ \operatorname{VaR}_\alpha(M_n) = v &= \mu + \sigma H_\xi^{-1} (\alpha) \quad &\square \end{align}\]
  • For ES it is hard to have explicit formula
    • Usually use MC to calculate.
    • Calculate the VaR
    • Use GEV to get a large sample $Y_i$
    • Find the mean of $Y_i > VaR$

2.3 Problem of Block Maxima/Minima method

• We need to choose a proper block size
• We need a large block size so that $M_n$ is actually the tail loss, and to ensure $M_n$ has iid.
• We need also large number of blocks in order to estimate the params of GEV via MLE

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3 Threshold Exceedance method

Data efficient way to model the tail distribution by keeping all tail data

3.1 Threshold

Set $u > 0$ be a constant threshold. We keep data with $X_i > u$

When $u$ is large, these sample are big losses

We want to estimate the tail distribution

\[\begin{align} F_u(x) &= \mathbb{P}(X-u <= x | X > u)\\ &= \frac{\mathbb{P}(u<X<=u+x)}{\mathbb{P}(X>u)} \\ &= \frac{F_X(u+x)-F_X(u)}{1-F_X(u)} \quad \square \end{align}\]

3.2 Generalized Pareto Distribution (GPD)

The distribution of $F_u(x)$ is model by GPD

\[\begin{align} G_{\xi,\beta}(x) = \begin{cases} 1-(1+\frac{\xi x}{\beta})^{-1/\xi} & \text{if } \xi \ne 0 \\ 1-e^{-x/\xi} & \text{if } \xi = 0 \\ \end{cases} \end{align}\]

if $F_X \in MDA(H)$, then $F_u(x)$ can also be approximate by GPD (Pickands-Balkema-deHann theorem)

3.3 Mean excess function

After model the excess loss $X-u$ by GPD, we want to estimate the mean excess loss which is similar to Expected Shortfall

\[\begin{align} e(u) &= \mathbb{E}[X-u|X>u] \\ \end{align}\]

By direct integration with the PDF of $G_{\xi,\beta}$ the integral converge to the same function (converge only when $\xi<1$)

\[e(u)= \frac{\beta}{1-\xi} \text{ for } \xi < 1\]

3.3.1 Relationship of to other threshold

We can get the mean excessed function of higher threshold $v>u$

\[\begin{align} \beta_v &=\beta_u+ \xi(v-u) \quad &\square \\ F_v(x) &= G_{\xi,\beta_u + \xi(v-u)}(x) \quad &\square\\ e(v) &= \frac{\beta_u+\xi(v-u)}{1-\xi} \quad &\square\\ \end{align}\]

this only works for $v>u$

3.4 Calculating VaR and ES with GPD

3.4.1 VaR

\[\operatorname{VaR}_\alpha(X)=v=((\frac{p}{1-\alpha})^\xi-1)\frac{\beta}{\xi}+u \quad \square\]

Proof:

\[\begin{align} \text{let } p = \mathbb{P}(X>u) \\ \text{let } v = \operatorname{VaR_\alpha(X)} \\ \\ F_u(x) &= \mathbb{P}(X < x+u | X>u) \\ &= \mathbb{P}(u<X<x+u) / \mathbb{P}(X>u) \\ \mathbb{P}(u<X<x+u) &= F_u(x) p \\ F_X(x+u)-F_X(u) &= F_u(x) p \\ F_X(x) &= F_u(x-u) p +F_X(u) \quad &\square \\ \\ \text{sub } v = x \\ F_X(v) &= F_u(v-u)p + (1-p) \\ \alpha &= F_u(v-u)p + (1-p) \\ F_u(v-u) &=\frac{\alpha-1}{p}+1 \\ G_{\xi,\beta}(v-u) &=1 - \frac{1-\alpha}{p} \\ (1+\frac{\xi(v-u)}{\beta})^{-1/\xi} &= \frac{1-\alpha}{p} \\ v&=((\frac{1-\alpha}{p})^{-\xi}-1)\frac{\beta}{\xi}+u \\ \operatorname{VaR}_\alpha(X)=v&=((\frac{p}{1-\alpha})^\xi-1)\frac{\beta}{\xi}+u \quad &\square \end{align}\]

3.4.2 ES

\[\operatorname{ES}_\alpha(X)=\frac{\beta+v-\xi u}{1-\xi} \quad \square\]

To find ES, we can do

1. direct integration of ES using alternative formula
2. Using Mean excess function of u

3.4.2.1 ES by alternative formula

\(\begin{align} \operatorname{ES}_\alpha(X) &= \frac{1}{1-\alpha}\int_\alpha^1{\operatorname{VaR}_x(X)}{dx} \\ &= \frac{1}{1-\alpha}\int_\alpha^1{((\frac{p}{1-x})^\xi-1)\frac{\beta}{\xi}+u}{dx} \\ &= \frac{1}{1-\alpha}(u-\frac{\beta}{\xi})(1-\alpha)+ \frac{1}{1-\alpha}\frac{\beta}{\xi}p^\xi \cdot \int_\alpha^1(1-x)^{-\xi} dx \\ \quad \text{let x'=1-x, dx=-dx', bounds=(}1-\alpha, 1-1=0\text{)}\\ &=(u-\frac{\beta}{\xi})+\frac{1}{1-\alpha}\frac{\beta}{\xi}p^\xi \cdot \int_0^{1-\alpha}(x')^{-\xi} dx' \\ &=(u-\frac{\beta}{\xi})+\frac{1}{1-\alpha}\frac{\beta}{\xi}p^\xi \cdot [\frac{1}{-\xi+1}(1-\alpha)^{-\xi+1}] \\ &=(u-\frac{\beta}{\xi})+\frac{\beta}{\xi}(\frac{p}{(1-\alpha)})^\xi \cdot \frac{1}{1-\xi} \\ &=\frac{1}{1-\xi}[(u-\frac{\beta}{\xi})+\frac{\beta}{\xi}(\frac{p}{(1-\alpha)})^\xi -\xi(u-\frac{\beta}{\xi}) ] \\ &=\frac{1}{1-\xi}[v -\xi u + \beta ] \\ &=\frac{\beta+v-\xi u}{1-\xi} \quad &\square \end{align}\)

3.4.2.2 ES by Mean excess function of u

\[\begin{align} \operatorname{ES}_\alpha(X) &= \mathbb{E}[X|X>v] \quad\text{ given v>u } \\ &=\mathbb{E}[X-v|X>v] + v \\ &= e_u(v) +v \\ &= \frac{\beta+\xi(v-u)}{1-\xi} + v \\ &= \frac{\beta+v-\xi u}{1-\xi} \quad &\square\\ \end{align}\]

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