Quant Risks - Loss Tail Analysis III - Multivariant dependence

1 Recall Probability

1.1 Single-variant

\[\begin{align} \mathbb{E}[A{X}+{b}] &= A\mathbb{E}[{X}] + {b} \\ \operatorname{COV}[A,B] &= \mathbb{E}[(A-\mathbb{E}[A])(B-\mathbb{E}[B])] \\ \operatorname{COV}[A+x,B+x] &= \operatorname{COV}[A,B] \\ \operatorname{COV}[cA,dB] &= cd \cdot \operatorname{COV}[A,B] \\ \operatorname{COV}[A+C,B] &= \operatorname{COV}[A,B] + \operatorname{COV}[C,B] \\ \end{align}\]

1.1.1 Normal distribution

Normal distribution ($X \sim N(\mu, \sigma)$)

\[\begin{align}X=\mu+\sigma Z\end{align}\] \[\begin{align} f_Z(x) = \phi(x) &= \frac{1}{\sqrt{2 \pi}}\exp(-\frac{x^2}{2}) \\ F_Z(x) &= N(x) \\ f_X(x) &= \frac{1}{\sqrt{2 \pi}\sigma}\exp(-\frac{(x-\mu)^2}{2\sigma^2}) \\ F_X(x) &= N(\frac{x-\mu}{\sigma}) \\ \mathbb{E}[X] &= \mu \\ \operatorname{VAR}[X] &= \sigma^2 \\ \end{align}\]

1.1.2 Student’s-t distribution

T distribution ($X \sim T_v(\mu, \lambda)$) is a bell-shape but heavy tail distribution

\[\begin{align} X &=\mu+\lambda T_v \\ T_v &= \frac{Z}{\sqrt{\chi^2_v/v}} \quad \text{, where } V\sim\chi^2(v) \\ \end{align}\] \[\begin{align} f_{T_v} &= ... \\ F_{T_v} &= ... \\ F_X(x) &= F_{T_v}(\frac{x-\mu}{\lambda}) \\ \mathbb{E}[X] &= \mu \\ \operatorname{VAR}[X] &= \lambda^2 \frac{v}{v-2} \\ \end{align}\]

1.2 Multi-variant

\[\begin{align} \vec{X} &= \begin{bmatrix} X_1 \\ X_2 \\ X_3 \end{bmatrix} \\ F_\vec{X}(x_1, x_2, x_3) &= \mathbb{P}(X_1<x_1, X_2<x_2,X_3<x_3) \\ \mathbb{E}[\vec{X}] &= \begin{bmatrix} \mathbb{E}[X_1] \\ \mathbb{E}[X_2] \\ \mathbb{E}[X_3] \end{bmatrix} \\ \mathbb{E}[A\vec{X}+\vec{b}] &= A\mathbb{E}[\vec{X}] + \vec{b} \\ COV[\vec{X}] = \Sigma &= \begin{bmatrix} COV(X_1, X_1) \quad COV(X_1, X_2) \quad COV(X_1, X_3) \\ COV(X_2, X_1) \quad COV(X_2, X_2) \quad COV(X_2, X_3) \\ COV(X_3, X_1) \quad COV(X_3, X_2) \quad COV(X_3, X_3) \end{bmatrix} \\ COV[A\vec{X}+\vec{b}] &= ACOV[\vec{X}]A^T \\ \end{align}\]

Independence: $COV(X_1, X_2) = 0 $

1.2.1 Normal distribution (Multi-variant)

Multi-variant Normal Distribution ($\vec{X}\sim N(\vec{\mu}, \Sigma$)

\[\begin{align} \vec{X}&=\vec{\mu}+A\vec{Z} \quad \text{ , where } \vec{Z} \sim N(0,I_n) \text{ and } AA^T=\Sigma \\ \mathbb{E}[\vec{X}] &= \vec{\mu} \\ \operatorname{COV[\vec{X}]} &= \Sigma \end{align}\]

If $Y = \vec{w}^T\vec{X}$,

\[\begin{align} Y &\sim N(\vec{w}^T\vec{\mu}, \vec{w}^T\Sigma\vec{w}) \\ Y &= (\vec{w}^T\vec{\mu}) + (\vec{w}^T\Sigma\vec{w})Z \end{align}\]

1.2.2 Student’s-t distribution (Multi-variant)

Multi-variant T Distribution ($\vec{X} \sim T_v({\vec{\mu}, \Lambda})$)

\[\begin{align} \vec{X} &= \vec{\mu}+\sqrt\frac{v}{\chi^2_v}(AZ) \quad \text{ , where } AZ \sim N(0,\Lambda) \text{ and } AA^T=\Lambda \\ \mathbb{E}[\vec{X}] &= \vec{\mu} \\ \operatorname{COV[\vec{X}]} &= \frac{v}{v-2}\Lambda \end{align}\]

If $Y = \vec{w}^T\vec{X}$,

\[\begin{align} Y &= \vec{w}^T\vec{\mu} + \sqrt\frac{v}{\chi^2_v} \vec{w}^T (AZ) \\ &= \vec{w}^T\vec{\mu} + \sqrt\frac{v}{\chi^2_v} (\vec{w}^T\Lambda\vec{w}) Z \\ &= \vec{w}^T\vec{\mu} + (\vec{w}^T\Lambda\vec{w}) T_v \\ Y &\sim T_v(\vec{w^T}\vec{\mu}, \vec{w^T}\Lambda\vec{w}) \end{align}\]

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2 Coupla - Joint dependencies

\[\begin{align} F_\vec{X}(x_1, x_2, x_3, ...) = \mathbb{P}(X_1<x_1, X_2<x_2, X_3<x_3) \\ \end{align}\]

Noted if $Y_i=F_{X_i}(X_i)$ it has uniform distribution to $[0,1]$, this gives us a “normalized” version to study the dependency without the mariginal distribution

Coupla:

\[\begin{align} C(u_1, u_2, u_3, ...) &= \mathbb{P}(F_{X_1}(X_1)<u_1,F_{X_2}(X_2)<u_2,F_{X_3}(X_3)<u_3,...) \\ &= \mathbb{P}(X_1<F_{X_1}^{-1}(u_1), X_2<F_{X_2}^{-1}(u_2), X_3<F_{X_3}^{-1}(u_3),...) \\ &= F_\vec{X}(F_{X_1}^{-1}(u_1), F_{X_2}^{-1}(u_2), F_{X_3}^{-1}(u_3),...) \quad\square \end{align}\]

Proof:

\[\begin{align} \mathbb{P}(Y<y) &= \mathbb{P}(F_{X}(X)<y) \\ &= \mathbb{P}(X < F_X^{-1}(y)) \\ &= F_X(F_X^{-1}(y)) \\ &= y \quad\text{ for } y\in[0,1] \end{align}\]

2.1 Property of Copula

1. $C(u_1, u_2, …, u_n)$ is the joint distribution on $\vec{U}$ where $U_i \in [0,1]$ is the CDF of $X_i$, and $U_i$ is uniform
2. $C(\vec{u}) =0 \text{ if } \exists c_i =0$
3. Marginal distribution: $\mathbb{P}(U_i<u_i)= C(1,1,1,u_i,1,1,1) = u_i$
4. $\mathbb{P}(a_1<=U_1<=b_1, a_2<=U_2<=b_2)= C(b_1, b_2) + C(a_1, a_2) - C(a_1, b_2) - C(b_1, a_2)$
5. Joint distribution = marginal + dependence, copula only study the dependence, because all marginal is standardized to uniform distribution
6. Let $\vec{Y}$, where $Y_i=h_i(X_i)$ and $h_i$ is stictly increasing function, then $\vec{X}$ and $\vec{Y}$ have the same copula.
   Need to be strictly increasing function, such that the ordering is preserved.

2.2 Common Copula

2.2.1 Independent

$C(u_1, u_2, …, u_n) = u_1u_2u_3…u_n$

2.2.2 Comonotonic (correlation coefficient=1)

$C(u_1, u_2) = max(0, u1+u2-1)$

2.2.3 Countermonotonic (correlation coefficient=-1)

$C(u_1, u_2, …, u_n) = max(u_1, u_2, …, u_n)$

2.2.4 Gaussian

For standardized $\vec{Y}$

\[\begin{align} Y_i~N(0, 1), \vec{Y}\sim N(0, \Sigma=\begin{bmatrix} 1 &\rho_{12} &\rho_{13} &\rho_{14} ...\\ \rho_{12} &1 &\rho_{23} &\rho_{24} ...\\ \rho_{13} &\rho_{23} & 1&\rho_{24} ...\\ ... &...&...& 1\\ \end{bmatrix}) \end{align}\] \[\begin{align} C(u_1, u_2, ..., u_n) &= \mathbb{P}(Y_1<F_{Y_1}^{-1}(u_1), Y_2<F_{Y_2}^{-1}(u_2), ...) \\ &= \mathbb{P}(Z_i<F_{Y_i}^{-1}(u_i), ...) \\ &= \mathbb{P}(Z_i<N^{-1}(u_i), ...) \\ &= N_n(N^{-1}(u_i), ...) \quad \text{ where } N_n \text{ is n-dim CDF of normal} \end{align}\]

For $X_i~N(\mu_i, \sigma_i)$, $\vec{X}\sim N(\vec{\mu}, \Sigma)$

we can define

\[Y_i = \frac{X_i-\mu_i}{\sigma_i}= h_i(X_i)\]

then

\[\begin{align} C(u_1, u_2, ..., u_n) &= N_n(N^{-1}(u_i), ...) \quad\text{ where } N_n \text{ is n-dim CDF of normal} \end{align}\]

2.2.5 T-copula

Similar to Normal Copula

\[\begin{align} Y_i~T_v(0, 1), \vec{Y}\sim T_v(0, \Lambda=\begin{bmatrix} 1 &\rho_{12} &\rho_{13} &\rho_{14} ...\\ \rho_{12} &1 &\rho_{23} &\rho_{24} ...\\ \rho_{13} &\rho_{23} & 1&\rho_{24} ...\\ ... &...&...& 1\\ \end{bmatrix}) \end{align}\] \[\begin{align} C(u_1, u_2, ..., u_n) &= F_\vec{Y}(F_{T_v}^{-1}(u_i), ...) \quad\text{ where } N_n \text{ is n-dim CDF of normal} \end{align}\]

2.2.6 Archimedean Copula

Archimedean Copula is a class of copula for bivariate RV, usful for a lot of financial problem.

We first define a generator function $\phi(t)$. It must fulfill

1. stictly decreasing
2. convex
3. continuous
4. $\phi(0)=1$ and $\lim_{t\to\infty} \phi(t)=0$

Then

\[C(u_1, u_2)=\phi(\phi^-1(u_1)+\phi^-1(u_2)\]

A summary table of different common Archimedean Copula

Copula	$\phi(t)$	$C(u_1,u_2)$	Property	Example situation
Gumbel	\(\phi(t)=e^{-t^{1/\theta}},\ \theta\ge 1\)	\(C(u_1,u_2)=\exp\!\left(-\left((-\ln u_1)^\theta+(-\ln u_2)^\theta\right)^{1/\theta}\right)\)	Upper tail dependence; $\lambda_U=2-2^{1/\theta}$, $\lambda_L=0$	Insurance losses caused by natural disasters; claim frequency and claim severity can be jointly large
Clayton	\(\phi(t)=(1+\theta t)^{-1/\theta},\ \theta\ge 0\)	\(C(u_1,u_2)=\left(u_1^{-\theta}+u_2^{-\theta}-1\right)^{-1/\theta}\)	Lower tail dependence; $\lambda_U=0$, $\lambda_L=2^{-1/\theta}$	Stock log-returns with strong co-movement in market downturns
Generalized Clayton	\(\phi(t)=(1+\theta t^{1/\delta})^{-1/\theta},\ \theta\ge 0,\ \delta\ge 0\)	\(C(u_1,u_2)=\left(1+\left((u_1^{-\theta}-1)^\delta+(u_2^{-\theta}-1)^\delta\right)^{1/\delta}\right)^{-1/\theta}\)	Both upper and lower tail dependence; $\lambda_U=2-2^{1/\delta}$, $\lambda_L=2^{-1/(\theta\delta)}$	Asset returns where both crash risk and boom risk may occur together
Frank	\(\phi(t)=-\frac{1}{\theta}\ln\!\left(1+(e^{-\theta}-1)e^{-t}\right),\ \theta\in\mathbb{R}\)	\(C(u_1,u_2)=-\frac{1}{\theta}\ln\!\left(1+\frac{(e^{-\theta u_1}-1)(e^{-\theta u_2}-1)}{e^{-\theta}-1}\right)\)	No tail dependence; $\lambda_U=0$, $\lambda_L=0$	Variables with general dependence but without strong tail co-movement

2.2.7 Bounds

For any copula,

• Comonontoic is always the upper bound
• Countermonotonic is always the lower bound

2.3 Sklar Theorem

Relationship of joint distribution and copula

Joint distribution is marginal distribution + copula

\[\begin{align} F_\vec{X}(x_1, x_2,...)=C(F_{X_1}(x_1), F_{X_2}(x_2), ...) \end{align}\]

To use Sklar theorem on a set of RV $X_i$ to find the joint distribution:

1. For each ${X_i}$, we use MLE to fit a mariginal distribution $F_{X_i}$ using empirical data on $X_i$
2. Assume a copula that capture the depencies $C(\vec{U})$
3. Obtain Joint distribution by Sklar theorem

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3 Dependence Measure

Copula is a functional form, hard to measure.

We want a scalar (similar to correlation coefficient) to easy measure the dependence

Two common indicators:

1. Rank Correlation (Kendall’s Tau, Spearman’s Rho)
2. Coefficient of Tail Dependence

3.1 Kendall’s Tau

This is measuring the direction of the movement

\[\begin{align} \rho_\tau &= \mathbb{P}((X_1-X_1')(X_2-X_2') > 0) - \mathbb{P}((X_1-X_1')(X_2-X_2') < 0) \\ &\text{ where } (X_1', X_2') \text{ is I.I.D (joint) as } (X_1, X_2) \end{align}\] \[-1 \le \rho_\tau \le 1\]

See this two parts:

• $\mathbb{P}((X_1-X_1’)(X_2-X_2’) > 0)$
  • concordant: moving in same direction
  • if we sample another sample of $(X_1, X_2)$, they will either
    • getting larger together: $X_1’ > X_1 \text{ and } X_2’ > X_2$
    • getter smaller together: $X_1’ < X_1 \text{ and } X_2’ < X_2$
• $\mathbb{P}((X_1-X_1’)(X_2-X_2’) < 0)$
  • discordant: moving in opposite directiontion
  • if we sample another sample of $(X_1, X_2)$, they will move in different direction either
    • $X_1’ > X_1 \text{ and } X_2’ < X_2$
    • $X_1’ < X_1 \text{ and } X_2’ > X_2$

To compute $\rho_\tau$, we use the Copula

\[\begin{align} \rho_\tau &= 4\mathbb{E}[C(U_1, U_2)] - 1\\ &= 4 \int_0^1 \int_0^1 C(u_1, u_2) \cdot c(u_1, u_2) du_1 du_2 - 1 \\ & \text{where } c(u_1, u_2)=\frac{\partial^2 C}{\partial u_1\partial u_2} \end{align}\]

Proof:

\[\begin{align} \rho_\tau &= \mathbb{P}((X_1-X_1')(X_2-X_2') > 0) - \mathbb{P}((X_1-X_1')(X_2-X_2') < 0) \\ &= \mathbb{P}((X_1-X_1')(X_2-X_2') > 0) - (1-\mathbb{P}((X_1-X_1')(X_2-X_2') > 0)) \\ &= 2\mathbb{P}((X_1-X_1')(X_2-X_2') > 0) - 1\\ &= 2(\mathbb{P}(X_1>X_1', X_2>X_2')+\mathbb{P}(X_1<X_1', X_2<X_2')) - 1\\ \because &(X_1, X_2) (X_1', X_2') \text{ are IID} \\ \therefore &\mathbb{P}(X_1>X_1', X_2>X_2') = \mathbb{P}(X_1<X_1', X_2<X_2') \\ &=4 \mathbb{P}(X_1<X_1', X_2<X_2') - 1 \\ &=4 \mathbb{P}(F_{X_1}(X_1)<F_{X_1}(X_1'), F_{X_2}(X_2)<F_{X_2}(X_2')) - 1 \\ &=4 \mathbb{P}(U_1<U_1', U_2<U_2) - 1 \\ &=4 \int_0^1 \int_0^1 \mathbb{P}(U_1<u_1, U_2<u_2|U_1'=u_1, U_2'=u_2) c(u_1, u_2) du_1 du_2 - 1\\ &=4 \int_0^1 \int_0^1 C(u_1, u_2) c(u_1, u_2) du_1 du_2 - 1\\ \rho_\tau &= 4\mathbb{E}[C(U_1, U_2)] - 1\\ \end{align}\]

For Archimedean Copula, we can use an the generator function to calculate

\[\begin{align} \rho_\tau &= 1+4\int_0^1 {\frac{\phi^{-1}(t)}{\frac{d}{dt}\phi^{-1}(t)}} dt \end{align}\]

3.2 Coefficient of Tail Dependence

If $(X_1, X_2)$ is a pair of RV, if $X_1$ falls into tail distribution (> VaR), what is the likelihood $X_2$ also falls into tail dependencies
example: portfolio loss: when diversification fails

Coefficient of upper tail dependence

\[\begin{align} \lambda_U &=\lim_{\alpha\to1^-} \mathbb{P}(X_2>F_{X_2}^{-1}(\alpha)|X_1>F_{X_1}^{-1}(\alpha)) \\ \lambda_L &=\lim_{\alpha\to0^+} \mathbb{P}(X_2<F_{X_2}^{-1}(\alpha)|X_1<F_{X_1}^{-1}(\alpha)) \\ \end{align}\]

For Copula:

\[\begin{align} \lambda_U &=\lim_{\alpha\to1^-} \frac{1-2\alpha+C(\alpha, \alpha)}{1-\alpha} \\ \lambda_L &=\lim_{\alpha\to0^+} \frac{C(\alpha,\alpha)}{\alpha} \\ \end{align}\]

For Archimedean Copula:

\[\begin{align} \lambda_U &=\lim_{\alpha\to1^-} \frac{1-2\alpha+\phi(2\phi^{-1}(\alpha))}{1-\alpha} \\ \lambda_L &=\lim_{\alpha\to0^+} \frac{\phi(2\phi^{-1}(\alpha))}{\alpha} \\ \end{align}\]

Proof:

\[\begin{align} \lambda_U &=\lim_{\alpha\to1^-} \mathbb{P}(X_2>F_{X_2}^{-1}(\alpha)|X_1>F_{X_1}^{-1}(\alpha)) \\ &=\lim_{\alpha\to1^-} \mathbb{P}(X_2>F_{X_2}^{-1}(\alpha) , X_1>F_{X_1}^{-1}(\alpha)) / \mathbb{P}(X_1>F_{X_1}^{-1}(\alpha))\\ & \text{note: } \{X_1>a,X_2>a\}^c= \{X_1>a \text{ or }X_2>a\} = \{\{X_1>a\} \text{ or } \{X_2>a\}\} - \{X_1<a,X_2<a\} \\ &=\lim_{\alpha\to1^-} (1-(\mathbb{P}(X_1>F_{X_1}^{-1}(\alpha)) + \mathbb{P}(X_2>F_{X_2}^{-1}(\alpha)) - \mathbb{P}(X<F_{X_1}^{-1}(\alpha), X_2<F_{X_2}^{-1}(\alpha)))) \frac{1}{1-\alpha}\\ &=\lim_{\alpha\to1^-} \frac{1}{1-\alpha} (1-2\alpha+C(\alpha, \alpha)) \\ &=\lim_{\alpha\to1^-} \frac{1-2\alpha+C(\alpha, \alpha)}{1-\alpha} \quad \square \\ \end{align}\] \[\begin{align} \lambda_L &=\lim_{\alpha\to0^+} \mathbb{P}(X_2<F_{X_2}^{-1}(\alpha)|X_1<F_{X_1}^{-1}(\alpha)) \\ &=\lim_{\alpha\to0^+} \mathbb{P}(X_2<F_{X_2}^{-1}(\alpha),X_1<F_{X_1}^{-1}(\alpha))/\mathbb{P}(X_1<F_{X_1}^{-1}(\alpha)) \\ &=\lim_{\alpha\to0^+} \frac{C(\alpha, \alpha)}{\alpha} \quad \square \end{align}\]

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