Quant Risks - Loss Tail Analysis I - VaR and ES Metrics

1 Loss analysis

1.1 Loss variable

Define $L_n$ is a loss variable

• When the $\Delta_t$ is small, we can just use $L_{n+1}=-(V_{n+1}-V_n)$ , with $V_n$ as our portfolio value, is $F_n$-measurable
• For long time period, the time value of money might also need to be accounted
  • Small time period \(L_{n+1}=-(V_{n+1}-V_n)\)
  • Simple effective rate \(L_{n+1}=-(\frac{V_{n+1}}{1+r_{n+1}}-V_n)\)
  • Continuous rate: \(L_{n+1}=-(V_{n+1}e^{-\Delta_t}-V_n)\)

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2 Risk Measure

2.1 Variance

More variance, more risk.

• $VAR = \mathbb{E}[(L_{t+1}-\mathbb{E}[L_{t+1}])^2]=\mathbb{E}[X^2]-\mathbb{E}[X]^2$
• $VAR=\vec{w}^T\mathbf{\Sigma}\vec{w}$

• 2.2 Value-at-Risk (VaR)

  Formal defintion:

  \[\operatorname{VaR}_\alpha(L)=\inf\{x: \mathbb{P}(L<=x)>=\alpha\}\]

• VaR is measuring the at $\alpha$ probabilty, the Loss <= VaR

• Typical formula (for continuous Loss):

  \[\begin{align} \mathbb{P}(L<\operatorname{VaR}_\alpha(L)) &= \alpha \\ F_L(\operatorname{VaR}_\alpha(L)) &=\alpha \\ \operatorname{VaR}_\alpha(L) &= F_L^-1(\alpha) \end{align}\]

2.3 Expected Shortfall (ES)

Formal defintion:

\[\operatorname{ES}_\alpha(L)=\mathbb{E}[L|L>\operatorname{VaR}_\alpha(L)]\]

• The VaR give an estimateion of the loss bound
• ES wants to know if the Loss > VaR, what is the expectation of Loss

• Typical formula

  \[\begin{align} \operatorname{ES}_\alpha(L) &= \int_{VaR}^{\infty}{x\cdot\frac{f_L(x)}{\mathbb{P}(L>\operatorname{VaR}_\alpha(L))}}{dx} \\ &= \int_{VaR}^{\infty}{x\cdot\frac{f_L(x)}{1-F_L(\operatorname{VaR})}}{dx} \\ &= \int_{VaR}^{\infty}{x\cdot\frac{f_L(x)}{1-\alpha}}{dx} \\ &= \frac{1}{1-\alpha}\int_{VaR}^{\infty}{x \cdot f_L(x)}{dx} \quad \square \end{align}\]

• Alternative formula

  \[\begin{align} \operatorname{ES}_\alpha(L) &= \frac{1}{1-\alpha}\int_{VaR}^{\infty}{x \cdot f_L(x)}{dx} \\ \text{let } y =F_L(x) \\ \text{ hence } dy=f_L(x)dx \\ \text{ and } x=F_L^-1(y) \\ \text{upper bound} = F_L(\infty) = 1 \\ \text{lower bound} = F_L(VaR) = \alpha \\ &= \frac{1}{1-\alpha} \int_{\alpha}^1 {F_L^-1(y)} {dy}\\ &= \frac{1}{1-\alpha} \int_{\alpha}^1 {\operatorname{VaR}_y(L)} {dy} \quad \square \end{align}\]

2.4 Some Example

2.4.1 Pareto Distribution

Pareto Distribution is a power law distribution, usually used for distribution of law. It has 80:20 rule.

\[L \sim Pareto(\alpha, \theta)\]

Pareto Type II - Lomax: Start from 0, have heavy tail,

\[\begin{align} f_L(x) &= \frac{\alpha\theta^\alpha}{(x+\theta)^{\alpha+1}}\\ F_L(x) &= 1 - (\frac{\theta}{x+\theta})^\alpha \\ \mathbb{E}[X] &= \frac{\theta}{\alpha-1} \\ \operatorname{VAR}[X] &= \frac{\alpha\theta^2}{(\alpha-1)^2(\alpha-2)} \\ \operatorname{VaR}_\beta[X] &= \theta[(1-\beta)^{-1/\alpha}-1] \\ \operatorname{ES}_\beta[X] &= \frac{\alpha\theta}{\alpha-1}(1-\beta)^{-1/\alpha}-\theta \\ \end{align}\]

2.4.2 Normal Distribution

Normal Distribution

\[L \sim N(\mu, \sigma)\]

\[\begin{align} L &= \mu+\sigma Z\\ f_L(x) &= \frac{1}{\sqrt{2\pi}\sigma}\exp(-{\frac{(x-\mu)^2}{2\sigma^2}}) \\ F_L(x) &= N(\frac{x-\mu}{\sigma}) \\ \mathbb{E}[X] &= \mu \\ \operatorname{VAR}[X] &= \sigma^2 \\ \operatorname{VaR}_\beta[X] &= \mu+\sigma N^{-1}(\beta) \\ \operatorname{ES}_\beta[X] &= \mu+\frac{\sigma}{1-\beta}\phi(N^{-1}(\beta)) \\ \end{align}\]

2.4.3 T-Distribution

T-distribution: heavy tailed bell-shape

\[L \sim T(\mu, \lambda, v)\] \[T_v = \frac{Z}{\sqrt{V / v}} \quad \text{, where } V \sim \chi^2(v)\]

\[\begin{align} L &= \mu+\lambda T_v\\ f_L(x) &= ...\\ F_L(x) &= F_T(\frac{x-\mu}{\lambda}) \\ \mathbb{E}[X] &= \mu \\ \operatorname{VAR}[X] &= \lambda^2 \frac{v}{v-2} \\ \operatorname{VaR}_\beta[X] &= \mu+\lambda F_{T_v}^{-1}(\beta) \\ \operatorname{ES}_\beta[X] &= \mu+\frac{\lambda}{1-\beta}f_{T_v}(F_{T_V}^{-1}(\beta)) \cdot \frac{v+(F_{T_V}^{-1}(\beta))^2}{v-1} \\ \end{align}\]

2.5 Criteria of a good Risk Metric

2.5.1 Coherent Metric

Coherent is a necessary condition for a good metric.

A good coherent metric must have these 4 properties

Property	Definition	Remarks
Monotonicity	if $L_1 \le L_2$, then $\gamma(L_1) \le \gamma(L_2)$	If $Port_2$ is worse than $Port_1$ for all scenario, then the risk measure of 2 is higher
Translation Invariance	$\gamma(L+c) = \gamma(L) + c$	adding deterministic loss just shift the loss (e.g. adding risk-free deposit just lower the risk by the same quantity)
Poisitive Homogeneity	$\gamma(\lambda L)=\lambda \gamma(L)$	Scaling the portfolio will result in the same scale of the loss
Subadditivity	$\gamma(L_1+L_2) \le \gamma(L_1) + \gamma(L_2)$	Diversification makes less risk

2.5.2 Convex

Convex is a necessary condition for a good metric.

Coherent implies Convex

Property	Definition	Remarks
Monotonicity	if $L_1 \le L_2$, then $\gamma(L_1) \le \gamma(L_2)$	If $Port_2$ is worse than $Port_1$ for all scenario, then the risk measure of 2 is higher
Translation Invariance	$\gamma(L+c) = \gamma(L) + c$	adding deterministic loss just shift the loss (e.g. adding risk-free deposit just lower the risk by the same quantity)
Conexity	$\gamma(\lambda L_1 + (1-\lambda) L_2 \le \lambda \gamma(L_1) + (1-\lambda) \gamma(L_2)$	Diversification makes less risk

2.5.3 Summary of Metric

Metric	Monotonicity	Translation Invariance	Poisitive Homogeneity	Subadditivity	Conexity	Coherent？	Convex ?
Variance	❌	❌	❌	✅	❌	❌	❌
VaR	✅	✅	✅	❌	❌	❌	❌
Expected Shortfall (ES)	✅	✅	✅	✅	✅	✅	✅

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3 How to estimate VaR and ES

3.1 Variance-Covariance Method

• Assume the $L$ have some distribution
• Fit $L \sim Dist$ and estimate the parameters via MLE (Maximum Log-Likelihood estimation )
• Mathematically get the formula of VaR and ES
• Pros: analytical way, can cover situation that havnt seen in empirical way
• Cons: Dist is hard to estimate, might need to use Monte-Carlo Distribution to get VaR and ES

3.2 Empirical estimation

• using empirical data, how the $\alpha$-percentile of the Loss
• Estimate the ES by taking mean of data > VaR
• Pros: model-free, does not need to estimate the distribution
• Cons: Need large sample size

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