# Value-at risk and tail-value-at-risk

In actuarial applications, an important focus is on developing loss distributions for insurance products. In such applications, it is desirable to employ risk measures to evaluate the exposure to risk. Such risk measures are indicators, often one or a small set of numbers, that inform actuaries and risk managers about the degree to which the risk bearing entity is subject to various aspects of risk. This post introduces two risk measures – value-at-risk (VaR) and tail-value-at-risk (TVaR).

Value-at-Risk

One natural question for a risk bearing entity (e.g. insurance companies and other enterprises) is: what is the chance of an adverse outcome? Value-at-risk (VaR) provides a ready answer to this question. Mathematically speaking, VaR is a quantile of the distribution of aggregate losses. For example, VaR at the 99% probability level indicates the level of adverse outcome such that the probability of exceeding this threshold is 1%. More broadly, VaR is the amount of capital required to ensure, with a high level of confidence, that the risk bearing entity does not become insolvent. The security level or probability level is chosen arbitrarily. In practice, it is usually a high number such as 95%, 99% or 99.5%. The preference is for a higher security level when evaluating the risk exposure for the entire enterprise. For a sub unit of the enterprise, the security level may be set to a lower number such as 95%.

Suppose that $X$ is a random variable that models the loss distribution in question. We assume that the support of $X$ is the set of positive real numbers or some appropriate subset. The value-at-risk (VaR) of $X$ at the $100p$th security level, denoted by $\text{VaR}_p(X)$, is the $100p$th percentile of $X$.

In the current discussion, we focus on loss distributions that are continuous random variables. Thus $\text{VaR}_p(X)$ is the value $\pi_p$ such that $P(X>\pi_p)=1-p$. In some ways, VaR is an attractive risk measure. Mathematically speaking, VaR has a clear and simple definition. For certain probability models, VaR can be evaluated in closed form. For those models that have no closed form for percentiles, VaR can be evaluated using software. However, VaR has limitations (this point will be briefly discussed below).

Example 1
Suppose that the loss $X$ is normally distributed with mean $\mu$ and variance $\sigma^2$. Then $\text{VaR}_p(X)=\mu+z_p \cdot \sigma$ where $z_p$ is the $100p$th percentile of the standard normal distribution (i.e. normal with mean 0 and standard deviation 1).

VaR for normal distribution is identical to the risk measure called standard deviation principle. For any loss $X$ with mean $\mu$ and variance $\sigma^2$, the quantity $\mu+k \cdot \sigma$ for some fixed constant $k$ is a risk measure called the standard deviation principle. The constant $k$ is usually chosen such that losses will exceed $\mu+k \cdot \sigma$ with a pre-determined small probability. For losses that are normally distributed, $k=1.645$ for security level $p=0.95$ and $k=2.326$ for security level $p=0.99$.

Comment
The value-at-risk discussed here is for gauging the exposure of risk with respect to insurance losses. Thus we assume that the random variable $X$ is one that takes on positive real values (or some appropriate subset of the positive real number line). In this context, the adverse outcomes would be the extremely high values of the positive real numbers. In other words, the adverse outcomes we wish to guard against would be the right tails of the loss distribution. Thus $\text{VaR}_p(X)$ is evaluated for high values for $p$ such as 0.95 or 0.99 or higher. In the actuarial loss point of view, VaR is the high right tail of the loss distribution.

VaR is also used extensively in banking and investment industry. The random variable $X$ in that setting is a profit and loss distribution, which can extend into the negative real numbers (when there are losses). The adverse outcomes for banking and investment applications would be the extreme left tails of the profit and loss distribution. Thus when VaR is evaluated at the security level 95%, we actually calculate the 5th percentile of the profit and loss distribution.

Tail-Value-at-Risk

Tail-value-at-risk (TVaR) is risk measure that is in many ways superior than VaR. The risk measure VaR is a merely a cutoff point and does not describe the tail behavior beyond the VaR threshold. We will see that TVaR reflects the shape of the tail beyond VaR threshold.

Suppose that $X$ is the random variable that models losses. As before we assume that $X$ takes on positive real numbers. The tail-value-at-risk at the $100p$th security level, denoted by $\text{TVaR}_p(X)$, is the expected loss on the condition that loss exceeds the $100p$th percentile of $X$.

Just as in the discussion of VaR, the discussion of TVaR focuses on continuous distributions. As before, we use $\pi_p$ to denote the $100p$th percentile of $X$. The quantity $\text{TVaR}_p(X)$ can be expressed as follows: $\displaystyle (1) \ \ \ \ \ \text{TVaR}_p(X)=E(X \lvert X> \pi_p)=\frac{\int_{\pi_p}^\infty x f(x) \ dx}{1-F(\pi_p)}=\frac{\int_{\pi_p}^\infty x f(x) \ dx}{1-p}$

The above is the formulation of TVaR is that based on the definition. To use (1) to obtain TVaR, evaluate VaR and then evaluate the integral on the right side. TVaR is obtained by dividing the integral result by $1-p$. However, there are other formulations that will give more insight into TVaR. The following is another formulation. $\displaystyle (2) \ \ \ \ \ \text{TVaR}_p(X)=\frac{\int_p^1 \text{VaR}_w(X) \ dw}{1-p}$

The above is derived by the substitution $w=F(x)$ where $F(x)$ is the cumulative distribution function (CDF) of $X$. From a computation standpoint, (2) is not easy to use since it is usually hard to integrate VaR. The value of (2) is in the insight. From (2), we see that TVaR can be viewed as the average of all VaR at the level $w$ above $p$. Thus TVaR tells us much more about the tail of the loss distribution than VaR for just one security level. The following is another formulation. \displaystyle \begin{aligned} (3) \ \ \ \ \ \text{TVaR}_p(X)&=\frac{\int_{\pi_p}^\infty x f(x) \ dx}{1-p} \\&=\pi_p+\frac{\int_{\pi_p}^\infty (x-\pi_p) f(x) \ dx}{1-p} \\&=\pi_p+e(\pi_p) \\&=\text{VaR}_p(X)+e(\text{VaR}_p(X))\end{aligned}

In (3), the function $e(x)$ is the mean excess loss function evaluated at $x$. Thus TVaR is the VaR plus the mean excess loss evaluated at VaR. Thus TVaR is VaR plus the average excess of all losses exceeding the threshold of VaR. As a result, TVaR gives information on the tail to the right of VaR. For those parametric loss distributions that have accessible formulations for the limited expectation $E(X \wedge x)$, the following is a useful formulation for TVaR. \displaystyle \begin{aligned} (4) \ \ \ \ \ \text{TVaR}_p(X)&=\text{VaR}_p(X)+e(\text{VaR}_p(X)) \\&=\text{VaR}_p(X)+\frac{E[X]-E[X \wedge \text{VaR}_p(X)]}{1-p} \end{aligned}

See this blog post in a companion blog for more information on mean excess loss function and limited expectation $E(X \wedge x)$.

Tail-value-at-risk is also known as conditional tail expectation (CTE) as well as tail conditional expectation (TCE). CTE and TCE are widely used in North America. In Europe, TVaR is also known as expected shortfall (ES).

Formulas for VaR and TVaR

Many distributions have CDFs that allow relatively easy computation of percentiles. Thus VaR is quite accessible for these parametric distributions. This link has a table of distributional information for parametric distributions that includes value-at-risk. Distributions in this table that have formulas for VaR include Burr distribution, inverse Burr distribution, Pareto distribution, inverse Pareto distribution, loglogistic distribution, paralogistic distribution, inverse paralogistic distribution, Weibull distribution, inverse Weibull distribution, exponential distribution, inverse exponential distribution.

The calculation for TVaR is not so accessible in the table in the given link. In fact, for some distributions that have heavy tails, TVaR is not even defined since TVaR involves an average value of the excess of losses above a threshold. In the remainder of the post, we present formulas for TVaR for four distributions – exponential, Pareto, normal and lognormal.

Distribution VaR TVaR
Exponential $-\theta \text{ln}(1-p)$ $\theta (1- \text{ln}(1-p))$
Pareto $\displaystyle \frac{\theta}{(1-p)^{1/\alpha}}-\theta$ $\displaystyle \text{VaR}_p(X)+\frac{\theta+\text{VaR}_p(X)}{\alpha-1}$ $\alpha>1$
Pareto $\displaystyle \text{VaR}_p(X) \ \frac{\alpha}{\alpha-1}+\frac{\theta}{\alpha-1}$ $\alpha>1$
Pareto $\displaystyle \frac{\theta}{\alpha-1} \biggl[1+ \frac{\alpha}{\theta} \ \text{VaR}_p(X) \biggr]$ $\alpha>1$
Normal $\mu+z_p \ \sigma$ $\displaystyle \mu+ \sigma \ \frac{\phi(z_p)}{1-p}$
Lognormal $\displaystyle e^{\mu+z_p \ \sigma}$ $\displaystyle e^{\mu+0.5 \ \sigma^2} \biggl[\frac{\Phi(\sigma-z_p)}{1-p} \biggr]$

The exponential distribution in the table has parametrized by a scale parameter $\theta$, which happens to be the mean of the distribution (see here for more information).

The Pareto distribution in the table is Pareto Type II Lomax distribution discussed here. It is parametrized by a shape parameter $\alpha$ and a scale parameter $\theta$. The table gives three formulations of TVaR. Note that in order for the mean to exist, the shape parameter must be greater than 1. Thus TVaR is not defined when $\alpha>1$.

For the normal distribution, the parameters are $\mu$ (mean) and $\sigma$ (standard deviation). The lognormal distribution are parametrized by $\mu$ and $\sigma$, meaning that the logarithm of the lognormal distribution is a normal distribution with mean $\mu$ and standard deviation $\sigma$. To use the formulas for TVaR, note that $\phi$ and $\Phi$ are the probability density function and the cumulative distribution function of the standard normal distribution, respectively. Thus $\phi$ and $\Phi$ are: $\displaystyle \phi(x)=\frac{1}{\sqrt{2 \pi}} \ e^{-0.5 x^2}$ $\displaystyle \Phi(x)=\int_{-\infty}^x \frac{1}{\sqrt{2 \pi}} \ e^{-0.5 t^2} \ dt$

Of course $\Phi(x)$ is evaluated by using a table or software and not by evaluating the integral.

Comment

The risk measure of value-at-risk, though simple in concept and calculation, has shortcomings. One undesirable aspect is that VaR does not possess certain desirable properties among risk measures. VaR is not a coherent risk measure. To be a coherent risk measure, it must satisfy four properties, one of which is subadditivity. When a risk measure is subadditive, the risk measure of two risks combined into one will not be greater than the risk measure for the two risks treated separately. It is desirable to have the benefits of diversification by combing several risks into one. When the risk measure is not subadditive, it cannot model the diversification of risks. It had been shown that VaR is not a subadditive risk measure. On the other hand, TVaR had been shown to be a coherent measure. It satisfies subadditivity and three other desirable properties for risk measures. The topic of risk measures is to be discussed in a subsequent post.

A Formula for a Mixture

A relatively easy to use formula for tail-value-at-risk for a mixture distribution is shown in next post.

Practice Problems

Practice problems are available in the companion blog to reinforce the concepts of value-at-risk and tail-value-at-risk.

actuarial
math

Daniel Ma
mathematics $\copyright$ 2017 – Dan Ma