Tail value at risk explained

In financial mathematics, tail value at risk (TVaR), also known as tail conditional expectation (TCE) or conditional tail expectation (CTE), is a risk measure associated with the more general value at risk. It quantifies the expected value of the loss given that an event outside a given probability level has occurred.

Background

There are a number of related, but subtly different, formulations for TVaR in the literature. A common case in literature is to define TVaR and average value at risk as the same measure. Under some formulations, it is only equivalent to expected shortfall when the underlying distribution function is continuous at

\operatorname{VaR} _{\alpha} (X) , the value at risk of level \alpha . Under some other settings, TVaR is the conditional expectation of loss above a given value, whereas the expected shortfall is the product of this value with the probability of it occurring. The former definition may not be a coherent risk measure in general, however it is coherent if the underlying distribution is continuous. The latter definition is a coherent risk measure. TVaR accounts for the severity of the failure, not only the chance of failure. The TVaR is a measure of the expectation only in the tail of the distribution.

Mathematical definition

The canonical tail value at risk is the left-tail (large negative values) in some disciplines and the right-tail (large positive values) in other, such as actuarial science. This is usually due to the differing conventions of treating losses as large negative or positive values. Using the negative value convention, Artzner and others define the tail value at risk as:

X which is the payoff of a portfolio at some future time and given a parameter 0 < \alpha < 1 then the tail value at risk is defined by$$\backslash operatorname\_(X)\; =\; \backslash operatorname\; [-X|X\; \backslash leq\; -\backslash operatorname\{VaR\}\_\{\backslash alpha\}(X)]\; =\; \backslash operatorname\; [-X\; |\; X\; \backslash leq\; x^\{\backslash alpha\}],$$where x ^{\alpha} is the upper \alpha -quantile given by x ^{\alpha} = inf \{x \in R : \Pr(X \leq x) > \alpha\} . Typically the payoff random variable X is in some L^p-space where p \geq 1 to guarantee the existence of the expectation. The typical values for \alpha are 5% and 1%.

Formulas for continuous probability distributions

Closed-form formulas exist for calculating TVaR when the payoff of a portfolio

X or a corresponding loss L = - X follows a specific continuous distribution. If X follows some probability distribution with the probability density function (p.d.f.) f and the cumulative distribution function (c.d.f.) F , the left-tail TVaR can be represented as

$$\backslash operatorname\_(X)\; =\; \backslash operatorname\; [-X|X\; \backslash leq\; -\backslash operatorname\{VaR\}\_\{\backslash alpha\}(X)]\; =\; -\backslash frac\; \backslash int\_0^\backslash alpha\; \backslash operatorname\_\backslash gamma(X)d\backslash gamma\; =\; -\backslash frac\backslash int\_^xf(x)dx.$$

For engineering or actuarial applications it is more common to consider the distribution of losses

L =- X , in this case the right-tail TVaR is considered (typically for \alpha 95% or 99%):

$$\backslash operatorname^\backslash text\_\backslash alpha(L)\; =\; E[L\backslash mid\; L\; \backslash geq\; \backslash operatorname\{VaR\}\_\{\backslash alpha\}(L)]\; =\; \backslash frac\; \backslash int^1\_\backslash alpha\; \backslash operatorname\_\backslash gamma(L)d\backslash gamma\; =\; \backslash frac\backslash int^\_yf(y)dy.$$

Since some formulas below were derived for the left-tail case and some for the right-tail case, the following reconciliations can be useful:

$$\backslash operatorname\_(X)\; =\; -\backslash fracE[X]+\backslash frac\backslash operatorname^\backslash text\_\backslash alpha(L)$$ and $$\backslash operatorname^\backslash text\_\backslash alpha(L)\; =\; \backslash fracE[L]+\backslash frac\backslash operatorname\_(X).$$

Normal distribution

If the payoff of a portfolio

X follows normal (Gaussian) distribution with the p.d.f. $$f(x)\; =\; \backslash frace^$$ then the left-tail TVaR is equal to $$\backslash operatorname\_(X)\; =\; -\backslash mu+\backslash sigma\backslash frac,$$ where $\backslash phi(x)\; =\; \backslash frace^$ is the standard normal p.d.f., \Phi(x) is the standard normal c.d.f., so \Phi ^{-1} (\alpha) is the standard normal quantile.^[1]

If the loss of a portfolio

L follows normal distribution, the right-tail TVaR is equal to^[2] $$\backslash operatorname^\backslash text\_\backslash alpha(L)\; =\; \backslash mu+\backslash sigma\backslash frac.$$

Generalized Student's t-distribution

If the payoff of a portfolio

X follows generalized Student's t-distribution with the p.d.f. $$f(x)\; =\; \backslash frac\backslash left(1+\backslash frac\backslash left(\backslash frac\backslash right)^2\backslash right)^$$ then the left-tail TVaR is equal to $$\backslash operatorname\_(X)\; =\; -\backslash mu+\backslash sigma\backslash frac\backslash frac,$$ where $$\backslash tau(x)=\backslash frac\backslash left(1+\backslash frac\backslash right)^$$ is the standard t-distribution p.d.f., \Tau(x) is the standard t-distribution c.d.f., so \Tau ^{-1} (\alpha) is the standard t-distribution quantile.

If the loss of a portfolio

L follows generalized Student's t-distribution, the right-tail TVaR is equal to $$\backslash operatorname^\backslash text\_\backslash alpha(L)\; =\; \backslash mu+\backslash sigma\backslash frac\backslash frac.$$

Laplace distribution

If the payoff of a portfolio

X follows Laplace distribution with the p.d.f. $$f(x)\; =\; \backslash frace^$$ and the c.d.f. then the left-tail TVaR is equal to \operatorname{TVaR} _{\alpha} (X) = - \mu + b( 1-ln2 \alpha) for \alpha \le 0 . 5 .

If the loss of a portfolio

L follows Laplace distribution, the right-tail TVaR is equal to

Logistic distribution

If the payoff of a portfolio

X follows logistic distribution with the p.d.f. $$f(x)\; =\; \backslash frace^\backslash left(1+e^\backslash right)^$$ and the c.d.f. $$F(x)\; =\; \backslash left(1+e^\backslash right)^$$ then the left-tail TVaR is equal to $$\backslash operatorname\_(X)\; =\; -\backslash mu+s\backslash ln\backslash frac.$$

If the loss of a portfolio

L follows logistic distribution, the right-tail TVaR is equal to $$\backslash operatorname^\backslash text\_\backslash alpha(L)\; =\; \backslash mu\; +\; s\backslash frac.$$

Exponential distribution

If the loss of a portfolio

L follows exponential distribution with the p.d.f. and the c.d.f. then the right-tail TVaR is equal to $$\backslash operatorname^\backslash text\_\backslash alpha(L)\; =\; \backslash frac.$$

Pareto distribution

If the loss of a portfolio

L follows Pareto distribution with the p.d.f. and the c.d.f. then the right-tail TVaR is equal to $$\backslash operatorname^\backslash text\_\backslash alpha(L)\; =\; \backslash frac.$$

Generalized Pareto distribution (GPD)

If the loss of a portfolio

L follows GPD with the p.d.f. $$f(x)\; =\; \backslash frac\; \backslash left(1+\backslash frac\; \backslash right)^$$ and the c.d.f. then the right-tail TVaR is equal to and the VaR is equal to

Weibull distribution

If the loss of a portfolio

L follows Weibull distribution with the p.d.f. and the c.d.f. then the right-tail TVaR is equal to $$\backslash operatorname^\backslash text\_\backslash alpha(L)\; =\; \backslash frac\; \backslash Gamma\backslash left(1+\backslash frac,-\backslash ln(1-\backslash alpha)\backslash right),$$ where \Gamma(s,x) is the upper incomplete gamma function.

Generalized extreme value distribution (GEV)

If the payoff of a portfolio

X follows GEV with the p.d.f. and the c.d.f. then the left-tail TVaR is equal to and the VaR is equal to where \Gamma(s,x) is the upper incomplete gamma function, li (x) = \int (dx)/(ln x) is the logarithmic integral function.^[3]

If the loss of a portfolio

L follows GEV, then the right-tail TVaR is equal to where \gamma(s,x) is the lower incomplete gamma function, y is the Euler-Mascheroni constant.

Generalized hyperbolic secant (GHS) distribution

If the payoff of a portfolio

X follows GHS distribution with the p.d.f. $$f(x)\; =\; \backslash frac\; \backslash operatorname\backslash left(\backslash frac\backslash frac\backslash right)$$and the c.d.f. $$F(x)\; =\; \backslash frac\backslash arctan\backslash left[\backslash exp\backslash left(\backslash frac\{\backslash pi\}\{2\}\backslash frac\{x-\backslash mu\}\{\backslash sigma\}\backslash right)\backslash right]$$ then the left-tail TVaR is equal to $$\backslash operatorname\_(X)\; =\; -\backslash mu\; -\; \backslash frac\; \backslash ln\backslash left(\backslash tan\; \backslash frac\; \backslash right)\; -\; \backslash fraci\backslash left[\backslash text\{Li\}\_2\backslash left(-i\backslash tan\backslash frac\{\backslash pi\backslash alpha\}\{2\}\backslash right)-\backslash text\{Li\}\_2\backslash left(i\backslash tan\backslash frac\{\backslash pi\backslash alpha\}\{2\}\backslash right)\backslash right],$$ where Li _{2} is the dilogarithm and i = \sqrt{ -1 } is the imaginary unit.

Johnson's SU-distribution

If the payoff of a portfolio

X follows Johnson's SU-distribution with the c.d.f. $$F(x)\; =\; \backslash Phi\backslash left[\backslash gamma+\backslash delta\backslash sinh^\{-1\}\backslash left(\backslash frac\{x-\backslash xi\}\{\backslash lambda\}\backslash right)\backslash right]$$ then the left-tail TVaR is equal to $$\backslash operatorname\_(X)\; =\; -\backslash xi\; -\; \backslash frac\; \backslash left[\backslash exp\backslash left(\backslash frac\{1-2\backslash gamma\backslash delta\}\{2\backslash delta^2\}\backslash right)\; \backslash Phi\backslash left(\backslash Phi^\{-1\}(\backslash alpha)-\backslash frac\{1\}\{\backslash delta\}\backslash right)\; -\; \backslash exp\backslash left(\backslash frac\{1+2\backslash gamma\backslash delta\}\{2\backslash delta^2\}\backslash right)\backslash Phi\backslash left(\backslash Phi^\{-1\}(\backslash alpha)+\backslash frac\{1\}\{\backslash delta\}\backslash right)\; \backslash right],$$ where \Phi is the c.d.f. of the standard normal distribution.^[4]

Burr type XII distribution

If the payoff of a portfolio

X follows the Burr type XII distribution with the p.d.f. $$f(x)\; =\; \backslash frac\backslash left(\backslash frac\backslash right)^\backslash left[1+\backslash left(\backslash frac\{x-\backslash gamma\}\{\backslash beta\}\backslash right)^c\backslash right]^$$ and the c.d.f. $$F(x)\; =\; 1-\backslash left[1+\backslash left(\backslash frac\{x-\backslash gamma\}\{\backslash beta\}\backslash right)^c\backslash right]^,$$ the left-tail TVaR is equal to $$\backslash operatorname\_(X)\; =\; -\backslash gamma\; -\backslash frac\backslash left((1-\backslash alpha)^-1\; \backslash right)^\; \backslash left[\backslash alpha\; -1+\{\_2F\_1\}\backslash left(\backslash frac\{1\}\{c\},k;1+\backslash frac\{1\}\{c\};1-(1-\backslash alpha)^\{-1/k\}\backslash right)\; \backslash right],$$ where _{2F} _{1} is the hypergeometric function. Alternatively, $$\backslash operatorname\_(X)\; =\; -\backslash gamma\; -\backslash frac\backslash frac\backslash left((1-\backslash alpha)^-1\; \backslash right)^\; \backslash left(1+\backslash frac,k+1;2+\backslash frac;1-(1-\backslash alpha)^\backslash right).$$

Dagum distribution

If the payoff of a portfolio

X follows the Dagum distribution with the p.d.f. $$f(x)\; =\; \backslash frac\backslash left(\backslash frac\backslash right)^\backslash left[1+\backslash left(\backslash frac\{x-\backslash gamma\}\{\backslash beta\}\backslash right)^c\backslash right]^$$ and the c.d.f. $$F(x)\; =\; \backslash left[1+\backslash left(\backslash frac\{x-\backslash gamma\}\{\backslash beta\}\backslash right)^\{-c\}\backslash right]^,$$ the left-tail TVaR is equal to $$\backslash operatorname\_(X)\; =\; -\backslash gamma\; -\backslash frac\backslash frac\backslash left(\backslash alpha^-1\; \backslash right)^\; \backslash left(k+1,k+\backslash frac;k+1+\backslash frac;-\backslash frac\backslash right),$$ where _{2F} _{1} is the hypergeometric function.

Lognormal distribution

If the payoff of a portfolio

X follows lognormal distribution, i.e. the random variable ln ( 1+ X) follows normal distribution with the p.d.f. $$f(x)\; =\; \backslash frace^,$$ then the left-tail TVaR is equal to $$\backslash operatorname\_(X)\; =\; 1-\backslash exp\backslash left(\backslash mu+\backslash frac\backslash right)\; \backslash frac,$$ where \Phi(x) is the standard normal c.d.f., so \Phi ^{-1} (\alpha) is the standard normal quantile.^[5]

Log-logistic distribution

If the payoff of a portfolio

X follows log-logistic distribution, i.e. the random variable ln ( 1+ X) follows logistic distribution with the p.d.f. $$f(x)\; =\; \backslash frace^\backslash left(1+e^\backslash right)^,$$ then the left-tail TVaR is equal to $$\backslash operatorname\_(X)\; =\; 1-\backslash fracI\_\backslash alpha(1+s,1-s)\backslash frac,$$ where I _{\alpha} is the regularized incomplete beta function, I _{\alpha(a,b) = (\Beta _{\alpha(a,b)})/(\Beta(a,b))} .

As the incomplete beta function is defined only for positive arguments, for a more generic case the left-tail TVaR can be expressed with the hypergeometric function: $$\backslash operatorname\_(X)\; =\; 1-\backslash frac\; (s,s+1;s+2;\backslash alpha).$$

If the loss of a portfolio

L follows log-logistic distribution with p.d.f. $$f(x)\; =\; \backslash frac$$ and c.d.f. $$F(x)\; =\; \backslash frac,$$ then the right-tail TVaR is equal to $$\backslash operatorname^\backslash text\_\backslash alpha(L)\; =\; \backslash frac\backslash left[\backslash frac\{\backslash pi\}\{b\}\backslash csc\; \backslash left(\backslash frac\{\backslash pi\}\{b\}\backslash right)-\backslash Beta\_\backslash alpha\backslash left(\backslash frac\{1\}\{b\}+1,1-\backslash frac\{1\}\{b\}\backslash right)\backslash right],$$ where B _{\alpha} is the incomplete beta function.

Log-Laplace distribution

If the payoff of a portfolio

X follows log-Laplace distribution, i.e. the random variable ln ( 1+ X) follows Laplace distribution the p.d.f. $$f(x)\; =\; \backslash frace^,$$ then the left-tail TVaR is equal to

Log-generalized hyperbolic secant (log-GHS) distribution

If the payoff of a portfolio

X follows log-GHS distribution, i.e. the random variable ln ( 1+ X) follows GHS distribution with the p.d.f. $$f(x)\; =\; \backslash frac\; \backslash operatorname\backslash left(\backslash frac\backslash frac\backslash right),$$ then the left-tail TVaR is equal to $$\backslash operatorname\_(X)\; =\; 1-\backslash frac\; \backslash left(\backslash tan\backslash frac\backslash exp\backslash frac\backslash right)^\; \backslash tan\backslash frac\; \backslash left(1,\backslash frac+\backslash frac;\backslash frac+\backslash frac;-\backslash tan\backslash left(\backslash frac\backslash right)^2\backslash right),$$ where _{2F} _{1} is the hypergeometric function.

Notes and References

1. Khokhlov. Valentyn. 2016. Conditional Value-at-Risk for Elliptical Distributions. Evropský časopis Ekonomiky a Managementu. 2. 6. 70–79.
2. Norton. Matthew. Khokhlov. Valentyn. Uryasev. Stan. 2018-11-27. Calculating CVaR and bPOE for Common Probability Distributions With Application to Portfolio Optimization and Density Estimation. 1811.11301. q-fin.RM.
3. 3200629. Conditional Value-at-Risk for Uncommon Distributions. Khokhlov. Valentyn. 2018-06-21. SSRN.
4. 1855986. Moment-Based CVaR Estimation: Quasi-Closed Formulas. Stucchi. Patrizia. 2011-05-31. SSRN.
5. 3197929. Conditional Value-at-Risk for Log-Distributions. Khokhlov. Valentyn. 2018-06-17. SSRN.