Expected shortfall

If $X\; \backslash in\; L^p(\backslash mathcal\{F\})$ (an L^p) is the payoff of a portfolio at some future time and then we define the expected shortfall as

:$\backslash operatorname\{ES\}\_\backslash alpha(X)\; =\; \backslash frac\{1\}\{\backslash alpha\}\; \backslash int\_0^\backslash alpha\; \backslash operatorname\{VaR\}\_\backslash gamma(X)\; \backslash ,\; d\backslash gamma$

where $\backslash operatorname\{VaR\}\_\backslash gamma$ is the value at risk. This can be equivalently written as

:$\backslash operatorname\{ES\}\_\backslash alpha(X)\; =\; -\backslash frac\{1\}\{\backslash alpha\}\; \backslash left(\backslash operatorname\; E[X\; \backslash \; 1\_\{\backslash \{X\; \backslash leq\; x\_\{\backslash alpha\}\backslash \}\}]\; +\; x\_\backslash alpha(\backslash alpha\; -\; P[X\; \backslash leq\; x\_\backslash alpha])\backslash right)$

where $x\_\backslash alpha\; =\; \backslash inf\backslash \{x\; \backslash in\; \backslash mathbb\{R\}:\; P(X\; \backslash leq\; x)\; \backslash geq\; \backslash alpha\backslash \}=-\backslash operatorname\{VaR\}\_\backslash alpha(X)$ is the lower $\backslash alpha$-quantile and is the indicator function.{{cite journal|

author = Carlo Acerbi|

author2= Dirk Tasche|

title = Expected Shortfall: a natural coherent alternative to Value at Risk|

journal = Economic Notes|

year = 2002|

volume = 31|

issue= 2|

pages = 379–388|

url = http://www.bis.org/bcbs/ca/acertasc.pdf|

access-date = April 25, 2012

|doi=10.1111/1468-0300.00091|

arxiv= cond-mat/0105191|

s2cid= 10772757}} Note, that the second term vanishes for random variables with continuous distribution functions.

The dual representation is

:$\backslash operatorname\; \{ES\}\_\backslash alpha(X)\; =\; \backslash inf\_\{Q\; \backslash in\; \backslash mathcal\{Q\}\_\backslash alpha\}\; E^Q[X]$

where $\backslash mathcal\{Q\}\_\backslash alpha$ is the set of probability measures which are absolutely continuous to the physical measure $P$ such that $\backslash frac\{dQ\}\{dP\}\; \backslash leq\; \backslash alpha^\{-1\}$ almost surely.{{cite journal|last1=Föllmer|first1=H.|last2=Schied|first2=A.|year=2008|title=Convex and coherent risk measures|url=http://wws.mathematik.hu-berlin.de/~foellmer/papers/CCRM.pdf|access-date=October 4, 2011}} Note that $\backslash frac\{dQ\}\{dP\}$ is the Radon–Nikodym derivative of $Q$ with respect to $P$.

Expected shortfall can be generalized to a general class of coherent risk measures on $L^p$ spaces (Lp space) with a corresponding dual characterization in the corresponding $L^q$ dual space. The domain can be extended for more general Orlicz Hearts.{{cite journal|

author = Patrick Cheridito|

author2= Tianhui Li|

title = Dual characterization of properties of risk measures on Orlicz hearts|

journal = Mathematics and Financial Economics|

year = 2008|

volume = 2|

pages = 2–29|

doi= 10.1007/s11579-008-0013-7|

s2cid= 121880657|

author2-link= Tianhui Li}}

If the underlying distribution for $X$ is a continuous distribution then the expected shortfall is equivalent to the tail conditional expectation defined by $\backslash operatorname\{TCE\}\_\{\backslash alpha\}(X)\; =\; E[-X\backslash mid\; X\; \backslash leq\; -\backslash operatorname\{VaR\}\_\{\backslash alpha\}(X)]$.{{cite web|url=https://statistik.ets.kit.edu/download/doc_secure1/7_StochModels.pdf|title=Average Value at Risk|access-date=February 2, 2011|url-status=dead|archive-url=https://web.archive.org/web/20110719222242/https://statistik.ets.kit.edu/download/doc_secure1/7_StochModels.pdf|archive-date=July 19, 2011}}

Informally, and non-rigorously, this equation amounts to saying "in case of losses so severe that they occur only alpha percent of the time, what is our average loss".

Expected shortfall can also be written as a distortion risk measure given by the distortion function

: {{cite web|title=Distortion Risk Measures: Coherence and Stochastic Dominance|author=Julia L. Wirch|author2=Mary R. Hardy|url=http://pascal.iseg.utl.pt/~cemapre/ime2002/main_page/papers/JuliaWirch.pdf|access-date=March 10, 2012|archive-url=https://web.archive.org/web/20160705041252/http://pascal.iseg.utl.pt/~cemapre/ime2002/main_page/papers/JuliaWirch.pdf|archive-date=July 5, 2016|url-status=dead}}{{Cite journal | last1 = Balbás | first1 = A. | last2 = Garrido | first2 = J. | last3 = Mayoral | first3 = S. | doi = 10.1007/s11009-008-9089-z | title = Properties of Distortion Risk Measures | journal = Methodology and Computing in Applied Probability | volume = 11 | issue = 3 | pages = 385 | year = 2008 | hdl = 10016/14071 | s2cid = 53327887 | url = http://e-archivo.uc3m.es/bitstream/10016/14071/1/properties_balbas_MCAP_2009_ps.pdf | hdl-access = free }}

Example 1. If we believe our average loss on the worst 5% of the possible outcomes for our portfolio is EUR 1000, then we could say our expected shortfall is EUR 1000 for the 5% tail.

Example 2. Consider a portfolio that will have the following possible values at the end of the period:

Now assume that we paid 100 at the beginning of the period for this portfolio. Then the profit in each case is (ending value−100) or:

From this table let us calculate the expected shortfall $\backslash operatorname\{ES\}\_q$ for a few values of $q$:

To see how these values were calculated, consider the calculation of $\backslash operatorname\{ES\}\_\{0.05\}$, the expectation in the worst 5% of cases. These cases belong to (are a subset of) row 1 in the profit table, which have a profit of −100 (total loss of the 100 invested). The expected profit for these cases is −100.

Now consider the calculation of $\backslash operatorname\{ES\}\_\{0.20\}$, the expectation in the worst 20 out of 100 cases. These cases are as follows: 10 cases from row one, and 10 cases from row two (note that 10+10 equals the desired 20 cases). For row 1 there is a profit of −100, while for row 2 a profit of −20. Using the expected value formula we get

:$\backslash frac\{\; \backslash frac\{10\}\{100\}(-100)+\backslash frac\{10\}\{100\}(-20)\; \}\{\; \backslash frac\{20\}\{100\}\}\; =\; -60.$

Similarly for any value of $q$. We select as many rows starting from the top as are necessary to give a cumulative probability of $q$ and then calculate an expectation over those cases. In general, the last row selected may not be fully used (for example in calculating $-\backslash operatorname\{ES\}\_\{0.20\}$ we used only 10 of the 30 cases per 100 provided by row 2).

As a final example, calculate $-\backslash operatorname\{ES\}\_1$. This is the expectation over all cases, or

:$0.1(-100)+0.3(-20)+0.4\backslash cdot\; 0+0.2\backslash cdot\; 50\; =\; -6.\; \backslash ,$

The value at risk (VaR) is given below for comparison.

The expected shortfall $\backslash operatorname\{ES\}\_q$ increases as $q$ decreases.

The 100%-quantile expected shortfall $\backslash operatorname\{ES\}\_\{1\}$ equals negative of the expected value of the portfolio.

For a given portfolio, the expected shortfall $\backslash operatorname\{ES\}\_q$ is greater than or equal to the Value at Risk $\backslash operatorname\{VaR\}\_q$ at the same $q$ level.

Expected shortfall, in its standard form, is known to lead to a generally non-convex optimization problem. However, it is possible to transform the problem into a linear program and find the global solution.{{Cite journal|last1=Rockafellar|first1=R. Tyrrell|last2=Uryasev|first2=Stanislav|date=2000|title=Optimization of conditional value-at-risk|url=https://sites.math.washington.edu/~rtr/papers/rtr179-CVaR1.pdf|journal=Journal of Risk|volume=2|issue=3|pages=21–42|doi=10.21314/JOR.2000.038|s2cid=854622 }} This property makes expected shortfall a cornerstone of alternatives to mean-variance portfolio optimization, which account for the higher moments (e.g., skewness and kurtosis) of a return distribution.

Suppose that we want to minimize the expected shortfall of a portfolio. The key contribution of Rockafellar and Uryasev in their 2000 paper is to introduce the auxiliary function $F\_\{\backslash alpha\}(w,\backslash gamma)$ for the expected shortfall:$$F\_\backslash alpha(w,\backslash gamma)\; =\; \backslash gamma\; +\; \{1\backslash over\{1-\backslash alpha\}\}\; \backslash int\_\{\backslash ell(w,x)\backslash geq\; \backslash gamma\}\; \backslash left[\backslash ell(w,x)-\backslash gamma\backslash right]\_\{+\}\; p(x)\; \backslash ,\; dx$$Where $\backslash gamma\; =\; \backslash operatorname\{VaR\}\_\backslash alpha(X)$ and $\backslash ell(w,x)$ is a loss function for a set of portfolio weights $w\backslash in\backslash mathbb\{R\}^p$ to be applied to the returns. Rockafellar/Uryasev proved that $F\_\backslash alpha(w,\backslash gamma)$ is convex with respect to $\backslash gamma$ and is equivalent to the expected shortfall at the minimum point. To numerically compute the expected shortfall for a set of portfolio returns, it is necessary to generate $J$ simulations of the portfolio constituents; this is often done using copulas. With these simulations in hand, the auxiliary function may be approximated by:$$\backslash widetilde\{F\}\_\backslash alpha(w,\backslash gamma)\; =\; \backslash gamma\; +\; \{1\backslash over\{(1-\backslash alpha)J\}\}\backslash sum\_\{j=1\}^J\; [\backslash ell(w,x\_j)\; -\; \backslash gamma]\_\{+\}$$This is equivalent to the formulation:$$\backslash min\_\{\backslash gamma,z,w\}\; \backslash ;\; \backslash gamma\; +\; \{1\backslash over\{(1-\backslash alpha)J\}\}\; \backslash sum\_\{j=1\}^J\; z\_j,\; \backslash quad\; \backslash text\{s.t.\; \}\; z\_j\; \backslash geq\; \backslash ell(w,x\_j)-\backslash gamma,\backslash ;\; z\_j\; \backslash geq\; 0$$ Finally, choosing a linear loss function $\backslash ell(w,x\_\{j\})\; =\; -w^T\; x\_j$ turns the optimization problem into a linear program. Using standard methods, it is then easy to find the portfolio that minimizes expected shortfall.

Closed-form formulas exist for calculating the expected shortfall when the payoff of a portfolio $X$ or a corresponding loss $L\; =\; -X$ follows a specific continuous distribution. In the former case, the expected shortfall corresponds to the opposite number of the left-tail conditional expectation below $-\backslash operatorname\{VaR\}\_\backslash alpha\; (X)$:

: $\backslash operatorname\; \{ES\}\_\backslash alpha(X)\; =\; E[-X\backslash mid\; X\; \backslash leq\; -\backslash operatorname\{VaR\}\_\backslash alpha(X)]\; =\; -\backslash frac\{1\}\{\backslash alpha\}\backslash int\_0^\backslash alpha\; \backslash operatorname\{VaR\}\_\backslash gamma(X)\; \backslash ,\; d\backslash gamma\; =\; -\backslash frac\{1\}\{\backslash alpha\}\; \backslash int\_\{-\backslash infty\}^\{-\backslash operatorname\{VaR\}\_\backslash alpha(X)\}\; xf(x)\; \backslash ,\; dx.$

Typical values of $\backslash alpha$ in this case are 5% and 1%.

For engineering or actuarial applications it is more common to consider the distribution of losses $L\; =\; -X$, the expected shortfall in this case corresponds to the right-tail conditional expectation above $\backslash operatorname\{VaR\}\_\backslash alpha\; (L)$ and the typical values of $\backslash alpha$ are 95% and 99%:

: $\backslash operatorname\; \{ES\}\_\backslash alpha(L)$

= \operatorname E[L\mid L \geq \operatorname{VaR}_\alpha(L)]

= \frac{1}{1-\alpha} \int^1_\alpha \operatorname{VaR}_\gamma(L) \, d\gamma

= \frac{1}{1-\alpha} \int^{+\infty}_{\operatorname{VaR}_\alpha(L)} 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:

: $\backslash operatorname\; \{ES\}\_\backslash alpha(X)$

= -\frac{1}{\alpha} \operatorname E[X] + \frac{1-\alpha}{\alpha} \operatorname {ES}_\alpha(L) \text{ and } \operatorname{ES}_\alpha(L)

= \frac{1}{1-\alpha} \operatorname E[L]+\frac{\alpha}{1-\alpha} \operatorname {ES}_\alpha(X).

If the payoff of a portfolio $X$ follows the normal (Gaussian) distribution with p.d.f. $f(x)\; =\; \backslash frac\{1\}\{\backslash sqrt\{2\backslash pi\}\backslash sigma\}e^\{-\backslash frac\{(x-\backslash mu)^2\}\{2\backslash sigma^2\}\}$ then the expected shortfall is equal to $\backslash operatorname\{ES\}\_\backslash alpha(X)\; =\; -\backslash mu+\backslash sigma\backslash frac\{\backslash varphi(\backslash Phi^\{-1\}(\backslash alpha))\}\{\backslash alpha\}$, where $\backslash varphi(x)=\backslash frac\{1\}\{\backslash sqrt\{2\backslash pi\}\}e^\{-\backslash frac\{x^2\}\{2\}\}$ is the standard normal p.d.f., $\backslash Phi(x)$ is the standard normal c.d.f., so $\backslash Phi^\{-1\}(\backslash alpha)$ is the standard normal quantile.{{Cite journal|last=Khokhlov|first=Valentyn|date=2016|title=Conditional Value-at-Risk for Elliptical Distributions|journal=Evropský časopis Ekonomiky a Managementu|volume=2|issue=6|pages=70–79}}

If the loss of a portfolio $L$ follows the normal distribution, the expected shortfall is equal to $\backslash operatorname\{ES\}\_\backslash alpha(L)\; =\; \backslash mu+\backslash sigma\backslash frac\{\backslash varphi(\backslash Phi^\{-1\}(\backslash alpha))\}\{1-\backslash alpha\}$.{{cite arXiv|last1=Norton|first1=Matthew|last2=Khokhlov|first2=Valentyn|last3=Uryasev|first3=Stan|date=2018-11-27|title=Calculating CVaR and bPOE for Common Probability Distributions With Application to Portfolio Optimization and Density Estimation|eprint=1811.11301|class=q-fin.RM}}

If the payoff of a portfolio $X$ follows the generalized Student's t-distribution with p.d.f. $f(x)\; =\; \backslash frac\{\backslash Gamma\backslash left(\backslash frac\{\backslash nu+1\}\{2\}\backslash right)\}\{\backslash Gamma\backslash left(\backslash frac\{\backslash nu\}\{2\}\; \backslash right)\; \backslash sqrt\{\backslash pi\backslash nu\}\; \backslash sigma\}\; \backslash left(1+\backslash frac\{1\}\{\backslash nu\}\backslash left(\backslash frac\{x-\backslash mu\}\{\backslash sigma\}\backslash right)^2\backslash right)^\{-\backslash frac\{\backslash nu+1\}\{2\}\}$ then the expected shortfall is equal to $\backslash operatorname\{ES\}\_\backslash alpha(X)\; =\; -\backslash mu+\backslash sigma\backslash frac\{\backslash nu+(\backslash Tau^\{-1\}(\backslash alpha))^2\}\{\backslash nu-1\}\backslash frac\{\backslash tau(\backslash Tau^\{-1\}(\backslash alpha))\}\{\backslash alpha\}$, where $\backslash tau(x)=\backslash frac\{\backslash Gamma\backslash bigl(\backslash frac\{\backslash nu+1\}\{2\}\backslash bigr)\}\{\backslash Gamma\backslash bigl(\backslash frac\{\backslash nu\}\{2\}\backslash bigr)\backslash sqrt\{\backslash pi\backslash nu\}\}\backslash Bigl(1+\backslash frac\{x^2\}\{\backslash nu\}\backslash Bigr)^\{-\backslash frac\{\backslash nu+1\}\{2\}\}$ is the standard t-distribution p.d.f., $\backslash Tau(x)$ is the standard t-distribution c.d.f., so $\backslash Tau^\{-1\}(\backslash alpha)$ is the standard t-distribution quantile.

If the loss of a portfolio $L$ follows generalized Student's t-distribution, the expected shortfall is equal to $\backslash operatorname\{ES\}\_\backslash alpha(L)\; =\; \backslash mu+\backslash sigma\backslash frac\{\backslash nu+(\backslash Tau^\{-1\}(\backslash alpha))^2\}\{\backslash nu-1\}\backslash frac\{\backslash tau(\backslash Tau^\{-1\}(\backslash alpha))\}\{1-\backslash alpha\}$.

If the payoff of a portfolio $X$ follows the Laplace distribution with the p.d.f.

: $f(x)\; =\; \backslash frac\{1\}\{2b\}e^\{-|x-\backslash mu|/b\}$

and the c.d.f.

: $F(x)\; =\; \backslash begin\{cases\}$

1 - \frac{1}{2} e^{-(x-\mu)/b} & \text{if }x \geq \mu,\\[4pt]

\frac{1}{2} e^{(x-\mu)/b} & \text{if }x < \mu.

\end{cases}

then the expected shortfall is equal to $\backslash operatorname\{ES\}\_\backslash alpha(X)\; =\; -\backslash mu\; +\; b(1\; -\; \backslash ln\; 2\backslash alpha)$ for $\backslash alpha\; \backslash le\; 0.5$.

If the loss of a portfolio $L$ follows the Laplace distribution, the expected shortfall is equal to

: $\backslash operatorname\{ES\}\_\backslash alpha(L)\; =\; \backslash begin\{cases\}$

\mu + b \frac{\alpha}{1-\alpha} (1-\ln2\alpha) & \text{if }\alpha < 0.5,\\[4pt]

\mu + b[1 - \ln(2(1-\alpha))] & \text{if }\alpha \ge 0.5.

\end{cases}

If the payoff of a portfolio $X$ follows the logistic distribution with p.d.f. $f(x)\; =\; \backslash frac\{1\}\{s\}\; e^\{-\backslash frac\{x-\backslash mu\}\{s\}\}\backslash left(1+e^\{-\backslash frac\{x-\backslash mu\}\{s\}\}\backslash right)^\{-2\}$ and the c.d.f. $F(x)\; =\; \backslash left(1+e^\{-\backslash frac\{x-\backslash mu\}\{s\}\}\backslash right)^\{-1\}$ then the expected shortfall is equal to $\backslash operatorname\{ES\}\_\backslash alpha(X)\; =\; -\backslash mu\; +\; s\; \backslash ln\backslash frac\{(1-\backslash alpha)^\{1-\backslash frac\{1\}\{\backslash alpha\}\}\}\{\backslash alpha\}$.

If the loss of a portfolio $L$ follows the logistic distribution, the expected shortfall is equal to $\backslash operatorname\{ES\}\_\backslash alpha(L)\; =\; \backslash mu\; +\; s\backslash frac\{-\backslash alpha\backslash ln\backslash alpha-(1-\backslash alpha)\backslash ln(1-\backslash alpha)\}\{1-\backslash alpha\}$.

If the loss of a portfolio $L$ follows the exponential distribution with p.d.f. and the c.d.f. then the expected shortfall is equal to $\backslash operatorname\{ES\}\_\backslash alpha(L)\; =\; \backslash frac\{-\backslash ln(1-\backslash alpha)+1\}\{\backslash lambda\}$.

If the loss of a portfolio $L$ follows the Pareto distribution with p.d.f. $f(x)\; =\; \backslash begin\{cases\}$

\frac{a x_m^a}{x^{a+1}} & \text{if }x \geq x_m,\\

0 & \text{if }x < x_m.

\end{cases} and the c.d.f. $F(x)\; =\; \backslash begin\{cases\}$

1 - (x_m/x)^a & \text{if }x \geq x_m,\\

0 & \text{if }x < x_m.

\end{cases} then the expected shortfall is equal to $\backslash operatorname\{ES\}\_\backslash alpha(L)\; =\; \backslash frac\{x\_m\; a\}\{(1-\backslash alpha)^\{1/a\}(a-1)\}$.

If the loss of a portfolio $L$ follows the GPD with p.d.f.

: $f(x)\; =\; \backslash frac\{1\}\{s\}\; \backslash left(\; 1+\backslash frac\{\backslash xi\; (x-\backslash mu)\}\{s\}\; \backslash right)^\{\backslash left(-\backslash frac\{1\}\{\backslash xi\}-1\backslash right)\}$

and the c.d.f.

: $F(x)\; =\; \backslash begin\{cases\}$

1 - \left(1+\frac{\xi(x-\mu)}{s}\right)^{-1 /\xi} & \text{if }\xi \ne 0,\\

1-\exp \left( -\frac{x-\mu}{s} \right) & \text{if }\xi = 0.

\end{cases}

then the expected shortfall is equal to

: $\backslash operatorname\{ES\}\_\backslash alpha(L)\; =\; \backslash begin\{cases\}$

\mu + s \left[ \frac{(1-\alpha)^{-\xi}}{1-\xi}+\frac{(1-\alpha)^{-\xi}-1}{\xi} \right] & \text{if }\xi \ne 0,\\

\mu + s \left[1 - \ln(1-\alpha) \right] & \text{if }\xi = 0,

\end{cases}

and the VaR is equal to

: $\backslash operatorname\{VaR\}\_\backslash alpha(L)\; =\; \backslash begin\{cases\}$

\mu + s \frac{(1-\alpha)^{-\xi}-1}{\xi} & \text{if }\xi \ne 0,\\

\mu - s \ln(1-\alpha) & \text{if }\xi = 0.

\end{cases}

If the loss of a portfolio $L$ follows the Weibull distribution with p.d.f. $f(x)\; =\; \backslash begin\{cases\}$

\frac{k}{\lambda} \left(\frac{x}{\lambda}\right)^{k-1} e^{-(x/\lambda)^k} & \text{if }x \geq 0,\\

0 & \text{if }x < 0.

\end{cases} and the c.d.f. $F(x)\; =\; \backslash begin\{cases\}$

1 - e^{-(x/\lambda)^k} & \text{if }x \geq 0,\\

0 & \text{if }x < 0.

\end{cases} then the expected shortfall is equal to $\backslash operatorname\{ES\}\_\backslash alpha(L)\; =\; \backslash frac\{\backslash lambda\}\{1-\backslash alpha\}\; \backslash Gamma\backslash left(1+\backslash frac\{1\}\{k\},-\backslash ln(1-\backslash alpha)\backslash right)$, where $\backslash Gamma(s,x)$ is the upper incomplete gamma function.

If the payoff of a portfolio $X$ follows the GEV with p.d.f. $f(x)\; =\; \backslash begin\{cases\}$

\frac{1}{\sigma} \left( 1+\xi \frac{ x-\mu}{\sigma} \right)^{-\frac{1}{\xi}-1} \exp\left[-\left( 1 + \xi \frac{x-\mu}{\sigma} \right)^{-{1}/{\xi}}\right] & \text{if } \xi \ne 0,\\

\frac{1}{\sigma}e^{-\frac{x-\mu}{\sigma}}e^{-e^{-\frac{x-\mu}{\sigma}}} & \text{if } \xi = 0.

\end{cases} and c.d.f. $F(x)\; =\; \backslash begin\{cases\}$

\exp\left(-\left(1+\xi\frac{x-\mu}{\sigma}\right)^{-{1}/{\xi}}\right) & \text{if }\xi \ne 0,\\

\exp\left(-e^{-\frac{x-\mu}{\sigma}}\right) & \text{if }\xi = 0.

\end{cases} then the expected shortfall is equal to $\backslash operatorname\{ES\}\_\backslash alpha(X)\; =\; \backslash begin\{cases\}$

-\mu - \frac{\sigma}{\alpha \xi} \big[ \Gamma(1-\xi,-\ln\alpha)-\alpha \big] & \text{if }\xi \ne 0,\\

-\mu - \frac{\sigma}{\alpha} \big[ \text{li}(\alpha) - \alpha \ln(-\ln \alpha) \big] & \text{if }\xi = 0.

\end{cases} and the VaR is equal to $\backslash operatorname\{VaR\}\_\backslash alpha(X)\; =\; \backslash begin\{cases\}$

-\mu - \frac{\sigma}{\xi} \left[(-\ln \alpha)^{-\xi}-1 \right] & \text{if }\xi \ne 0,\\

-\mu + \sigma \ln(-\ln\alpha) & \text{if }\xi = 0.

\end{cases}, where $\backslash Gamma(s,x)$ is the upper incomplete gamma function, $\backslash mathrm\{li\}(x)\; =\; \backslash int\; \backslash frac\{dx\}\{\backslash ln\; x\}$ is the logarithmic integral function.{{Cite journal|ssrn=3200629|title=Conditional Value-at-Risk for Uncommon Distributions|last=Khokhlov|first=Valentyn|date=2018-06-21|doi=10.2139/ssrn.3200629 |s2cid=219371851 }}

If the loss of a portfolio $L$ follows the GEV, then the expected shortfall is equal to $\backslash operatorname\{ES\}\_\backslash alpha(X)\; =\; \backslash begin\{cases\}$

\mu + \frac{\sigma}{(1-\alpha) \xi} \bigl[ \gamma(1-\xi,-\ln\alpha)-(1-\alpha) \bigr] & \text{if }\xi \ne 0,\\

\mu + \frac{\sigma}{1-\alpha} \bigl[y - \text{li}(\alpha) + \alpha \ln(-\ln \alpha) \bigr] & \text{if }\xi = 0.

\end{cases}, where $\backslash gamma(s,x)$ is the lower incomplete gamma function, $y$ is the Euler-Mascheroni constant.

If the payoff of a portfolio $X$ follows the GHS distribution with p.d.f. $f(x)\; =\; \backslash frac\{1\}\{2\; \backslash sigma\}\; \backslash operatorname\{sech\}\backslash left(\backslash frac\{\backslash pi\}\{2\}\; \backslash frac\{x-\backslash mu\}\{\backslash sigma\}\backslash right)$and the c.d.f. $F(x)\; =\; \backslash frac\{2\}\{\backslash pi\}\backslash arctan\backslash left[\backslash exp\backslash left(\backslash frac\{\backslash pi\}\{2\}\backslash frac\{x-\backslash mu\}\{\backslash sigma\}\backslash right)\backslash right]$ then the expected shortfall is equal to $\backslash operatorname\{ES\}\_\backslash alpha(X)\; =$

- \mu - \frac{2\sigma}{\pi} \ln\left( \tan \frac{\pi\alpha}{2} \right)

- \frac{2\sigma}{\pi^2\alpha}i\left[\operatorname{Li}_2\left(-i\tan\frac{\pi\alpha}{2}\right)-\operatorname{Li}_2\left(i\tan\frac{\pi\alpha}{2}\right)\right], where $\backslash operatorname\{Li\}\_2$ is the dilogarithm and $i=\backslash sqrt\{-1\}$ is the imaginary unit.

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 expected shortfall is equal to $\backslash operatorname\{ES\}\_\backslash alpha(X)\; =$

-\xi - \frac{\lambda}{2\alpha}

\left[

\exp\left(\frac{1-2\gamma\delta}{2\delta^2}\right) \;

\Phi\left(\Phi^{-1}(\alpha)-\frac{1}{\delta}\right)

- \exp\left(\frac{1+2\gamma\delta}{2\delta^2}\right) \;

\Phi\left(\Phi^{-1}(\alpha)+\frac{1}{\delta}\right)

\right], where $\backslash Phi$ is the c.d.f. of the standard normal distribution.{{Cite journal|ssrn=1855986|title=Moment-Based CVaR Estimation: Quasi-Closed Formulas|last=Stucchi|first=Patrizia|date=2011-05-31|doi=10.2139/ssrn.1855986 |s2cid=124145569 }}

If the payoff of a portfolio $X$ follows the Burr type XII distribution the p.d.f. $f(x)\; =\; \backslash frac\{ck\}\{\backslash beta\}$

\left(\frac{x-\gamma}{\beta}\right)^{c-1}

\left[1+\left(\frac{x-\gamma}{\beta} \right)^c\right]^{-k-1} and the c.d.f. $F(x)\; =\; 1-\backslash left[1+\backslash left(\backslash frac\{x-\backslash gamma\}\{\backslash beta\}\; \backslash right)^c\; \backslash right]^\{-k\}$, the expected shortfall is equal to $\backslash operatorname\{ES\}\_\backslash alpha(X)\; =$

- \gamma

- \frac{\beta}{\alpha}

\left( (1-\alpha)^{-1/k}-1 \right)^{1/c}

\left[ \alpha -1+{_2F_1}\left(\frac{1}{c},k;1+\frac{1}{c};1-(1-\alpha)^{-1/k}\right) \right], where $\_2F\_1$ is the hypergeometric function. Alternatively, $\backslash operatorname\{ES\}\_\backslash alpha(X)\; =$

- \gamma

- \frac{\beta}{\alpha}

\frac{ck}{c+1}

\left( (1-\alpha)^{-1/k}-1 \right)^{1+\frac{1}{c}}

{_2F_1}\left(1+\frac{1}{c}, k+1;2+\frac{1}{c};1-(1-\alpha)^{-1/k}\right) .

If the payoff of a portfolio $X$ follows the Dagum distribution with p.d.f. $f(x)\; =$

\frac{ck}{\beta}

\left(\frac{x-\gamma}{\beta}\right)^{ck-1}

\left[1+\left(\frac{x-\gamma}{\beta}\right)^c\right]^{-k-1} and the c.d.f. $F(x)\; =\; \backslash left[1+\backslash left(\backslash frac\{x-\backslash gamma\}\{\backslash beta\}\backslash right)^\{-c\}\backslash right]^\{-k\}$, the expected shortfall is equal to $\backslash operatorname\{ES\}\_\backslash alpha(X)\; =$

- \gamma

- \frac{\beta}{\alpha}

\frac{ck}{ck+1}

\left( \alpha^{-1/k}-1 \right)^{-k-\frac{1}{c}}

{_2F_1}\left(k+1,k+\frac{1}{c};k+1+\frac{1}{c};-\frac{1}{\alpha^{-1/k}-1}\right) , where $\_2F\_1$ is the hypergeometric function.

If the payoff of a portfolio $X$ follows lognormal distribution, i.e. the random variable $\backslash ln(1+X)$ follows the normal distribution with p.d.f. $f(x)\; =\; \backslash frac\{1\}\{\backslash sqrt\{2\backslash pi\}\backslash sigma\}e^\{-\backslash frac\{(x-\backslash mu)^2\}\{2\backslash sigma^2\}\}$, then the expected shortfall is equal to $\backslash operatorname\{ES\}\_\backslash alpha(X)\; =\; 1\; -\; \backslash exp\backslash left(\backslash mu+\backslash frac\{\backslash sigma^2\}\{2\}\backslash right)\; \backslash frac\{\backslash Phi\backslash left(\backslash Phi^\{-1\}(\backslash alpha)-\backslash sigma\backslash right)\}\{\backslash alpha\}$, where $\backslash Phi(x)$ is the standard normal c.d.f., so $\backslash Phi^\{-1\}(\backslash alpha)$ is the standard normal quantile.{{Cite SSRN|ssrn=3197929|title=Conditional Value-at-Risk for Log-Distributions|last=Khokhlov|first=Valentyn|date=2018-06-17}}

If the payoff of a portfolio $X$ follows log-logistic distribution, i.e. the random variable $\backslash ln(1+X)$ follows the logistic distribution with p.d.f. $f(x)\; =\; \backslash frac\{1\}\{s\}\; e^\{-\backslash frac\{x-\backslash mu\}\{s\}\}\; \backslash left(1+e^\{-\backslash frac\{x-\backslash mu\}\{s\}\}\backslash right)^\{-2\}$, then the expected shortfall is equal to $\backslash operatorname\{ES\}\_\backslash alpha(X)\; =\; 1-\backslash frac\{e^\backslash mu\}\{\backslash alpha\}I\_\backslash alpha(1+s,1-s)\backslash frac\{\backslash pi\; s\}\{\backslash sin\backslash pi\; s\}$, where $I\_\backslash alpha$ is the regularized incomplete beta function, $I\_\backslash alpha(a,b)=\backslash frac\{\backslash Beta\_\backslash alpha(a,b)\}\{\backslash Beta(a,b)\}$.

As the incomplete beta function is defined only for positive arguments, for a more generic case the expected shortfall can be expressed with the hypergeometric function: $\backslash operatorname\{ES\}\_\backslash alpha(X)\; =\; 1-\backslash frac\{e^\backslash mu\; \backslash alpha^s\}\{s+1\}\; \{\_2F\_1\}(s,s+1;s+2;\backslash alpha)$.

If the loss of a portfolio $L$ follows log-logistic distribution with p.d.f. $f(x)\; =\; \backslash frac\{\backslash frac\{b\}\{a\}(x/a)^\{b-1\}\}\{(1+(x/a)^b)^2\}$ and c.d.f. $F(x)\; =\; \backslash frac\{1\}\{1+(x/a)^\{-b\}\}$, then the expected shortfall is equal to $\backslash operatorname\{ES\}\_\backslash alpha(L)\; =$

\frac{a}{1-\alpha}

\left[

\frac{\pi}{b} \csc\left(\frac{\pi}{b}\right)

- \Beta_\alpha \left(\frac{1}{b}+1,1-\frac{1}{b}\right)

\right], where $B\_\backslash alpha$ is the incomplete beta function.

If the payoff of a portfolio $X$ follows log-Laplace distribution, i.e. the random variable $\backslash ln(1+X)$ follows the Laplace distribution the p.d.f. $f(x)\; =\; \backslash frac\{1\}\{2b\}e^\{-\backslash frac$

: $\backslash operatorname\{ES\}\_\backslash alpha(X)\; =\; \backslash begin\{cases\}$

1 - \frac{e^\mu (2\alpha)^b}{b+1} & \text{if }\alpha \le 0.5,\\

1 - \frac{e^\mu 2^{-b}}{\alpha(b-1)} \left[(1-\alpha)^{(1-b)}-1\right] & \text{if } \alpha > 0.5.

\end{cases}

If the payoff of a portfolio $X$ follows log-GHS distribution, i.e. the random variable $\backslash ln(1+X)$ follows the GHS distribution with p.d.f. $f(x)\; =\; \backslash frac\{1\}\{2\; \backslash sigma\}\; \backslash operatorname\{sech\}\; \backslash left(\backslash frac\{\backslash pi\}\{2\}\backslash frac\{x-\backslash mu\}\{\backslash sigma\}\backslash right)$, then the expected shortfall is equal to

: $\backslash operatorname\{ES\}\_\backslash alpha(X)\; =\; 1\; -\; \backslash frac\{1\}\{\backslash alpha(\backslash sigma+\{\backslash pi/2\})\}\; \backslash left(\backslash tan\backslash frac\{\backslash pi\; \backslash alpha\}\{2\}\backslash exp\backslash frac\{\backslash pi\; \backslash mu\}\{2\backslash sigma\}\backslash right)^\{2\backslash sigma/\backslash pi\}\; \backslash tan\backslash frac\{\backslash pi\; \backslash alpha\}\{2\}\; \{\_2F\_1\}\backslash left(1,\backslash frac\{1\}\{2\}+\backslash frac\{\backslash sigma\}\{\backslash pi\};\backslash frac\{3\}\{2\}+\backslash frac\{\backslash sigma\}\{\backslash pi\};-\backslash tan\backslash left(\backslash frac\{\backslash pi\; \backslash alpha\}\{2\}\backslash right)^2\backslash right),$

where $\_2F\_1$ is the hypergeometric function.

The conditional version of the expected shortfall at the time t is defined by

:$\backslash operatorname\{ES\}\_\backslash alpha^t(X)\; =\; \backslash operatorname\{ess\backslash sup\}\_\{Q\; \backslash in\; \backslash mathcal\{Q\}\_\{\backslash alpha\}^t\}\; E^Q[-X\; \backslash mid\; \backslash mathcal\{F\}\_t]$

where $\backslash mathcal\{Q\}\_\{\backslash alpha\}^t\; =\; \backslash left\backslash \{Q\; =\; P\backslash ,\backslash vert\_\{\backslash mathcal\{F\}\_t\}:\; \backslash frac\{dQ\}\{dP\}\; \backslash leq\; \backslash alpha\_t^\{-1\}\; \backslash text\{\; a.s.\}\backslash right\backslash \}$.{{cite journal|title=Conditional and dynamic convex risk measures|first1=Kai|last1=Detlefsen|first2=Giacomo|last2=Scandolo|journal=Finance Stoch.|volume=9|issue=4|pages=539–561|year=2005|url=http://www.dmd.unifi.it/scandolo/pdf/Scandolo-Detlefsen-05.pdf|access-date=October 11, 2011|doi=10.1007/s00780-005-0159-6|citeseerx=10.1.1.453.4944|s2cid=10579202}}{{Dead link|date=January 2012}}{{cite journal|title=Dynamic convex risk measures |first1=Beatrice |last1=Acciaio |first2=Irina |last2=Penner |year=2011 |url=http://wws.mathematik.hu-berlin.de/~penner/Acciaio_Penner.pdf |access-date=October 11, 2011 |url-status=dead |archive-url=https://web.archive.org/web/20110902182345/http://wws.mathematik.hu-berlin.de/~penner/Acciaio_Penner.pdf |archive-date=September 2, 2011 }}

This is not a time-consistent risk measure. The time-consistent version is given by

:$\backslash rho\_\{\backslash alpha\}^t(X)\; =\; \backslash operatorname\{ess\backslash sup\}\_\{Q\; \backslash in\; \backslash tilde\{\backslash mathcal\{Q\}\}\_\{\backslash alpha\}^t\}\; E^Q[-X\backslash mid\backslash mathcal\{F\}\_t]$

such that{{cite journal|first1=Patrick|last1=Cheridito|first2=Michael|last2=Kupper|title=Composition of time-consistent dynamic monetary risk measures in discrete time|journal=International Journal of Theoretical and Applied Finance|date=May 2010|url=http://wws.mathematik.hu-berlin.de/~kupper/papers/comp2010.pdf|access-date=February 4, 2011|url-status=dead|archive-url=https://web.archive.org/web/20110719042954/http://wws.mathematik.hu-berlin.de/~kupper/papers/comp2010.pdf|archive-date=July 19, 2011}}

:$\backslash tilde\{\backslash mathcal\{Q\}\}\_\{\backslash alpha\}^t\; =\; \backslash left\backslash \{Q\; \backslash ll\; P:\; \backslash operatorname\{E\}\backslash left[\backslash frac\{dQ\}\{dP\}\; \backslash mid\; \backslash mathcal\{F\}\_\{\backslash tau+1\}\; \backslash right]\; \backslash leq\; \backslash alpha\_t^\{-1\}\; \backslash operatorname\{E\}\backslash left[\backslash frac\{dQ\}\{dP\}\; \backslash mid\; \backslash mathcal\{F\}\_\{\backslash tau\}\backslash right]\; \backslash ;\; \backslash forall\; \backslash tau\; \backslash geq\; t\; \backslash text\{\; a.s.\}\backslash right\backslash \}.$

• Coherent risk measure
• EMP for stochastic programming – solution technology for optimization problems involving ES and VaR
• Entropic value at risk
• Value at risk

Methods of statistical estimation of VaR and ES can be found in Embrechts et al.Embrechts P., Kluppelberg C. and Mikosch T., Modelling Extremal Events for Insurance and Finance. Springer (1997). and Novak.Novak S.Y., Extreme value methods with applications to finance. Chapman & Hall/CRC Press (2011). {{ISBN|978-1-4398-3574-6}}. When forecasting VaR and ES, or optimizing portfolios to minimize tail risk, it is important to account for asymmetric dependence and non-normalities in the distribution of stock returns such as auto-regression, asymmetric volatility, skewness, and kurtosis.{{cite journal|last1=Low|first1=R.K.Y.|last2=Alcock|first2=J.|last3=Faff|first3=R.|last4=Brailsford|first4=T.|title=Canonical vine copulas in the context of modern portfolio management: Are they worth it?|journal=Journal of Banking & Finance|date=2013|volume=37|issue=8|pages=3085–3099|doi=10.1016/j.jbankfin.2013.02.036|s2cid=154138333|url=https://espace.library.uq.edu.au/view/UQ:297895/EC15UQ297895.pdf}}

{{Reflist}}

• [http://www.ise.ufl.edu/uryasev/files/2011/11/CVaR1_JOR.pdf Rockafellar, Uryasev: Optimization of conditional Value-at-Risk, 2000.]
• [https://arxiv.org/abs/cond-mat/0104295 C. Acerbi and D. Tasche: On the Coherence of Expected Shortfall, 2002.]
• [http://www.ise.ufl.edu/uryasev/files/2011/11/cvar2_jbf.pdf Rockafellar, Uryasev: Conditional Value-at-Risk for general loss distributions, 2002.]
• [https://web.archive.org/web/20070721133000/http://www.finance-and-physics.org/susinno/acerbi1.pdf Acerbi: Spectral measures of risk, 2005]
• [https://editorialexpress.com/cgi-bin/conference/download.cgi?db_name=QMF2004&paper_id=142: Phi-Alpha optimal portfolios and extreme risk management, Best of Wilmott, 2003]
• "[https://onlinelibrary.wiley.com/doi/10.1111/1467-9965.00068 Coherent measures of Risk]", Philippe Artzner, Freddy Delbaen, Jean-Marc Eber, and David Heath

Category:Financial risk modeling

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