Expected shortfall

Short description: Measure of financial risk

Expected shortfall (ES) is a risk measure—a concept used in the field of financial risk measurement to evaluate the market risk or credit risk of a portfolio. The "expected shortfall at q% level" is the expected return on the portfolio in the worst q% of cases. ES is an alternative to value at risk that is more sensitive to the shape of the tail of the loss distribution.

Expected shortfall is also called conditional value at risk (CVaR),[1] average value at risk (AVaR), expected tail loss (ETL), and superquantile.[2]

ES estimates the risk of an investment in a conservative way, focusing on the less profitable outcomes. For high values of q it ignores the most profitable but unlikely possibilities, while for small values of q it focuses on the worst losses. On the other hand, unlike the discounted maximum loss, even for lower values of q the expected shortfall does not consider only the single most catastrophic outcome. A value of q often used in practice is 5%.[citation needed]

Expected shortfall is considered a more useful risk measure than VaR because it is a coherent spectral measure of financial portfolio risk. It is calculated for a given quantile-level q and is defined to be the mean loss of portfolio value given that a loss is occurring at or below the q-quantile.

## Contents

* 1 Formal definition
* 2 Examples
* 3 Properties
* 4 Optimization of expected shortfall
* 5 Formulas for continuous probability distributions
* 5.1 Normal distribution
* 5.2 Generalized Student's t-distribution
* 5.3 Laplace distribution
* 5.4 Logistic distribution
* 5.5 Exponential distribution
* 5.6 Pareto distribution
* 5.7 Generalized Pareto distribution (GPD)
* 5.8 Weibull distribution
* 5.9 Generalized extreme value distribution (GEV)
* 5.10 Generalized hyperbolic secant (GHS) distribution
* 5.11 Johnson's SU-distribution
* 5.12 Burr type XII distribution
* 5.13 Dagum distribution
* 5.14 Lognormal distribution
* 5.15 Log-logistic distribution
* 5.16 Log-Laplace distribution
* 5.17 Log-generalized hyperbolic secant (log-GHS) distribution
* 6 Dynamic expected shortfall
* 7 See also
* 8 References
* 9 External links

## Formal definition

If X∈Lp(ℱ) (an Lp) is the payoff of a portfolio at some future time and 0<α<1 then we define the expected shortfall as

ESα(X)=−1α∫0αVaRγ(X)dγ

where VaRγ is the value at risk. This can be equivalently written as

ESα(X)=−1α(E⁡[X 1{X≤xα}]+xα(α−P[X≤xα]))

where xα=inf⁡{x∈ℝ:P(X≤x)≥α}=−VaRα(X) is the lower α-quantile and 1A(x)={1if x∈A0else is the indicator function.[3] Note, that the second term vanishes for random variables with continuous distribution functions.

The dual representation is

ESα(X)=infQ∈𝒬αEQ[X]

where 𝒬α is the set of probability measures which are absolutely continuous to the physical measure P such that dQdP≤α−1 almost surely.[4] Note that dQdP 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 Lp spaces (Lp space) with a corresponding dual characterization in the corresponding Lq dual space. The domain can be extended for more general Orlicz Hearts.[5]

If the underlying distribution for X is a continuous distribution then the expected shortfall is equivalent to the tail conditional expectation defined by TCEα(X)=E[−X∣X≤−VaRα(X)].[6]

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

g(x)={x1−αif 0≤x<1−α,1if 1−α≤x≤1.[7][8]

## Examples

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:

probability
of event  | ending value
of the portfolio

|

10%  | 0
30%  | 80
40%  | 100
20%  | 150

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:

probability
of event  | profit

|

10%  | −100
30%  | −20
40%  | 0
20%  | 50

From this table let us calculate the expected shortfall ESq for a few values of q:

q | expected shortfall ESq

|

5%  | 100
10%  | 100
20%  | 60
30%  | 46.6
40%  | 40
50%  | 32
60%  | 26.6
80%  | 20
90%  | 12.2
100%  | 6

To see how these values were calculated, consider the calculation of ES0.05, 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 ES0.20, 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

10100(−100)+10100(−20)20100=−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 −ES0.20 we used only 10 of the 30 cases per 100 provided by row 2).

As a final example, calculate −ES1. This is the expectation over all cases, or

0.1(−100)+0.3(−20)+0.4⋅0+0.2⋅50=−6.

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

q | VaRq

|

0%≤q<10% | −100
10%≤q<40% | −20
40%≤q<80% | 0
80%≤q≤100% | 50

## Properties

The expected shortfall ESq increases as q decreases.

The 100%-quantile expected shortfall ES1 equals negative of the expected value of the portfolio.

For a given portfolio, the expected shortfall ESq is greater than or equal to the Value at Risk VaRq at the same q level.

## Optimization of expected shortfall

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.[9] 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 Fα(w,γ) for the expected shortfall:Fα(w,γ)=γ+11−α∫ℓ(w,x)≥γ[ℓ(w,x)−γ]p(x)dxWhere γ=VaRα(X) and ℓ(w,x) is a loss function for a set of portfolio weights w∈ℝp to be applied to the returns. Rockafellar/Uryasev proved that Fα(w,γ) is convex with respect to γ 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:F~α(w,γ)=γ+1(1−α)J∑j=1J[ℓ(w,xj)−γ]+This is equivalent to the formulation:minγ,z,wγ+1(1−α)J∑j=1Jzj,s.t. zj≥ℓ(w,xj)−γ,zj≥0 Finally, choosing a linear loss function ℓ(w,xj)=−wTxj turns the optimization problem into a linear program. Using standard methods, it is then easy to find the portfolio that minimizes expected shortfall.

## Formulas for continuous probability distributions

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 −VaRα(X):

ESα(X)=E[−X∣X≤−VaRα(X)]=−1α∫0αVaRγ(X)dγ=−1α∫−∞−VaRα(X)xf(x)dx.

Typical values of α 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 VaRα(L) and the typical values of α are 95% and 99%:

ESα(L)=E⁡[L∣L≥VaRα(L)]=11−α∫α1VaRγ(L)dγ=11−α∫VaRα(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:

ESα(X)=−1αE⁡[X]+1−ααESα(L) and ESα(L)=11−αE⁡[L]+α1−αESα(X).

### Normal distribution

If the payoff of a portfolio X follows the normal (Gaussian) distribution with p.d.f. f(x)=12πσe−(x−μ)22σ2 then the expected shortfall is equal to ESα(X)=−μ+σφ(Φ−1(α))α, where φ(x)=12πe−x22 is the standard normal p.d.f., Φ(x) is the standard normal c.d.f., so Φ−1(α) is the standard normal quantile.[10]

If the loss of a portfolio L follows the normal distribution, the expected shortfall is equal to ESα(L)=μ+σφ(Φ−1(α))1−α.[11]

### Generalized Student's t-distribution

If the payoff of a portfolio X follows the generalized Student's t-distribution with p.d.f. f(x)=Γ(ν+12)Γ(ν2)πνσ(1+1ν(x−μσ)2)−ν+12 then the expected shortfall is equal to ESα(X)=−μ+σν+(T−1(α))2ν−1τ(T−1(α))α, where τ(x)=Γ(ν+12)Γ(ν2)πν(1+x2ν)−ν+12 is the standard t-distribution p.d.f., T(x) is the standard t-distribution c.d.f., so T−1(α) is the standard t-distribution quantile.[10]

If the loss of a portfolio L follows generalized Student's t-distribution, the expected shortfall is equal to ESα(L)=μ+σν+(T−1(α))2ν−1τ(T−1(α))α.[11]

### Laplace distribution

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

f(x)=12be−|x−μ|/b

and the c.d.f.

F(x)={1−12e−(x−μ)/bif x≥μ,12e(x−μ)/bif x<μ.

then the expected shortfall is equal to ESα(X)=−μ+b(1−ln⁡2α) for α≤0.5.[10]

If the loss of a portfolio L follows the Laplace distribution, the expected shortfall is equal to[11]

ESα(L)={μ+bα1−α(1−ln⁡2α)if α<0.5,μ+b[1−ln⁡(2(1−α))]if α≥0.5.

### Logistic distribution

If the payoff of a portfolio X follows the logistic distribution with p.d.f. f(x)=1se−x−μs(1+e−x−μs)−2 and the c.d.f. F(x)=(1+e−x−μs)−1 then the expected shortfall is equal to ESα(X)=−μ+sln⁡(1−α)1−1αα.[10]

If the loss of a portfolio L follows the logistic distribution, the expected shortfall is equal to ESα(L)=μ+s−αln⁡α−(1−α)ln⁡(1−α)1−α.[11]

### Exponential distribution

If the loss of a portfolio L follows the exponential distribution with p.d.f. f(x)={λe−λxif x≥0,0if x<0. and the c.d.f. F(x)={1−e−λxif x≥0,0if x<0. then the expected shortfall is equal to ESα(L)=−ln⁡(1−α)+1λ.[11]

### Pareto distribution

If the loss of a portfolio L follows the Pareto distribution with p.d.f. f(x)={axmaxa+1if x≥xm,0if x<xm. and the c.d.f. F(x)={1−(xm/x)aif x≥xm,0if x<xm. then the expected shortfall is equal to ESα(L)=xma(1−α)1/a(a−1).[11]

### Generalized Pareto distribution (GPD)

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

f(x)=1s(1+ξ(x−μ)s)(−1ξ−1)

and the c.d.f.

F(x)={1−(1+ξ(x−μ)s)−1/ξif ξ≠0,1−exp⁡(−x−μs)if ξ=0.

then the expected shortfall is equal to

ESα(L)={μ+s[(1−α)−ξ1−ξ+(1−α)−ξ−1ξ]if ξ≠0,μ+s[1−ln⁡(1−α)]if ξ=0,

and the VaR is equal to[11]

VaRα(L)={μ+s(1−α)−ξ−1ξif ξ≠0,μ−sln⁡(1−α)if ξ=0.

### Weibull distribution

If the loss of a portfolio L follows the Weibull distribution with p.d.f. f(x)={kλ(xλ)k−1e−(x/λ)kif x≥0,0if x<0. and the c.d.f. F(x)={1−e−(x/λ)kif x≥0,0if x<0. then the expected shortfall is equal to ESα(L)=λ1−αΓ(1+1k,−ln⁡(1−α)), where Γ(s,x) is the upper incomplete gamma function.[11]

### Generalized extreme value distribution (GEV)

If the payoff of a portfolio X follows the GEV with p.d.f. f(x)={1σ(1+ξx−μσ)−1ξ−1exp⁡[−(1+ξx−μσ)−1/ξ]if ξ≠0,1σe−x−μσe−e−x−μσif ξ=0. and c.d.f. F(x)={exp⁡(−(1+ξx−μσ)−1/ξ)if ξ≠0,exp⁡(−e−x−μσ)if ξ=0. then the expected shortfall is equal to ESα(X)={−μ−σαξ[Γ(1−ξ,−ln⁡α)−α]if ξ≠0,−μ−σα[li(α)−αln⁡(−ln⁡α)]if ξ=0. and the VaR is equal to VaRα(X)={−μ−σξ[(−ln⁡α)−ξ−1]if ξ≠0,−μ+σln⁡(−ln⁡α)if ξ=0., where Γ(s,x) is the upper incomplete gamma function, li(x)=∫dxln⁡x is the logarithmic integral function.[12]

If the loss of a portfolio L follows the GEV, then the expected shortfall is equal to ESα(X)={μ+σ(1−α)ξ[γ(1−ξ,−ln⁡α)−(1−α)]if ξ≠0,μ+σ1−α[y−li(α)+αln⁡(−ln⁡α)]if ξ=0., where γ(s,x) is the lower incomplete gamma function, y is the Euler-Mascheroni constant.[11]

### Generalized hyperbolic secant (GHS) distribution

If the payoff of a portfolio X follows the GHS distribution with p.d.f. f(x)=12σsech⁡(π2x−μσ)and the c.d.f. F(x)=2πarctan⁡[exp⁡(π2x−μσ)] then the expected shortfall is equal to ESα(X)=−μ−2σπln⁡(tan⁡πα2)−2σπ2αi[Li2(−itan⁡πα2)−Li2(itan⁡πα2)], where Li2 is the dilogarithm and i=−1 is the imaginary unit.[12]

### Johnson's SU-distribution

If the payoff of a portfolio X follows Johnson's SU-distribution with the c.d.f. F(x)=Φ[γ+δsinh−1(x−ξλ)] then the expected shortfall is equal to ESα(X)=−ξ−λ2α[exp⁡(1−2γδ2δ2)Φ(Φ−1(α)−1δ)−exp⁡(1+2γδ2δ2)Φ(Φ−1(α)+1δ)], where Φ is the c.d.f. of the standard normal distribution.[13]

### Burr type XII distribution

If the payoff of a portfolio X follows the Burr type XII distribution the p.d.f. f(x)=ckβ(x−γβ)c−1[1+(x−γβ)c]−k−1 and the c.d.f. F(x)=1−[1+(x−γβ)c]−k, the expected shortfall is equal to ESα(X)=−γ−βα((1−α)−1/k−1)1/c[α−1+2F1(1c,k;1+1c;1−(1−α)−1/k)], where 2F1 is the hypergeometric function. Alternatively, ESα(X)=−γ−βαckc+1((1−α)−1/k−1)1+1c2F1(1+1c,k+1;2+1c;1−(1−α)−1/k).[12]

### Dagum distribution

If the payoff of a portfolio X follows the Dagum distribution with p.d.f. f(x)=ckβ(x−γβ)ck−1[1+(x−γβ)c]−k−1 and the c.d.f. F(x)=[1+(x−γβ)−c]−k, the expected shortfall is equal to ESα(X)=−γ−βαckck+1(α−1/k−1)−k−1c2F1(k+1,k+1c;k+1+1c;−1α−1/k−1), where 2F1 is the hypergeometric function.[12]

### Lognormal distribution

If the payoff of a portfolio X follows lognormal distribution, i.e. the random variable ln⁡(1+X) follows the normal distribution with p.d.f. f(x)=12πσe−(x−μ)22σ2, then the expected shortfall is equal to ESα(X)=1−exp⁡(μ+σ22)Φ(Φ−1(α)−σ)α, where Φ(x) is the standard normal c.d.f., so Φ−1(α) is the standard normal quantile.[14]

### Log-logistic distribution

If the payoff of a portfolio X follows log-logistic distribution, i.e. the random variable ln⁡(1+X) follows the logistic distribution with p.d.f. f(x)=1se−x−μs(1+e−x−μs)−2, then the expected shortfall is equal to ESα(X)=1−eμαIα(1+s,1−s)πssin⁡πs, where Iα is the regularized incomplete beta function, Iα(a,b)=Bα(a,b)B(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: ESα(X)=1−eμαss+12F1(s,s+1;s+2;α).[14]

If the loss of a portfolio L follows log-logistic distribution with p.d.f. f(x)=ba(x/a)b−1(1+(x/a)b)2 and c.d.f. F(x)=11+(x/a)−b, then the expected shortfall is equal to ESα(L)=a1−α[πbcsc⁡(πb)−Bα(1b+1,1−1b)], where Bα is the incomplete beta function.[11]

### Log-Laplace distribution

If the payoff of a portfolio X follows log-Laplace distribution, i.e. the random variable ln⁡(1+X) follows the Laplace distribution the p.d.f. f(x)=12be−|x−μ|b, then the expected shortfall is equal to

ESα(X)={1−eμ(2α)bb+1if α≤0.5,1−eμ2−bα(b−1)[(1−α)(1−b)−1]if α>0.5.[14]

### 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 the GHS distribution with p.d.f. f(x)=12σsech⁡(π2x−μσ), then the expected shortfall is equal to

ESα(X)=1−1α(σ+π/2)(tan⁡πα2exp⁡πμ2σ)2σ/πtan⁡πα22F1(1,12+σπ;32+σπ;−tan⁡(πα2)2),

where 2F1 is the hypergeometric function.[14]

## Dynamic expected shortfall

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

ESαt(X)=esssupQ∈𝒬αtEQ[−X∣ℱt]

where 𝒬αt={Q=P|ℱt:dQdP≤αt−1 a.s.}.[15][16]

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

ραt(X)=esssupQ∈𝒬~αtEQ[−X∣ℱt]

such that[17]

𝒬~αt={Q≪P:E⁡[dQdP∣ℱτ+1]≤αt−1E⁡[dQdP∣ℱτ]∀τ≥t a.s.}.

## See also

* 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.[18] and Novak.[19] 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.[20]

## References

1. ↑ Rockafellar, R. Tyrrell; Uryasev, Stanislav (2000). "Optimization of conditional value-at-risk". Journal of Risk 2 (3): 21–42. doi:10.21314/JOR.2000.038. https://sites.math.washington.edu/~rtr/papers/rtr179-CVaR1.pdf.
2. ↑ Rockafellar, R. Tyrrell; Royset, Johannes (2010). "On Buffered Failure Probability in Design and Optimization of Structures". Reliability Engineering and System Safety 95 (5): 499–510. doi:10.1016/j.ress.2010.01.001. https://sites.math.washington.edu/~rtr/papers/rtr211-BufferedProb.pdf.
3. ↑ Carlo Acerbi; Dirk Tasche (2002). "Expected Shortfall: a natural coherent alternative to Value at Risk". Economic Notes 31 (2): 379–388. doi:10.1111/1468-0300.00091. http://www.bis.org/bcbs/ca/acertasc.pdf. Retrieved April 25, 2012.
4. ↑ Föllmer, H.; Schied, A. (2008). Convex and coherent risk measures. http://wws.mathematik.hu-berlin.de/~foellmer/papers/CCRM.pdf. Retrieved October 4, 2011.
5. ↑ Patrick Cheridito; Tianhui Li (2008). "Dual characterization of properties of risk measures on Orlicz hearts". Mathematics and Financial Economics 2: 2–29. doi:10.1007/s11579-008-0013-7.
6. ↑ "Average Value at Risk". https://statistik.ets.kit.edu/download/doc_secure1/7_StochModels.pdf.
7. ↑ Julia L. Wirch; Mary R. Hardy. "Distortion Risk Measures: Coherence and Stochastic Dominance". http://pascal.iseg.utl.pt/~cemapre/ime2002/main_page/papers/JuliaWirch.pdf.
8. ↑ Balbás, A.; Garrido, J.; Mayoral, S. (2008). "Properties of Distortion Risk Measures". Methodology and Computing in Applied Probability 11 (3): 385. doi:10.1007/s11009-008-9089-z. http://e-archivo.uc3m.es/bitstream/10016/14071/1/properties_balbas_MCAP_2009_ps.pdf.
9. ↑ Rockafellar, R. Tyrrell; Uryasev, Stanislav (2000). "Optimization of conditional value-at-risk". Journal of Risk 2 (3): 21–42. doi:10.21314/JOR.2000.038. https://sites.math.washington.edu/~rtr/papers/rtr179-CVaR1.pdf.
10. ↑ 10.0 10.1 10.2 10.3 Khokhlov, Valentyn (2016). "Conditional Value-at-Risk for Elliptical Distributions". Evropský časopis Ekonomiky a Managementu 2 (6): 70–79.
11. ↑ 11.00 11.01 11.02 11.03 11.04 11.05 11.06 11.07 11.08 11.09 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". arXiv:1811.11301 [q-fin.RM].
12. ↑ 12.0 12.1 12.2 12.3 Khokhlov, Valentyn (2018-06-21). Conditional Value-at-Risk for Uncommon Distributions. doi:10.2139/ssrn.3200629.
13. ↑ Stucchi, Patrizia (2011-05-31). Moment-Based CVaR Estimation: Quasi-Closed Formulas. doi:10.2139/ssrn.1855986.
14. ↑ 14.0 14.1 14.2 14.3 Khokhlov, Valentyn (2018-06-17). "Conditional Value-at-Risk for Log-Distributions". SSRN 3197929.
15. ↑ Detlefsen, Kai; Scandolo, Giacomo (2005). "Conditional and dynamic convex risk measures". Finance Stoch. 9 (4): 539–561. doi:10.1007/s00780-005-0159-6. http://www.dmd.unifi.it/scandolo/pdf/Scandolo-Detlefsen-05.pdf. Retrieved October 11, 2011. [|permanent dead link|dead link}}]
16. ↑ Acciaio, Beatrice; Penner, Irina (2011). Dynamic convex risk measures. http://wws.mathematik.hu-berlin.de/~penner/Acciaio_Penner.pdf. Retrieved October 11, 2011.
17. ↑ Cheridito, Patrick; Kupper, Michael (May 2010). "Composition of time-consistent dynamic monetary risk measures in discrete time". International Journal of Theoretical and Applied Finance. http://wws.mathematik.hu-berlin.de/~kupper/papers/comp2010.pdf. Retrieved February 4, 2011.
18. ↑ Embrechts P., Kluppelberg C. and Mikosch T., Modelling Extremal Events for Insurance and Finance. Springer (1997).
19. ↑ Novak S.Y., Extreme value methods with applications to finance. Chapman & Hall/CRC Press (2011). ISBN 978-1-4398-3574-6.
20. ↑ Low, R.K.Y.; Alcock, J.; Faff, R.; Brailsford, T. (2013). "Canonical vine copulas in the context of modern portfolio management: Are they worth it?". Journal of Banking & Finance 37 (8): 3085–3099. doi:10.1016/j.jbankfin.2013.02.036. https://espace.library.uq.edu.au/view/UQ:297895/EC15UQ297895.pdf.

## External links

* Rockafellar, Uryasev: Optimization of conditional Value-at-Risk, 2000.
* C. Acerbi and D. Tasche: On the Coherence of Expected Shortfall, 2002.
* Rockafellar, Uryasev: Conditional Value-at-Risk for general loss distributions, 2002.
* Acerbi: Spectral measures of risk, 2005
* Phi-Alpha optimal portfolios and extreme risk management, Best of Wilmott, 2003
* "Coherent measures of Risk", Philippe Artzner, Freddy Delbaen, Jean-Marc Eber, and David Heath

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