Expected shortfall

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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 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), average value at risk (AVaR), and expected tail loss (ETL).

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

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

Formal definition[edit]

If (an Lp space) is the payoff of a portfolio at some future time and then we define the expected shortfall as

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

where is the lower -quantile and is the indicator function.[1] The dual representation is

where is the set of probability measures which are absolutely continuous to the physical measure such that almost surely.[2] Note that is the Radon–Nikodym derivative of with respect to .

Expected shortfall can be generalized to a general class of coherent risk measures on spaces (Lp space) with a corresponding dual characterization in the corresponding dual space. The domain can be extended for more general Orlitz Hearts.[3]

If the underlying distribution for is a continuous distribution then the expected shortfall is equivalent to the tail conditional expectation defined by .[4]

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

[5][6]

Examples[edit]

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 ending value
of event 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 for a few values of :

expected shortfall
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 , 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 , 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

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

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

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

−100
−20
0
50

Properties[edit]

The expected shortfall increases as decreases.

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

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

Optimization of expected shortfall[edit]

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.[7] 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 for the expected shortfall:

Where and is a loss function for a set of portfolio weights to be applied to the returns. Rockafellar/Uryasev proved that is convex with respect to 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 simulations of the portfolio constituents; this is often done using copulas. With these simulations in hand, the auxiliary function may be approximated by:
This is equivalent to the formulation:
Finally, choosing a linear loss function 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[edit]

Closed-form formulas exist for calculating the expected shortfall when the payoff of a portfolio or a corresponding loss follows a specific continuous distribution. In the former case the expected shortfall corresponds to the opposite number of the left-tail conditional expectation below :

Typical values of in this case are 5% and 1%.

For engineering or actuarial applications it is more common to consider the distribution of losses , the expected shortfall in this case corresponds to the right-tail conditional expectation above and the typical values of are 95% and 99%:

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

Normal distribution[edit]

If the payoff of a portfolio follows normal (Gaussian) distribution with the p.d.f. then the expected shortfall is equal to , where is the standard normal p.d.f., is the standard normal c.d.f., so is the standard normal quantile.[8]

If the loss of a portfolio follows normal distribution, the expected shortfall is equal to .[9]

Generalized Student's t-distribution[edit]

If the payoff of a portfolio follows generalized Student's t-distribution with the p.d.f. then the expected shortfall is equal to , where is the standard t-distribution p.d.f., is the standard t-distribution c.d.f., so is the standard t-distribution quantile.[8]

If the loss of a portfolio follows generalized Student's t-distribution, the expected shortfall is equal to .[9]

Laplace distribution[edit]

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

and the c.d.f.

then the expected shortfall is equal to for .[8]

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

[9]

Logistic distribution[edit]

If the payoff of a portfolio follows logistic distribution with the p.d.f. and the c.d.f. then the expected shortfall is equal to .[8]

If the loss of a portfolio follows logistic distribution, the expected shortfall is equal to .[9]

Exponential distribution[edit]

If the loss of a portfolio follows exponential distribution with the p.d.f. and the c.d.f. then the expected shortfall is equal to .[9]

Pareto distribution[edit]

If the loss of a portfolio follows Pareto distribution with the p.d.f. and the c.d.f. then the expected shortfall is equal to .[9]

Generalized Pareto distribution (GPD)[edit]

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

and the c.d.f.

then the expected shortfall is equal to

and the VaR is equal to

[9]

Weibull distribution[edit]

If the loss of a portfolio follows Weibull distribution with the p.d.f. and the c.d.f. then the expected shortfall is equal to , where is the upper incomplete gamma function.[9]

Generalized extreme value distribution (GEV)[edit]

If the payoff of a portfolio follows GEV with the p.d.f. and the c.d.f. then the expected shortfall is equal to and the VaR is equal to , where is the upper incomplete gamma function, is the logarithmic integral function.[10]

If the loss of a portfolio follows GEV, then the expected shortfall is equal to , where is the lower incomplete gamma function, is the Euler-Mascheroni constant.[9]

Generalized hyperbolic secant (GHS) distribution[edit]

If the payoff of a portfolio follows GHS distribution with the p.d.f. and the c.d.f. then the expected shortfall is equal to , where is the Spence's function, is the imaginary unit.[10]

Johnson's SU-distribution[edit]

If the payoff of a portfolio follows Johnson's SU-distribution with the c.d.f. then the expected shortfall is equal to , where is the c.d.f. of the standard normal distribution.[11]

Burr type XII distribution[edit]

If the payoff of a portfolio follows the Burr type XII distribution with the p.d.f. and the c.d.f. , the expected shortfall is equal to , where is the hypergeometric function. Alternatively, .[10]

Dagum distribution[edit]

If the payoff of a portfolio follows the Dagum distribution with the p.d.f. and the c.d.f. , the expected shortfall is equal to , where is the hypergeometric function.[10]

Lognormal distribution[edit]

If the payoff of a portfolio follows lognormal distribution, i.e. the random variable follows normal distribution with the p.d.f. , then the expected shortfall is equal to , where is the standard normal c.d.f., so is the standard normal quantile.[12]

Log-logistic distribution[edit]

If the payoff of a portfolio follows log-logistic distribution, i.e. the random variable follows logistic distribution with the p.d.f. , then the expected shortfall is equal to , where is the regularized incomplete beta function, .

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: .[12]

If the loss of a portfolio follows log-logistic distribution with p.d.f. and c.d.f. , then the expected shortfall is equal to , where is the incomplete beta function.[9]

Log-Laplace distribution[edit]

If the payoff of a portfolio follows log-Laplace distribution, i.e. the random variable follows Laplace distribution the p.d.f. , then the expected shortfall is equal to .[12]

Log-generalized hyperbolic secant (log-GHS) distribution[edit]

If the payoff of a portfolio follows log-GHS distribution, i.e. the random variable follows GHS distribution with the p.d.f. , then the expected shortfall is equal to , where is the hypergeometric function.[12]

Dynamic expected shortfall[edit]

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

where .[13][14]

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

such that

[15]

See also[edit]

Methods of statistical estimation of VaR and ES can be found in Embrechts et al.[16] and Novak.[17] 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.[18]

References[edit]

  1. ^ Carlo Acerbi; Dirk Tasche (2002). "Expected Shortfall: a natural coherent alternative to Value at Risk" (PDF). Economic Notes. 31 (2): 379–388. arXiv:cond-mat/0105191. doi:10.1111/1468-0300.00091. Retrieved April 25, 2012.
  2. ^ Föllmer, H.; Schied, A. (2008). "Convex and coherent risk measures" (PDF). Retrieved October 4, 2011. Cite journal requires |journal= (help)
  3. ^ 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.
  4. ^ "Average Value at Risk" (PDF). Archived from the original (PDF) on July 19, 2011. Retrieved February 2, 2011.
  5. ^ Julia L. Wirch; Mary R. Hardy. "Distortion Risk Measures: Coherence and Stochastic Dominance" (PDF). Archived from the original (PDF) on July 5, 2016. Retrieved March 10, 2012.
  6. ^ Balbás, A.; Garrido, J.; Mayoral, S. (2008). "Properties of Distortion Risk Measures" (PDF). Methodology and Computing in Applied Probability. 11 (3): 385. doi:10.1007/s11009-008-9089-z. hdl:10016/14071.
  7. ^ Rockafellar, R. Tyrrell; Uryasev, Stanislav (2000). "Optimization of conditional value-at-risk" (PDF). Journal of Risk. 2: 21–42.
  8. ^ a b c d Khokhlov, Valentyn (2016). "Conditional Value-at-Risk for Elliptical Distributions". Evropský časopis Ekonomiky a Managementu. 2 (6): 70–79.
  9. ^ a b c d e f g h i j 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].
  10. ^ a b c d Khokhlov, Valentyn (2018-06-21). "Conditional Value-at-Risk for Uncommon Distributions". SSRN 3200629. Cite journal requires |journal= (help)
  11. ^ Stucchi, Patrizia (2011-05-31). "Moment-Based CVaR Estimation: Quasi-Closed Formulas". SSRN 1855986. Cite journal requires |journal= (help)
  12. ^ a b c d Khokhlov, Valentyn (2018-06-17). "Conditional Value-at-Risk for Log-Distributions". SSRN 3197929. Cite journal requires |journal= (help)
  13. ^ Detlefsen, Kai; Scandolo, Giacomo (2005). "Conditional and dynamic convex risk measures" (PDF). Finance Stoch. 9 (4): 539–561. CiteSeerX 10.1.1.453.4944. doi:10.1007/s00780-005-0159-6. Retrieved October 11, 2011.[dead link]
  14. ^ Acciaio, Beatrice; Penner, Irina (2011). "Dynamic convex risk measures" (PDF). Archived from the original (PDF) on September 2, 2011. Retrieved October 11, 2011. Cite journal requires |journal= (help)
  15. ^ Cheridito, Patrick; Kupper, Michael (May 2010). "Composition of time-consistent dynamic monetary risk measures in discrete time" (PDF). International Journal of Theoretical and Applied Finance. Archived from the original (PDF) on July 19, 2011. Retrieved February 4, 2011.
  16. ^ Embrechts P., Kluppelberg C. and Mikosch T., Modelling Extremal Events for Insurance and Finance. Springer (1997).
  17. ^ Novak S.Y., Extreme value methods with applications to finance. Chapman & Hall/CRC Press (2011). ISBN 978-1-4398-3574-6.
  18. ^ 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.

External links[edit]