Expected shortfall

  (Redirected from Average value at 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 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 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 definitionEdit

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.[3] The dual representation is

 

where   is the set of probability measures which are absolutely continuous to the physical measure   such that   almost surely.[4] 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 Orlicz Hearts.[5]

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

 [7][8]

ExamplesEdit

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   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

PropertiesEdit

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 shortfallEdit

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   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 distributionsEdit

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 distributionEdit

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.[10]

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

Generalized Student's t-distributionEdit

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.[10]

If the loss of a portfolio   follows generalized Student's t-distribution, the expected shortfall is equal to  .[11]

Laplace distributionEdit

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  .[10]

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

 

Logistic distributionEdit

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  .[10]

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

Exponential distributionEdit

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  .[11]

Pareto distributionEdit

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  .[11]

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[11]

 

Weibull distributionEdit

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.[11]

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.[12]

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.[11]

Generalized hyperbolic secant (GHS) distributionEdit

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.[12]

Johnson's SU-distributionEdit

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.[13]

Burr type XII distributionEdit

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,  .[12]

Dagum distributionEdit

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.[12]

Lognormal distributionEdit

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.[14]

Log-logistic distributionEdit

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:  .[14]

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.[11]

Log-Laplace distributionEdit

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  .[14]

Log-generalized hyperbolic secant (log-GHS) distributionEdit

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.[14]

Dynamic expected shortfallEdit

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

 

where  .[15][16]

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

 

such that[17]

 

See alsoEdit

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]

ReferencesEdit

  1. ^ Rockafellar, R. Tyrrell; Uryasev, Stanislav (2000). "Optimization of conditional value-at-risk" (PDF). Journal of Risk. 2 (3): 21–42. doi:10.21314/JOR.2000.038.
  2. ^ Rockafellar, R. Tyrrell; Royset, Johannes (2010). "On Buffered Failure Probability in Design and Optimization of Structures" (PDF). Reliability Engineering and System Safety. 95 (5): 499–510. doi:10.1016/j.ress.2010.01.001.
  3. ^ 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. S2CID 10772757. Retrieved April 25, 2012.
  4. ^ Föllmer, H.; Schied, A. (2008). "Convex and coherent risk measures" (PDF). Retrieved October 4, 2011. Cite journal requires |journal= (help)
  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. S2CID 121880657.
  6. ^ "Average Value at Risk" (PDF). Archived from the original (PDF) on July 19, 2011. Retrieved February 2, 2011.
  7. ^ 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.
  8. ^ 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. S2CID 53327887.
  9. ^ Rockafellar, R. Tyrrell; Uryasev, Stanislav (2000). "Optimization of conditional value-at-risk" (PDF). Journal of Risk. 2 (3): 21–42. doi:10.21314/JOR.2000.038.
  10. ^ a b c d Khokhlov, Valentyn (2016). "Conditional Value-at-Risk for Elliptical Distributions". Evropský časopis Ekonomiky a Managementu. 2 (6): 70–79.
  11. ^ 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].
  12. ^ a b c d Khokhlov, Valentyn (2018-06-21). "Conditional Value-at-Risk for Uncommon Distributions". SSRN 3200629. Cite journal requires |journal= (help)
  13. ^ Stucchi, Patrizia (2011-05-31). "Moment-Based CVaR Estimation: Quasi-Closed Formulas". SSRN 1855986. Cite journal requires |journal= (help)
  14. ^ a b c d Khokhlov, Valentyn (2018-06-17). "Conditional Value-at-Risk for Log-Distributions". SSRN 3197929. Cite journal requires |journal= (help)
  15. ^ 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. S2CID 10579202. Retrieved October 11, 2011.[dead link]
  16. ^ 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)
  17. ^ 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.
  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?" (PDF). Journal of Banking & Finance. 37 (8): 3085–3099. doi:10.1016/j.jbankfin.2013.02.036. S2CID 154138333.

External linksEdit