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 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 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.
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 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
|
Formulas for continuous probability distributions
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
If the payoff of a portfolio follows the normal (Gaussian) distribution with 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 the normal distribution, the expected shortfall is equal to .[11]
Generalized Student's t-distribution
If the payoff of a portfolio follows the generalized Student's t-distribution with 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 distribution
If the payoff of a portfolio follows the 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 the Laplace distribution, the expected shortfall is equal to[11]
Logistic distribution
If the payoff of a portfolio follows the logistic distribution with p.d.f. and the c.d.f. then the expected shortfall is equal to .[10]
If the loss of a portfolio follows the logistic distribution, the expected shortfall is equal to .[11]
Exponential distribution
If the loss of a portfolio follows the exponential distribution with p.d.f. and the c.d.f. then the expected shortfall is equal to .[11]
Pareto distribution
If the loss of a portfolio follows the Pareto distribution with p.d.f. and the c.d.f. then the expected shortfall is equal to .[11]
Generalized Pareto distribution (GPD)
If the loss of a portfolio follows the GPD with p.d.f.
and the c.d.f.
then the expected shortfall is equal to
and the VaR is equal to[11]
Weibull distribution
If the loss of a portfolio follows the Weibull distribution with 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)
If the payoff of a portfolio follows the GEV with p.d.f. and 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 the 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) distribution
If the payoff of a portfolio follows the GHS distribution with p.d.f. and the c.d.f. then the expected shortfall is equal to , where is the dilogarithm and is the imaginary unit.[12]
Johnson's SU-distribution
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 distribution
If the payoff of a portfolio follows the Burr type XII distribution the p.d.f. and the c.d.f. , the expected shortfall is equal to , where is the hypergeometric function. Alternatively, .[12]
Dagum distribution
If the payoff of a portfolio follows the Dagum distribution with p.d.f. and the c.d.f. , the expected shortfall is equal to , where is the hypergeometric function.[12]
Lognormal distribution
If the payoff of a portfolio follows lognormal distribution, i.e. the random variable follows the normal distribution with 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 distribution
If the payoff of a portfolio follows log-logistic distribution, i.e. the random variable follows the logistic distribution with 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 distribution
If the payoff of a portfolio follows log-Laplace distribution, i.e. the random variable follows the Laplace distribution the p.d.f. , then the expected shortfall is equal to
- [14]
Log-generalized hyperbolic secant (log-GHS) distribution
If the payoff of a portfolio follows log-GHS distribution, i.e. the random variable follows the GHS distribution with p.d.f. , then the expected shortfall is equal to
where is the hypergeometric function.[14]