Expected shortfall

Overview

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 %

{\displaystyle 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), average value at risk (AVaR), tail value at risk (TVaR), conditional tail expectation (CTE), expected tail loss (ETL), and superquantile. These names are often used interchangeably, although several definitions exist in the literature. These definitions coincide in many cases, but may differ for certain types of loss distributions.

Background

Risk measures are used both in mathematical finance and in actuarial science, and the value-at-risk and expected shortfall measures are often expressed using different sign conventions and tail conventions in these disciplines. The discussion that follows takes the mathematical finance point of view. In mathematical finance, risk measures arise when considering the profit/loss distribution, i.e., payoff, for a financial portfolio, modeled as a random variable

X

{\displaystyle X}

. This can take positive or negative values, and downside risk corresponds to quantiles with

α

{\displaystyle \alpha }

close to 0. A risk threshold

α ∈ [ 0 , 1 ]

{\displaystyle \alpha \in [0,1]}

is selected, and

VaR

α

⁡ ( X )

{\displaystyle \operatorname {VaR} _{\alpha }(X)}

is defined to be the absolute value of the

α

{\displaystyle \alpha }

quantile of

X

{\displaystyle X}

Formal definition

If

X

{\displaystyle X}

is an integrable random variable representing the payoff of a portfolio at some future time and

0 < α ≤ 1

{\displaystyle 0<\alpha \leq 1}

then the expected shortfall of

X

{\displaystyle X}

at level

α

{\displaystyle \alpha }

is

ES

α

⁡ ( X ) =

1 α

∫

0

α

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:

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

ES

q

{\displaystyle \operatorname {ES} _{q}}

for a few values of

q

{\displaystyle q}

:

To see how these values were calculated, consider the calculation of

ES

0.05

{\displaystyle \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

ES

0.20

{\displaystyle \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

10 100

Properties

The expected shortfall

ES

q

{\displaystyle \operatorname {ES} _{q}}

increases as

q

{\displaystyle q}

decreases. The 100%-quantile expected shortfall

ES

1

{\displaystyle \operatorname {ES} _{1}}

equals negative of the expected value of the portfolio. For a given portfolio, the expected shortfall

ES

q

{\displaystyle \operatorname {ES} _{q}}

is greater than or equal to the Value at Risk

VaR

q

{\displaystyle \operatorname {VaR} _{q}}

at the same

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. 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

α

( w , γ )

{\displaystyle F_{\alpha }(w,\gamma )}

for the expected shortfall:

F

α

( w , γ ) = γ +

1

1 − α

∫

ℓ ( w , x ) ≥ γ

[

ℓ ( w , x ) − γ

]

+

p ( x )

d x

{\displaystyle F_{\alpha }(w,\gamma )=\gamma +{1 \over {1-\alpha }}\int _{\ell (w,x)\geq \gamma }\left[\ell (w,x)-\gamma \right]_{+}p(x)\,dx}

Formulas for continuous probability distributions

Closed-form formulas exist for calculating the expected shortfall when the payoff of a portfolio

X

{\displaystyle X}

or a corresponding loss

L = − X

{\displaystyle 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

−

VaR

α

⁡ ( X )

{\displaystyle -\operatorname {VaR} _{\alpha }(X)}

:

ES

α

⁡ ( X ) = E [ − X ∣ X ≤ −

VaR

α

⁡ ( X ) ] = −

1 α

Normal distribution

If the payoff of a portfolio

X

{\displaystyle X}

follows the normal (Gaussian) distribution with p.d.f.

f ( x ) =

1

2 π

σ

e

−

( x − μ

)

2

2

σ

2

{\displaystyle f(x)={\frac {1}{{\sqrt {2\pi }}\sigma }}e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}}

then the expected shortfall is equal to

ES

α

Generalized Student's t-distribution

If the payoff of a portfolio

X

{\displaystyle X}

follows the generalized Student's t-distribution with p.d.f.

f ( x ) =

Γ

(

ν + 1

2

)

Γ

(

ν 2

)

π ν

σ

(

1 +

1 ν

(

Laplace distribution

If the payoff of a portfolio

X

{\displaystyle X}

follows the Laplace distribution with the p.d.f.

f ( x ) =

1

2 b

e

−

|

x − μ

|

/

b

{\displaystyle f(x)={\frac {1}{2b}}e^{-|x-\mu |/b}}

and the c.d.f.

F ( x ) =

{

1 −

1 2

Logistic distribution

If the payoff of a portfolio

X

{\displaystyle X}

follows the logistic distribution with p.d.f.

f ( x ) =

1 s

e

−

x − μ

s

(

1 +

e

−

x − μ

s

)

− 2

{\displaystyle f(x)={\frac {1}{s}}e^{-{\frac {x-\mu }{s}}}\left(1+e^{-{\frac {x-\mu }{s}}}\right)^{-2}}

and the c.d.f.

Exponential distribution

If the loss of a portfolio

L

{\displaystyle L}

follows the exponential distribution with p.d.f.

f ( x ) =

{

λ

e

− λ x

if

x ≥ 0 ,

0

if

x < 0.

{\displaystyle f(x)={\begin{cases}\lambda e^{-\lambda x}&{\text{if }}x\geq 0,\\0&{\text{if }}x<0.\end{cases}}}

and the c.d.f.

F ( x ) =

{

1 −

e