Entropic value at risk

Overview

In financial mathematics and stochastic optimization, the concept of risk measure is used to quantify the risk involved in a random outcome or risk position. Many risk measures have hitherto been proposed, each having certain characteristics. The entropic value at risk (EVaR) is a coherent risk measure introduced by Ahmadi-Javid, which is an upper bound for the value at risk (VaR) and the conditional value at risk (CVaR), obtained from the Chernoff inequality. The EVaR can also be represented by using the concept of relative entropy. Because of its connection with the VaR and the relative entropy, this risk measure is called "entropic value at risk". The EVaR was developed to tackle some computational inefficiencies of the CVaR. Getting inspiration from the dual representation of the EVaR, Ahmadi-Javid developed a wide class of coherent risk measures, called g-entropic risk measures. Both the CVaR and the EVaR are members of this class.

Definition

Let

( Ω ,

F

, P )

{\displaystyle (\Omega ,{\mathcal {F}},P)}

be a probability space with

Ω

{\displaystyle \Omega }

a set of all simple events,

F

{\displaystyle {\mathcal {F}}}

a

σ

{\displaystyle \sigma }

-algebra of subsets of

Ω

{\displaystyle \Omega }

and

P

{\displaystyle P}

Properties

The EVaR is a coherent risk measure. The moment-generating function

M

X

( z )

{\displaystyle M_{X}(z)}

can be represented by the EVaR: for all

X ∈

L

M

+

{\displaystyle X\in \mathbf {L} _{M^{+}}}

and

z > 0

{\displaystyle z>0}

For

X , Y ∈

L

M

{\displaystyle X,Y\in \mathbf {L} _{M}}

,

Examples

For

X ∼ N ( μ ,

σ

2

) ,

{\displaystyle X\sim N(\mu ,\sigma ^{2}),}

For

X ∼ U ( a , b ) ,

{\displaystyle X\sim U(a,b),}

Figures 1 and 2 show the comparing of the VaR, CVaR and EVaR for

N ( 0 , 1 )

{\displaystyle N(0,1)}

and

U ( 0 , 1 )

{\displaystyle U(0,1)}

.

Optimization

Let

ρ

{\displaystyle \rho }

be a risk measure. Consider the optimization problem

where

w

∈

W

⊆

R

n

{\displaystyle {\boldsymbol {w}}\in {\boldsymbol {W}}\subseteq \mathbb {R} ^{n}}

is an

n

{\displaystyle n}

-dimensional real decision vector,

ψ

{\displaystyle {\boldsymbol {\psi }}}

is an

m

Generalization (g-entropic risk measures)

Drawing inspiration from the dual representation of the EVaR given in (3), one can define a wide class of information-theoretic coherent risk measures, which are introduced in. Let

g

{\displaystyle g}

be a convex proper function with

g ( 1 ) = 0

{\displaystyle g(1)=0}

and

β

{\displaystyle \beta }

be a non-negative number. The

g

{\displaystyle g}

-entropic risk measure with divergence level

β

{\displaystyle \beta }

is defined as

where

ℑ = { Q ≪ P :

H

g

Disciplined Convex Programming Framework

The disciplined convex programming framework of sample EVaR was proposed by Cajas and has the following form:

where

z

{\displaystyle z}

,

t

{\displaystyle t}

and

u

{\displaystyle u}

are variables;

K

exp

{\displaystyle K_{\text{exp}}}

is an exponential cone; and

T

{\displaystyle T}

is the number of observations. If we define

w

{\displaystyle w}

See also

Stochastic optimization Risk measure Coherent risk measure Value at risk Conditional value at risk Expected shortfall Entropic risk measure Kullback–Leibler divergence Generalized relative entropy