Entropic value at risk

Let ${\displaystyle (\Omega ,{\mathcal {F}},P)}$ be a probability space with ${\displaystyle \Omega }$ a set of all simple events, ${\displaystyle {\mathcal {F}}}$ a ${\displaystyle \sigma }$-algebra of subsets of ${\displaystyle \Omega }$ and ${\displaystyle P}$ a probability measure on ${\displaystyle {\mathcal {F}}}$. Let ${\displaystyle X}$ be a random variable and ${\displaystyle \mathbf {L} _{M^{+}}}$ be the set of all Borel measurable functions ${\displaystyle X:\Omega \to \mathbb {R} }$ whose moment-generating function ${\displaystyle M_{X}(z)}$ exists for all ${\displaystyle z\geq 0}$. The entropic value at risk (EVaR) of ${\displaystyle X\in \mathbf {L} _{M^{+}}}$ with confidence level ${\displaystyle 1-\alpha }$ is defined as follows:

${\displaystyle {\text{EVaR}}_{1-\alpha }(X):=\inf _{z>0}\left\{z^{-1}\ln \left({\frac {M_{X}(z)}{\alpha }}\right)\right\}.}$

(1)

In finance, the random variable ${\displaystyle X\in \mathbf {L} _{M^{+}},}$ in the above equation, is used to model the losses of a portfolio.

Consider the Chernoff inequality

${\displaystyle \Pr(X\geq a)\leq e^{-za}M_{X}(z),\quad \forall z>0.}$

(2)

Solving the equation ${\displaystyle e^{-za}M_{X}(z)=\alpha }$ for ${\displaystyle a,}$ results in

${\displaystyle a_{X}(\alpha ,z):=z^{-1}\ln \left({\frac {M_{X}(z)}{\alpha }}\right).}$

By considering the equation (1), we see that

${\displaystyle {\text{EVaR}}_{1-\alpha }(X):=\inf _{z>0}\{a_{X}(\alpha ,z)\},}$

which shows the relationship between the EVaR and the Chernoff inequality. It is worth noting that ${\displaystyle a_{X}(1,z)}$ is the entropic risk measure or exponential premium, which is a concept used in finance and insurance, respectively.

Let ${\displaystyle \mathbf {L} _{M}}$ be the set of all Borel measurable functions ${\displaystyle X:\Omega \to \mathbb {R} }$ whose moment-generating function ${\displaystyle M_{X}(z)}$ exists for all ${\displaystyle z}$. The dual representation (or robust representation) of the EVaR is as follows:

${\displaystyle {\text{EVaR}}_{1-\alpha }(X)=\sup _{Q\in \Im }(E_{Q}(X)),}$

(3)

where ${\displaystyle X\in \mathbf {L} _{M},}$ and ${\displaystyle \Im }$ is a set of probability measures on ${\displaystyle (\Omega ,{\mathcal {F}})}$ with ${\displaystyle \Im =\{Q\ll P:D_{KL}(Q||P)\leq -\ln \alpha \}}$. Note that

${\displaystyle D_{KL}(Q||P):=\int {\frac {dQ}{dP}}\left(\ln {\frac {dQ}{dP}}\right)dP}$

is the relative entropy of ${\displaystyle Q}$ with respect to ${\displaystyle P,}$ also called the Kullback–Leibler divergence. The dual representation of the EVaR discloses the reason behind its naming.

• The EVaR is a coherent risk measure.

• The moment-generating function ${\displaystyle M_{X}(z)}$ can be represented by the EVaR: for all ${\displaystyle X\in \mathbf {L} _{M^{+}}}$ and ${\displaystyle z>0}$

${\displaystyle M_{X}(z)=\sup _{0<\alpha \leq 1}\{\alpha \exp(z{\text{EVaR}}_{1-\alpha }(X))\}.}$

(4)

• For ${\displaystyle X,Y\in \mathbf {L} _{M}}$, ${\displaystyle {\text{EVaR}}_{1-\alpha }(X)={\text{EVaR}}_{1-\alpha }(Y)}$ for all ${\displaystyle \alpha \in ]0,1]}$ if and only if ${\displaystyle F_{X}(b)=F_{Y}(b)}$ for all ${\displaystyle b\in \mathbb {R} }$.

• The entropic risk measure with parameter ${\displaystyle \theta ,}$ can be represented by means of the EVaR: for all ${\displaystyle X\in \mathbf {L} _{M^{+}}}$ and ${\displaystyle \theta >0}$

${\displaystyle \theta ^{-1}\ln M_{X}(\theta )=a_{X}(1,\theta )=\sup _{0<\alpha \leq 1}\{{\text{EVaR}}_{1-\alpha }(X)+\theta ^{-1}\ln \alpha \}.}$

(5)

• The EVaR with confidence level ${\displaystyle 1-\alpha }$ is the tightest possible upper bound that can be obtained from the Chernoff inequality for the VaR and the CVaR with confidence level ${\displaystyle 1-\alpha }$;

${\displaystyle {\text{VaR}}(X)\leq {\text{CVaR}}(X)\leq {\text{EVaR}}(X).}$

(6)

• The following inequality holds for the EVaR:

${\displaystyle {\text{E}}(X)\leq {\text{EVaR}}_{1-\alpha }(X)\leq {\text{esssup}}(X)}$

(7)

where ${\displaystyle {\text{E}}(X)}$ is the expected value of ${\displaystyle X}$ and ${\displaystyle {\text{esssup}}(X)}$ is the essential supremum of ${\displaystyle X}$, i.e., ${\displaystyle \inf _{t\in \mathbb {R} }\{t:\Pr(X\leq t)=1\}}$. So do hold ${\displaystyle {\text{EVaR}}_{0}(X)={\text{E}}(X)}$ and ${\displaystyle \lim _{\alpha \to 0}{\text{EVaR}}_{1-\alpha }(X)={\text{esssup}}(X)}$.

Comparing the VaR, CVaR and EVaR for the standard normal distribution

Comparing the VaR, CVaR and EVaR for the uniform distribution over the interval (0,1)

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

${\displaystyle {\text{EVaR}}_{1-\alpha }(X)=\mu +\sigma {\sqrt {-2\ln \alpha }}.}$

(8)

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

${\displaystyle {\text{EVaR}}_{1-\alpha }(X)=\inf _{t>0}\left\lbrace t\ln \left(t{\frac {e^{t^{-1}b}-e^{t^{-1}a}}{b-a}}\right)-t\ln \alpha \right\rbrace .}$

(9)

Figures 1 and 2 show the comparing of the VaR, CVaR and EVaR for ${\displaystyle N(0,1)}$ and ${\displaystyle U(0,1)}$.

Let ${\displaystyle \rho }$ be a risk measure. Consider the optimization problem

${\displaystyle \min _{{\boldsymbol {w}}\in {\boldsymbol {W}}}\rho (G({\boldsymbol {w}},{\boldsymbol {\psi }})),}$

(10)

where ${\displaystyle {\boldsymbol {w}}\in {\boldsymbol {W}}\subseteq \mathbb {R} ^{n}}$ is an ${\displaystyle n}$-dimensional real decision vector, ${\displaystyle {\boldsymbol {\psi }}}$ is an ${\displaystyle m}$-dimensional real random vector with a known probability distribution and the function ${\displaystyle G({\boldsymbol {w}},.):\mathbb {R} ^{m}\to \mathbb {R} }$ is a Borel measurable function for all values ${\displaystyle {\boldsymbol {w}}\in {\boldsymbol {W}}.}$ If ${\displaystyle \rho ={\text{EVaR}},}$ then the optimization problem (10) turns into:

${\displaystyle \min _{{\boldsymbol {w}}\in {\boldsymbol {W}},t>0}\left\{t\ln M_{G({\boldsymbol {w}},{\boldsymbol {\psi }})}(t^{-1})-t\ln \alpha \right\}.}$

(11)

Let ${\displaystyle {\boldsymbol {S}}_{\boldsymbol {\psi }}}$ be the support of the random vector ${\displaystyle {\boldsymbol {\psi }}.}$ If ${\displaystyle G(.,{\boldsymbol {s}})}$ is convex for all ${\displaystyle {\boldsymbol {s}}\in {\boldsymbol {S}}_{\boldsymbol {\psi }}}$, then the objective function of the problem (11) is also convex. If ${\displaystyle G({\boldsymbol {w}},{\boldsymbol {\psi }})}$ has the form

${\displaystyle G({\boldsymbol {w}},{\boldsymbol {\psi }})=g_{0}({\boldsymbol {w}})+\sum _{i=1}^{m}g_{i}({\boldsymbol {w}})\psi _{i},\qquad g_{i}:\mathbb {R} ^{n}\to \mathbb {R} ,i=0,1,\ldots ,m,}$

(12)

and ${\displaystyle \psi _{1},\ldots ,\psi _{m}}$ are independent random variables in ${\displaystyle \mathbf {L} _{M}}$, then (11) becomes

${\displaystyle \min _{{\boldsymbol {w}}\in {\boldsymbol {W}},t>0}\left\lbrace g_{0}({\boldsymbol {w}})+t\sum _{i=1}^{m}\ln M_{g_{i}({\boldsymbol {w}})\psi _{i}}(t^{-1})-t\ln \alpha \right\rbrace .}$

(13)

which is computationally tractable. But for this case, if one uses the CVaR in problem (10), then the resulting problem becomes as follows:

${\displaystyle \min _{{\boldsymbol {w}}\in {\boldsymbol {W}},t\in \mathbb {R} }\left\lbrace t+{\frac {1}{\alpha }}{\text{E}}\left[g_{0}({\boldsymbol {w}})+\sum _{i=1}^{m}g_{i}({\boldsymbol {w}})\psi _{i}-t\right]_{+}\right\rbrace .}$

(14)

It can be shown that by increasing the dimension of ${\displaystyle \psi }$, problem (14) is computationally intractable even for simple cases. For example, assume that ${\displaystyle \psi _{1},\ldots ,\psi _{m}}$ are independent discrete random variables that take ${\displaystyle k}$ distinct values. For fixed values of ${\displaystyle {\boldsymbol {w}}}$ and ${\displaystyle t,}$ the complexity of computing the objective function given in problem (13) is of order ${\displaystyle mk}$ while the computing time for the objective function of problem (14) is of order ${\displaystyle k^{m}}$. For illustration, assume that ${\displaystyle k=2,m=100}$ and the summation of two numbers takes ${\displaystyle 10^{-12}}$ seconds. For computing the objective function of problem (14) one needs about ${\displaystyle 4\times 10^{10}}$ years, whereas the evaluation of objective function of problem (13) takes about ${\displaystyle 10^{-10}}$ seconds. This shows that formulation with the EVaR outperforms the formulation with the CVaR (see [2] for more details).

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.[1][2] Let ${\displaystyle g}$ be a convex proper function with ${\displaystyle g(1)=0}$ and ${\displaystyle \beta }$ be a non-negative number. The ${\displaystyle g}$-entropic risk measure with divergence level ${\displaystyle \beta }$ is defined as

${\displaystyle {\text{ER}}_{g,\beta }(X):=\sup _{Q\in \Im }{\text{E}}_{Q}(X)}$

(15)

where ${\displaystyle \Im =\{Q\ll P:H_{g}(P,Q)\leq \beta \}}$ in which ${\displaystyle H_{g}(P,Q)}$ is the generalized relative entropy of ${\displaystyle Q}$ with respect to ${\displaystyle P}$. A primal representation of the class of ${\displaystyle g}$-entropic risk measures can be obtained as follows:

${\displaystyle {\text{ER}}_{g,\beta }(X)=\inf _{t>0,\mu \in \mathbb {R} }\left\lbrace t\left[\mu +{\text{E}}_{P}\left(g^{*}\left({\frac {X}{t}}-\mu +\beta \right)\right)\right]\right\rbrace }$

(16)

where ${\displaystyle g^{*}}$ is the conjugate of ${\displaystyle g}$. By considering

${\displaystyle g(x)={\begin{cases}x\ln x&x>0\\0&x=0\\+\infty &x<0\end{cases}}}$

(17)

with ${\displaystyle g^{*}(x)=e^{x-1}}$ and ${\displaystyle \beta =-\ln \alpha }$, the EVaR formula can be deduced. The CVaR is also a ${\displaystyle g}$-entropic risk measure, which can be obtained from (16) by setting

${\displaystyle g(x)={\begin{cases}0&0\leq x\leq {\frac {1}{\alpha }}\\+\infty &{\text{otherwise}}\end{cases}}}$

(18)

with ${\displaystyle g^{*}(x)={\tfrac {1}{\alpha }}\max\{0,x\}}$ and ${\displaystyle \beta =0}$ (see [1][3] for more details).

For more results on ${\displaystyle g}$-entropic risk measures see.[4]

The disciplined convex programming framework of sample EVaR was proposed by Cajas[5] and has the following form:

${\displaystyle {\begin{aligned}{\text{EVaR}}_{\alpha }(X)&=\left\{{\begin{array}{ll}{\underset {z,\,t,\,u}{\text{min}}}&t+z\ln \left({\frac {1}{\alpha T}}\right)\\{\text{s.t.}}&z\geq \sum _{j=1}^{T}u_{j}\\&(X_{j}-t,z,u_{j})\in K_{\text{exp}}\;\forall \;j=1,\ldots ,T\\\end{array}}\right.\end{aligned}}}$

(19)

where ${\displaystyle z}$, ${\displaystyle t}$ and ${\displaystyle u}$ are variables; ${\displaystyle K_{\text{exp}}}$ is an exponential cone;[6] and ${\displaystyle T}$ is the number of observations. If we define ${\displaystyle w}$ as the vector of weights for ${\displaystyle N}$ assets, ${\displaystyle r}$ the matrix of returns and ${\displaystyle \mu }$ the mean vector of assets, we can posed the minimization of the expected EVaR given a level of expected portfolio return ${\displaystyle {\bar {\mu }}}$ as follows.

${\displaystyle {\begin{aligned}&{\underset {w,\,z,\,t,\,u}{\text{min}}}&&t+z\ln \left({\frac {1}{\alpha T}}\right)\\&{\text{s.t.}}&&\mu w^{\tau }\geq {\bar {\mu }}\\&&&\sum _{i=1}^{N}w_{i}=1\\&&&z\geq \sum _{j=1}^{T}u_{j}\\&&&(-r_{j}w^{\tau }-t,z,u_{j})\in K_{\text{exp}}\;\forall \;j=1,\ldots ,T\\&&&w_{i}\geq 0\;;\;\forall \;i=1,\ldots ,N\\\end{aligned}}}$

(20)

Applying the disciplined convex programming framework of EVaR to uncompounded cumulative returns distribution, Cajas[5] proposed the entropic drawdown at risk(EDaR) optimization problem. We can posed the minimization of the expected EDaR given a level of expected return ${\displaystyle {\bar {\mu }}}$ as follows:

${\displaystyle {\begin{aligned}&{\underset {w,\,z,\,t,\,u,\,d}{\text{min}}}&&t+z\ln \left({\frac {1}{\alpha T}}\right)\\&{\text{s.t.}}&&\mu w^{\tau }\geq {\bar {\mu }}\\&&&\sum _{i=1}^{N}w_{i}=1\\&&&z\geq \sum _{j=1}^{T}u_{j}\\&&&(d_{j}-R_{j}w^{\tau }-t,z,u_{j})\in K_{\text{exp}}\;\forall \;j=1,\ldots ,T\\&&&d_{j}\geq R_{j}w^{\tau }\;\forall \;j=1,\ldots ,T\\&&&d_{j}\geq d_{j-1}\;\forall \;j=1,\ldots ,T\\&&&d_{j}\geq 0\;\forall \;j=1,\ldots ,T\\&&&d_{0}=0\\&&&w_{i}\geq 0\;;\;\forall \;i=1,\ldots ,N\\\end{aligned}}}$

(21)

where ${\displaystyle d}$ is a variable that represent the uncompounded cumulative returns of portfolio and ${\displaystyle R}$ is the matrix of uncompounded cumulative returns of assets.

For other problems like risk parity, maximization of return/risk ratio or constraints on maximum risk levels for EVaR and EDaR, you can see [5] for more details.

The advantage of model EVaR and EDaR using a disciplined convex programming framework, is that we can use softwares like CVXPY [7] or MOSEK[8] to model this portfolio optimization problems. EVaR and EDaR are implemented in the python package Riskfolio-Lib.[9]

• 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