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 formula_1 be a probability space with formula_2 a set of all simple events, formula_3 a formula_4-algebra of subsets of formula_2 and formula_6 a probability measure on formula_3. Let formula_8 be a random variable and formula_9 be the set of all Borel measurable functions formula_10 whose moment-generating function formula_11 exists for all formula_12. The entropic value at risk (EVaR) of formula_13 with confidence level formula_14 is defined as follows:
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In finance, the random variable formula_15 in the above equation, is used to model the "losses" of a portfolio.
Consider the Chernoff inequality
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Solving the equation formula_16 for formula_17 results in 
formula_18
By considering the equation (1), we see that 
formula_19
which shows the relationship between the EVaR and the Chernoff inequality. It is worth noting that formula_20 is the "entropic risk measure" or "exponential premium", which is a concept used in finance and insurance, respectively.
Let formula_21 be the set of all Borel measurable functions formula_22 whose moment-generating function formula_11 exists for all formula_24. The dual representation (or robust representation) of the EVaR is as follows:
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where formula_25 and formula_26 is a set of probability measures on formula_27 with formula_28. Note that 
formula_29 
is the relative entropy of formula_30 with respect to formula_31 also called the Kullback–Leibler divergence. The dual representation of the EVaR discloses the reason behind its naming.
Properties.
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where formula_45 is the expected value of formula_8 and formula_47 is the essential supremum of formula_8, i.e., formula_49. So do hold formula_50 and formula_51.
Examples.
For formula_52
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For formula_53
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Figures 1 and 2 show the comparing of the VaR, CVaR and EVaR for formula_54 and formula_55.
Optimization.
Let formula_56 be a risk measure. Consider the optimization problem
where formula_57 is an formula_58-dimensional real decision vector, formula_59 is an formula_60-dimensional real random vector with a known probability distribution and the function formula_61 is a Borel measurable function for all values formula_62 If formula_63 then the optimization problem (10) turns into:
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Let formula_64 be the support of the random vector formula_65 If formula_66 is convex for all formula_67, then the objective function of the problem (11) is also convex. If formula_68 has the form
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and formula_69 are independent random variables in formula_70, then (11) becomes
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which is computationally tractable. But for this case, if one uses the CVaR in problem (10), then the resulting problem becomes as follows:
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It can be shown that by increasing the dimension of formula_71, problem (14) is computationally intractable even for simple cases. For example, assume that formula_69 are independent discrete random variables that take formula_73 distinct values. For fixed values of formula_74 and formula_75 the complexity of computing the objective function given in problem (13) is of order formula_76 while the computing time for the objective function of problem (14) is of order formula_77. For illustration, assume that formula_78 and the summation of two numbers takes formula_79 seconds. For computing the objective function of problem (14) one needs about formula_80 years, whereas the evaluation of objective function of problem (13) takes about formula_81 seconds. This shows that formulation with the EVaR outperforms the formulation with the CVaR (see for more details).
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 formula_82 be a convex proper function with formula_83 and formula_84 be a non-negative number. The formula_82-entropic risk measure with divergence level formula_84 is defined as
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where formula_87 in which formula_88 is the generalized relative entropy of formula_30 with respect to formula_6. A primal representation of the class of formula_82-entropic risk measures can be obtained as follows:
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where formula_92 is the conjugate of formula_82. By considering
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with formula_94 and formula_95, the EVaR formula can be deduced. The CVaR is also a formula_82-entropic risk measure, which can be obtained from (16) by setting
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with formula_97 and formula_98 (see for more details).
For more results on formula_82-entropic risk measures see.
Disciplined Convex Programming Framework.
The disciplined convex programming framework of sample EVaR was proposed by Cajas and has the following form:
 ;templatestyles src="Numbered block/styles.css" / ; &amp; t + z \ln \left ( \frac{1}{\alpha T} \right ) \\
 \text{s.t.} &amp; z \geq \sum^{T}_{j=1} u_{j} \\
 &amp; (X_{j}-t, z, u_{j}) \in K_{\text{exp}} \; \forall \; j =1, \ldots, T \\
\end{array} \right .
where formula_24, formula_101 and formula_102 are variables; formula_103 is an exponential cone; and formula_104 is the number of observations. If we define formula_105 as the vector of weights for formula_106 assets, formula_107 the matrix of returns and formula_108 the mean vector of assets, we can posed the minimization of the expected EVaR given a level of expected portfolio return formula_109 as follows.
 ;templatestyles src="Numbered block/styles.css" / ; &amp; &amp; t + z \ln \left ( \frac{1}{\alpha T} \right ) \\
&amp; \text{s.t.} &amp; &amp; \mu w^{\tau} \geq \bar{\mu} \\
&amp; &amp; &amp; \sum_{i=1}^{N} w_i = 1 \\
&amp; &amp; &amp; z \geq \sum^{T}_{j=1} u_{j} \\
&amp; &amp; &amp; (-r_{j}w^{\tau}-t, z, u_{j}) \in K_{\text{exp}} \; \forall \; j=1, \ldots, T \\
&amp; &amp; &amp; w_i \geq 0 \; ; \; \forall \; i =1, \ldots, N \\
Applying the disciplined convex programming framework of EVaR to uncompounded cumulative returns distribution, Cajas proposed the entropic drawdown at risk(EDaR) optimization problem. We can posed the minimization of the expected EDaR given a level of expected return formula_109 as follows:
 ;templatestyles src="Numbered block/styles.css" / ; &amp; &amp; t + z \ln \left ( \frac{1}{\alpha T} \right ) \\
&amp; \text{s.t.} &amp; &amp; \mu w^{\tau} \geq \bar{\mu} \\
&amp; &amp; &amp; \sum_{i=1}^{N} w_i = 1 \\
&amp; &amp; &amp; z \geq \sum^{T}_{j=1} u_{j} \\
&amp; &amp; &amp; (d_{j} - R_{j} w^{\tau} - t, z, u_{j}) \in K_{\text{exp}} \; \forall \; j =1, \ldots, T \\
&amp; &amp; &amp; d_{j} \geq R_{j} w^{\tau} \; \forall \; j=1, \ldots, T \\ 
&amp; &amp; &amp; d_{j} \geq d_{j-1} \; \forall \; j=1, \ldots, T \\ 
&amp; &amp; &amp; d_{j} \geq 0 \; \forall \; j=1, \ldots, T \\ 
&amp; &amp; &amp; d_{0} = 0 \\ 
&amp; &amp; &amp; w_{i} \geq 0 \; ; \; \forall \; i =1, \ldots, N \\
where formula_111 is a variable that represent the uncompounded cumulative returns of portfolio and formula_112 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 for more details.
The advantage of model EVaR and EDaR using a disciplined convex programming framework, is that we can use softwares like CVXPY or MOSEK to model this portfolio optimization problems. EVaR and EDaR are implemented in the python package Riskfolio-Lib.
References.
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