Entropic value at risk

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,[1][2] 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[clarification needed] of the CVaR. Getting inspiration from the dual representation of the EVaR, Ahmadi-Javid[1][2] 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

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Let   be a probability space with   a set of all simple events,   a  -algebra of subsets of   and   a probability measure on  . Let   be a random variable and   be the set of all Borel measurable functions   whose moment-generating function   exists for all  . The entropic value at risk (EVaR) of   with confidence level   is defined as follows:

  (1)

In finance, the random variable   in the above equation, is used to model the losses of a portfolio.

Consider the Chernoff inequality

  (2)

Solving the equation   for   results in

 

By considering the equation (1), we see that

 

which shows the relationship between the EVaR and the Chernoff inequality. It is worth noting that   is the entropic risk measure or exponential premium, which is a concept used in finance and insurance, respectively.

Let   be the set of all Borel measurable functions   whose moment-generating function   exists for all  . The dual representation (or robust representation) of the EVaR is as follows:

  (3)

where   and   is a set of probability measures on   with  . Note that

 

is the relative entropy of   with respect to   also called the Kullback–Leibler divergence. The dual representation of the EVaR discloses the reason behind its naming.

Properties

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  • The EVaR is a coherent risk measure.
  • The moment-generating function   can be represented by the EVaR: for all   and  
  (4)
  • For  ,   for all   if and only if   for all  .
  • The entropic risk measure with parameter   can be represented by means of the EVaR: for all   and  
  (5)
  • The EVaR with confidence level   is the tightest possible upper bound that can be obtained from the Chernoff inequality for the VaR and the CVaR with confidence level  ;
  (6)
  • The following inequality holds for the EVaR:
  (7)
where   is the expected value of   and   is the essential supremum of  , i.e.,  . So do hold   and  .

Examples

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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  

  (8)

For  

  (9)

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

Optimization

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Let   be a risk measure. Consider the optimization problem

  (10)

where   is an  -dimensional real decision vector,   is an  -dimensional real random vector with a known probability distribution and the function   is a Borel measurable function for all values   If   then the optimization problem (10) turns into:

  (11)

Let   be the support of the random vector   If   is convex for all  , then the objective function of the problem (11) is also convex. If   has the form

  (12)

and   are independent random variables in  , then (11) becomes

  (13)

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

  (14)

It can be shown that by increasing the dimension of  , problem (14) is computationally intractable even for simple cases. For example, assume that   are independent discrete random variables that take   distinct values. For fixed values of   and   the complexity of computing the objective function given in problem (13) is of order   while the computing time for the objective function of problem (14) is of order  . For illustration, assume that   and the summation of two numbers takes   seconds. For computing the objective function of problem (14) one needs about   years, whereas the evaluation of objective function of problem (13) takes about   seconds. This shows that formulation with the EVaR outperforms the formulation with the CVaR (see [2] for more details).

Generalization (g-entropic risk measures)

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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   be a convex proper function with   and   be a non-negative number. The  -entropic risk measure with divergence level   is defined as

  (15)

where   in which   is the generalized relative entropy of   with respect to  . A primal representation of the class of  -entropic risk measures can be obtained as follows:

  (16)

where   is the conjugate of  . By considering

  (17)

with   and  , the EVaR formula can be deduced. The CVaR is also a  -entropic risk measure, which can be obtained from (16) by setting

  (18)

with   and   (see [1][3] for more details).

For more results on  -entropic risk measures see.[4]

Disciplined Convex Programming Framework

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

  (19)

where  ,   and   are variables;   is an exponential cone;[6] and   is the number of observations. If we define   as the vector of weights for   assets,   the matrix of returns and   the mean vector of assets, we can posed the minimization of the expected EVaR given a level of expected portfolio return   as follows.

  (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   as follows:

  (21)

where   is a variable that represent the uncompounded cumulative returns of portfolio and   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]

See also

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References

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  1. ^ a b c d Ahmadi-Javid, Amir (2011). "An information-theoretic approach to constructing coherent risk measures". 2011 IEEE International Symposium on Information Theory Proceedings. St. Petersburg, Russia: Proceedings of IEEE International Symposium on Information Theory. pp. 2125–2127. doi:10.1109/ISIT.2011.6033932. ISBN 978-1-4577-0596-0. S2CID 8720196.
  2. ^ a b c d Ahmadi-Javid, Amir (2012). "Entropic value-at-risk: A new coherent risk measure". Journal of Optimization Theory and Applications. 155 (3): 1105–1123. doi:10.1007/s10957-011-9968-2. S2CID 46150553.
  3. ^ Ahmadi-Javid, Amir (2012). "Addendum to: Entropic Value-at-Risk: A New Coherent Risk Measure". Journal of Optimization Theory and Applications. 155 (3): 1124–1128. doi:10.1007/s10957-012-0014-9. S2CID 39386464.
  4. ^ Breuer, Thomas; Csiszar, Imre (2013). "Measuring Distribution Model Risk". arXiv:1301.4832v1 [q-fin.RM].
  5. ^ a b c Cajas, Dany (24 February 2021). "Entropic Portfolio Optimization: a Disciplined Convex Programming Framework". doi:10.2139/ssrn.3792520. S2CID 235319743. {{cite journal}}: Cite journal requires |journal= (help)
  6. ^ Chares, Robert (2009). "Cones and interior-point algorithms for structured convex optimization involving powers andexponentials". S2CID 118322815. {{cite web}}: Missing or empty |url= (help)
  7. ^ "CVXPY".
  8. ^ "MOSEK".
  9. ^ "Riskfolio-Lib". GitHub. 18 July 2022.