In
financial mathematics
Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets.
In general, there exist two separate branches of finance that require ...
and
stochastic optimization
Stochastic optimization (SO) methods are optimization methods that generate and use random variables. For stochastic problems, the random variables appear in the formulation of the optimization problem itself, which involves random objective functi ...
, the concept of
risk measure
In financial mathematics, a risk measure is used to determine the amount of an asset or set of assets (traditionally currency) to be kept in reserve. The purpose of this reserve is to make the risks taken by financial institutions, such as bank ...
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
In the fields of actuarial science and financial economics there are a number of ways that risk can be defined; to clarify the concept theoreticians have described a number of properties that a risk measure might or might not have. A coherent ris ...
introduced by Ahmadi-Javid,
which is an upper bound for the
value at risk
Value at risk (VaR) is a measure of the risk of loss for investments. It estimates how much a set of investments might lose (with a given probability), given normal market conditions, in a set time period such as a day. VaR is typically used by ...
(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
Relative may refer to:
General use
*Kinship and family, the principle binding the most basic social units society. If two people are connected by circumstances of birth, they are said to be ''relatives''
Philosophy
*Relativism, the concept that ...
. 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 measure
In the fields of actuarial science and financial economics there are a number of ways that risk can be defined; to clarify the concept theoreticians have described a number of properties that a risk measure might or might not have. A coherent ris ...
s, called
g-entropic risk measures. Both the CVaR and the EVaR are members of this class.
Definition
Let
be a
probability space
In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models t ...
with
a set of all simple events,
a
-algebra of subsets of
and
a
probability measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as ''countable additivity''. The difference between a probability measure and the more ge ...
on
. Let
be a
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
and
be the set of all
Borel measurable
In mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all Borel sets). Some authors require additional restrictions on the measure, as described below.
...
functions
whose
moment-generating function
In probability theory and statistics, the moment-generating function of a real-valued random variable is an alternative specification of its probability distribution. Thus, it provides the basis of an alternative route to analytical results compare ...
exists for all
. The entropic value at risk (EVaR) of
with confidence level
is defined as follows:
In finance, the
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
in the above equation, is used to model the ''losses'' of a portfolio.
Consider the Chernoff inequality
Solving the equation
for
results in
:
By considering the equation (), we see that
:
which shows the relationship between the EVaR and the Chernoff inequality. It is worth noting that
is the ''
entropic risk measure
In financial mathematics (concerned with mathematical modeling of financial markets), the entropic risk measure is a risk measure which depends on the risk aversion of the user through the exponential utility function. It is a possible alternat ...
'' 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
In mathematics, if is a group and is a linear representation of it on the vector space , then the dual representation is defined over the dual vector space as follows:
: is the transpose of , that is, = for all .
The dual representation ...
(or robust representation) of the EVaR is as follows:
where
and
is a set of probability measures on
with
. Note that
:
is the
relative entropy
Relative may refer to:
General use
*Kinship and family, the principle binding the most basic social units society. If two people are connected by circumstances of birth, they are said to be ''relatives''
Philosophy
*Relativism, the concept that ...
of
with respect to
also called the
Kullback–Leibler divergence
In mathematical statistics, the Kullback–Leibler divergence (also called relative entropy and I-divergence), denoted D_\text(P \parallel Q), is a type of statistical distance: a measure of how one probability distribution ''P'' is different fr ...
. The dual representation of the EVaR discloses the reason behind its naming.
Properties
* The EVaR is a coherent risk measure.
* The moment-generating function
can be represented by the EVaR: for all
and
* 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
* 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
;
* The following inequality holds for the EVaR:
:where
is the
expected value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a ...
of
and
is the
essential supremum
In mathematics, the concepts of essential infimum and essential supremum are related to the notions of infimum and supremum, but adapted to measure theory and functional analysis, where one often deals with statements that are not valid for ''all' ...
of
, i.e.,
. So do hold
and
.
Examples
For
For
Figures 1 and 2 show the comparing of the VaR, CVaR and EVaR for
and
.
Optimization
Let
be a risk measure. Consider the optimization problem
where
is an
-dimensional real
decision vector
Decision may refer to:
Law and politics
*Judgment (law), as the outcome of a legal case
*Landmark decision, the outcome of a case that sets a legal precedent
* ''Per curiam'' decision, by a court with multiple judges
Books
* ''Decision'' (novel ...
,
is an
-dimensional real
random vector
In probability, and statistics, a multivariate random variable or random vector is a list of mathematical variables each of whose value is unknown, either because the value has not yet occurred or because there is imperfect knowledge of its value ...
with a known
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...
and the function
is a Borel measurable function for all values
If
then the optimization problem () turns into:
Let
be the
support of the random vector If
is
convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytop ...
for all
, then the objective function of the problem () is also convex. If
has the form
and
are
independent random variables in
, then () becomes
which is computationally
tractable. But for this case, if one uses the CVaR in problem (), then the resulting problem becomes as follows:
It can be shown that by increasing the dimension of
, problem () is computationally intractable even for simple cases. For example, assume that
are independent
discrete random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
s that take
distinct values. For fixed values of
and
the
complexity
Complexity characterises the behaviour of a system or model whose components interact in multiple ways and follow local rules, leading to nonlinearity, randomness, collective dynamics, hierarchy, and emergence.
The term is generally used to ch ...
of computing the objective function given in problem () is of order
while the computing time for the objective function of problem () is of order
. For illustration, assume that
and the summation of two numbers takes
seconds. For computing the objective function of problem () one needs about
years, whereas the evaluation of objective function of problem () takes about
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 (), one can define a wide class of information-theoretic coherent risk measures, which are introduced in.[ Let be a convex ]proper function
In mathematics, a function between topological spaces is called proper if inverse images of compact subsets are compact. In algebraic geometry, the analogous concept is called a proper morphism.
Definition
There are several competing definit ...
with and be a non-negative number. The -entropic risk measure with divergence level is defined as
where in which is the generalized relative entropy Generalized relative entropy (\epsilon-relative entropy) is a measure of dissimilarity between two quantum states. It is a "one-shot" analogue of quantum relative entropy and shares many properties of the latter quantity.
In the study of quantum i ...
of with respect to . A primal representation of the class of -entropic risk measures can be obtained as follows:
where is the conjugate of . By considering
with and , the EVaR formula can be deduced. The CVaR is also a -entropic risk measure, which can be obtained from () by setting
with and (see for more details).
For more results on -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:
& t + z \ln \left ( \frac \right ) \\
\text & z \geq \sum^_ u_ \\
& (X_-t, z, u_) \in K_ \; \forall \; j =1, \ldots, T \\
\end \right .
\end
,
where , and are variables; is an exponential cone; 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.
& & t + z \ln \left ( \frac \right ) \\
& \text & & \mu w^ \geq \bar \\
& & & \sum_^ w_i = 1 \\
& & & z \geq \sum^_ u_ \\
& & & (-r_w^-t, z, u_) \in K_ \; \forall \; j=1, \ldots, T \\
& & & w_i \geq 0 \; ; \; \forall \; i =1, \ldots, N \\
\end
,
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 as follows:
& & t + z \ln \left ( \frac \right ) \\
& \text & & \mu w^ \geq \bar \\
& & & \sum_^ w_i = 1 \\
& & & z \geq \sum^_ u_ \\
& & & (d_ - R_ w^ - t, z, u_) \in K_ \; \forall \; j =1, \ldots, T \\
& & & d_ \geq R_ w^ \; \forall \; j=1, \ldots, T \\
& & & d_ \geq d_ \; \forall \; j=1, \ldots, T \\
& & & d_ \geq 0 \; \forall \; j=1, \ldots, T \\
& & & d_ = 0 \\
& & & w_ \geq 0 \; ; \; \forall \; i =1, \ldots, N \\
\end
,
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 ][ 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.]
See also
* Stochastic optimization
Stochastic optimization (SO) methods are optimization methods that generate and use random variables. For stochastic problems, the random variables appear in the formulation of the optimization problem itself, which involves random objective functi ...
* Risk measure
In financial mathematics, a risk measure is used to determine the amount of an asset or set of assets (traditionally currency) to be kept in reserve. The purpose of this reserve is to make the risks taken by financial institutions, such as bank ...
* Coherent risk measure
In the fields of actuarial science and financial economics there are a number of ways that risk can be defined; to clarify the concept theoreticians have described a number of properties that a risk measure might or might not have. A coherent ris ...
* Value at risk
Value at risk (VaR) is a measure of the risk of loss for investments. It estimates how much a set of investments might lose (with a given probability), given normal market conditions, in a set time period such as a day. VaR is typically used by ...
* Conditional value at risk
* Expected shortfall
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 wor ...
* Entropic risk measure
In financial mathematics (concerned with mathematical modeling of financial markets), the entropic risk measure is a risk measure which depends on the risk aversion of the user through the exponential utility function. It is a possible alternat ...
* Kullback–Leibler divergence
In mathematical statistics, the Kullback–Leibler divergence (also called relative entropy and I-divergence), denoted D_\text(P \parallel Q), is a type of statistical distance: a measure of how one probability distribution ''P'' is different fr ...
* Generalized relative entropy Generalized relative entropy (\epsilon-relative entropy) is a measure of dissimilarity between two quantum states. It is a "one-shot" analogue of quantum relative entropy and shares many properties of the latter quantity.
In the study of quantum i ...
References
{{reflist
Financial risk modeling
Utility