Multivariate Optimized Certainty Equivalent Risk Measures and their Numerical Computation Sarah Kaaka Anis MatoussiAchraf Tamtalini

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Multivariate Optimized Certainty Equivalent Risk

Measures and their Numerical Computation∗

Sarah Kaaka¨ı †Anis Matoussi ‡Achraf Tamtalini §

Abstract

We present a framework for constructing multivariate risk measures that is inspired from

univariate Optimized Certainty Equivalent (OCE) risk measures. We show that this new

class of risk measures veriﬁes the desirable properties such as convexity, monotonocity and

cash invariance. We also address numerical aspects of their computations using stochastic

algorithms instead of using Monte Carlo or Fourier methods that do not provide any error

of the estimation.

Keywords: Multivariate risk measures, Optimized certainty equivalent, Numerical methods,

stochastic algorithms, risk allocations.

Introduction

One of the major concerns in ﬁnance is how to assess or quantify the risk associated with

a random cashﬂow in the future. Starting with the pioneering work of Markowitz (1952),

the risk associated with a random outcome was quantiﬁed by its variance. Then, Artzner

et al. (1999) published their famous seminal paper in which they introduce the theory of risk

measures. In their paper, risk measures were deﬁned as a map verifying certain properties,

which are called “axioms”, namely: Subadditivity, translation invariance, monotonocity and

∗Acknowledgements: The authors research is part of the ANR project DREAMeS (ANR-21-CE46-0002) and

beneﬁted from the support of the ”Chair Risques Emergents en Assurance” under the aegis of Fondation du Risque,

a joint initiative by Le Mans University and Cov´ea.

†Laboratoire Manceau de Math´ematiques & FR CNRS No2962, Institut du Risque et de l’Assurance,

Le Mans University.

‡Laboratoire Manceau de Math´ematiques & FR CNRS No2962, Institut du Risque et de l’Assurance,

Le Mans University and Ecole Polytechnique.

§Laboratoire Manceau de Math´ematiques & FR CNRS No2962, Institut du Risque et de l’Assurance,

Le Mans University.

arXiv:2210.13825v2 [math.OC] 6 Dec 2022

positive homogeneity. Such risk measures are called coherent risk measures. Many extensions

have been proposed and studied in the literature after the introduction of the axiomatic ap-

proach. One important extension is the notion of convex risk measure developed by F¨ollmer

and Schied (2002) and Frittelli and Gianin (2002) where the subadditivity and positive ho-

mogeneity properties were replaced by the weaker property of convexity. The latter reﬂects

the fact that diversiﬁcation decreases the risk. In the banking industry, one of the most

popular risk measures is the Value at Risk (VaR in short). This is due ﬁrst, to its ﬁnancial

interpretation and second, to its easy and fast implementation. Indeed, VaR is deﬁned as

the minimal cash amount that needed to be added to a ﬁnancial position in order to have a

probability of losses below a certain threshold. Its computation amounts to the calculation

of a quantile of the portfolio distribution. Nevertheless, VaR suﬀers from one drawback: it

does not verify the convexity property. This has prompted the search for new examples of

risk measures, the most prominent being the Conditional Value at Risk (CVaR), the entropic

risk measure and the utility based risk measure (also known as shortfall risk measure).

Some decision making problem based on utility functions are closely related to risk measures.

One can cite the optimized certainty equivalent (OCE) that was ﬁrst introduced by Ben-Tal

and Teboulle (1986). The idea behind the deﬁnition of OCE is as follows: Assume that a

decision maker, with some utility function u, is expecting a random income Xin the future

and can consume a part of it at present. If he chooses to consume mdollars, the resulting

present value of Xis then P(X, m):=m+E[u(X−m)]. Hence, one can deﬁne the sure

present value of X(i.e., its certainty equivalent) as the result of an optimal allocation of X

between present and future consumption, that is the decision maker will try to ﬁnd mthat

maximizes P(X, m). The main properties of the OCE were studied in Ben-Tal and Teboulle

(2007) where it is showed that the opposite of the OCE provides a wide family of risk mea-

sures that veriﬁes the axiomatic formalism of convex risk measures. They also proved that

several risk measures, such as CVaR and the entropic risk measure, can be derived as special

cases of the OCE by using particular utility functions (see also Cherny and Kupper (2007)).

From a systemic point of view, the ﬁnancial crisis of 2008 has demonstrated the need for

novel approaches that capture the risk of a system of ﬁnancial institutions. More precisely,

given a network/system of d∈Ndiﬀerent but dependent portfolios X:= (X1, ..., Xd), we

are interested in measuring/quantifying the risk carried by this system of portfolios. A

classical approach consists in ﬁrst aggregating the portfolios using some aggregation func-

tion Λ : Rd→Rand then apply some univariate risk measure applied to the aggregated

portfolio. In practice, most of the times the aggregation function is just the sum of the

components, i.e., Λ(x) = Pd

i=1 xi. This will result in having a systemic risk measure of

the form: R(X) = η(Λ(X)) = η(Pd

i=1 Xi), where ηis a univariate risk measure, such as

the VaR, CVaR, entropic risk measure, etc. The mechanism behind this approach is also

known as “Aggregate then Inject Cash” mechanism (see Biagini et al. (2019)). However,

this approach suﬀers from one major drawback: While it quantiﬁes the systemic risk carried

by the whole system, it does not provide risk levels of each portfolio, and thus, one could

not have a ranking of portfolios in terms of their systemic riskiness. One way to remedi-

ate to this, is to consider the reverse mechanism, that is to “Inject Cash then Aggregate”.

This consists in associating to each portfolio a risk measure and summing up the resulting

risk levels. This results in considering systemic risk measures R(X) of the following form:

R(X) = Pd

i=1 ηi(Xi), where ηi’s are the univariate risk measures associated to each portfolio.

Obviously, one could use the same univariate for all portfolios, that is ηi=η, ∀i∈ {1, ..., d}.

However, by doing so, we are assuming that the system is made of “isolated” portfolios with

no interdependence structure, and hence, we might be overestimating or underestimating

the systemic risk. This led several authors to look for approaches that address simultane-

ously the design of an overall risk measure and the allocation of this risk measure among the

diﬀerent components of the system. In this spirit, an extension of shortfall risk measures,

introduced in F¨ollmer and Schied (2002), has been studied in Armenti et al. (2018) based

on multivariate loss functions. However, one should note that, to ensure the existence of

optimal allocation problem, these loss functions must verify a key property: permutation

invariance. In other words, each component of the system is treated as if it has the same risk

proﬁle as all the other components and thus one cannot discriminate a particular component

against one another. Moreover, classical risk measures such that the CVaR and the entropic

risk measure cannot be recovered using multivariate shortfall risk measures, which limit their

use in practice. We will see that, with our multivariate extension of OCE risk measure, the

permutation invariance condition is no longer needed and by choosing the appropriate loss

functions, we can retrieve most of the classical risk measures.

One of the major issues that arises when studying risk measures is their numerical approxima-

tion. The standard VaR can be computed by inverting the simulated empirical distribution

of the ﬁnancial position using Monte Carlo (see Glasserman (2004) and Glasserman et al.

(2008)). An alternative method for computing VaR and CVaR is to use stochastic algorithms

(SA). The rational idea behind this perspective comes from the fact that both VaR and CVaR

are the solutions and the value of the same convex optimization problem as pointed out in

Rockafellar and Uryasev (2002) and the fact that the objective function is expressed as an

expectation. This was done in Bardou et al. (2009), where they prove the consistency and

the asymptotic normality of the estimators. In the same direction, in Kaaka¨ı et al. (2022),

we extended the work of Dunkel and Weber (2010) to approximate multivariate shortfall risk

measures using stochastic algorithms. In Neufeld (2008), they developed numerical schemes

for the computations of univariate OCE using Fourier transform methods.

The outline of this paper is as follows: in section 1, we give the deﬁnition of multivariate

OCE by introducing ﬁrst the class of appropriate loss functions. Then, we show that this

class of risk measures veriﬁes the desirable properties. We also characterize the optimal so-

lutions, give a dual representation and study the sensitivity with respect to external shocks.

Finally, section 2treats the computational aspects of approximating multivariate OCE using

1. Multivariate OCE

a deterministic scheme and a stochastic one.

1 Multivariate OCE

Let (Ω,F, P ) a probability space and we denote by L0(Rd) the space of F- measurable

random vectors taking values in Rd. For x, y in Rd, we denote by || · || the Euclidean norm

and x·y=Pxiyi. For a function f:Rd→[−∞,∞], we deﬁne f∗the convex conjugate of

fas f∗(y) = supx{x·y−f(x)}. The space L0(Rd) inherits the lattice structure of Rdand

hence, we can use the classical notations in Rdin a P-almost-surely sens. We will say for

example, for X, Y ∈L0(Rd) that X≥Yif P(X≥Y) = 1. To alleviate the notations, we

will drop the reference to Rdin L0(Rd) whenever it is unnecessary. For Q= (Q1, ..., Qd) a

vector of probabilities, we will write QPif for all i= 1, ..., d, we have QiP. In this

section, we introduce the notion of multivariate Optimized Certainty Equivalent (OCE) and

give its main properties. The latter is an extension of univariate OCE that was introduced

and studied in details in Ben-Tal and Teboulle (2007). First, we start by giving the deﬁnition

of a multivariate loss function that will be used in the rest of the paper. For the rest of the

paper, the random vector X= (X1, ..., Xd)∈L0represents proﬁts and losses of dportfolios.

Deﬁnition 1.1. A function l:Rd7→ (−∞,∞]is called a loss function, if it satisﬁes the

following properties:

1. lis nondecreasing, that is if x≤ycomponentwise, then l(x)≤l(y).

2. lis lower-semicontinuous and convex.

3. l(0) = 0 and l(x)>Pd

i=1 xi,∀x6= 0.

For integrability reasons, we will work in the multivariate Orlicz heart deﬁned as:

Mθ:={X∈L0:E[θ(λX)] <∞,∀λ > 0},

where θ(x) = l(|x|), x ∈Rd. On this space, we deﬁne the Luxembourg norm as:

||X||θ:=λ > 0, E θ|X|

λ≤1.

Under the Luxembourg norm, Mθis a Banach lattice and its dual with respect to this norm

is given by the Orlicz space Lθ∗:

Lθ∗:={X∈L0, E[θ∗(λX)] <∞,for some λ > 0}.

We also introduce the set of d-dimensional measure densities in Lθ∗, that is:

Qθ∗:=dQ

dP := (Z1, ..., Zd), Z ∈Lθ∗, Zk≥0 and E[Zk] = 1.

1. Multivariate OCE

Note that for Q∈Qθ∗and X∈Mθ,dQ

dP ·X∈L1, thanks to Fenchel inequality and for the

sake of simplicity, we will write EQ[X]:=E[dQ/dP ·X]. We refer to Appendix B in Armenti

et al. (2018) for more details about multivariate Orlicz spaces.

Deﬁnition 1.2. Assume lis a loss function. The multivariate OCE risk measure is deﬁned

for every X∈Mθas:

R(X) = inf

w∈Rd(d

i=1

wi+E[l(−X−w)]).(1.1)

Example 1. When d= 1, we can recover some important convex risk measures such CVaR

(also called Expected Shortfall or Average Value at Risk) and Entropic risk measure.

1. CVaR: Let α∈(0,1) and take l(x) = 1

1−αx+, then the associated risk measure is the

CVaR (see Rockafellar and Uryasev (2002)).

2. Polynomial loss function: For an integer γ > 1, the polynomial loss function is deﬁned

by: l(x) = ([1+x]+)γ−1

γ. When γ= 2, the corresponding risk measure is the Monotone

Mean-Variance (see ˇ

Cern`y et al. (2012)).

3. Entropic risk measure: Fix λ > 0and let l(x):=exp(λx)−1

λ. Then, the problem in (1.1)

can be explicitly solved and the optimal w∗and R(X)are given by:

w∗=1

λlog(E[e−λX ]), R(X) = w∗=1

λlog(E[e−λX ]).

Using univariate loss functions, we can construct multivariate loss functions in the follow-

ing way: Given l1, ..., ldunivariate loss functions and a nonnegative, convex and lower-

semicontinuous function Λ : R→R+with Λ(0) = 0, one can deﬁne a multivariate loss

function as follows:

l(x):=

i=1

li(xi) + Λ(x).(1.2)

It is easy to see that lveriﬁes all the conditions in the deﬁnition 1.1. Note that by taking

Λthe null function, the corresponding multivariate OCE boils down to a sum of univariate

OCE. It is in this function Λwhere the dependence between the diﬀerent components in the

system is taken into account. In this paper, we will focus on the following multivariate loss

functions inspired from the univariate risk measures above:

l(x) =

i=1

eλixi−1

λi

+αePd

i=1 λixi, λi>0, α ≥0,(1.3)

l(x) =

i=1

([1 + xi]+)θi−1

θi

+αX

i<j

([1 + xi]+)θi

θi

([1 + xi]+)θj

θj

, θi>1, α ≥0,(1.4)

l(x) =

i=1

1−βi

+αX

i<j

1−βi

1−βj

,0< βi<1, α ≥0.(1.5)

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MultivariateOptimizedCertaintyEquivalentRiskMeasuresandtheirNumericalComputation*SarahKaakaAnisMatoussiAchrafTamtalini§AbstractWepresentaframeworkforconstructingmultivariateriskmeasuresthatisinspiredfromunivariateOptimizedCertaintyEquivalent(OCE)riskmeasures.Weshowthatthisnewclassofriskmeasuresv...

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Multivariate Optimized Certainty Equivalent Risk Measures and their Numerical Computation Sarah Kaaka Anis MatoussiAchraf Tamtalini

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