VaR as the CVaR sensitivity: Applications in riskoptimization

Alejandro Balbása, Beatriz Balbásb, Raquel Balbásc

aUniversity Carlos III of Madrid. C/ Madrid, 126. 28903 Getafe (Madrid, Spain). alejan-

dro.balbas@uc3m.es.

bUniversity of Castilla La Mancha. Avda. Real Fábrica de Seda, s/n. 45600 Talavera

(Toledo, Spain). beatriz.balbas@uclm.es.

cUniversity Complutense of Madrid. Somosaguas. 28223 Pozuelo de Alarcón (Madrid,

Spain). raquel.balbas@ccee.ucm.es.

Abstract VaR minimization is a complex problem playing a critical role in many actuarial and

�nancial applications of mathematical programming. The usual methods of convex programming

do not apply due to the lack of sub-additivity. The usual methods of di¤erentiable programming

do not apply either, due to the lack of continuity. Taking into account that the CVaR may be

given as an integral of VaR, one has that VaR becomes a �rst order mathematical derivative of

CVaR. This property will enable us to give accurate approximations in VaR optimization, since

the optimization VaR and CVaR will become quite closely related topics. Applications in both

�nance and insurance will be given.

Key words VaR Optimization, CVaR Sensitivity, Approximation Methods, Optimality Condi-

tions, Actuarial and Financial Applications.

A.M.S. Classi�cation 90C59, 90C30, 91B30, 91G10, 90B50.

J.E.L. Classi�cation C02, C61, G11, G22.

1 Introduction

V aR has many applications in �nance and insurance. Risk management, capital require-

ments, �nancial reporting, asset allocation, bonus-malus systems, optimal reinsurance, etc.

just compose a brief list of topics closely related to V aR. Beyond V aR, risk measurement

1

is an open problem provoking a growing interest and discussion in recent years. Since

Artzner et al. ( 1999) introduced their coherent measures of risk much more approaches

have been proposed. Very important examples are the expectation bounded measures of

risk (Rockafellar et al., 2006), consistent risk measures (Goovaerts et al., 2004 ), actuarial

risk measures (Goovaerts and Laeven, 2008), indices of riskiness (Aumann and Serrano,

2008, Foster and Hart, 2009, Bali et al., 2011), etc.

The existence of alternative risk measures implies that many risk-linked problems may be

studied without dealing with V aR. Moreover, V aR is not sub-additive (Artzner et al.,

1999), it is di¢ cult to optimize (Gaivoronski and P�ug, 2005) and it presents some more

drawbacks which may recommend to deal with other risk measures such as CV aR (Rock-

afellar and Uryasev, 2000). Nevertheless, for several reasons V aR still plays a critical role

for many practitioners, institutions and researchers. Firstly, regulation (Basel for banks,

Solvency for insurers, etc.) still assigns a vital role to V aR. Secondly, V aR never becomes

in�nity, while the rest of usual risk measures may attain this value. For instance, CV aR

becomes in�nity for random risks whose expected losses equal in�nity too (for instance,

positive random variables with unbounded expectation). In�nite values may provoke an-

alytical and mathematical problems quite di¢ cult to overcome, specially if several heavy

tails are simultaneously involved (Chavez-Demoulin et al., 2006). Heavy tails are usual in

some actuarial topics (Zajdenwebe, 1996), some operational risk topics (Mitra et al., 2015)

and other issues. Thirdly, sub-additivity may be undesirable for some actuarial and �nan-

cial problems, as pointed out by Dhaene et al. (2008), who suggested the use of V aR for

some merger-linked problems, for instance. Fourthly, for very important �nancial problems

V aR often provides valuable solutions from both theoretical (Basak and Shapiro, 2001,

Assa, 2015) and empirical (Annaert et al., 2009) viewpoints, and V aR also facilitates the

use probabilities in both the objective function and/or the constraints of several �nancial

optimization problems (Dupacová and Kopa, 2014, Zhao and Xiao, 2016, etc.).

The optimization of V aR is much more complicated than the optimization of other risk

measures (Rockafellar and Uryasev, 2000, Larsen et al., 2002, Gaivoronski and P�ug, 2005,

Shaw, 2011, Wozabal, 2012, etc.). Since V aR is neither convex nor di¤erentiable, one may

face the existence of many local minima, and they may become undetectable by means

of the standard optimization methods. There are many and quite di¤erent approaches

addressing the optimization of V aR (Larsen et al., 2002, Gaivoronski and P�ug, 2005,

2

Shaw, 2011, Wozabal, 2012, etc.). All of them yield interesting algorithms or optimality

conditions allowing us to �nd adequate solutions under di¤erent assumptions, but non of

them solves the problem in an exhaustive manner. There are many cases which cannot be

treated with the existent methodologies.

A very interesting approach may be found in Wozabal et al. (2010) and Wozabal (2012).

The authors deal with discrete probability spaces composed of �nitely many atoms, and

they prove that V aR equals the di¤erence of two convex functions. This property allows

them to provide e¢ cient optimizing algorithms. Nevertheless, it is easy to show that the

property above does not hold for general probability spaces. Since there are many problems

involving V aR and continuous random variables (Shaw, 2011, Zhao and Xiao, 2016, etc.),

further extensions containing general probability spaces should be welcome.

This paper deals with a very simple idea. If the CV aR (also called AV aR, or average

value at risk) may be given as an integral of V aR, then V aR must become a �rst order

mathematical derivative of CV aR. Consequently, an approximation of V aR must be given

by the change in CV aR over the change in level of con�dence (or, in other words, by a

quotient of increments). Hence, an approximation of V aR must be given by the di¤erence

of two convex functionals, and the result of Wozabal (2012) will become true in general

probability spaces if one takes a limit.

Ideas above will be formalized in Section 2, where it will be proved that V aR is the limit

of the di¤erence of convex functionals. We will also explain why one does not need to

take any limit in the discrete case. In Section 3 we will consider a sequence of optimization

problems whose objective function has a limit, and we will analyze the relationship between

the sequence of solutions and the solution optimizing the limit. As a consequence, we will

establish conditions under which the optimization of V aR may be solved by optimizing the

di¤erence of two convex functionals. In Section 4 we will focus on a methodology proposed

in Balbás et al. ( 2010a) and we will address the minimization of the di¤erence of two convex

functionals in arbitrary probability spaces. Several optimality conditions will be found.

Applications in �nance (optimal investment) and insurance (optimal reinsurance) will be

given in Section 5. Though the purpose of Section 5 is merely illustrative, these examples

will be general enough, since they will apply in both static and dynamic frameworks and

for discrete or continuous price/claim processes. Section 6 will summarize the paper.

3

,F , σ− F

≤ p <∞ Lp

Lp Lp ,F , y |y|p < ∞

Lq Lp

< q ≤ ∞ /p /q L

Lp

y p |y|p/p

≤ p <∞ y Ess Sup |y| Ess Sup

≤ p ≤ p ≤ ∞ Lp ⊃ Lp L ⊃ Lp ⊃ L ≤ p ≤

∞ ≤ p ≤ ∞ Lp

σ Lp, Lq < p <∞

Lp σ Lp, Lq −

Lp

≤ p ≤ ∞ Lp Lp ≤ p ≤ p ≤ ∞

Lp

L

L

L d y, z Min , |y − z|

L ⊂ L

− μ ∈ , y ∈ L

V aR1−μ (y) y

y

V aR1−μ (y) := Sup {x ∈ IR; IP (y ≤ x) < 1− μ} .

V aR μ y y

V aR μ y −Inf {x ∈ y ≤ x > μ} ,

y ∈ L ⊂ L CV aR μ y

CV aR μ yμ

μ

V aR t y dt.

CV aR μ y

CV aR μ y Max {− yz ≤ z ≤ /μ, z } ,

μ {z ∈ L ≤ z ≤ /μ, z } ,

y CV aR μ− Lq

≤ q ≤ ∞ σ Lq, Lp − < q ≤ ∞

−CV aR μ y Min { yz ≤ z ≤ /μ, z }

y ∈ L L −

L y → CV aR μ y ∈ ,

σ L ,L −

y ∈ L

, t→ V aR t y ∈

Lp y → V aR1−μ (y) ∈ IR p = 0 1 ≤ p <∞

μ = 0.1 Ω = (0, 1) F σ− IP

(0, 1) ω → yn (ω) =

⎧⎨⎩ −1, if 0 < ω < 0.1 + 1/(2n)

0, otherwise

n = 1, 2, ...

(0, 1) ω → y0 (ω) =

⎧⎨⎩ −1, if 0 < ω < 0.1

0, otherwise

Limn→∞ (yn) = y0 Lp 1 ≤ p < ∞ L0

V aR1−μ (y0) = 0 V aR1−μ (yn) = 1 n > 0

,

, μ→ ϕy μ μCV aR μ y ∈ ,

ϕy μμ

V aR t y dt,

ϕy μ V aR μ y

μ ∈ , ϕy ϕy

n ∈

V aR μ y ≈μ /n CV aR μ /n y − μCV aR μ y

/n,

i.e.

V aR μ y ≈ nμ CV aR μ /n y − nμCV aR μ y .

V aR μ y Limn nμ CV aR μ /n y − nμCV aR μ y

y ∈ L Y ⊂ L

Min {V aR μ y y ∈ Y }

Minnμ

nμCV aR μ /n y − CV aR μ y y ∈ Y

A fn n A i.e.

fn A→ n , , , ... fn n → f A

xn ∈ A Min {fn x x ∈ A} n ≥

a f x ≥ Lim Supn fn xn x ∈ A Lim Supn fn xn

fn xn n

b E C A ⊂ C ⊂ E

fn A → C n , , ...

i.e. fn λx λfn x x ∈ C λ ≥ n , , , ... Inf

f x x ∈ λ λA > −∞ f x ≥ Lim Supn fn xn x ∈

λ λA

a x ∈ A ε > f x ≥ −ε

Lim Supn fn xn n ∈ |fn x − f x | < ε

n ≥ n f x ≥ fn x − ε ≥ fn xn − ε n ≥ n

b x ∈ A f x ≥

Inf f z z ∈λ

λA ≥ Inf {λf x λ > } −∞,

f f x ≥

x ∈ λ λA Lim Supn fn xn <

Lim Supn fn xn ≥ λ ≥

λLim Supn fn xn ≥ Lim Supn fn xn .

z λx x ∈ A a f x ≥ Lim Supn fn xn

f z λf x ≥ λLim Supn fn xn ≥ Lim Supn fn xn

A fn n A

fn n → f A xn ∈ A Min {fn x x ∈ A}

n ≥

a x ∈ A A x

xn n x Limn xn f

x f x Lim Supn fn xn x Min {f x x ∈ A}

b a Min {f x x ∈ A}

A f A xn n

x ∈ A Min {f x x ∈ A} f x

Lim Supn fn xn Min {f x x ∈ A}

Lim Supn fn xn

c E C A ⊂ C ⊂ E

fn A → C n , , ...

Inf f x x ∈ λ λA > −∞ x ∈ A

A x xn n f

x f x Lim Supn fn xn x Min

f x x ∈ λ> λA

a ε > V x n ∈

f x > f x − ε x ∈ V |fn x − f x | < ε x ∈ A

n ≥ n x ∈ V n ≥ n

f x < f x ε < fn x ε.

n m ≥

n k ≥ m xk ∈ V f x < fk xk ε

f x ≤ Lim Supn fn xn ε f x ≤

Lim Supn fn xn a

b {∅} ∪ f ∪ f x,∞ x ∈

A A f A

xn n x ∈ A

f x Lim Supn fn xn Min {f x x ∈ A} a

A

A ⊂x

f x,∞ fx

x,∞ f Infxx,∞ ,

Infxx a ∈ A f a Min {f x x ∈ A}

a ∈ f Infxx,∞ f a > Infxx x,∞

f a > x a ∈ f x,∞ A ⊂

f x,∞ f x ≥ f a > x x ∈ A

c b f x ≤ Lim Supn fn xn

ε ε > a V ⊂ A x

n ∈ f x > f x − ε x ∈ V |fn x − f x | < ε

x ∈ A n ≥ n x ∈ V n ≥ n

a m ≥ n k ≥ m xk ∈ V

f x < fk xk ε

Y ⊂ L yn ∈ Y n ∈

V aR μ y ≥ Lim Supn nμ CV aR μ /n yn − nμCV aR μ yn

y ∈ Y

a

Y ⊂ L Y

n ∈ yn ∈ Y n ≥ n

a y ∈ Y Y y

yn n n0Y y → V aR μ y ∈

y y V aR μ y

b < p < ∞ Y ⊂ Lp Y y → V aR μ y ∈

σ Lp, Lq − Y yn n n0y ∈ Y

V aR μ y

c yn n n0y ∈ Y L − Y y

V aR μ y

a a b a

Y σ Lp, Lq − σ Lp, Lq −

yn n n0⊂ Y σ Lp, Lq −

y ∈ Y c a

Y y → nμ CV aR μ /n y − nμCV aR μ y ∈

L − Y y → V aR μ y ∈

Y

b c V aR

Y ⊂ L Inf V aR μ y y ∈ λ λY >

−∞ n ∈ yn ∈ Y n ≥ n

a

Min V aR μ y y ∈λ

λY

b Y y ∈ Y

Y y yn n Y y → V aR μ y ∈

y y V aR μ y

c Y < p < ∞ Y ⊂ Lp

Y y → V aR μ y ∈ σ Lp, Lq − Y

yn n n0y ∈ Y V aR μ y

d Y yn n n0y ∈ Y

L − Y y V aR μ y

V aR μ nμ CV aR μ /n − nμCV aR μ L

a

b b a c d b

b c

a

f a, b → < h ≤ b− a

≤ f a −h

a h

a

f t dt ≤ f a − f a h .

b y ∈ L n , , , ...

≤ V aR μ y − nμ CV aR μ /n y − nμCV aR μ y

≤ V aR μ y − V aR μ /n y .

c Y ⊂ L

Limn V aR μ /n y V aR μ y

y ∈ Y y ∈ Y

a hf a ≥a h

af t dt ≥ hf a h f f a ≥

h

a h

af t dt ≥ f a h ≤ f a −

h

a h

af t dt

−h

a h

af t dt ≤ −f a h f a −

h

a h

af t dt ≤ f a − f a h

b a hn

h

μ h

μ

V aR t y dt nμ CV aR μ /n y − nμCV aR μ y .

c

C ,ω ⊂ ω ∈

ω /∈ j ωj ≤ μ ω j ωj > μ C

∅ y

V aR μ y ,ω ∈ C

{ω y ,ω y , ...,ωk y } y ω y ≤ y ω y ≤ .... ≤ y ωk y

{ω y ,ω y , ...,ωj y }ji ωi y ≤ μ j

i ωi y > μ

ω ωj y −V aR μ y y ω μ /n <ji ωi y −V aR μ /n y y ω i.e. V aR μ y V aR μ /n y

C n ∈ n ≥ n ,ω ∈ C

ω j ωj > μ /n V aR μ y V aR μ /n y

n ≥ n y ∈ L L L

n ≥ n

y ∈ L y

n ≥ n

L

Y ⊂ L

n ≥ n

V aR μ

k > ν >

− μ− ν ∈ , Y ⊂ L

Min {kCV aR μ ν y − CV aR μ y y ∈ Y } .

CV aR

a

a

b a b

y ∈ Y y θ ∈ z ∈ L

y , θ , z

Min θ yz

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

θ k yw ≥ , ∀w ∈ μ ν

z

z ≥

z ≤ /μ

y ∈ Y

y, θ, z ∈ L × × L

y, θ, z θ kCV aR μ ν y

y z −CV aR μ y

y y, θ, z θ

kCV aR μ ν y z ∈ μ y z −CV aR μ y

y , θ , z y, θ, z

θ ≥ kCV aR μ ν y −CV aR μ y ≤ yz

θ yz ≥ kCV aR μ ν y − CV aR μ y ≥

kCV aR μ ν y − CV aR μ y θ y z .

y , θ , z y ∈ Y

z ∈ μ yz −CV aR μ y

y, θ kCV aR μ ν y , z .

kCV aR μ ν y − CV aR μ y θ yz ≥ θ y z .

θ kCV aR μ ν y .

θ kCV aR μ ν y < θ

θ y z > kCV aR μ ν y y z

y z −CV aR μ y .

y z > −CV aR μ y z ∈ μ

y , θ , z y z −CV aR μ y < y z θ

y z > θ y z

y

θ− y, z −

y z

Y y ∈ Y

w , z ,α,α ,αμ ∈ μ ν × μ × × L × L⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

α z

αμ /μ− z

y α α − αμ

α ≥ , αμ ≥

y z − kw ≤ y z − kw , ∀y ∈ Y

y w ≤ y w , ∀w ∈ μ ν

≤ p ≤ ∞ y ∈ Lp α ,αμ ∈ Lp × Lp

y , θ , z z ∈ μ y , θ

Min θ yz

⎧⎪⎪⎨⎪⎪⎩

θ k yw ≥ , ∀w ∈ μ ν

y ∈ Y

y, θ ∈ L ×

a⎧⎨⎩ Max w Inf { y z − kw y ∈ Y }

w ∈ μ ν

,

θ , z

Min θ y z

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

θ k y w ≥ , ∀w ∈ μ ν

z

z ≥

z ≤ /μ

θ, z ∈ × L

a

L z, w,α ,αμ y z − k y w − zα dω zαμ dω ,

α ≥ αμ ≥ L z,w,α ,αμ

L z, w,α ,αμ

{z ∈ L z } α ∈

y z − zα dω zαμ dω α z .

z ∈ L y −α αμ α

α

⎧⎨⎩ y α, z < /μ

, otherwiseand αμ

⎧⎨⎩ α− y , z >

, otherwise,

α ,αμ ∈ Lp × Lp y ∈ Lp

V aR

ORP

PCAA

a

u ≥

T ur

uc u ur uc ur ≥ uc ≥ ORP

stop loss uc u− U Max {u− U, }

U ≥

X

x ,∞ →

x Sup {|x t | t ≥ } u σu u

u < ∞ σu < ∞ u ,∞

i.e. B u ∈ B B ⊂ ,∞

J X → L

J x tt

x s ds,

J

|J x t | ≤t

x ds x t

x ∈ X t ≥

J x dx ≤ x t dx x σu u

x ∈ X

x

u u t J t t

t ≥ x ∈ X ur J x uc J − x

x ≥ h

h ∈ X h ≥ stop − loss

h stop− loss

h ≥ ε ε >

C >

C uc C J − x

W x − J x − C J − x

V aR μ W x

ORP ⎧⎪⎪⎨⎪⎪⎩Max W x

Min V aR μ W x

x ∈ X, h ≤ x ≤

C > ∈ C J ∈ V aR μ

⎧⎪⎪⎨⎪⎪⎩Min − J x

Min V aR μ −J x − C J x

x ∈ X, h ≤ x ≤

W > J x V aR μ −J x − C J x

ORP V aR μ −J x − C W J x

ORP W C W > V aR μ

⎧⎨⎩ Min V aR μ W J x − J x

x ∈ X, h ≤ x ≤

Y {W J x − J x x ∈ X, h ≤ x ≤ } .

Y ⊂ L

Y Y b

c Y c

Limn V aR μ /n W J x − J x V aR μ W J x − J x

x ∈ X, h ≤ x ≤ V aR ν − ν ∈

,

Limn V aR μ /n −J x V aR μ −J x

x ∈ X, h ≤ x ≤ h ≤ x ≤ J x J − x

V aR ν

− ν ∈ ,

V aR μ /n −J V aR μ /n −J x V aR μ /n −J − x

V aR μ −J V aR μ −J x V aR μ −J − x

V aR μ −J x − V aR μ /n −J x

V aR μ −J − V aR μ −J − x

− V aR μ /n −J − V aR μ /n −J − x

V aR μ −J − V aR μ /n −J

− V aR μ −J − x − V aR μ /n −J − x ≤

V aR μ −J − V aR μ /n −J

y J

b c

w , z ,α,α ,αμ ∈ μ ν × μ × × L × L z

z − kw⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

α z

αμ /μ− z

−J x W J x α α − αμ

α ≥ , αμ ≥

W J x − J x z ≤ W J x − J x z , ∀h ≤ x ≤

W J x − J x w ≤ W J x − J x w , ∀w ∈ μ ν

b

CV aR

PCAA b

SDF

, T T

,F , y L

i.e. y ∈ L

b

L y → y

z y ∈ z ∈ L SDF z >

z

PCAA R >

R

PCAA

Min V aR μ y

⎧⎨⎩ y ≥ R, z y ≤

y ∈ L

y y −R R V aR μ y −R V aR μ y −R y ≥ R ⇔ y −R ≥

z y ≤ ⇔ z y −R ≤ −R y y −R

y

Min V aR μ y

⎧⎨⎩ y ≥ , z y ≤ −α

y ∈ L

α R − >

b

M y y ≤ M

Min V aR μ y

⎧⎨⎩ y ≥ , z y ≤ −α

y ∈ L , y ≤M

σ L ,L −

λy λ ≥ y a

c

c d

V aR

authors have justi�ed the usefulness of V aR in many applications.

The optimization of V aR is much more complicated than the optimization of other risk

measures. Since V aR is neither convex nor di¤erentiable, the standard methods of mathe-

matical programming are frequently di¢ cult to apply. There are many and quite di¤erent

approaches addressing the optimization of V aR. All of them yield interesting algorithms or

optimality conditions, but non of them solves the problem in an exhaustive manner. There

are many cases which cannot be treated with the existent methodologies.

This paper has proved that a V aR approximation may be given with a linear combination

of two CV aRs with di¤erent con�dence level. More accurately, V aR is a CV aR derivative,

and therefore it is the limit of a sequence of linear combination of CV aRs with di¤erent

con�dence level. This property has been used in order to provide new methods to optimize

both V aR and linear combinations of CV aRs in general probability spaces. Applications

in �nance (optimal investment) and insurance (optimal reinsurance) have been given. They

show the practical e¤ectiveness of the provided new methodologies.

Acknowledgments. This research was partially supported by �Ministerio de Economía�

(Spain, Grant ECO2012− 39031− C02− 01). The usual caveat applies.

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