1_xxx_gupta.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 3, No. 5, 2010, 779-785

ISSN 1307-5543 – www.ejpam.com

Convex Ordering of Random Variables and its Applications in

Econometrics and Actuarial Science

Arjun K. Gupta∗, Mohammad A. S. Aziz

Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, Ohio

43403, U.S.A

Abstract. It is well known that in economics and finance, the data usually have “fat tail” and in this

case the Normal distribution is not a good model to use. The skew normal distributions recently draw

considerable attention as an alternative model. Unfortunately, the distribution of the sum of log-skew

normal random variables does not have a closed form. In this work, we discuss the use of lower

convex order of random variables to approximate this distribution. Further, two application of this

approximate distribution are given : first to describe the final wealth of a series of payments, and

second to describe the present value of a series of payments.

2000 Mathematics Subject Classifications: Primary 62E17; Secondary 62P05

Key Words and Phrases: Log-skew normal random variable, lower convex order bound, comono-

tonocity

1. Preliminaries

Definition 1. Consider two random variables X and Y such that E[φ(X )] ≤ E[φ(Y )] , for all

convex functions φ, provided expectation exist.Then X is said to be smaller than Y in the convex

order denoted as X ≤cx Y .

Definition 2 (Convex order definition using stop-loss premium). Consider two random vari-

ables X and Y. Then X is said to precede Y in convex order sense if and only if

E[X ] = E[Y ]

E[(X − d)+]≤ E[(Y − d)+], I(−∝,∝)(d)

where

(X − d)+ = max(X − d , 0)

∗Corresponding author.

Email address: gupta�bgsu.edu (Arjun K. Gupta)

http://www.ejpam.com 779 c© 2010 EJPAM All rights reserved.



A Gupta, M Aziz / Eur. J. Pure Appl. Math, 3 (2010), 779-785 780

An equivalent definition can be derived from the following relation

E[(X − d)+]− E[(d − X )+] = E(X )− d

For the random variable Y the same relation is given by,

E[(Y − d)+]− E[(d − Y )+] = E(Y )− d

Now assume X ≤cx Y , which implies that

E[X ] = E[Y ]

and

E[(X − d)+]≤ E[(Y − d)+], I(−∝,∝)(d)

Hence

E[(d − X )+]≤ E[(d − Y )+]

Therefore, a definition equivalent to the definition here is

E[X ] = E[Y ]

E[(d − X )+]≤ E[(d − Y )+]

1.1. Properties of Convex Ordering of Random Variables

1. If X precedes Y in convex order sense i.e if X ≤cx Y , then E[X ] = E[Y ] and

Var[X ] ≤ Var[Y ]

2. If X ≤cx Y and Z is independent of X and Y then X + Z ≤cx Y + Z

3. Let X and Y be two random variables, then X ≤cx Y ⇔−X ≤cx −Y

4. Let X and Y be two random variables such that E[X ] = E[Y ]. Then X ≤cx Y if and only

if E|X − a| ≤cx E|Y − a|,∀ a ∈ ℜ
5. The convex order is closed under mixtures: Let X, Y and Θ be random variables such

that [X |Θ = θ] ≤cx [Y |Θ= θ]∀θ in the support of Θ. Then X ≤cx Y .

6. The convex order is closed under convolution: Let X1, X2, . . . , Xm be a set of independent

random variables and Y1, Y2, . . . , Yn be another set of independent random variables. If

X i ≤cx Yi, for i = 1, ...., m, then
∑m

j=1 X j ≤cx

∑m
j=1 Yj

7. Let X be a random variable with finite mean. Then X + E[X ]≤cx 2X

8. Let X1, X2, . . . , Xm and Y be (n+1) random variables. If X i ≤cx Y , i = 1, ...., n, then∑n
i=1 aiX i ≤cx Y , whenever ai ≥ 0,i = 1, . . . , n and

∑n
i=1 ai = 1



A Gupta, M Aziz / Eur. J. Pure Appl. Math, 3 (2010), 779-785 781

9. Let X and Y be independent random variables. Then X i ≤cx Yi if and only if

E[φ(X , Y )]≤ E[φ(Y, X )]∀φ ∈ ℘cx , where

℘cx = {φ :ℜ2 −→ℜ : φ(X , Y )−φ(Y, X ) is convex for all x ∈ y}.

10. Let X1 and X2 be a pair of independent random variables and let Y1 and Y2 be another

pair of independent random variables. If X i ≤cx Yi , i = 1,2 then X1X2 ≤cx Y1Y2.

2. Main Result of Convex Ordering

Theorem 1. For any random vector X = (X1, X2, ....., Xm) and any random variable Λ, which is

assumed to be a function of X, we have,

n∑

i=1

E[X i|Λ]≤cx

n∑

i=1

X i

Proof. From the definition 1 we have, X ≤cx Y if and only if E[φ(X )] ≤ E[φ(Y )]. In

accordance with this definition we need to show that

EΛ[φ(

n∑

i=1

E[X i|Λ])]≤cx E[φ(

n∑

i=1

X i)]

Now,

E[φ(

n∑

i=1

X i)] = EΛE[φ(

n∑

i=1

X i)|Λ]≥ EΛ[φ(E(

n∑

i=1

X i|Λ))] = EΛ[φ(

n∑

i=1

E[X i|Λ])]

The last inequality was obtained by Jensen’s inequality, which states that for any convex

function φ,

φ(E(X ))≤ E(φ(X ))

Therefore,

EΛ[φ(

n∑

i=1

E[X i|Λ])]≤ E[φ(

n∑

i=1

X i)]

Hence,
n∑

i=1

E[X i|Λ]≤cx

n∑

i=1

X i

which completes the proof.



A Gupta, M Aziz / Eur. J. Pure Appl. Math, 3 (2010), 779-785 782

2.1. Lower Bound Approximations of the Distribution Sum of Random Variables

with Convex Ordering

In this section we will describe two examples [follow from 1] that show how distribution

function of the sum of random variables can be approximated by convex order of random

variable.

Example 1 (Approximation of distribution sum of two independent standard normal random

variables). Suppose X and Y be independent N(0,1) random variables. We want to derive lower

bounds for S = X + Y . In this case we know the exact distribution of S, i.e S ∼ N(0,2). Let us

see how lower bound approximation works in this case.

Let Z = X + aY for some real a . Then Z ∼ N(0,1+ a2). The conditional distribution of S|Z
is

N[µS +
ρS,ZσS

σZ

(Z −µZ ),σ
2
S(1−ρ2

S,Z)]

Here Cov(X+Y, X+aY) = 1.1cov(X , X )+1.acov(Y, Y ) = 1+a and ρS,Z =
1+ap

2
p

1+a2
. Therefore,

S|Z ∼ N[Z
1+ a

1+ a2
,
(1− a)2

1+ a2
]

Hence

E[S|Z] = 1+ a

1+ a2
Z ∼ N[0,

(1+ a)2

1+ a2
]

Therefore, for some choices of a, we get the following distribution for the lower bound for S:

a = 0 gives N(0,1)≤cx S = X + Y ∼ N(0,2)

a = 1 gives N(0,2)≤cx S = X + Y ∼ N(0,2)

a = −1 gives N(0,0)≤cx S = X + Y ∼ N(0,2)

Thus in this case best lower bound is obtained for a = 1 which is the exact distribution. The

variance of the lower bound can be seen to have a maximum at a = 1 and a minimum at

a = −1.

Example 2 (Approximation of distribution sum of two lognormal random variables). Suppose

Y1 and Y2 be independent N(0,1). Define X1 = eY1 ⇒ X1 ∼ lognormal(0,1) and X2 = eY1+Y2 ⇒
X2 ∼ lognormal(0,2). We want to find the lower bound for the distribution of S = X1 + X2. Let

Z = Y1 + Y2. As shown in example 1, the conditional distribution of Y1 given Z is,

Y1|(Y1+ Y2) = Z ∼ N(
1

2
Z ,

1

2
)

Therefore, E[X1 = eY1 |(Y1+ Y2) = z] = MY (1, 1

2
Z , 1

2
), where Y ∼ N(µ,σ2)

= ex p(
1

2
Z +

1

4
)



A Gupta, M Aziz / Eur. J. Pure Appl. Math, 3 (2010), 779-785 783

We also observe that E[X2|(Y1+ Y2) = z] = eZ .Therefore the lower bound for approximating the

distribution of S = X1 + X2 is Sl = E[(X1 + X2)|Z] = ex p(1

2
Z + 1

4
). It can be easily verified

that E(Sl) = E[ex p(1

2
Z + 1

4
)] = e

1

2 + e and E((Sl)
2) = e

3

2 + 2e
5

2 + e4. Thus the variance of the

lower bound is 64.374 and is close to the variance of S = 67.281. The idea is to obtain lower

convex bound in such a way that the variance of the lower bound gets as close as possible to the

variance of the sum. With this view in mind considering more general form of the conditioning

variable as Z = Y1 + aY2, it could be shown that optimal lower bound is reached for a = 1.27

and the variance of Sl in this case is 66.082. Thus choosing the conditioning variable is crucial

in determining the lower convex order bound.

3. Application of Convex Order of Random Variables in Finance and Economics

Let α0,α1,α2, . . . ,αn−1 be non-negative real numbers. Let Y = (Y1, Y2, . . . , Yn)
T be a mul-

tivariate skew normal random vector with specified mean vector and variance-covariance ma-

trix and satisfying additive properties. Define, Zi =
∑n

k=i+1 Yk, i = 0,1, . . . , n− 1, that is, Zi ’s

are linear combinations of the components (Y1, Y2, . . . , Yn). With the components so defined,

consider the sum

S =

n−1∑

i=0

αie
Zi =

n−1∑

i=0

αie
Yi+1+,...,+Yn (1)

From economic or actuarial point of view, the sum S could be interpreted as the final wealth

or terminal wealth or the accumulated value of a series of deterministic saving amounts or

alternatively the accumulated value of a series of payments. In this situation, αi (i = 0, . . . , n−
1) represents yearly saving in period i or amount invested in period i, Yi+1 refers to the random

rate of return in period i for i = 0, . . . , n−1. The term Yk = log
Pk

Pk−1
= logPk− logPk−1 i.e eYk

=
Pk

Pk−1
, where Pk is the price of the asset at period k , for k = 0, . . . , n; is called the random

log-return in period k and Zi denote the sum of stochastic or random returns in period i,

i = 0, . . . , n− 1. With suitable adjustment, S could also be referred as the present value of

a series of payments. To be more precise, if −Zi denotes the stochastic log-return over the

period [0, i], then eZi ,represents the stochastic discount factor over the period [0, i]. In this

situation, the sum S is the present value of αi [4].

The sum defined in (1) plays a central role in the actuarial and financial theory because it

allows computation of risk measures such as value at risk or stop-loss premium. To calculate

the risk measures we need to evaluate the distribution function of S. Unfortunately, the dis-

tribution of the sum S (of log-normally or log-skew normally distributed random variables)

is not available in the closed-form. It is possible to use Monte Carlo simulation method to

approximate the distribution function. However, Monte Carlo simulation of the distribution

is often time-consuming. Thus one has to find alternate way to approximate the distribution

of the sum. Among the proposed solutions, moment matching methods, lognormal and in-

verse gamma approximations are commonly used. Both methods approximate the unknown

distribution function by a given one such that the first two moments coincide.

Kaas, et al. [2] and Dhaene, et al. [1] propose to approximate the distribution function



REFERENCES 784

of S by so called “convex lower bound”. The underlying idea of convex lower order bound is

to replace an unknown or too complex distribution ( for which no explicit form is found) by

another one which is easier to determine. In this approach, the real distribution is known to

be bounded in terms of convex ordering to the approximated distribution. To be more precise,

by Theorem 1, the distribution function of S =
∑n−1

i=0 αie
Zi is approximated by the distribution

function of Sl , where Sl is defined by,

Sl =

n−1∑

i=0

αi E(e
Zi |Λ) (2)

An appropriate choice of the conditioning random variable Λ is required. This approach has

two-fold advantages. Firstly, use of this approach transforms the multidimensionality problem

caused by (Z0, Z2, . . . , Zn−1) to a single dimension caused by Λ. Secondly, an appropriate

choice of Λ (that makes the expectation in (2) non-decreasing or non-increasing function of

the conditioning random variable Λ) will make a comonotonic sum i.e the elements of the sum

in (2) posses the so called comonotonic dependence structure. Using additivity properties of

sum of comonotonic random variables risk measures related to the distribution function of S is

then approximated by the corresponding risk measures of Sl . According to [2], comonotonic

upper bound for the sum in convex order sense can also be derived using the result

n−1∑

i=0

X i ≤cx

n−1∑

i=0

FX i
(U),

where U is the uniform random variable over (0,1). However, comonotonic upper bounds

generally provide too conservative estimates of the cumulative distribution function [3]. Thus

we only discussed convex lower bound here.

Remark 1. In general, the random vector (E(X0|Λ), E(X1|Λ), . . . , E(Xn−1|Λ)) does not have the

same marginal distribution as (X0, X1, . . . , Xn−1). However, if the conditioning random variable

Λ is chosen in such a way that all random variables E(X i|Λ), (i = 0,1,2, . . . , n− 1) are non-

decreasing functions of Λ (or non-increasing functions of Λ), then the sum
∑n−1

i=0 E[X i|Λ] is a

sum of n comonotonous random variables and can be referred to as comonotonic lower bound.

Hence risk measures for the sum could easily be obtained by summing the corresponding risk

measures for the marginals involved.

References

[1] J Dhaene, M Denuit, M Gooverts, R Kass, and D Vyncke. The concept of comonotonicity

in actuarial science and finance: Applications. Insurance: Mathematics and Economics,

31:133–161, 2002.

[2] J Dhaene R Kaas, M Goovaerts and M Denuit. Modern Actuarial Risk Theory. Kluwer

Academic Publishers, Dordrecht, 2001.



REFERENCES 785

[3] O Roch and E Valdez. Lower convex order bound approximations for sums of log-skew

normal random variables. Working Paper (2008).

[4] S Vanduffel, X Chen, J Dhaene, M Goovaerts, L Henrard, and R Kass. Optimal approxima-

tions for risk measures of sums of lognormals based on conditional expectations. Journal

of Computational and Applied Mathematics, 221:202–218, 2008.