Insurance: Mathematics and Economics 27 (2000) 151–168
Upper and lower bounds for sums of random variables夽
Rob Kaas a,b,∗ , Jan Dhaene a,b , Marc J. Goovaerts a,b
a
Institute of Actuarial Science, University of Amsterdam, Roetersstraat 11, 1018 WB Amsterdam, Netherlands
b CRIR, KU Leuven, Belgium
Received March 2000; received in revised form July 2000
Abstract
In this contribution, the upper bounds for sums of dependent random variables X1 + X2 + · · · + Xn derived by using
comonotonicity are sharpened for the case when there exists a random variable Z such that the distribution functions of the
Xi , given Z = z, are known. By a similar technique, lower bounds are derived. A numerical application for the case of
lognormal random variables is given. © 2000 Elsevier Science B.V. All rights reserved.
Keywords: Dependent risks; Comonotonicity; Convex order; Cash-flows; Present values; Stochastic annuities
1. Introduction
In some recent articles, Goovaerts, Denuit, Dhaene, Müller and several others have applied theory originally
studied by Fréchet in the previous century to derive upper bounds for sums S = X1 + X2 + · · · + Xn of random
variables X1 , X2 , . . . , Xn of which the marginal distribution is known, but the joint distribution of the random
vector X1 , X2 , . . . , Xn is either unspecified or too cumbersome to work with. These upper bounds are actually
suprema in the sense of convex order. The concept of convex order is closely related to the notion of stop-loss order
which is more familiar in actuarial circles. Both express which of two risks is the more risky one. Assuming that
only the marginal distributions of the Xi are given (or used), the riskiest instance Su of S occurs when the risks
X1 , X2 , . . . , Xn are comonotonous. This means that they are all non-decreasing functions of one uniform (0, 1)
random variable U , and since the marginal distribution must be Pr[Xi ≤ x] = Fi (x), the comonotonous distribution
is that of the vector F1−1 (U ), F2−1 (U ), . . . , Fn−1 (U ).
In this contribution we assume that the marginal distribution of each random variable X1 , X2 , . . . , Xn is known.
In addition, we assume that there exists some random variable Z, with a known distribution function, such that for
any i and for any z in the support of Z, the conditional distribution function of Xi , given Z = z, is known. We will
derive upper and lower bounds in convex order for S = X1 + X2 + · · · + Xn , based on these conditional distribution
functions. Two extreme situations are possible here. One is that Z = S, or some one-to-one function of it. Then
the convex lower bound for S, which equals E[S|Z], will just be S itself. The other is that Z is independent of all
X1 , X2 , . . . , Xn . In this case we actually do not have any extra information at all and the upper bound for S is just
the same comonotonous bound as before, while the lower bound reduces to the trivial bound E[S]. But in some
夽
Paper presented at the Fourth Conference on Insurance: Mathematics and Economics, Barcelona, 24–26 July 2000.
Corresponding author.
E-mail address: [email protected] (R. Kaas).
∗
0167-6687/00/$ – see front matter © 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 1 6 7 - 6 6 8 7 ( 0 0 ) 0 0 0 6 0 - 3
152
R. Kaas et al. / Insurance: Mathematics and Economics 27 (2000) 151–168
cases, and the lognormal discount process of Section 5 is a good example, a random variable Z can be found with
the property that by conditioning on it we can actually compute a non-trivial lower bound and a sharper upper bound
than Su for S.
In Section 2, we will present a short exposition of the theory we need. Section 3 gives upper bounds, Section 4
improved upper bounds, as well as lower bounds, both applied to the case of lognormal distributions in Section 5.
Section 6 gives numerical examples of the performance of these bounds, and Section 7 concludes.
2. Some theory on comonotonous random variables
Let F1 , F2 , . . . , Fn be univariate cumulative distribution functions (cdfs in short). Fréchet studied the class of all
n-dimensional cdfs FX of random vectors X ≡ (X1 , X2 , . . . , Xn ) with given marginal cdfs F1 , F2 , . . . , Fn , where
for any real number x we have Pr[Xi ≤ x] = Fi (x), i = 1, 2, . . . , n. In this paper, we will consider the problem
of determining stochastic lower and upper bounds for the cdf of the random variable X1 + X2 + · · · + Xn , without
restricting to independence between the terms Xi . We will always assume that the marginals cdfs of the Xi are
given, and that all cdfs involved have a finite mean.
The stochastic bounds for random variables will be in terms of “convex order”, which is defined as follows:
Definition 1. Consider two random variables X and Y . Then X is said to precede Y in the convex order sense,
notation X ≤cx Y , if and only if for all convex real functions v such that the expectations exist, we have
E[v(X)] ≤ E[v(Y )].
It can be proven, see e.g. Shaked and Shanthikumar (1994), that the condition in this definition is equivalent with
the following condition:
E[X] = E[Y ],
E[X − d]+ ≤ E[Y − d]+
for all d,
where E[Z]+ is a notation for E[max{Z, 0}].
Using an integration by parts, the ordering condition between the stop-loss premiums E[X − d]+ and E[Y − d]+
can also be expressed as
Z ∞
Z ∞
(1 − FX (x)) dx ≤
(1 − FY (x)) dx for all d.
d
d
In case X ≤cx Y , extreme values are more likely for Y than for X. In terms of utility theory, X ≤cx Y entails
that loss X is preferred to loss Y by all risk averse decision makers, i.e. E[u(−X)] ≥ E[u(−Y )] for all concave
non-decreasing utility functions u. This means that replacing the (unknown) distribution function of a loss X by
the distribution function of a loss Y can be considered as an actuarially prudent strategy, e.g. when determining
reserves.
From the above relation, we see immediately that
d
{E[X − x]+ − E[Y − x]+ } = FX (x) − FY (x).
dx
Thus, two random variables X and Y with equal mean are convex ordered if their cdfs cross once. This last condition
can be observed to hold in most conceivable examples, but it is easy to construct instances with X ≤cx Y where the
cdfs cross more than once.
It follows immediately that X ≤cx Y implies Var[X] ≤ Var[Y ]. The reverse implication does not hold in general;
for a counterexample, see e.g. Brockett and Garven (1998). Also note that X ≤cx Y is equivalent to −X ≤cx −Y .
This means that it makes no difference if we interpret the random variables as losses or as gains.
R. Kaas et al. / Insurance: Mathematics and Economics 27 (2000) 151–168
153
For any random vector X with marginal cdfs F1 , F2 , . . . , Fn the following convex order relation holds:
X1 + X2 + · · · + Xn ≤cx Su = F1−1 (U ) + F2−1 (U ) + · · · + Fn−1 (U ),
def
where U is a uniform (0, 1) random variable, and where the pth quantile of a random variable X with cdf FX is, as
usual, defined by
FX−1 (p) = inf{x ∈ R|FX (x) ≥ p},
def
p ∈ [0,1].
Goovaerts et al. (2000) prove this order relation directly, while Müller (1997) derives it as a special case of the concept
of supermodular ordering. This relation can be interpreted as follows: the most risky random vector with given
marginals (in the sense that the sum of their components is largest in the convex order sense) has the comonotonous
joint distribution, which means that it has the joint distribution of F1−1 (U ), F2−1 (U ), . . . , Fn−1 (U ). The components
of this random vector are maximally dependent, all components being non-decreasing functions of the same random
variable.
The inverse cdf of a sum of comonotonous random variables can easily be computed. Indeed, if Su =d F1−1 (U ) +
F2−1 (U ) + · · · + Fn−1 (U ), where =d means equality in distribution, then
(p) =
FS−1
u
n
X
Fi−1 (p),
p ∈ [0,1].
i=1
Recently, the concept of comonotonicity has been considered in many actuarial papers, see e.g. Müller (1997),
Wang and Dhaene (1998) and Dhaene et al. (1998). Dependence in portfolios and related stochastic orders are also
considered in Dhaene and Goovaerts (1996), Denuit and Lefèvre (1997), Dhaene and Goovaerts (1997), Bäuerle
and Müller (1998), Wang and Young (1998), Goovaerts and Redant (1999), Denuit et al. (1999), Dhaene and Denuit
(1999) and others.
3. Comonotonous upper bounds for sums of random variables
The usual definition of the inverse of a cdf is the left-continuous function FX−1 (p) = inf{x ∈ R|FX (x) ≥ p}.
But if FX (x) = p holds for an interval of values for x, any element of it could serve as FX−1 (p). In this paper, we
introduce a more sophisticated definition which enables us to choose that particular inverse cdf with the property
that for a certain d, the relation FX−1 (FX (d)) = d holds.
For p ∈ [0, 1], a possible choice for the inverse of FX in p is any point in the interval
[inf{x ∈ R|FX (x) ≥ p}; sup{x ∈ R|FX (x) ≤ p}].
Here we take inf ∅ = +∞ and sup ∅ = −∞. Taking the left-hand side border of this interval to be the value of the
inverse cdf at p, we get FX−1 (p). Similarly, we define FX−1• (p) as the right-hand side border of the interval
FX−1• (p) = sup{x ∈ R|FX (x) ≤ p},
p ∈ [0,1].
Note that FX−1 (0) = −∞ and FX−1• (1) = +∞, while FX−1 (p) and FX−1• (p) are finite for all p ∈ (0, 1). For any α
in [0, 1], we define the α-inverse of FX as follows:
−1(α)
FX
(p) = αFX−1 (p) + (1 − α)FX−1• (p),
p ∈ (0,1).
For a comonotonous random vector (X1 , X2 , . . . , Xn ), it follows that for all α in [0, 1]
−1(α)
FX1 +X2 +···+Xn (p) =
n
X
−1(α)
FXi (p),
i=1
p ∈ (0,1).
154
R. Kaas et al. / Insurance: Mathematics and Economics 27 (2000) 151–168
The following result was already mentioned in Section 1. We give a new proof for it, based on the α-inverse just
introduced, because this method of proof leads to new results that we will need in the sequel of this paper.
Proposition 1. Let U be a uniform (0, 1) random variable. For any random vector (X1 , X2 , . . . , Xn ) with marginal
cdfs F1 , F2 , . . . , Fn , we have
X1 + X2 + · · · + Xn ≤cx F1−1 (U ) + F2−1 (U ) + · · · + Fn−1 (U ).
Proof. Let S and Su be defined by S = X1 + X2 + · · · + Xn and Su = F1−1 (U ) + F2−1 (U ) + · · · + Fn−1 (U ),
respectively, with U uniform (0, 1). Then obviously E[S] = E[Su ]. To prove the stop-loss inequalities needed
to establish convex order, consider an arbitrary fixed real number d, with 0 < FSu (d) < 1. Let α ∈ [0, 1] be
determined such that
−1(α)
FSu
(FSu (d)) = d.
Then we have
−1(α)
E[S − d]+ = E[S − FSu (FSu (d))]+
" n
#
n
X
X
−1(α)
−1(α)
=E
(Xi − Fi
(FSu (d)))
≤
E[Xi − Fi
(FSu (d))]+ .
i=1
i=1
+
On the other hand we find
Z
(U ) − d]+ =
E[Su − d]+ = E[FS−1
u
=
n Z
X
1
1
0
(FS−1
(p) − d)+ dp =
u
−1(α)
i=1 FSu (d)
(Fi−1 (p) − Fi
−1(α)
−1(α)
1
FSu (d)
−1(α)
(FS−1
(p) − FSu
u
(FSu (d))) dp
(FSu (d))) dp.
One can verify that for any p ∈ (FSu (d); Fi (Fi
Fi−1 (p) = Fi
Z
(FSu (d)))) we have
(FSu (d)).
This implies
E[Su − d]+ =
=
=
n Z
X
1
−1(α)
−1(α)
(FSu (d)))
Fi (Fi
i=1
n Z 1
X
i=1 0
n
X
i=1
(Fi−1 (p) − Fi
−1(α)
(Fi−1 (p) − Fi
−1(α)
E[Xi − Fi
(FSu (d))) dp
(FSu (d)))+ dp =
n
X
−1(α)
E[Fi−1 (U ) − Fi
(FSu (d))]+
i=1
(FSu (d))]+ ,
so we have proven that E[S − d]+ ≤ E[Su − d]+ holds for all retentions d with 0 < FSu (d) < 1.
As the stop-loss transform is a continuous, non-increasing function of the retention d, we find that the result
above implies
(0)]+ ≤ E[Su − FS−1•
(0)]+ ,
E[S − FS−1•
u
u
R. Kaas et al. / Insurance: Mathematics and Economics 27 (2000) 151–168
155
as well as
(1)]+ ≤ E[Su − FS−1
(1)]+ .
E[S − FS−1
u
u
So E[S − d]+ ≤ E[Su − d]+ also holds for retentions d with FSu (d) = 0 or FSu (d) = 1.
(0), FS−1
(1)), then 0 < FSu (d) < 1, so we find the following corollary from the proof of
If d ∈ (FS−1•
u
u
Proposition 1.
(0), FS−1
(1)), then the
Corollary 1. Let U be uniform (0, 1) and let Su = F1−1 (U ) + · · · + Fn−1 (U ). If d ∈ (FS−1•
u
u
stop-loss premium at retention d of Su is given by
E[Su − d]+ =
n
X
−1(α)
E[Xi − Fi
(FSu (d))]+
i=1
−1(α)
with α ∈ [0, 1] determined by FSu
(FSu (d)) = d.
The expression for the stop-loss premiums of a comonotonous sum Su can also be written in terms of the usual
(0), FS−1
(1)), we have
inverse cdfs. Indeed, for any retention d ∈ (FS−1•
u
u
−1(α)
E[Xi − Fi
−1(α)
(FSu (d))]+ = E[Xi − Fi−1 (FSu (d))]+ − (Fi
(FSu (d)) − Fi−1 (FSu (d)))(1 − FSu (d)).
Summing over i, and taking into account the definition of α, we find the expression derived in Dhaene et al. (1998),
where the random variables are assumed to be non-negative
E[Su − d]+ =
n
X
E[Xi − Fi−1 (FSu (d))]+ − (d − FS−1
(FSu (d)))(1 − FSu (d)).
u
i=1
From Corollary 1, we can conclude that any stop-loss premium of a sum of comonotonous random variables can
be written as the sum of stop-loss premiums for the individual random variables involved. Corollary 1 provides an
algorithm for directly computing stop-loss premiums of sums of comonotonous random variables, without having
to compute the entire cdf of the sum itself. Indeed, in order to compute the stop-loss premium with retention d, we
only need to know FSu (x) for x equal to d. The cdf at x follows from
(
)
n
X
−1
FSu (x) = sup p ∈ [0,1]| Fi (p) ≤ x .
i=1
Now assume that the marginal cdfs Fi are continuous on R and strictly increasing on (Fi−1• (0), Fi−1 (1)). Then one
(0), FS−1
(1)), and that FS−1
is strictly
can verify that FSu is also continuous on R and strictly increasing on (FS−1•
u
u
u
−1•
−1
increasing and continuous on (0, 1). Hence, for any x ∈ (FSu (0), FSu (1)), the value FSu (x) can be obtained
unambiguously from
n
X
Fi−1 (FSu (x)) = x.
i=1
In this case, we also find
E[Su − d]+ =
n
X
E[Xi − Fi−1 (FSu (d))]+ ,
i=1
(0), FS−1
(1)).
which holds for any retention d ∈ (FS−1•
u
u
Corollary 1 can be used for deriving upper bounds for the price of an Asian option, see Simon et al. (2000).
156
R. Kaas et al. / Insurance: Mathematics and Economics 27 (2000) 151–168
4. Improved bounds for sums of random variables
4.1. Upper bounds
As (F1−1 (U ), F2−1 (U ), . . . , Fn−1 (U )) is a random vector with marginals F1 , . . . , Fn , the upper bound Su =
F1−1 (U ) + F2−1 (U ) + · · · + Fn−1 (U ) is the best that can be derived under the conditions stated in Proposition 1; it is
a supremum in terms of convex order. Let us now assume that we have complete (or partial) information, more than
just the marginal distributions, concerning the dependence structure of the random vector (X1 , X2 , . . . , Xn ), but
that exact computation of the cdf of the sum S = X1 + X2 + · · · + Xn is not feasible. In this case, we will show that
it is possible to derive improved upper bounds for S, and also non-trivial lower bounds, based on the information we
have on the dependence structure. This is accomplished by conditioning on a random variable Z which is assumed
to be some function of the random vector X. We will assume that we know the distribution
P of Z, and also the
conditional cdfs, given Z = z, of the random variables Xi . A suitable example is to use Z = log Xi when the Xi
are lognormal. In the following proposition, we introduce the notation FX−1
(U ) for the random variable fi (U, Z),
i |Z
where the function fi is defined by fi (u, z) = FX−1
(u).
i |Z=z
Proposition 2. Let U be uniform (0, 1), and consider a random variable Z which is independent of U. Then we
have
(U ) + FX−1
(U ) + · · · + FX−1
(U ).
X1 + X2 + · · · + Xn ≤cx Su0 = FX−1
n |Z
1 |Z
2 |Z
def
Proof. From Proposition 1, we get for any convex function v,
Z ∞
E[v(X1 + · · · + Xn )|Z = z] dFZ (z)
E[v(X1 + · · · + Xn )] =
−∞
Z ∞
≤
E[v(f1 (U, z) + · · · + fn (U, z))] dFZ (z) = E[v(f1 (U, Z) + · · · + fn (U, Z))]
−∞
from which the stated result follows directly.
(U ), FX−1
(U ), . . . , FX−1
(U )) has marginals F1 , F2 , . . . , Fn , because
Note that the random vector (FX−1
n |Z
1 |Z
2 |Z
Z ∞
Z ∞
Pr[Xi ≤ x|Z = z] dFZ (z) =
Pr[FX−1
(U ) ≤ x] dFZ (z)
Fi (x) = Pr[Xi ≤ x] =
i |Z=z
−∞
−∞
Z ∞
Pr[fi (U, z) ≤ x] dFZ (z) = Pr[fi (U, Z) ≤ x].
=
−∞
In view of Proposition 1 this implies
(U ) + FX−1
(U ) + · · · + FX−1
(U ) ≤cx F1−1 (U ) + F2−1 (U ) + · · · + Fn−1 (U ).
FX−1
n |Z
1 |Z
2 |Z
The left-hand side of this relation is Su0 ; the right-hand side is Su . In order to obtain the distribution function of Su0 ,
observe that given the event Z = z, this random variable is a sum of comonotonous random variables. Hence
n
X
FX−1
(p), p ∈ [0,1].
FS−1
0 |Z=z (p) =
i |Z=z
u
i=1
If the marginal cdfs FXi |Z=z are strictly increasing and continuous, so is FSu0 |Z=z , and then FSu0 |Z=z (x) follows by
solving
n
X
FX−1
(FSu0 |Z=z (x)) = x.
i |Z=z
i=1
R. Kaas et al. / Insurance: Mathematics and Economics 27 (2000) 151–168
157
The cdf of Su0 then follows from
Z ∞
FSu0 |Z=z (x) dFZ (z).
FSu0 (x) =
−∞
Application of Proposition 2 to lognormal marginals Xi is considered in Section 5, but see also the simple examples
with n = 2 at the end of this section. Note that if Z is independent of all X1 , X2 , . . . , Xn , upper bound Su0 reduces
to Su .
4.2. Lower bounds
Let X be a random vector with marginals F1 , F2 , . . . , Fn , and assume that we want to find a lower bound Sl , in
the sense of convex order, for S = X1 + X2 + · · · + Xn . We can obtain such a bound by conditioning on some
random variable Z, again assumed to be a function of the random vector X.
Proposition 3. For any random vector X and random variable Z, we have
def
Sl = E[X1 |Z] + E[X2 |Z] + · · · + E[Xn |Z] ≤cx X1 + X2 + · · · + Xn .
Proof. By Jensen’s inequality, we find that for any convex function v, the following inequality holds:
E[v(X1 + X2 + · · · + Xn )] = EZ E[v(X1 + X2 + · · · + Xn )|Z] ≥ EZ [v(E[X1 + X2 + · · · + Xn |Z])]
= EZ [v(E[X1 |Z] + · · · + E[Xn |Z])].
This proves the stated result.
Note that if Z and S are mutually independent, Proposition 3 leads to the trivial lower bound E[S] ≤cx S. On the
other hand, if Z and S have a one-to-one relation, the lower bound in Proposition 3 coincides with S. Note further that
E[E[Xi |Z]] = E[Xi ] always holds, but Var[E[Xi |Z]] < Var[Xi ] unless E[Var[Xi |Z]] = 0 which means that Xi ,
given Z = z, is a constant for each z. This implies that the random vector (E[X1 |Z], E[X2 |Z], . . . , E[Xn |Z]) will
in general not have F1 , F2 , . . . , Fn as its marginal distribution functions. But if the conditioning random variable
Z has the property that all random variables E[Xi |Z] are non-increasing functions of Z (or all are non-decreasing
functions of Z), the lower bound in Proposition 3 has the form of a sum of n comonotonous random variables. The
cdf of this sum is then obtained by the results of Section 2. An application of Proposition 3 in the case of lognormal
marginals Xi is considered in Section 5.
With S = X1 + X2 + · · · + Xn , the lower bound Sl in Proposition 3 can be written as E[S|Z]. To judge the
quality of this stochastic bound, we might look at its variance. To maximize it, the mean value of Var[S|Z = z]
should be minimized. Thus, for the best lower bound, Z and S should be as alike as possible.
Let us further assume that the random variable Z is such that all E[Xi |Z] are non-increasing continuous functions
of Z. The quantiles of the random variable E[S|Z] then follow from
−1
(p) =
FE[S|Z]
n
n
X
X
−1
FE[X
(p)
=
E[Xi |Z = FZ−1 (1 − p)],
i |Z]
i=1
p ∈ (0,1).
i=1
In order to derive the above result, we used the fact that for a non-increasing continuous function f , we have
−1
Ff−1
(S) (p) = f (FS (1 − p)),
p ∈ (0,1).
Similarly, for a non-decreasing continuous function f , we have
−1
Ff−1
(S) (p) = f (FS (p)),
p ∈ (0,1).
158
R. Kaas et al. / Insurance: Mathematics and Economics 27 (2000) 151–168
If we now in addition assume that the cdfs of the random variables E[Xi |Z] are strictly increasing and con−1•
(0),
tinuous, then the cdf of E[S|Z] is also strictly increasing and continuous, and we get for all x ∈ (FE[S|Z]
−1
(1)),
FE[S|Z]
n
X
−1
FE[X
(FE[S|Z] (x)) = x,
i |Z]
i=1
or equivalently,
n
X
E[Xi |Z = FZ−1 (1 − FE[S|Z] (x))] = x,
i=1
which unambiguously determines the cdf of the convex order lower bound E[S|Z] for S in case all E[Xi |Z = z]
are non-increasing in z.
The stop-loss premiums of E[S|Z] can be computed as follows:
E[E[S|Z] − d]+ =
n
X
{E[Xi |Z] − E[Xi |Z = FZ−1 (1 − FE[S|Z] (d))]}+ ,
i=1
−1•
−1
(0), FE[S|Z]
(1)).
which holds for all retentions d ∈ (FE[S|Z]
The technique for deriving lower bounds as explained in this section is also considered (for some special cases)
in Vyncke et al. (2000). The idea of this technique stems from mathematical physics, and was applied by Rogers
and Shi (1995) to derive approximate values for the value of Asian options.
4.3. Some simple examples
Let X, Y be independent N(0, 1) random variables, and consider random variables of the type Z = X + aY for
some real a. We want to derive stochastic bounds for S = X + Y . The conditional distribution of X, given Z = z,
is, as is well-known,
ρX,Z σX
z
a2
2
2
(z − µZ ), σX (1 − ρX,Z ) = N
,
.
N µX +
σZ
1 + a2 1 + a2
−1
(U ), with U uniform
But this means that for the conditional expectation E[X|Z] and for the random variable FX|Z
(0, 1) and independent of Z, we get
E[X|Z] =
Z
,
1 + a2
−1
FX|Z
(U ) = E[X|Z] +
|a|Φ −1 (U )
.
(1 + a 2 )1/2
In line with E[X + aY|Z] ≡ Z, we also get
E[Y |Z] =
aZ
,
1 + a2
FY−1
|Z (U ) = E[Y |Z] +
Φ −1 (U )
.
(1 + a 2 )1/2
−1
(U ) and FY−1
It can be shown that both FX|Z
|Z (U ) have N (0, 1) distributions. Their U -dependent parts are comonotonous. For the lower and upper bounds derived above we get
S = X + Y ∼ N(0,2),
Sl = E[X + Y |Z] =
1+a
(1 + a)2
Z
∼
N
0,
,
1 + a2
1 + a2
R. Kaas et al. / Insurance: Mathematics and Economics 27 (2000) 151–168
159
1+a
1 + |a|
(1 + a)2 + (1 + |a|)2
0
−1
Su =
Z+
Φ (U ) ∼ N 0,
,
1 + a2
(1 + a 2 )1/2
1 + a2
Su =d 2X ∼ N(0,4).
For some special choices of a, we get the following distributions for the lower and upper bounds Sl and Su0 :
a=0:
N(0,1) ≤cx S ≤cx N(0,2),
a=1:
N(0,2) ≤cx S ≤cx N(0,4),
a = −1 :
|a| → ∞ :
N(0,0) ≤cx S ≤cx N(0,2),
N(0,1) ≤cx S ≤cx N(0,2).
Note that the actual distribution of S is N(0, 2), so the best convex lower bound (a = 1) and the best upper bound
(a ≤ 0 or a → ∞) coincide with S. Of course taking |a| → ∞ gives the same results as taking Z = Y . The
variance of Sl can be seen to have a maximum at a = +1, a minimum at a = −1. On the other hand, Var[Su0 ] also
has a maximum at a = 1, and minima at a ≤ 0 and a → ∞. So the best lower bound in this case is attained for
Z = S, the worst for Z and S independent. The best improved upper bound is found by taking Z = X, Z = Y , or
any a < 0, including the case a = −1 with Z and S independent; the worst, however, by taking Z = S.
To compare the variance of the stochastic upper bound Su0 with the variance of S boils down to comparing
−1
(U ), FY−1
cov(FX|Z
|Z (U )) with cov(X, Y ). It is clear that, in general, the optimal choice for the conditioning random
variable Z will depend on the correlation of X and Y . If this correlation equals 1, any Z results in S =d Su0 =d S. In
our case where X and Y are mutually independent, the optimal choice proves to be taking Z ≡ X or Z ≡ Y , thus
ensuring that S and Su0 coincide.
As a second example, consider a simple special case of the theory dealt in the next section. We present it here for
the reader’s convenience, just as an illustration. Take Y1 and Y2 be independent N (0, 1) random variables. Look at
the sum of X1 = eY1 ∼ lognormal(0, 1), and X2 = eY1 +Y2 ∼ lognormal(0, 2). Take Z = Y1 + Y2 . For the lower
bound Sl , note that E[X2 |Z] = eZ , while Y1 |Y1 + Y2 = z ∼ N ( 21 z, 21 ), hence
E[eY1 |Y1 + Y2 = z] = m(1; 21 z, 21 ),
where m(t; µ, σ 2 ) = exp(µt + 21 σ 2 t 2 ) is the N(µ, σ 2 ) moment generating function. This leads to
E[eY1 |Z] = exp( 21 Z + 41 ).
So the lower bound is
Sl = E[X1 + X2 |Z] = exp( 21 Z + 41 ) + eZ .
√
0
Upper bound Su has (X1 , X2 ) =d (eW , e 2W
√ ) for W ∼ N (0, 1). The improved upper bound Su has as a second
1
1
Z
term again e , and as first term exp( 2 Z + 2 2W ), with Z and W mutually independent. All terms occurring in the
bounds given above are lognormal random variables, so the variances of the bounds are easy to compute. Note that
to compare variances is meaningful when comparing stop-loss premiums of stop-loss ordered random variables, see,
e.g., Kaas et al. (1994, p. 68). The following relation, which can be proven using partial integration, links variances
and stop-loss premiums:
Z ∞
1
{E[X − t]+ − (E[X] − t)+ } dt,
Var[X] =
2
−∞
from which we deduce that if X ≤cx Y , thus E[Y − t]+ ≥ E[X − t]+ for all t, then
Z ∞
1
{E[Y − t]+ − E[X − t]+ } dt
{Var[Y ] − Var[X]} =
2
−∞
160
R. Kaas et al. / Insurance: Mathematics and Economics 27 (2000) 151–168
Thus, half the variance difference between two convex ordered random variables equals the integrated difference
of their stop-loss premiums. This implies that if X ≤cx Y and in addition Var[X] = Var[Y ], then X and Y must
necessarily be equal in distribution. Moreover, the ratio of the variances is roughly equal to the ratio of the stop-loss
premiums, minus their minimal possible value for random variables with the same mean. We have, as the reader
may verify,
E[S]2 = e1 + 2e5/2 + e4 ,
E[Sl2 ] = e3/2 + 2e5/2 + e4 ,
0
E[S 2 ] = E[Su2 ] = e2 + 2e5/2 + e4 ,
√
2
E[Su2 ] = e2 + 2e3/2+
+ e4 .
Hence
Var[E[S]] = 0,
Var[Sl ] = 1.763,
Var[S] = Var[Su0 ] = 4.671,
Var[Su ] = 17.174.
So an improved stochastic lower bound Sl for S is obtained by conditioning on Y1 + Y2 , and the improved upper
bound Su0 for this case proves to be very good indeed, having in fact the same distribution as S.
5. Present values — lognormal discount process
5.1. General result
Consider a series of deterministic payments α1 , α2 , . . . , αn , of arbitrary sign, that are due at times 1, 2, . . . , n,
respectively. The present value of this series of payments equals
S=
n
X
αi exp(−(Y1 + Y2 + · · · + Yi )).
i=1
Assume that (Y1 , Y2 , . . . , Yn ) has a multivariate normal distribution. We introduce the random variables Xi and
Y (i) defined by
Y (i) = Y1 + Y2 + · · · + Yi ;
Xi = αi e−Y (i) .
then S = X1 + X2 + · · · + Xn .
For some given choice of the βi , consider a conditioning random variable Z defined as follows:
Z=
n
X
βi Yi ,
i=1
For a multivariate normal distribution, every linear function of its components has a univariate normal distribution,
so Z is normally distributed. Also, (Y (i), Z) has a bivariate normal distribution. Conditionally given Z = z, Y (i)
has a univariate normal distribution with mean and variance given by
σY (i)
(z − E[Z]),
E[Y (i)|Z = z] = E[Y (i)] + ρi
σZ
and
Var[Y (i)|Z = z] = σY2(i) (1 − ρi2 ),
where ρi is the correlation between Z and Y (i).
R. Kaas et al. / Insurance: Mathematics and Economics 27 (2000) 151–168
161
Proposition 4. Let S, Sl , Su0 and Su be defined as follows:
S=
n
X
αi exp(−(Y1 + Y2 + · · · + Yi )),
i=1
n
X
αi exp(−E[Y (i)] − ρi σY (i) Φ −1 (U ) + 21 (1 − ρi2 )σY2(i) ),
Sl =
i=1
Su0
n
q
X
=
αi exp(−E[Y (i)] − ρi σY (i) Φ −1 (U ) + sign(αi ) 1 − ρi2 σY (i) Φ −1 (V )),
i=1
n
X
αi exp(−E[Y (i)] + sign(αi )σY (i) Φ −1 (U )),
Su =
i=1
where U and V are mutually independent uniform (0, 1) random variables, and Φ is the cdf of the N (0, 1) distribution.
Then we have
Sl ≤cx S ≤cx Su0 ≤cx Su .
Proof.
P
1. If a random variable X is lognormal(µ, σ 2 ), then E[X] = exp(µ + 21 σ 2 ). Hence, for Z = ni=1 βi Yi , we find
that, taking U = Φ((Z − E[Z])/σZ ), so U ∼ uniform(0, 1),
E[Xi |Z] = αi exp(−E[Y (i)] − ρi σY (i) Φ −1 (U ) + 21 (1 − ρi2 )σY2(i) ),
From Proposition 3, we find Sl ≤cx S.
−1
(p) = α exp(µ + sign(α)σ Φ −1 (p)). Hence, we
2. If a random variable X is lognormal(µ, σ 2 ), then we have FαX
find that
q
−1
(p)
=
α
exp(−E[Y
(i)]
−
ρ
σ
Φ
(U
)
+
sign(α
)
1 − ρi2 σY (i) Φ −1 (p)).
FX−1
i
i
i
Y
(i)
i |Z
From Proposition 2 we find that S ≤cx Su0 .
3. The stochastic inequality Su0 ≤cx Su follows from Proposition 1.
P
In order to compare the cdf of S = ni=1 αi exp(−(Y1 + Y2 + · · · + Yi )) with the cdfs of Sl , Su0 and Su , especially
their variances, we need the correlations of the different random variables involved. We find the following results
for the lognormal discount process considered in this section:
corr[Xi , Xj ] =
corr[E[Xi |Z], E[Xj |Z]] =
(U ), FX−1
(U )] =
corr[FX−1
i |Z
j |Z
ecov[Y (i),Y (j )] − 1
σ2
(eσY (i) − 1)1/2 (e Y (j ) − 1)1/2
eρi ρj σY (i) σY (j ) − 1
2
;
;
2 2
ρ2σ 2
(eρi σY (i) − 1)1/2 (e j Y (j ) − 1)1/2
exp([ρi ρj + sign(αi αj )(1 − ρi2 )1/2 (1 − ρj2 )1/2 ]σY (i) σY (j ) ) − 1
(eσY (i) − 1)1/2 (e
esign(αi αj )σY (i) σY (j ) − 1
−1
(U
),
F
(U
)]
=
.
corr[FX−1
X
2
i
j
σ2
(eσY (i) − 1)1/2 (e Y (j ) − 1)1/2
2
σY2(j )
− 1)1/2
;
162
R. Kaas et al. / Insurance: Mathematics and Economics 27 (2000) 151–168
From these correlations, we can for instance deduce that if all payments αi are positive and corr[Y (i), Y (j )] = 1
for all i and j , then S =d Su . In practice, the discount factors will not be perfectly correlated. But for any realistic
discount process, corr[Y (i), Y (j )] = corr[Y1 + · · · + Yi , Y1 + · · · + Yj ] will be close to 1 provided that i and j
are close to each other. This gives an indication that the cdf of Su might perform well as approximation for the cdf
of S for such processes. This is indeed the case in the numerical illustrations in Goovaerts et al. (2000). A similar
reasoning leads to the conclusion that the cdf of Su will not perform well as a convex upper bound for the cdf of
S if the payments αi have mixed signs. This phenomenon will indeed be observed in the numerical illustrations in
Section 6.
It remains to derive expressions for the cdfs of Sl , Su0 and Su .
5.2. The cdf and the stop-loss premiums of Su
The quantiles of Su follow from Goovaerts et al. (2000)
FS−1
(p) =
u
n
X
αi exp(−E[Y (i)] + sign(αi )σY (i) Φ −1 (p)),
p ∈ (0,1).
i=1
Also, FSu (x) follows implicitly from solving
n
X
αi exp(−E[Y (i)] + sign(αi )σY (i) Φ −1 (FSu (x))) = x.
i=1
It is straightforward to derive expressions for the stop-loss premiums in this case
E[Su − d]+ =
n
X
|αi | e−E[Y (i)] E[sign(αi )(Zi − exp(sign(αi )σY (i) Φ −1 (FSu (d))))]+ ,
i=1
where the Zi are lognormal(0, σY2(i) ) random variables.
In order to derive an explicit expression for the stop-loss premiums E[Su − d]+ , we first mention the following
result, which can easily be proven, e.g. by using (d/dt)E[X − t]+ = FX (t) − 1.
Proposition 5. If Y is lognormal (µ, σ 2 ), then for any d > 0 we have
E[Y − d]+ = exp(µ + 21 σ 2 )Φ(d1 ) − dΦ(d2 ),
E[Y − d]− = exp(µ + 21 σ 2 )Φ(−d1 ) − dΦ(−d2 ),
where d1 and d2 are determined by
d1 =
µ + σ 2 − ln(d)
,
σ
d2 = d1 − σ.
At d ≤ 0, the stop-loss premiums are trivially equal to E[Y ] − d. The following expression results for the
stop-loss premiums at d > 0:
E[Su − d]+ =
n
X
αi e−E[Y (i)] {exp( 21 σY2(i) )Φ(sign(αi )di,1 )
i=1
−exp(sign(αi )σY (i) Φ −1 (FSu (d)))Φ(sign(αi )di,2 )}
with di,1 and di,2 given by
di,1 = σY (i) − sign(αi )Φ −1 (FSu (d)),
di,2 = −sign(αi )Φ −1 (FSu (d)).
R. Kaas et al. / Insurance: Mathematics and Economics 27 (2000) 151–168
163
Using the implicit definition for FSu (d) leads to the following expression for the stop-loss premiums:
E[Su − d]+ =
n
X
αi exp(−E[Y (i)] + 21 σY2(i) )Φ[sign(αi )σY (i) − Φ −1 (FSu (d))] − d(1 − FSu (d)).
i=1
5.3. The cdf and the stop-loss premiums of Sl
In general, Sl will not be a sum of n comonotonous random variables. But in the remainder of this subsection,
we assume that all αi ≥ 0 and all ρi = cov[Y (i), Z]/σY (i) σZ ≥ 0. These conditions ensure that Sl is the sum of n
comonotonous random variables.P
Taking into account that Z = ni=1 βi Yi is normally distributed, we find that
FZ−1 (1 − p) = E[Z] − σZ Φ −1 (p),
and hence
(p) =
FS−1
l
n
n
X
X
−1
FE[Xi |Z] (p) =
E[Xi |Z = FZ (1 − p)]
i=1
i=1
n
X
αi exp(−E[Y (i)] + ρi σY (i) Φ −1 (p) + 21 σY2(i) (1 − ρi2 )),
=
p ∈ (0,1).
i=1
FSl (x) can be obtained from
n
X
αi exp(−E[Y (i)] + ρi σY (i) Φ −1 (FSl (x)) + 21 σY2(i) (1 − ρi2 )) = x.
i=1
We have
E[Sl − d]+ =
n
X
−1
E[E[Xi |Z] − FE[X
(FSl (d))]+ .
i |Z]
i=1
After some straightforward computations, one finds that an explicit expression for the stop-loss premiums is given by
n
X
αi exp(−E[Y (i)] + 21 σY2(i) )Φ[ρi σY (i) − Φ −1 (FSl (d))] − d(1 − FSl (d)).
E[Sl − d]+ =
i=1
5.4. The cdf of Su0
Since FSu0 |U =u is a sum of n comonotonous random variables, we have
FS−1
0 |U =u (p) =
u
n
n
q
X
X
−1
FX−1
(p)
=
α
exp(−E[Y
(i)]
−
ρ
σ
Φ
(u)
+
sign(α
)
1 − ρi2 σY (i) σ Φ −1 (p)).
i
i
i
Y
(i)
i |U =u
i=1
i=1
FSu0 |U =u also follows implicitly from
n
q
X
αi exp(−E[Y (i)] − ρi σY (i) Φ −1 (u) + sign(αi ) 1 − ρi2 σY (i) Φ −1 (FSu0 |U =u (x))) = x.
i=1
The cdf of Su0 then follows from
Z 1
FSu0 |U =u (x) du.
FSu0 (x) =
0
164
R. Kaas et al. / Insurance: Mathematics and Economics 27 (2000) 151–168
6. Numerical illustration
P
In this section, we will numerically illustrate the bounds we derived for S = ni=1 αi exp(−(Y1 + Y2 + · · · + Yi )).
We will take n = 20. In order to be able to compare the distribution functions of the stochastic bounds Sl , Su0 and Su
with the distribution function of S, we will completely specify the multivariate distribution function of the random
vector (Y1 , Y2 , . . . , Y20 ). In particular, we will assume that the random variables Yi are i.i.d. and N (µ, σ 2 ). This
will enable us to simulate the cdfs in case there is no way to compute them analytically. The conditioning random
variable Z is defined as before
20
X
βi Yi .
Z=
i=1
In this case, we find
Var[Y (i)] = iσ 2 ,
E[Y (i)] = iµ,
Var[Z] = σ 2
20
X
k=1
βk2 ,
ρi =
Pi
cov[Y (i), Z]
k=1 βk
=qP
.
σY (i) σZ
2
i 20
β
k=1 k
In our numerical illustrations, we will choose the parameters of the normal distribution involved as follows:
µ = 0.07,
σ = 0.1.
We will compute the lower bound and the upper bound for the following choice of the parameters βi :
βi =
20
X
αj e−j µ ,
i = 1, . . . , 20.
j =i
By this choice, the lower bound will perform well in these cases. This is due to the fact that this choice makes Z a
linear transformation of a first-order approximation to S. This can be seen from the following computation, which
depends on σ , and hence Yi − µ, being “small”:




j
j
n
n
X
X
X
X
αj exp −j µ −
(Yi − µ) ≈
αj e−j µ 1 −
(Yi − µ)
S=
j =1
=C −
i=1
j =1
n
X
αj e−j µ
j
X
n
n
X
X
Yi
αj e−j µ ,
j =1
i=1
i=1
Yi = C −
i=1
j =i
where C is the appropriate constant. By the remarks in Section 4, Sl will then be “close” to S.
Fig. 1 shows the cdfs of S, Sl , Su0 and Su for the following payments:
αk = 1,
k = 1, . . . , 20.
Since Sl ≤cx S ≤cx Su0 ≤cx Su , and the same ordering holds for the tails of their distribution functions which can be
observed to cross only once, we can easily identify the cdfs. We see that the cdf of Sl is very close to the distribution
of S, which was expected because of the choice of Z. Note that in this case Sl is a sum of comonotonous random
variables, so its quantiles can be computed easily. The cdf of Su also performs rather well, as was observed in
Goovaerts et al. (2000). We find that the improved upper bound Su0 is very close to the comonotonous upper bound
(U ), FX−1
(U )) is close to cov(Xi , Xj ) for any pair (i, j ) with i 6= j .
Su . This is due to the fact that cov(FX−1
i |Z
j |Z
R. Kaas et al. / Insurance: Mathematics and Economics 27 (2000) 151–168
165
Fig. 1. Payments: 20 × 1; Z is such that the lower bound is optimized.
Fig. 2 shows the cdfs of S, Sl , Su0 and Su for the following payments:
−1, k = 1, . . . , 5,
αk =
1,
k = 6, . . . , 20.
Note that the cdf of the lower bound Sl cannot be computed exactly in this case; it is obtained by simulation. In this
case, we see that the lower bound Sl still performs very well. The comonotonous upper bound Su performs very
Fig. 2. Payments: 5 × (−1), 15 × 1; Z is such that the lower bound is optimized.
166
R. Kaas et al. / Insurance: Mathematics and Economics 27 (2000) 151–168
Fig. 3. Payments: 5 × (−1), 15 × 1; Z is such that it is an approximation to the discounted total of the negative payments.
badly in this case, as was to be expected from the observations in Section 5.1. The improved upper bound performs
better.
In Fig. 3, we consider the same series of payments as in Fig. 2. We consider the cdf of the improved upper bound
for a different choice of the conditioning random variable Z. We choose Z such that it is an approximation to the
discounted total of the five negative payments:
P5
−j µ , i = 1, . . . , 5,
j =i αj e
βi =
0,
i = 6, . . . , 20.
The (simulated) cdf of S is the dotted line. Note that the upper bound Su0 is much improved, the lower bound is
worse.
7. Conclusions and related research
In this contribution we considered the problem of deriving stochastic lower and upper bounds, in the sense of
convex order, for a sum S = X1 + X2 + · · · + Xn of possibly dependent random variables X1 , X2 , . . . , Xn . We
assumed that, as is often the case, the marginal distribution of each random variable X1 , X2 , . . . , Xn is known. The
problem of deriving a convex upper bound without using additional information about the dependency structure
was considered in Müller (1997) and Goovaerts et al. (2000). In this paper, we additionally assumed that there
exists some random variable Z, with a computable distribution, such that for any i and for any z in the support
of Z, the conditional distribution function of Xi , given Z = z, is also computable. Based on this, we derived
random variables Sl and Su0 , the cdfs of which are known to be less and larger than the one of S in convex order,
meaning that the tails of Sl are thinner, the ones of Su0 are thicker in general. Though it is not guaranteed that
two convex ordered cdfs cross only once, in the majority of examples they do so. Thus, we obtain a band of
possible values of Pr[S ≤ x] which might provide more, and more reliable, information than a point estimate
as obtained from a number of simulations. This is especially the case when the inverse cdf is sought, such as
R. Kaas et al. / Insurance: Mathematics and Economics 27 (2000) 151–168
167
when one wants to determine fair values and supervisory values. But note that Pr[S ≤ x] cannot be guaranteed
to be between Pr[Sl ≤ x] and Pr[Su0 ≤ x]. It has been argued before, see e.g. Kaas (1994), that actuaries should
not be focused on probabilities and quantiles, but rather on stop-loss premiums, since it is not the probability of
exceeding a threshold d that matters, but the amount by which this happens, of which the expected value is just the
stop-loss premium at d. And for stop-loss premiums, the property E[Sl − d]+ ≤ E[S − d]+ ≤ E[Su0 − d]+ does
hold.
It should be noted that the upper bound Su0 is no longer a supremum (in the sense of convex order) over the set of
all random vectors with fixed marginals, and that the lower bound Sl is not a sum of terms with the proper marginal
distributions. This follows from the fact that the bounds that we derived take into account the dependency structure
of the random vector under consideration.
It should also be noted that our results actually do not require the complete dependency structure, but only the
distribution of Z and the conditional distributions of Xi given Z = z. In Section 6 we chose an example where
the distribution of the random vector was completely known, in order to be able to compare the bounds with the
(simulated) exact cdf.
A topic for future research is the determination of the optimal conditioning random variable Z for the improved
upper bound Su0 , in the spirit of the remarks made at the end of Section 4.3. Another item for future research is the
extension of the results of this paper to the case where also the cash flows are stochastic, hence to find improved
upper bounds and lower bounds for S = X1 Y1 + X2 Y2 + · · · + Xn Yn . Another idea that we intend to pursue is
conditioning on more than one random variable Z.
Acknowledgements
The authors would like to thank Etienne De Vylder for helpful and insightful comments on an earlier version
of this paper, and David Vyncke for providing the numerical illustrations of Section 6. Marc Goovaerts and Jan
Dhaene acknowledge the financial support of the Onderzoeksfonds K.U. Leuven (grant OT/97/6) and of FWO (grant
“Actuarial ordering of dependent risks”).
References
Bäuerle, N., Müller, A., 1998. Modeling and comparing dependencies in multivariate risk portfolios. ASTIN Bulletin 28, 59–76.
Brockett, P., Garven, J., 1998. A reexamination of the relationship between preferences and moment orderings by rational risk averse investors.
Geneva Papers on Risk and Insurance Theory 23, 127–137.
Denuit, M., Lefèvre, C., 1997. Stochastic product orderings, with applications in actuarial sciences. Bulletin Français d’Actuariat 1, 61–82.
Denuit, M., De Vylder, F., Lefèvre, C., 1999. Extrema with respect to s-convex orderings in moment spaces: a general solution. Insurance:
Mathematics and Economics 24, 201–217.
Dhaene, J., Denuit, M., 1999. The safest dependency structure among risks. Insurance: Mathematics and Economics 25, 11–21.
Dhaene, J., Goovaerts, M., 1996. Dependency of risks and stop-loss order. ASTIN Bulletin 26, 201–212.
Dhaene, J., Goovaerts, M.J., 1997. On the dependency of risks in the individual life model. Insurance: Mathematics and Economics 19, 243–253.
Dhaene, J., Wang, S., Young, V., Goovaerts, M.J., 1998. Comonotonicity and maximal stop-loss premiums. Research Report 9730. DTEW, KU
Leuven, p. 13, submitted for publication.
Goovaerts, M.J., Redant, R., 1999. On the distribution of IBNR reserves. Insurance: Mathematics and Economics 25, 1–9.
Goovaerts, M.J., Dhaene, J., De Schepper, A., 2000. Stochastic upper bounds for present value functions. Journal of Risk and Insurance Theory
67.1, 1–14.
Kaas, R., 1994. How to (and how not to) compute stop-loss premiums in practice. Insurance: Mathematics and Economics 13, 241–254.
Kaas, R., Van Heerwaarden, A.E., Goovaerts, M.J., 1994. Ordering of actuarial risks. Institute for Actuarial Science and Econometrics, University
of Amsterdam, Amsterdam, p. 144.
Müller, A., 1997. Stop-loss order for portfolios of dependent risks. Insurance: Mathematics and Economics 21, 219–223.
Rogers, L.C.G., Shi, Z., 1995. The value of an Asian option. Journal of Applied Probability 32, 1077–1088.
Shaked, M., Shanthikumar, J.G., 1994. Stochastic Orders and Their Applications. Academic Press, New York, p. 545.
Simon, S., Goovaerts, M.J., Dhaene, J., 2000. An easy computable upper bound for the price of an arithmetic Asian option. Insurance:
Mathematics and Economics 26.2-3, 175–184.
168
R. Kaas et al. / Insurance: Mathematics and Economics 27 (2000) 151–168
Vyncke, D., Goovaerts, M.J., Dhaene, J., 2000. Convex upper and lower bounds for present value functions, Research Report 0025, Dept. of
Applied Economics, KU Leuven, Belgium.
Wang, S., Dhaene, J., 1998. Comonotonicity, correlation order and stop-loss premiums. Insurance: Mathematics and Economics 22 (3), 235–243.
Wang, S., Young, V., 1998. Ordering risks: expected utility versus Yaari’s dual theory of choice under risk. Insurance: Mathematics and
Economics 22, 145–162.