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
= e
1
+ 2e
52
+ e
4
, E
[S
2 l
] = e
32
+ 2e
52
+ e
4
, E
[S
2
] = E[S
′
2 u
] = e
2
+ 2e
52
+ e
4
, E
[S
2 u
] = e
2
+ 2e
32+ √
2
+ e
4
. Hence
Var[E[S]] = 0, Var[S
l
] = 1.763, Var[S] = Var[S
′ u
] = 4.671, Var[S
u
] = 17.174. So an improved stochastic lower bound S
l
for S is obtained by conditioning on Y
1
+ Y
2
, and the improved upper bound S
′ u
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 α
1
, α
2
, . . . , α
n
, of arbitrary sign, that are due at times 1, 2, . . . , n, respectively. The present value of this series of payments equals
S =
n
X
i =1
α
i
exp−Y
1
+ Y
2
+ · · · + Y
i
. Assume that Y
1
, Y
2
, . . . , Y
n
has a multivariate normal distribution. We introduce the random variables X
i
and Y i
defined by Y i
= Y
1
+ Y
2
+ · · · + Y
i
; X
i
= α
i
e
−Y i
. then S = X
1
+ X
2
+ · · · + X
n
. For some given choice of the β
i
, consider a conditioning random variable Z defined as follows: Z
=
n
X
i =1
β
i
Y
i
, 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
E [Y i|Z = z] = E[Y i] + ρ
i
σ
Y i
σ
Z
z − E[Z],
and Var[Y i|Z = z] = σ
2 Y i
1 − ρ
2 i
, where ρ
i
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, S
l
, S
′ u
and S
u
be defined as follows: S
=
n
X
i =1
α
i
exp−Y
1
+ Y
2
+ · · · + Y
i
, S
l
=
n
X
i =1
α
i
exp−E[Y i] − ρ
i
σ
Y i
Φ
−1
U +
1 2
1 − ρ
2 i
σ
2 Y i
, S
′ u
=
n
X
i =1
α
i
exp−E[Y i] − ρ
i
σ
Y i
Φ
−1
U + signα
i
q 1 − ρ
2 i
σ
Y i
Φ
−1
V ,
S
u
=
n
X
i =1
α
i
exp−E[Y i] + signα
i
σ
Y i
Φ
−1
U , 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
S
l
≤
cx
S ≤
cx
S
′ u
≤
cx
S
u
.
Proof. 1. If a random variable X is lognormalµ, σ
2
, then E[X] = expµ +
1 2
σ
2
. Hence, for Z = P
n i
=1
β
i
Y
i
, we find that, taking U = ΦZ − E[Z]σ
Z
, so U ∼ uniform0, 1, E
[X
i
|Z] = α
i
exp−E[Y i] − ρ
i
σ
Y i
Φ
−1
U +
1 2
1 − ρ
2 i
σ
2 Y i
, From Proposition 3, we find S
l
≤
cx
S .
2. If a random variable X is lognormalµ, σ
2
, then we have F
−1 αX
p = α expµ + signασ Φ
−1
p . Hence, we
find that F
−1 X
i
|Z
p = α
i
exp−E[Y i] − ρ
i
σ
Y i
Φ
−1
U + signα
i
q 1 − ρ
2 i
σ
Y i
Φ
−1
p. From Proposition 2 we find that S ≤
cx
S
′ u
. 3. The stochastic inequality S
′ u
≤
cx
S
u
follows from Proposition 1. In order to compare the cdf of S =
P
n i
=1
α
i
exp−Y
1
+ Y
2
+ · · · + Y
i
with the cdfs of S
l
, S
′ u
and S
u
, 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[X
i
, X
j
] = e
cov[Y i,Y j ]
− 1 e
σ
2 Y i
− 1
12
e
σ
2 Y j
− 1
12
; corr[E[X
i
|Z], E[X
j
|Z]] = e
ρ
i
ρ
j
σ
Y i
σ
Y j
− 1 e
ρ
2 i
σ
2 Y i
− 1
12
e
ρ
2 j
σ
2 Y j
− 1
12
; corr[F
−1 X
i
|Z
U , F
−1 X
j
|Z
U ] =
exp[ρ
i
ρ
j
+ signα
i
α
j
1 − ρ
2 i
12
1 − ρ
2 j
12
]σ
Y i
σ
Y j
− 1 e
σ
2 Y i
− 1
12
e
σ
2 Y j
− 1
12
; corr[F
−1 X
i
U , F
−1 X
j
U ] =
e
signα
i
α
j
σ
Y i
σ
Y j
− 1 e
σ
2 Y i
− 1
12
e
σ
2 Y j
− 1
12
.
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 α
i
are positive and corr[Y i, Y j ] = 1 for all i and j , then S =
d
S
u
. In practice, the discount factors will not be perfectly correlated. But for any realistic discount process, corr[Y i, Y j ] = corr[Y
1
+ · · · + Y
i
, Y
1
+ · · · + Y
j
] will be close to 1 provided that i and j are close to each other. This gives an indication that the cdf of S
u
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 S
u
will not perform well as a convex upper bound for the cdf of S
if the payments α
i
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 S
l
, S
′ u
and S
u
. 5.2. The cdf and the stop-loss premiums of S
u
The quantiles of S
u
follow from Goovaerts et al. 2000 F
−1 S
u
p =
n
X
i =1
α
i
exp−E[Y i] + signα
i
σ
Y i
Φ
−1
p, p
∈ 0,1. Also, F
S
u
x follows implicitly from solving
n
X
i =1
α
i
exp−E[Y i] + signα
i
σ
Y i
Φ
−1
F
S
u
x = x.
It is straightforward to derive expressions for the stop-loss premiums in this case E
[S
u
− d]
+
=
n
X
i =1
|α
i
| e
−E[Y i]
E [signα
i
Z
i
− expsignα
i
σ
Y i
Φ
−1
F
S
u
d ]
+
, where the Z
i
are lognormal0, σ
2 Y i
random variables. In order to derive an explicit expression for the stop-loss premiums E[S
u
− d]
+
, we first mention the following result, which can easily be proven, e.g. by using ddtE[X − t]
+
= F
X
t − 1.
Proposition 5. If Y is lognormal
µ, σ
2
, then for any d 0 we have E
[Y − d]
+
= expµ +
1 2
σ
2
Φd
1
− dΦd
2
, E
[Y − d]
−
= expµ +
1 2
σ
2
Φ −d
1
− dΦ−d
2
, where d
1
and d
2
are determined by d
1
= µ
+ σ
2
− lnd σ
, d
2
= d
1
− 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
[S
u
− d]
+
=
n
X
i =1
α
i
e
−E[Y i]
{exp
1 2
σ
2 Y i
Φ signα
i
d
i, 1
−expsignα
i
σ
Y i
Φ
−1
F
S
u
dΦ signα
i
d
i, 2
} with d
i, 1
and d
i, 2
given by d
i, 1
= σ
Y i
− signα
i
Φ
−1
F
S
u
d, d
i, 2
= −signα
i
Φ
−1
F
S
u
d.
R. Kaas et al. Insurance: Mathematics and Economics 27 2000 151–168 163
Using the implicit definition for F
S
u
d leads to the following expression for the stop-loss premiums:
E [S
u
− d]
+
=
n
X
i =1
α
i
exp−E[Y i] +
1 2
σ
2 Y i
Φ [signα
i
σ
Y i
− Φ
−1
F
S
u
d ] − d1 − F
S
u
d. 5.3. The cdf and the stop-loss premiums of S
l
In general, S
l
will not be a sum of n comonotonous random variables. But in the remainder of this subsection, we assume that all α
i
≥ 0 and all ρ
i
= cov[Y i, Z]σ
Y i
σ
Z
≥ 0. These conditions ensure that S
l
is the sum of n comonotonous random variables.
Taking into account that Z = P
n i
=1
β
i
Y
i
is normally distributed, we find that F
−1 Z
1 − p = E[Z] − σ
Z
Φ
−1
p, and hence
F
−1 S
l
p =
n
X
i =1
F
−1 E
[X
i
|Z]
p =
n
X
i =1
E [X
i
|Z = F
Z
1 − p] =
n
X
i =1
α
i
exp−E[Y i] + ρ
i
σ
Y i
Φ
−1
p +
1 2
σ
2 Y i
1 − ρ
2 i
, p
∈ 0,1. F
S
l
x can be obtained from
n
X
i =1
α
i
exp−E[Y i] + ρ
i
σ
Y i
Φ
−1
F
S
l
x +
1 2
σ
2 Y i
1 − ρ
2 i
= x. We have
E [S
l
− d]
+
=
n
X
i =1
E [E[X
i
|Z] − F
−1 E
[X
i
|Z]
F
S
l
d ]
+
. After some straightforward computations, one finds that an explicit expression for the stop-loss premiums is given by
E [S
l
− d]
+
=
n
X
i =1
α
i
exp−E[Y i] +
1 2
σ
2 Y i
Φ [ρ
i
σ
Y i
− Φ
−1
F
S
l
d ] − d1 − F
S
l
d. 5.4. The cdf of S
′ u
Since F
S
′ u
|U =u
is a sum of n comonotonous random variables, we have F
−1 S
′ u
|U =u
p =
n
X
i =1
F
−1 X
i
|U =u
p =
n
X
i =1
α
i
exp−E[Y i] − ρ
i
σ
Y i
Φ
−1
u + signα
i
q 1 − ρ
2 i
σ
Y i
σ Φ
−1
p. F
S
′ u
|U =u
also follows implicitly from
n
X
i =1
α
i
exp−E[Y i] − ρ
i
σ
Y i
Φ
−1
u + signα
i
q 1 − ρ
2 i
σ
Y i
Φ
−1
F
S
′ u
|U =u
x = x.
The cdf of S
′ u
then follows from F
S
′ u
x =
Z
1
F
S
′ u
|U =u
x du.
164 R. Kaas et al. Insurance: Mathematics and Economics 27 2000 151–168
6. Numerical illustration
