164 R. Kaas et al. Insurance: Mathematics and Economics 27 2000 151–168
6. Numerical illustration
In this section, we will numerically illustrate the bounds we derived for S = P
n i
=1
α
i
exp−Y
1
+ Y
2
+ · · · + Y
i
. We will take n = 20. In order to be able to compare the distribution functions of the stochastic bounds S
l
, S
′ u
and S
u
with the distribution function of S, we will completely specify the multivariate distribution function of the random vector Y
1
, Y
2
, . . . , Y
20
. In particular, we will assume that the random variables Y
i
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
Z =
20
X
i =1
β
i
Y
i
. In this case, we find
E [Y i] = iµ,
Var[Y i] = iσ
2
, Var[Z] = σ
2 20
X
k =1
β
2 k
, ρ
i
= cov[Y i, Z]
σ
Y i
σ
Z
= P
i k
=1
β
k
q i
P
20 k
=1
β
2 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 β
i
: β
i
=
20
X
j =i
α
j
e
−j µ
, i
= 1, . . . , 20. 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 Y
i
− µ, being “small”: S
=
n
X
j =1
α
j
exp
−j µ −
j
X
i =1
Y
i
− µ
≈
n
X
j =1
α
j
e
−j µ
1 −
j
X
i =1
Y
i
− µ
= C −
n
X
j =1
α
j
e
−j µ j
X
i =1
Y
i
= C −
n
X
i =1
Y
i n
X
j =i
α
j
e
−j µ
, where C is the appropriate constant. By the remarks in Section 4, S
l
will then be “close” to S. Fig. 1 shows the cdfs of S, S
l
, S
′ u
and S
u
for the following payments: α
k
= 1, k
= 1, . . . , 20. Since S
l
≤
cx
S ≤
cx
S
′ u
≤
cx
S
u
, 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 S
l
is very close to the distribution of S, which was expected because of the choice of Z. Note that in this case S
l
is a sum of comonotonous random variables, so its quantiles can be computed easily. The cdf of S
u
also performs rather well, as was observed in Goovaerts et al. 2000. We find that the improved upper bound S
′ u
is very close to the comonotonous upper bound S
u
. This is due to the fact that covF
−1 X
i
|Z
U , F
−1 X
j
|Z
U is close to covX
i
, X
j
for any pair i, j with i 6= j.
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, S
l
, S
′ u
and S
u
for the following payments: α
k
= −1, k = 1, . . . , 5,
1, k
= 6, . . . , 20. Note that the cdf of the lower bound S
l
cannot be computed exactly in this case; it is obtained by simulation. In this case, we see that the lower bound S
l
still performs very well. The comonotonous upper bound S
u
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: β
i
= P
5 j
=i
α
j
e
−j µ
, i
= 1, . . . , 5, 0,
i = 6, . . . , 20.
The simulated cdf of S is the dotted line. Note that the upper bound S
′ u
is much improved, the lower bound is worse.
7. Conclusions and related research
