S. Haberman et al. Insurance: Mathematics and Economics 27 2000 237–259 255
for M, as previously, is as large as possible. Therefore, for σ ¯ σ , M
∗
= M
1
which decreases when i rises but remains the same when j changes. For σ ¯
σ M
∗
is shorter. Given the initial funding level of 100, the optimal spread period is more sensitive to changes in the valuation rate of interest. For example, when σ = 0.01, j = 3
and θ = 1, M
∗
= 26.3 and k
∗
= 0.0288 for i = 5, but M
∗
= 218.5 and k
∗
= 0.0099 for i = 3. Therefore, when the valuation rate of interest i is equal to the rate of interest used in the discounting term j see Tables 16, 17
and 20, changes in the optimal choice are attributable to changes in i. For the specific cases tabulated, when θ = 1, i = 3 and j = 5, ¯
σ = 0.21.
3. Further comments
3.1. Minimising the solvency risk If θ = 0, we are minimising the solvency risk. The degree of security will depend on the speed with which the
shortfall is removed by means of special contributions. In this case, the best course of action would be to pay the full amount of the shortfall as it arises without spreading any payments into the future i.e. M
∗
= 1, k
∗
= 1. But this may not always be attractive, or even possible, from the sponsoring employer’s point of view.
However, Tables 2–6 show that when the initial assets are much less than the initial liability e.g. F = 0, the
optimal spread period is much longer, especially, for low values of σ . In particular, for σ ¯ σ , the optimal choice is
the maximum feasible spread period M
1
which decreases as the mean return i increases. When σ ¯ σ , the optimal
spread period is approximately equal to 1 and k
∗
∼ = 1, for all values investigated for the parameters i, j and z.
The results of Sections 2.5 and 2.7 show that as F is increased, the optimal choices for minimising the solvency
risk are M
∗
∼ = 1 and k
∗
∼ = 1.
3.2. Minimising the contribution rate risk If θ = 1, we are minimising the contribution rate risk. We are concerned with stabilising the contribution rate by
spreading the unfunded liability over as long a period as possible. Stable contributions enable the employer to plan more effectively future cash flows. Therefore, in order to make the call on the employer’s resources more stable,
the actuary should choose the period for the extinguishing of the unfunded liability to be as long as possible and making the corresponding value of k as small as possible, otherwise the range of variation of the contribution rates
is increased.
The length of the spread period decreases as σ increases. For σ ¯ σ , the optimal choice is M
1
. For σ ¯ σ , M
∗
becomes shorter and k
∗
larger according to the particular combinations of i, j and z. When the initial funding level represented by z decreases, the contribution rate required rises. If our objective is one of minimising the
contribution rate risk, then the optimal spread period increases, M
∗
, and the optimal choice of k decreases. For the σ ¯
σ cases, the optimal choice M
∗
= M
1
does not change whatever the value of the interest rate used in the discounting process M
1
does not depend on j . For σ ¯ σ , a higher value of j leads to a longer optimal
choice for M
∗
and a lower value for k
∗
. An increase in j means that greater emphasis is being placed on the shorter-term state of the pension fund. For the case of an initial funding deficit low value of F
, this means that a higher adjustment to the contribution rate is required. If we are concerned with minimising the contribution rate
risk, a higher value of M
∗
should be chosen or lower value of k
∗
so as to reduce the variation in the contribution rate. The higher is the initial funding deficit, the greater is the impact of j on the optimal choice.
The results are also sensitive to changes in the interest rate i. The optimal choice M
∗
decreases and k
∗
increases when i increases for each value of σ . For σ ¯
σ , the changes in M
∗
arising from changes in i correspond to Table 1. For the σ ¯
σ cases, the extent to which the results are affected by changes in the investment assumption depends on the initial level of assets. If the pension fund had no assets initially F
= 0, the impact on M
∗
and k
∗
would be less. If the initial funding level were high, an increase in i would lead to a greater level of interest income obtained
256 S. Haberman et al. Insurance: Mathematics and Economics 27 2000 237–259
on the assets and consequently to a lower contribution and a smaller M
∗
. Hence, increasing i has a larger impact on the optimal choice, M
∗
, when the initial funding level as represented by z is high. Dufresne 1988 considers the trade-off between the limiting variances of the contribution rate risk and of the
fund level, recognising that improved security may imply regularly adjusted contribution rates and, conversely, stable contribution rates may be achieved by a greater variation in the fund level. He minimises the ultimate level
of these variances and finds a region for M 1, M
∗
, where M
∗
=
−log1 − dk
∗
log1 + i ,
i 6= 0, 1 +
1 σ
2
, i = 0,
k
∗
= if y 1,
1 − 1y if y 1.
19 and y = 1 + i
2
+ σ
2
. He calls this region an optimal one, in the sense that, for M M
∗
, both limiting variances increase with increasing M and, for M ≤ M
∗
, the limiting variance of the contribution rate risk decreases and the limiting variance of the fund level increases with increasing M. We note that this line of research has been taken further by Cairns and
Parker 1997 and Owadally and Haberman 1999. Therefore, we may consider as a measure of the contribution rate risk the variance of Ct in the limit, i.e.
Var C∞. We recall Eq. 14: Var C∞ = k
2
b AL
2
1 − a. Dufresne 1988 shows that k
∗
as defined by Eq. 19 is the value for which Var C∞ or αk = k
2
AL
2
1 − 1 − k
2
y is minimised. According to our formulation, for θ = 1, the contribution rate risk is defined as J
∞
= P
∞ t =0
w
t
Var Ct , where w is the discounting factor and Var Ct = k
2
Var F t . So J
∞
= k
2
bv 1 − wa
z
2
q
2
1 − wq
2
+ AL
2
1 − w +
2z AL q 1 − wq
. Hence, we must find the values of k for which J
∞
or βk =
k
2
[z
2
q
2
1 − w1 − wq + AL
2
1 − wq
2
1 − wq + 2z AL q1 − wq
2
1 − w] [1 − w1 − k
2
y]1 − wq
2
1 − wq 20
is minimised. We consider the case of z = −AL for convenience. Fig. 7 shows that the spread period M which minimises J
∞
is longer than the spread period which Dufresne defines as the optimal choice for minimising Var C.
Fig. 7. Graph of Var C∞ and J
∞
when i = 1 and σ = 0.01.
S. Haberman et al. Insurance: Mathematics and Economics 27 2000 237–259 257
In our formulation of the problem, the criterion of minimising the contribution rate risk is defined as a time-weighted sum of the Var Ct . Hence, the discounting factor is the weight applied to the variance which means that for j 0
w 1, more emphasis is placed on the shorter-term variances. On the other hand, minimising Var C∞ means that we consider only the ultimate situation t → ∞.
We note also, that in the special case where z = 0, so that F = AL, 20 simplifies to
βk = k
2
AL
2
1 − w1 − k
2
y ,
which, in a similar vein to Dufresne’s result 19, has a minimum at k = 1 − 1wy. 3.3. Minimising the risk: the general case 0 θ 1
Complete security or complete stability is not always an over-riding principle 0 θ 1. In this more general case, it is necessary to consider how quickly a particular contribution arrangement would meet the underlying
pension scheme liability, in order to build up the security of the benefits. We note that typically Var F t is much larger that Var Ct for M 1 or k 1. Thus, the discussion of Section
1.5 and Eq. 11 demonstrate that Var Ct = k
2
Var F t . This means that the most interesting values of θ lie in the range 0.9, 1. The results in Sections 2.3–2.7 bear out this feature, showing a relative insensitivity to the value
of θ when θ is below 0.9. An alternative formulation based on scaling Var F t e.g. by AL
2
and Var Ct e.g. by NC
2
leads to a similar problem in that the latter is then much larger than the former for low values of M.
When ¯ σ exists, we observe that, for σ ¯
σ , the optimal choice is M
1
, so that we want to spread the unfunded liability for as long as possible. For σ ¯
σ , M
∗
is much shorter and depends on the particular combinations of i, j, z and θ .
From the tables in Section 2 it can be seen that, when θ increases, the optimal spread period increases and the optimal choice of k decreases because we are more concerned with the criterion of stability. The sensitivity of the
risk to θ depends on the particular values of the other parameters. Tables 2–6 indicate that, for low values of σ , the optimal choice = M
1
is independent of θ F = 0. Table 10 shows that M
∗
increases and k
∗
decreases when θ increases for each value of θ F
= 0.50. Table 17 and unpublished tables for the cases F = 0.25 and F
= 0.75 show similar features.
When the initial funding level F rises, the risk as represented by J
∞
is minimised for shorter spread periods. We consider Eq. 17 as a function of z, −AL ≤ z ≤ 0. J
∞
is an increasing function of z and is more sensitive to changes in z for high values of q i.e. high values of M and low values of k. For low values of q, the risk is
approximately constant. So, when the initial funding level rises, the risk J
∞
remains approximately constant for low values of M, but it increases to a substantial extent for high values of M. With the objective of minimising the
risk J
∞
, the optimal spread period becomes shorter. The extent to which the optimal choice is decreased, when the initial funding level rises, depends on the particular combination of the other parameters. From a comparison of the
results see, e.g. Tables 4 and 11, it can be observed that, for low values of σ , a lower initial funding level makes the optimal spread period jump from a low value to become equal to the maximum feasible spread period M
1
. For higher values of σ , the effect of the initial funding level is of lesser significance.
The higher is the initial level of assets, the greater is the impact of the assumed rate of return i on the optimal choice. Because of the interest earned on the plan’s high initial funding level, the criterion of security is satisfied
when a small spread period is chosen, without leading to variations in contribution rate. Hence, the values of the optimal period range from 1.5 to 5.1 and k
∗
ranges from 0.1936 to 0.6459 when z = 0, as Tables 16–20 indicate for the range of parameters investigated.
We note that, in the special case where z = 0, the form of gq simplifies to gq =
θ 1 − qv
2
+ 1 − θ AL
2
1 − wq
2
1 + σ
2
v
2
258 S. Haberman et al. Insurance: Mathematics and Economics 27 2000 237–259
a ratio of two polynomials of degree 2 in q. The solution of ddqgq = 0 then becomes the simpler matter of solving a quadratic equation in q.
With the objective of placing more emphasis on the shorter-term state of the pension fund a higher value of j , the results become more dependent on j , the lower is the initial funding level and for σ ¯
σ for σ ¯ σ , M
∗
= M
1
which is independent of j . When the short-term state of the pension fund is to be emphasised and a large initial unfunded liability exists, minimisation of the risk J
∞
can be achieved by small changes to the contribution rate. This means that longer spread periods and smaller values of k should be chosen, as illustrated by Tables 4 and 6.
4. Conclusions
