Risk as a time-weighted mechanism

242 S. Haberman et al. Insurance: Mathematics and Economics 27 2000 237–259

2. Risk as a time-weighted mechanism

2.1. Introduction We wish to solve Eq. 18 and find the values of q at which J ∞ is minimised. For all the calculations, we assume for convenience and without loss of generality that AL = 1. As noted above, solutions are restricted to values such that q q max 1, q max = 1 + b −12 , b = σ 2 v 2 . We verify that the chosen values of q are the minimum points by detailed numerical calculations, and demonstrate some of the results by the relevant graphs of J ∞ plotted against q. The results indicate that, in some cases, there is a single global minimum and, in others, there is a global minimum and a local minimum. It is also possible that, in some cases, the minimum value of J ∞ confined to the range 0, q max occurs at one of the endpoints. In any particular case, calculation of the minimising values of q allows us to find the corresponding values of k and M from q = 1 + i1 − k, M = −log1 − d1 − qv log1 + i . The tables in Section 2.3 onwards provide the optimal values of k and M as a function of i, σ , j , z and θ to the nearest integer and the values of k and M which are marked with † correspond to the minimum value of J occurring at q max . Detailed investigations have been carried out for a range of values of z and θ but detailed results are reported here only for the cases z = −AL, − 1 2 AL and 0 and selected values of θ . 2.2. The maximum feasible values of the spread period As noted earlier, the requirement a 1 for convergence places a restriction on the choice of q. So the optimal values of q must be restricted to values such that q q max . Table 1 provides rounded values of the maximum spread period M 1 which correspond to q max for different combinations of σ and i and indicates the extent to which M 1 decreases as σ and i each increase. Similarly, there is a minimum value of k, k 1 corresponding to q max and M 1 . 2.3. Initial funding level of 0 Tables 2–6 provide the optimal values of k and the spread period M for F = 0 i.e. z = −1 and selected combinations of σ, i, j and θ , corresponding to the global and local minima of gq as appropriate. The top panel of each table presents the global minima and the bottom panel the local minima. Tables 2–6 and other results not tabulated here for reasons of space indicate that: 1. There is a value of k or M k ∗ and M ∗ , respectively which leads to a global minimum in J ∞ and hence gq. 2. For certain combinations of θ and σ see below, there is a local minimum for J ∞ . Table 1 Maximum spread period, M 1 , such that a 1 i σ 0.01 0.03 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.01 535 318 223 112 66 42 30 22 17 0.03 218 145 111 68 46 33 25 19 15 0.05 144 99 78 51 37 28 21 17 14 S. Haberman et al. Insurance: Mathematics and Economics 27 2000 237–259 243 Table 2 Optimal values of k ∗ and M ∗ : when F = 0, i = 1, j = 1 θ σ 0.01 0.03 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Global minimum of gq 0.0099535.0† 0.0103318.1† 0.99001.01 0.99011.01 0.99021.01 0.99041.01 0.99061.01 0.99081.01 0.99111.01 0.5 0.0099535.0† 0.0103318.1† 0.61031.64 0.61241.64 0.61591.65 0.62071.62 0.62661.61 0.63371.58 0.64171.56 0.75 0.0099535.0† 0.0103318.1† 0.111222.8† 0.42932.35 0.43392.32 0.44022.29 0.44822.24 0.45772.20 0.46862.15 0.85 0.0099535.0† 0.0103318.1† 0.111222.8† 0.33613.01 0.34122.96 0.34842.90 0.35742.82 0.36822.47 0.38062.65 0.95 0.0099535.0† 0.0103318.1† 0.111222.8† 0.19955.12 0.20564.96 0.21404.76 0.22484.53 0.23794.27 0.25304.01 1 0.0099535.0† 0.0103318.1† 0.111222.8† 0.0147112.2† 0.022757.7 0.039429.1 0.059618.3 0.083012.8 0.10929.55 Local minimum of gq 0.99001.01 0.99001.01 0.0111222.8† 0.0147112.2† NA NA NA NA NA 0.5 0.60961.65 0.60981.65 0.0111222.8† 0.0147112.2† NA NA NA NA NA 0.75 0.42562.37 0.42592.36 0.42652.36 0.0147112.2† NA NA NA NA NA 0.85 0.33203.04 0.33233.04 0.33303.03 0.0147112.2† NA NA NA NA NA 0.95 0.19465.25 0.19505.24 0.19585.21 0.0147112.2† NA NA NA NA NA 1 NA NA NA NA NA NA NA NA NA 3. When σ is small, M ∗ is very large and equal to its maximum permitted value, so that a 1, M 1 . Similarly k ∗ is close to its minimum permitted value k 1 , so that a 1. 4. As σ is increased, there is a dramatic change in the optimal values with M ∗ decreasing and k ∗ increasing. For example, we note that when i = 1, j = 1 and θ = 0, M ∗ = 318.1 and k ∗ = 0.0103 when σ = 0.03, but for σ ≥ 0.05, M ∗ = 1.01 and k ∗ = 0.9900. For the permitted range of values of M or k, J ∞ has a turning point but the choice of the location of the global minimum depends on the relationship between the value of J ∞ at the turning point and at the boundary values M 1 or k 1 . For small σ , the global minimum occurs at M 1 or k 1 i.e. to make M as large as possible, or k as small as possible. As σ is increased further the shape of J ∞ changes, so that there is only one minimum point. We have noted in Section 2.2, that when σ increases, M 1 decreases and k 1 increases. Further analysis shows that there is a critical value of σ, ¯ σ for which J ∞ has a minimum at the two points M 2 ≪ M 3 and k 2 ≫ k 3 , where k 3 Table 3 Optimal values of k ∗ and M ∗ : when F = 0, i = 3, j = 3 θ σ 0.01 0.03 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Global minimum of gq 0.0099218.5† 0.0103144.6† 0.0111110.9† 0.96961.03 0.97001.03 0.97051.03 0.97111.03 0.97181.03 0.97261.03 0.5 0.0099218.5† 0.0103144.6† 0.0111110.9† 0.014567.8† 0.59041.69 0.59561.67 0.60201.66 0.60971.64 0.61831.61 0.75 0.0099218.5† 0.0103144.6† 0.0111110.9† 0.014567.8† 0.40342.46 0.41052.42 0.41942.37 0.42992.31 0.44182.25 0.85 0.0099218.5† 0.0103144.6† 0.0111110.9† 0.014567.8† 0.30743.22 0.31573.14 0.32603.04 0.33822.93 0.35202.82 0.95 0.0099218.5† 0.0103144.6† 0.0111110.9† 0.014567.8† 0.020245.8† 0.17565.58 0.18955.18 0.20544.79 0.22314.41 1 0.0099218.5† 0.0103144.6† 0.0111110.9† 0.014567.8† 0.020245.8† 0.028132.8† 0.039623.5 0.063814.9 0.090210.6 Local minimum of gq 0.96931.03 0.96931.03 0.96941.03 0.014567.8† 0.020245.8† NA NA NA NA 0.5 0.58351.71 0.58371.71 0.58421.71 0.58661.70 0.020245.8† NA NA NA NA 0.75 0.39402.52 0.39432.52 0.39502.52 0.39822.49 0.020245.8† NA NA NA NA 0.85 0.29643.34 0.29683.33 0.29763.32 0.30133.28 0.020245.8† NA NA NA NA 0.95 0.14696.64 0.14766.61 0.14896.55 0.15486.31 0.16395.97 NA NA NA NA 1 NA NA NA NA NA NA NA NA NA 244 S. Haberman et al. Insurance: Mathematics and Economics 27 2000 237–259 Table 4 Optimal values of k ∗ and M ∗ : when F = 0, i = 3, j = 5 θ σ 0.01 0.03 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Global minimum of gq 0.0099218.5† 0.0103144.6† 0.0111110.9† 0.014567.8† 0.94881.05 0.94961.05 0.95071.05 0.95191.05 0.95331.05 0.5 0.0099218.5† 0.0103144.6† 0.0111110.9† 0.014567.8† 0.020245.8† 0.56561.76 0.57301.74 0.58171.71 0.59141.69 0.75 0.0099218.5† 0.0103144.6† 0.0111110.9† 0.014567.8† 0.020245.8† 0.37492.65 0.38532.58 0.39752.50 0.41102.42 0.85 0.0099218.5† 0.0103144.6† 0.0111110.9† 0.014567.8† 0.020245.8† 0.27593.58 0.28873.42 0.30333.26 0.31933.10 0.95 0.0099218.5† 0.0103144.6† 0.0111110.9† 0.014567.8† 0.020245.8† 0.028132.8† 0.14296.82 0.16625.89 0.18855.20 1 0.0099218.5† 0.0103144.6† 0.0111110.9† 0.014567.8† 0.020245.8† 0.028132.8† 0.037824.5† 0.049419.0† 0.066014.4 Local minimum of gq 0.94761.06 0.94771.06 0.94771.05 0.94811.05 0.020245.8† 0.028132.8† NA NA NA 0.5 0.55181.81 0.55211.80 0.55271.80 0.55531.79 0.55971.78 0.028132.8† NA NA NA 0.75 0.35512.79 0.35552.79 0.35642.78 0.36022.75 0.36652.71 0.028132.8† NA NA NA 0.85 0.25043.94 0.25103.93 0.25213.91 0.25723.72 0.265332.8 0.028132.8† NA NA NA 0.95 NA NA NA NA NA 0.11068.75 NA NA NA 1 NA NA NA NA NA NA NA NA NA and M 3 are determined by q 3 = 1 + ¯ σ 2 v 2 −12 , q 3 = 1 + i1 − k 3 , M 3 = − log1 − dk 3 log1 + i , so that M 3 is the maximum feasible value for the spread period corresponding to ¯ σ and k 3 is the corresponding minimum feasible value for k. For M 2 M M 3 or k 3 k k 2 , J ∞ is higher than at either of the end points of this interval. So when the choice of σ makes M 1 M 3 i.e. σ ¯ σ , J ∞ is only minimised at M 2 . When M is allowed to exceed M 3 , J ∞ decreases. Hence, when the choice of σ makes M 1 M 3 σ ¯ σ , the optimal choice becomes M 1 . Approximate values of ¯ σ for different values of j, θ and i are shown in Table 7. We note the dependence of ¯ σ on these parameters. Table 5 Optimal values of k ∗ and M ∗ : when F = 0, i = 5, j = 3 θ σ 0.01 0.03 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Global minimum of gq 0.0099143.6† 0.010398.7† 0.011078.1† 0.96961.03 0.97001.03 0.97041.03 0.97101.03 0.97171.03 0.97251.03 0.5 0.0099143.6† 0.010398.7† 0.011078.1† 0.014451.1† 0.593216.7 0.59831.66 0.60461.64 0.61201.62 0.62041.60 0.75 0.0099143.6† 0.010398.7† 0.011078.1† 0.014451.1† 0.40582.42 0.41282.38 0.42152.33 0.43192.27 0.44362.22 0.85 0.0099143.6† 0.010398.7† 0.011078.1† 0.014451.1† 0.30923.14 0.31753.06 0.32782.97 0.33992.81 0.35362.76 0.95 0.0099143.6† 0.010398.7† 0.011078.1† 0.014451.1† 0.019936.6† 0.17605.35 0.19044.97 0.20674.60 0.22474.25 1 0.0099143.6† 0.010398.7† 0.011078.1† 0.014451.1† 0.019936.6† 0.027427.5† 0.043418.6 0.070212.3 0.09779.17 Local minimum of gq 0.96931.03 0.96931.03 0.96941.03 0.014451.1† 0.019936.6† NA NA NA NA 0.5 0.58661.69 0.58681.69 0.58731.69 0.58961.68 0.019936.6† NA NA NA NA 0.75 0.39652.47 0.39692.47 0.39762.46 0.40072.44 0.019936.6† NA NA NA NA 0.85 0.29823.25 0.29863.24 0.29943.24 0.30313.20 0.019936.6† NA NA NA NA 0.95 0.14566.38 0.14636.35 0.14776.29 0.15406.05 0.16375.72 NA NA NA NA 1 NA NA NA NA NA NA NA NA NA S. Haberman et al. Insurance: Mathematics and Economics 27 2000 237–259 245 Table 6 Optimal values of k ∗ and M ∗ : when F = 0, i = 5, j = 5 θ σ 0.01 0.03 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Global minimum of gq 0.0099143.6† 0.010398.7† 0.011078.1† 0.014451.1† 0.94871.05 0.94961.05 0.95061.05 0.95181.05 0.95321.05 0.5 0.0099143.6† 0.010398.7† 0.011078.1† 0.014451.1† 0.019936.6† 0.56801.74 0.57521.72 0.58361.70 0.59311.67 0.75 0.0099143.6† 0.010398.7† 0.011078.1† 0.014451.1† 0.019936.6† 0.37622.60 0.38652.53 0.39852.46 0.41202.38 0.85 0.0099143.6† 0.010398.7† 0.011078.1† 0.014451.1† 0.019936.6† 0.027427.5† 0.28883.35 0.30353.19 0.31963.04 0.95 0.0099143.6† 0.010398.7† 0.011078.1† 0.014451.1† 0.019936.6† 0.027427.5† 0.13846.78 0.16405.71 0.18755.04 1 0.0099143.6† 0.010398.7† 0.011078.1† 0.014451.1† 0.019936.6† 0.027427.5† 0.036821.4† 0.048017.1† 0.069312.4 Local minimum of gq 0.94761.05 0.94761.05 0.94771.05 0.94811.05 0.019936.6† 0.027427.5† 0.036821.4† NA NA 0.5 0.55461.78 0.55491.78 0.55541.78 0.55801.77 0.56221.76 0.027425.5† 0.036821.4† NA NA 0.75 0.35662.74 0.35702.74 0.35792.13 0.36172.70 0.36792.65 0.027425.5† 0.036821.4† NA NA 0.85 0.25003.84 0.25053.83 0.25173.82 0.25693.74 0.26513.63 0.27593.50 0.036821.4† NA NA 0.95 NA NA NA NA NA NA 0.036821.4† NA NA 1 NA NA NA NA NA NA NA NA NA Tables 2–6 also indicate that for σ ¯ σ , this optimal choice of M or k does not depend on θ as neither M 1 nor k 1 depends on θ . On the other hand, for σ ¯ σ , an increase in θ has a dramatic effect on M ∗ and k ∗ . For example, when i = 1, j = 1 and σ = 0.05, the optimal spread period M ∗ = 1.64 with k ∗ = 0.6103 for θ = 0.5, but M ∗ = 222.8 and k ∗ = 0.0111 for θ = 0.85. The results thus indicate that, for σ large enough, there is a minimising value of M ∗ or k ∗ . If σ is small, then there are two minima and the global minimum corresponds to the choice M 1 and k 1 , where k 1 is close to d. We can interpret this feature as follows: if F = 0 and k is chosen to be equal to d, then from 10, r = 0 and EFt = 0 for all t and from 12, Var F t = 0 for all t. Thus, we would be operating a pay-as-you-go system with ECt = NC + d AL = B. Table 7 Critical values of ¯ σ : when F = 0 i j θ 0.25 0.5 0.75 0.85 0.95 1 0.01 0.01 0.04 0.043 0.046 0.051 0.055 0.064 0.14 0.03 0.01 0.045 0.046 0.047 0.053 0.056 0.066 0.13 0.03 0.03 0.097 0.105 0.11 0.125 0.135 0.16 0.245 0.03 0.05 0.14 0.155 0.167 0.186 0.205 0.23 0.33 0.05 0.03 0.098 0.105 0.115 0.12 0.135 0.16 0.24 0.05 0.05 0.147 0.157 0.17 0.19 0.205 0.24 0.33 0.05 0.10 0.255 0.273 0.295 0.33 0.345 0.38 0.5 246 S. Haberman et al. Insurance: Mathematics and Economics 27 2000 237–259 So, the case where σ is small leads to a funding policy with a very small fund and the sponsor paying the interest on the unfunded liability in addition to NC. So the analysis implies the existence of two possible funding positions, with the combination of parameters determining which is optimal. A funding policy based on a high value for M ∗ low value of k ∗ would aim at keeping the contributions stable by ignoring positive or negative fluctuations in investment returns and hence in F t . This would work in the short term i.e. small values of t but, in due course, the level of F t would need to be taken into account and the fluctuations in fund size will require compensating changes in Ct to control the system. On the other hand, a funding policy based on a value of M ∗ close to 1 and k ∗ close to 1 implies dealing immediately with any surplus or deficiencies as they emerge. Thus, the resulting Ct will be volatile in the short term but in the long term the relative stability of F t will compensate and lead to lower values of Var Ct Dufresne, 1988; Cairns and Parker, 1997; Owadally and Haberman, 1999. The balance between these two extremes will depend on the value of σ, j representing the relative weight to be given to the short term versus the long term in the overall measure J ∞ and θ representing the relative weight to be given to fund stability and contribution stability. We next consider Eq. 17 as a function of θ , 0 θ 1. We recall that θ controls the balance between the solvency risk and the contribution rate risk. The risk as represented by J ∞ is a decreasing function of θ : ∂J ∞ ∂θ αk 2 − 1 0, but this decrease in risk is significant only for large values of q i.e. large values of M or small values of k. So, when θ increases, the risk decreases but this downward shift in risk is not smooth. J ∞ decreases markedly when M is large k is small and remains approximately constant when M is small k is large, thereby, making the optimal spread period M ∗ larger and k ∗ smaller. Tables 3–6 indicate that for σ ¯ σ , when a higher discounting factor, w, is used a lower j the optimal choices M ∗ and k ∗ remain the same equal to M 1 and k 1 , respectively. This is explained as neither M 1 nor k 1 depends on j . On the other hand, for σ ¯ σ , we observe that the optimal choice M ∗ becomes smaller and k ∗ larger when j is decreased. For example, when σ = 0.15, θ = 0.75 and i = 3, M ∗ = 45.8 and k ∗ = 0.0202 when j = 5, but M ∗ = 2.46 and k ∗ = 0.4034 when j = 3. Fig. 1 shows the graph of J ∞ for the corresponding case when θ = 0.25. We observe that when j rises, the risk as represented by J ∞ decreases, with less weight being given to the elements of formula 15 corresponding to higher values of t . This downward shift in risk is much more significant for large values of M, making the optimal spread period longer. Hence, when j = 3 the risk J ∞ is minimised for Fig. 1. Graph of J ∞ when σ = 0.15, θ = 0.25 and i = 3. S. Haberman et al. Insurance: Mathematics and Economics 27 2000 237–259 247 Fig. 2. Graph of J ∞ j when σ = 0.15, θ = 0.25 and i = 3. M ∼ = 1. When j = 5 more emphasis is placed on the shorter-term state of the pension fund, the risk remains approximately the same for M ∼ = 1 but decreases considerably for M ∼ = 46 and the optimal spread period becomes M ∗ ∼ = 46 = M 1 . We can explain these results in a different way. We consider Eq. 17 as a function of j and choose for convenience the particular parameter values i = 3, σ = 0.15 and θ = 0.25. Fig. 2 shows the graph of this function for different values of M, selected carefully in the light of the earlier results. Fig. 2 demonstrates what we have already claimed. The risk as represented by J ∞ is a decreasing function of j and this decrease in risk becomes more significant as the values of M become larger. From Tables 2–6, it can also be seen that an increase in the assumed rate of return i causes a significant decrease in M ∗ when σ ¯ σ and a slight decrease in M ∗ when σ ¯ σ . We recall that when σ ¯ σ , the optimal choice is M 1 which depends on i, and which changes in the way shown by Table 1. When σ ¯ σ , the optimal choice for M is smaller and for k is longer and remains the same or decreases slightly or increases for k when i increases. For example, when j = 5, θ = 1 and σ = 0.35, M ∗ = 14.4 and k ∗ = 0.0660 for i = 3 but M ∗ = 12.4 and k ∗ = 0.0693 for i = 5. Fig. 3 illustrates these two cases. Fig. 3 shows that J ∞ remains approximately the same for low values of M and slightly increases for high values of M when i is increased. We consider Eq. 17 as a function of i and choose for convenience the particular parameter values, σ = 0.35, θ = 1 and j = 5. Fig. 4 shows this function for different values of M. Fig. 4 demonstrates the sensitivity of J ∞ to changes in i, indicating that the optimal choice is influenced only slightly when the assumed rate of return changes. It also indicates that for the case when the rate of interest used for discounting j is equal to the valuation rate of interest i see Tables 2, 3 and 6, the changes in J ∞ are principally attributable to changes in the discounting rate of interest. 2.4. Initial funding level of 25 The detailed results for this case of F = 0.25 z = −0.75 are not presented here. However, these show that, for a higher initial funding level, the optimal choice for M, spread period is much lower and much higher for k for many combinations of σ, i, j and θ . For example, when i = 1, j = 1, σ = 0.01 and θ = 0 the optimal choice is M ∗ = 535.0 and k ∗ = 0.0099 when F = 0, and M ∗ = 1.01 and k ∗ = 0.9925 when F = 0.25. The initial funding level of 25 leads to a shorter optimal choice of spread period M ∗ and larger value of k ∗ when 248 S. Haberman et al. Insurance: Mathematics and Economics 27 2000 237–259 Fig. 3. Graph of J ∞ when σ = 0.35, θ = 1 and j = 5. the other parameters are such that we do not have the case of M 1 . For example, for the combination of j = 10, i = 5, σ = 0.01, the higher initial funding level of 25 is not sufficient to change the optimal choice which remains M ∗ = 143.6 and k ∗ = 0.0099. Hence, for this case, the effect of the high value of j is more significant than the magnitude of the initial funding level. When the optimal spread period is shorter than M 1 σ ¯ σ , an increase in j leads to a higher optimal choice. For example, when i = 5, θ = 0.25 and σ = 0.01, M ∗ = 1.01 for j = 5 but M ∗ = 143.6 for j = 10. Fig. 5 shows J ∞ plotted against M. Fig. 5 illustrates what we have already claimed. The risk as represented by J ∞ is a decreasing function of j and is more sensitive to changes in j for the higher values of M see also Fig. 2, and the correspondingly lower values of k. Therefore, when we are more interested in the shorter-term position of the pension fund j = 10, the risk decreases to a greater extent for M ∼ = 144 than for M ∼ = 1 and the optimal spread period becomes M ∗ = 143.6. For an initial funding level of 25 and for the σ ¯ σ cases, the detailed results show that an increase in the assumed rate of return leads to the same results as for a zero initial funding level. The optimal choice for M does Fig. 4. Graph of J ∞ i when σ = 0.35, θ = 1 and j = 5. S. Haberman et al. Insurance: Mathematics and Economics 27 2000 237–259 249 Fig. 5. Graph of J ∞ when σ = 0.01, θ = 0.25 and i = 5. not change or decreases slightly. Therefore, when the valuation rate of interest i is equal to the rate of interest used in discounting, j , changes in M and k are principally in response to changes in j . As for the case F = 0, the results show that the increase in θ causes a significant change in the optimal choices for M and k. The reason for this change has already been discussed. The risk as represented by J ∞ is a decreasing function of θ and the level of this decrease is considerable only for the case of high M or low k. Therefore, when θ increases, the risk decreases for high values of M low values of k and the optimal choice becomes larger for M and lower for k. For example, when i = 1 and σ = 0.01, M ∗ = 1.01 for θ = 0, but M ∗ = 535.0 for θ = 1. Table 8 presents the critical values of ¯ σ where they exist for certain combinations of parameters. The dependence of ¯ σ on these parameters is clear. 2.5. Initial funding level of 50 Tables 9–13 provide the optimal values of k and M for F = 0.50 z = −0.50 and selected combinations of θ, σ, j and i, corresponding to the global and local minimum of gq as appropriate. The results presented by Tables 9–13 show less dramatic variation than the corresponding results in Section 2.3: this arises because of the higher initial funding level. For low values of θ , we observe that the optimal choice of M is not affected by changes in σ, i or j . For example, when θ = 0.5, M ∗ ∼ = 2 and k lies in the range 0.581, 0.646 for each value of i, j and σ investigated. Table 8 Critical values of ¯ σ : when F = 0.25 i j θ 0.25 0.5 0.75 0.85 0.95 1 0.01 0.01 – – – – – 0.125 0.03 0.01 – – – – – – 0.11 0.03 0.03 – – – 0.02 0.055 0.095 0.215 0.03 0.05 – – 0.055 0.095 0.12 0.17 0.32 0.05 0.03 – – – 0.025 0.06 0.096 0.195 0.05 0.05 – – 0.06 0.99 0.125 0.175 0.29 0.05 0.10 0.12 0.155 0.185 0.23 0.26 0.305 0.45 250 S. Haberman et al. Insurance: Mathematics and Economics 27 2000 237–259 Table 9 []Optimal values of k ∗ and M ∗ : when F = 1 2 AL, i = 1, j = 1 θ σ 0.01 0.03 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Global minimum of gq 0.99501.01 0.99501.01 0.99501.01 0.99511.01 0.99511.00 0.99521.00 0.99531.00 0.99541.00 0.99561.00 0.5 0.61461.63 0.61481.63 0.61521.63 0.61731.63 0.62071.62 0.62541.60 0.63121.59 0.63811.57 0.64601.55 0.75 0.43092.34 0.43122.33 0.43182.33 0.43452.32 0.43902.29 0.44522.26 0.45312.22 0.46242.18 0.47312.13 0.85 0.33762.99 0.33802.99 0.33862.98 0.34162.96 0.34662.91 0.35362.85 0.36242.78 0.37292.70 0.38502.62 0.95 0.20125.08 0.20145.07 0.20215.05 0.20564.96 0.21144.82 0.21964.64 0.23004.42 0.24274.19 0.25753.94 1 0.0099535.0† 0.0103318.1† 0.0111222.8† 0.0147112.2† 0.026646.8 0.042826.4 0.062817.2 0.086112.3 0.11219.29 Local minimum of gq NA NA NA NA NA NA NA NA NA 0.5 NA NA NA NA NA NA NA NA NA 0.75 NA NA NA NA NA NA NA NA NA 0.85 0.0099535.0† NA NA NA NA NA NA NA NA 0.95 0.0099535.0† NA NA NA NA NA NA NA NA 1 NA NA NA NA NA NA NA NA NA When θ increases except for the cases of θ = 0.95 and θ = 1, the optimal choice for M increases slightly and for k decreases. Given the initial funding level of 50, changes in j andor σ do not cause any dramatic change in M ∗ or k ∗ . Similarly the range of the optimal values is reduced relative to the earlier cases of F = 0 or F = 0.25. For higher values of θ θ = 0.95 or θ = 1 and when σ ¯ σ , an increase in j leads to a higher optimal choice for M and a lower choice for k. For example, when i = 5, θ = 0.95 and σ = 0.1, M ∗ = 5.15 and k ∗ = 0.1833 for j = 3, M ∗ = 6.03 and k ∗ = 0.1547 for j = 5 but M ∗ = 51.1 and k ∗ = 0.0144 not tabulated for j = 10. When σ ¯ σ , a change in value of j does not cause any changes in M ∗ = M 1 for these cases unlike the assumed rate of return which affects the optimal choice as noted earlier. So, when i increases, the maximum feasible spread period decreases, as can be seen from Table 1. For σ ¯ σ , M ∗ does not change markedly in response to changes in i. Hence, in the case of i = j see Tables 9, 10 and 13, the optimal choice is principally affected by j . Table 14 shows the values of ¯ σ where they exist for combinations of i, j and θ =0.95 and 1.0. Table 10 Optimal values of k ∗ and M ∗ : when F = 1 2 AL, i = 3, j = 3 θ σ 0.01 0.03 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Global minimum of gq 0.98481.02 0.98481.02 0.98481.02 0.98501.02 0.98511.02 0.98541.01 0.98571.01 0.98601.01 0.98641.01 0.5 0.59961.66 0.59991.66 0.60031.66 0.60251.65 0.60601.65 0.61081.63 0.61691.62 0.62401.60 0.63211.58 0.75 0.41232.41 0.41262.41 0.41322.40 0.41612.39 0.42092.36 0.42742.33 0.43552.38 0.44532.23 0.45642.18 0.85 0.31693.12 0.31723.12 0.31793.11 0.32123.08 0.32653.03 0.33392.97 0.34322.89 0.35432.80 0.36702.70 0.95 0.17635.56 0.17675.54 0.17765.52 0.18165.40 0.18835.21 0.19744.97 0.20894.71 0.22274.42 0.23844.13 1 0.0099218.5† 0.0103144.6† 0.0111110.9† 0.014567.8† 0.020245.8† 0.031429.4 0.051318.3 0.074312.8 0.10009.64 Local minimum of gq 0.0099218.5† NA NA NA NA NA NA NA NA 0.5 0.0099218.5† 0.0103144.6† NA NA NA NA NA NA NA 0.75 0.0099218.5† 0.0103144.6† NA NA NA NA NA NA NA 0.85 0.0099218.5† 0.0103144.6† NA NA NA NA NA NA NA 0.95 0.0099218.5† 0.0103144.6† NA NA NA NA NA NA NA 1 NA NA NA NA NA NA NA NA NA S. Haberman et al. Insurance: Mathematics and Economics 27 2000 237–259 251 Table 11 Optimal values of k ∗ and M ∗ : when F = 1 2 AL, i = 3, j = 5 θ σ 0.01 0.03 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Global minimum of gq 0.97431.03 0.97431.03 0.97431.03 0.97451.03 0.97481.03 0.97531.03 0.97581.02 0.97641.02 0.97701.02 0.5 0.58101.72 0.58121.71 0.58171.71 0.58401.71 0.58781.70 0.59301.68 0.59941.66 0.60711.64 0.61571.62 0.75 0.39032.54 0.39062.54 0.39132.54 0.39442.52 0.39952.49 0.40642.44 0.41522.39 0.42552.34 0.43732.27 0.85 0.29303.37 0.29343.37 0.29423.36 0.29773.22 0.30363.26 0.31163.18 0.32163.08 0.33342.97 0.34692.86 0.95 0.14906.55 0.14956.53 0.15056.48 0.15536.29 0.15296.00 0.17325.65 0.18585.28 0.20054.90 0.21724.53 1 0.0099218.5† 0.0103144.6† 0.0111110.9† 0.014567.8† 0.020245.8† 0.028132.8† 0.037824.5† 0.054117.4 0.079112.1 Local minimum of gq 0.0099218.5† 0.0103144.6† NA NA NA NA NA NA NA 0.5 0.0099218.5† 0.0103144.6† NA NA NA NA NA NA NA 0.75 0.0099218.5† 0.0103144.6† 0.0111110.9† NA NA NA NA NA NA 0.85 0.0099218.5† 0.0103144.6† 0.0111110.9† NA NA NA NA NA NA 0.95 0.0099218.5† 0.0103144.6† 0.0111110.9† NA NA NA NA NA NA 1 NA NA NA NA NA NA NA NA NA 2.6. Initial funding level of 75 As in Section 2.4, the detailed results for this case of F = 0.75 z = −0.25 are not presented here. With an initial funding level of 75, the results are similar to those for 50 funding. Except for the case of θ = 1, an increase in j andor in θ does not cause any dramatic change in M ∗ for all values of σ . The effect of the valuation rate of interest i on M ∗ is also of minor significance except for the case of θ = 1. When θ = 1, the results do indicate some dramatic changes in M ∗ . For example, when σ = 0.01, M ∗ = 43.8 and k ∗ = 0.0211 for i = 3 and j = 1 and M ∗ = 535.0 and k ∗ = 0.0099 for i = 1 and j = 1. The higher initial funding level, means that the effect of the valuation rate of interest is more significant than for lower initial funding levels. This is also illustrated in Fig. 6 which presents the variation of J ∞ viewed as a function of i. Table 12 Optimal values of k ∗ and M ∗ : when F = 1 2 AL, i = 5, j = 3 θ σ 0.01 0.03 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Global minimum of gq 0.98481.02 0.98481.02 0.98481.02 0.98491.02 0.98511.01 0.98541.01 0.98561.01 0.98601.01 0.98641.01 0.5 0.60291.64 0.60311.64 0.60361.64 0.60561.64 0.60911.63 0.61371.62 0.61961.60 0.62651.58 0.63441.56 0.75 0.41532.36 0.41562.36 0.41622.36 0.41912.34 0.42372.32 0.43012.28 0.43812.24 0.44772.20 0.45862.15 0.85 0.31943.04 0.31983.04 0.32053.03 0.3237300 0.32902.96 0.33642.89 0.34562.82 0.35662.74 0.36922.65 0.95 0.17785.30 0.17825.28 0.17915.26 0.18335.15 0.19024.97 0.19964.75 0.21134.50 0.22524.24 0.24123.98 1 0.0099143.6† 0.010398.7† 0.011078.1† 0.014451.1† 0.020835.1 0.040119.9 0.061413.8 0.084810.4 0.11028.22 Local minimum of gq 0.0099143.6† 0.010398.7† NA NA NA NA NA NA NA 0.5 0.0099143.6† 0.010398.7† NA NA NA NA NA NA NA 0.75 0.0099143.6† 0.010398.7† NA NA NA NA NA NA NA 0.85 0.0099143.6† 0.010398.7† NA NA NA NA NA NA NA 0.95 0.0099143.6† 0.010398.7† NA NA NA NA NA NA NA 1 NA NA NA NA NA NA NA NA NA 252 S. Haberman et al. Insurance: Mathematics and Economics 27 2000 237–259 Table 13 Optimal values of k ∗ and M ∗ : when F = 1 2 AL, i = 5, j = 5 θ σ 0.01 0.03 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Global minimum of gq 0.97431.03 0.97431.03 0.97431.03 0.97451.03 0.97481.03 0.97521.03 0.97571.02 0.97631.02 0.97691.02 0.5 0.58411.70 0.58431.69 0.58481.69 0.58701.69 0.59071.68 0.59571.66 0.60201.65 0.60941.63 0.61781.60 0.75 0.39272.49 0.39312.49 0.39372.49 0.39682.47 0.40182.44 0.40872.40 0.41732.35 0.42752.30 0.43902.24 0.85 0.29473.29 0.29513.28 0.29583.27 0.29943.24 0.30523.18 0.31323.10 0.32323.01 0.33492.91 0.34832.80 0.95 0.14806.28 0.14866.26 0.14976.22 0.15476.03 0.16275.75 0.17345.42 0.18655.06 0.20164.71 0.21874.36 1 0.0099143.6† 0.010398.7† 0.011078.1† 0.014451.1† 0.019936.6† 0.027427.5† 0.039120.3 0.062013.7 0.087410.1 Local minimum of gq 0.0099143.6† 0.010398.7† NA NA NA NA NA NA NA 0.5 0.0099143.6† 0.010398.7† 0.011078.1† NA NA NA NA NA NA 0.75 0.0099143.6† 0.010398.7† 0.011078.1† NA NA NA NA NA NA 0.85 0.0099143.6† 0.010398.7† 0.011078.1† NA NA NA NA NA NA 0.95 0.0099143.6† 0.010398.7† 0.011078.1† NA NA NA NA NA NA 1 NA NA NA NA NA NA NA NA NA Fig. 6 demonstrates that the risk as represented by J ∞ is an increasing function of i for high values of M. Therefore, an increase in i leads to an upwards shift in the risk for the long spread periods, making the optimal choice of M shorter. When θ increases, the results show that the optimal choice increases. When θ = 1, for low values of σ , the risk is minimised when M ∗ = M 1 . For higher values of σ , the optimal choice for M decreases and for k increases. Table 15 indicates the critical values of ¯ σ when θ = 1. It is clear that the influence of the high initial funding level is more significant than any of the other parameters, making the optimal choice shorter for most cases, when compared with the F = 0 and F = 0.50 cases discussed earlier. We also observe that the higher is the initial funding level, the lower is the value of ¯ σ see Tables 7, 8, 14 and 15. Table 14 Critical values of ¯ σ : when F = 0.50 θ i j ¯ σ 0.95 0.01 0.01 – 0.03 0.01 – 0.03 0.03 – 0.03 0.05 – 0.05 0.03 – 0.05 0.05 – 0.05 0.1 0.105 1 0.01 0.01 0.12 0.03 0.01 0.07 0.03 0.03 0.17 0.03 0.05 0.25 0.05 0.03 0.145 0.05 0.05 0.24 0.05 0.1 0.43 S. Haberman et al. Insurance: Mathematics and Economics 27 2000 237–259 253 Fig. 6. Graph of J ∞ i when σ = 0.01, θ = 1 and j = 1. Table 15 Critical values of ¯ σ : when θ = 1 and F = 0.75 i j ¯ σ 0.01 0.01 0.095 0.03 0.03 0.13 0.03 0.05 0.24 0.05 0.05 0.17 0.05 0.1 0.37 2.7. Initial funding level of 100 Tables 16–20 provide the optimal values of k and M ∗ when F = 1 and for selected combinations of θ, σ, j and i. We note in this case, that there is only a global minimum for gq. When the initial unfunded liability is zero, a different value of the interest rate j used for discounting does not lead to a markedly different optimal choice for k or M — except for the case of θ = 1. From Tables 16–20, it can be seen that, for low values of θ , the results for k and M depend little on σ , i or j . For example, when θ = 0.5, M ∗ lies in the range 1.52, 1.63 and k ∗ in the range 0.6134, 0.6511 for the values of σ, i and j investigated. When θ increases, there is a slight increase in the optimal choice for M, as for the other cases discussed in Sections 2.3–2.6. When we are only concerned with stabilising the contribution rate θ = 1, the optimal choice Table 16 Optimal values of k ∗ and M ∗ : when F = AL, i = 1, j = 1 θ σ 0.01 0.03 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Global minimum of gq 11 11 11 11 11 11 11 11 11 0.5 0.62081.62 0.62101.62 0.62151.61 0.62351.61 0.62681.62 0.63131.59 0.63701.57 0.64371.56 0.65131.54 0.75 0.43792.30 0.43822.30 0.43872.29 0.44142.28 0.44572.26 0.45182.23 0.45942.19 0.46842.15 0.47882.10 0.85 0.34502.93 0.34532.92 0.34602.92 0.34892.89 0.35372.85 0.36042.80 0.36892.73 0.37912.66 0.39102.58 0.95 0.20914.88 0.20944.87 0.21014.85 0.21354.77 0.21904.65 0.22674.49 0.23674.29 0.24904.08 0.26333.85 1 0.0099535.0† 0.0108252.6 0.0123163.6 0.019571.2 0.031338.3 0.047338.3 0.067116.1 0.090211.7 0.11618.96 254 S. Haberman et al. Insurance: Mathematics and Economics 27 2000 237–259 Table 17 Optimal values of k ∗ and M ∗ : when F = AL, i = 3, j = 3 θ σ 0.01 0.03 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Global minimum of gq 11 11 11 11 11 11 11 11 11 0.5 0.61881.61 0.61901.61 0.61941.61 0.62141.60 0.62461.60 0.62901.59 0.63451.57 0.64111.56 0.64861.54 0.75 0.43402.29 0.43432.29 0.43482.29 0.43742.27 0.44172.25 0.44762.22 0.45512.19 0.46402.14 0.47422.10 0.85 0.34002.91 0.34042.91 0.34102.91 0.34392.88 0.34862.84 0.35532.79 0.36362.73 0.37372.66 0.38542.58 0.95 0.20254.85 0.20294.84 0.20364.83 0.20694.75 0.21254.63 0.22024.47 0.23024.28 0.24244.07 0.25673.84 1 0.0099218.5† 0.0107121.6 0.012233.7 0.019148.7 0.030530.2 0.045920.4 0.065014.6 0.087311.0 0.11248.61 Table 18 Optimal values of k ∗ and M ∗ : when F = AL, i = 3, j = 5 θ σ 0.01 0.03 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Global minimum of gq 11 11 11 11 11 11 11 11 11 0.5 0.61341.63 0.61361.62 0.61401.62 0.61601.62 0.61921.61 0.62371.60 0.62931.58 0.63591.57 0.64341.55 0.75 0.42692.33 0.42722.33 0.42772.32 0.43042.31 0.43462.29 0.44052.26 0.44802.22 0.45692.18 0.46712.13 0.85 0.33222.98 0.33252.98 0.33312.97 0.33602.95 0.34072.91 0.34732.85 0.35572.79 0.36572.71 0.37732.63 0.95 0.19365.07 0.19395.06 0.19465.04 0.19794.96 0.20334.83 0.21094.66 0.22074.46 0.23274.23 0.24684.00 1 0.0099218.5† 0.0103144.6† 0.0111110.9† 0.014567.8† 0.020245.8† 0.028132.8† 0.046820.0 0.069613.7 0.095210.1 Table 19 Optimal values of k ∗ and M ∗ : when F = AL, i = 5, j = 3 θ σ 0.01 0.03 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Global minimum of gq 11 11 11 11 11 11 11 11 11 0.5 0.62221.59 0.62241.59 0.62281.59 0.62471.59 0.62781.58 0.63211.57 0.63751.56 0.64381.54 0.65111.52 0.75 0.43752.25 0.43772.24 0.43832.24 0.44082.23 0.44502.21 0.45082.18 0.45812.15 0.46692.11 0.47692.07 0.85 0.34342.84 0.34372.83 0.34432.83 0.34722.81 0.35192.77 0.35842.72 0.36682.66 0.37672.59 0.38822.52 0.95 0.20594.62 0.20624.61 0.20704.59 0.21034.52 0.21604.41 0.22394.27 0.23404.09 0.24633.90 0.26063.69 1 0.028826.3 0.029625.8 0.031024.8 0.037521.1 0.048217.0 0.062813.5 0.080910.8 0.10218.81 0.12597.29 Table 20 Optimal values of k ∗ and M ∗ : when F = AL, i = 5, j = 5 θ σ 0.01 0.03 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Global minimum of gq 11 11 11 11 11 11 11 11 11 0.5 0.61681.61 0.61701.61 0.61741.61 0.61931.60 0.62251.59 0.62681.58 0.63221.57 0.63861.55 0.64591.94 0.75 0.43022.28 0.43052.28 0.43102.28 0.43362.26 0.43782.24 0.44362.22 0.45092.18 0.45972.14 0.46972.10 0.85 0.33532.90 0.33562.90 0.33622.90 0.33902.87 0.34382.83 0.35032.78 0.35862.72 0.36852.65 0.38002.57 0.95 0.19644.83 0.19674.82 0.19744.80 0.20074.73 0.20634.61 0.21414.45 0.22414.26 0.23624.06 0.25043.84 1 0.0099143.6† 0.010784.7 0.012164.4 0.018838.6 0.029725.7 0.044618.2 0.063013.5 0.084610.4 0.10898.31 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