American Journal of Theoretical and Applied Statistics
Volume 5, Issue 3, May 2016, Pages: 123-131

Actuarial Analysis of Single Life Status and Multiple Life Statuses

Abonongo John1, *, Luguterah Albert2

1Department of Mathematics, Kwame Nkrumah University of Science and Technology, Kumasi, Ghana

2Department of Statistics, University for Development Studies, Navrongo, Ghana

Email address:

(A. John)

*Corresponding author

To cite this article:

Abonongo John, Luguterah Albert. Actuarial Analysis of Single Life Status and Multiple Life Statuses. American Journal of Theoretical and Applied Statistics. Vol. 5, No. 3, 2016, pp. 123-131. doi: 10.11648/j.ajtas.20160503.17

Received: April 12, 2016; Accepted: April 22, 2016; Published: May 10, 2016


Abstract: Actuaries frequently employ probability models to analyse situations involving uncertainty. They are also not simply interested in modelling the future states of a subject but also model cash flows associated with future states. This study compared single life status and multiple life statuses using life functions. The expected time until death, annuity payments, insurance payable and premiums were estimated using age as a risk factor. The analysis also employed the De Moirve’s law on mortality in estimating the rate of mortality. The analysis revealed that, the expected time until death for single life status and multiple life statuses are all increasing functions of age.It was realized also that, the premium for single life status was increasing with age and the same with multiple life statuses. But the premium for single life was higher than multiple life statuses. In the case of the multiple life statuses, it was revealed that, premium for joint life was higher than the last survivor and that a change in the interest rate or force of interest and the benefit did not changed the trend in premium payments.

Keywords: Single Life Status, Multiple Life Statuses, Annuity, Insurance and Premium


1. Introduction

Actuarial practice indorses that its scientific base is extensively applicable in life insurance. Therefore actuaries have established a large range of models and varieties of methods and techniques in order to carry out professed actuarial calculations. One of the most important reasons for actuarial modelling is to introduce reliable methods for the practical pricing of insurance contracts, i.e. for the calculation of premium, which the insured life should pay to the insurer, so that the latter will pay his or her next-of-kin the insured amount on the occurrence of the insured event. Another actuarial calculation is the valuation of an insurance contract, thus the determination of its value during the lifetime of the contract; insurance reserve, for which special requirements apply with regard to how the insurer can invest the assets backing it and which forms the base for assessing the creditworthiness of the insurer; its ability to meet its liabilities now and in the future. A traditional assumption in the theory of multiple life contingencies is that the remaining life times of the lives involved are mutually independent. Computational feasibility rather than practicality seems to be the main reason for making this assumption. Such effects may have a significant influence on present values related to multiple life actuarial functions.

[8] and [3] showed alternative ways of modelling dependence of times of death of coupled lives. They released a significant degree of positive correlation between lifetimes. This implies that, joint life annuities were under-priced while last survivor annuities are over-priced. [2] presented boundaries of single premiums for last survivor annuities. [7] and [6] studied bounds of single premiums. These studies showed the impact of dependency of two remaining lifetimes on the pricing of life insurance products on the lives concerned. Dependency, however, also affected the valuation of such contracts over time. The reserves were based on laws of mortality which apply to the policy valuation date. If the remaining lifetimes of a couple are dependent at the outset of a policy, then any of the two lives’ survival probabilities may depend on the life status of the partner. Moreover, the joint distribution of remaining lifetimes, given the survival of both partners to a certain date, is affected as well. [16] showed that a lot of well-known relationships between probabilities and single premiums in multiple life contingencies are not valid in case of dependent lifetimes. They established that, the validity of those relationships can be restored if the definition of individual survival probabilities allows for the life status of the partner.

Standard actuarial theory of multiple life insurance assume the independence of the future lifetimes of the insured lives. This may occur when a policy is issued to a married couple. Numerous clinical studies have showed that broken heart syndrome may cause an increase in the mortality rate after the death of a spouse [14] and [11]. Also in insurance and annuities, multiple life model play an important role and the application of multiple life actuarial models are common. The investment income from a fund can be paid to a group of beneficiaries as long as at least one of the group survives [1].

[10] extended the classical analysis of the endowment contract on a single life to multiple lives, covering the joint-life and the last survivorship status. The results indicated that, the independence assumption overestimated the joint-life net single and level premiums and underestimates the last survivor net single and level premiums. [15] investigated the condition under which multiple life models can be replaced by single life models, they proved that in a survivorship group, the force of mortality of the group must follow Gompertz law provided that, for the joint-life status of very two lives, one can find a single-life status whose time until-death’s distribution is equal the joint-life status. Hence, the assumption that, the force of mortality follows Gompertz’s law is the necessary and sufficient criterion to guarantee that every joint-life status survival pattern can be replaced by a single-life status in the group.

[4] used the Frechet-Hoeffding bounds and Norberg’s Markov model in determining the effect of dependence of future lifetimes of couples with much emphases on the actuarial present values of widow’s pension benefit. Their results showed an economically significant positive dependence between joint lives: in Norbergs model, the amount of premium were reduced approximately 10% compared to the standard model that assumes independence. Also, [5] showed that the effect of a possible dependence was rather moderate for classical multiple life contracts at about 5%.

Moreover, other models can be used to incorporate dependencies between life times, for example, the frailty models described by [13] or Markov models as described by [12]. [9] re-investigated joint mortality functions and the assertion that relates the joint life and last survivor random variables. They realized that the common assertion that the sum of the lifetime of joint life and last survivor were equal to the sum of the lifetimes of the single statuses was true and modified the definition of the statuses so that this common assertion holds. They used copula model to indicate that the life insurance premiums with spousal status classification are lower than those without the classification and that the percentage differences are higher for older spouses and for higher interest rates.

The purpose of this paper is investigate the pricing of insurance for single-life status and multiple-life statuses using age as a risk factor. This is to provide insurers with the concept of insurance pricing using age as a risk factor and to give the insured information on the nature premium or annuity payments to be made pertaining to age.

2. Materials and Methods of Analysis

2.1. Data Source

This study employed ages of Ghanaians with reference from the mortality table of the Indian Institute of Actuaries.

2.2. Methods of Data Analysis

The mortality table is a tool which in a practical way represents a model of mortality and belongs to the basic mathematical toolbox in life insurance. Most often used in life insurance are complete mortality tables, which contain separate figures for each integer age, where   is the assumed maximum age, which someone may attain.

2.2.1. Single Life Theory

De Moivre’s law states that for all ages  such that, the expected number of survivors at age  and constant force of mortality are given by equation (1) and (2);

(1)

where,  is the expected number of survivors at age ,  is the terminal age and  is the attained age.

(2)

The survival function is given by;

(3)

where is the terminal age.

The expected time until first death is given by;

(4)

where is the survival at time t.

The Actuarial Present Value (APV) is given by;

(5)

where is the annuity payment and  is the interest rate.

The annuity-insurance relation is given by;

(6)

where is the discounting factor and  is the APV.

The premium is given by;

(7)

where is the benefit,  is the APV and  is the annuity.

For a continuous whole life insurance, the APV using a constant force model is given by;

(8)

where is the constant force of mortality and  is the force of interest. The annuity for a continuous whole life insurance is given by;

(9)

where is the constant force of mortality and  is the force of interest.

2.2.2. Multiple Life Theory

In multiple life theory consisting of two lives, the joint probability distribution of can be in two distinct forms; namely

a    Joint Life Status

The status is said to fail at the first time of failure of one of the component lives or fails upon the death of one of the component lives. The waiting time  until failure of the status is given by;

The status survives  years from now if .

b    Last Survivor Status

The status is said to fail at the last time of failure of the component lives or fails upon the death of the last component lives. The waiting time  until failure of the status is given by;

This status fails within the next  years if  (both lives have died in  years). The status is surviving in  years if  (second death has not occurred by time).

The Joint Life Status Force of Mortality Function with Independent Lives is given by;

(10)

where  is the force of mortality of a person aged  and  is the force of mortality of a person aged

2.2.3. Expected Time Until Death for Multiple Life Statuses

The expected time until first death of the component lives is given by;

(11)

where is the constant force of mortality for  and  is the constant force of mortality for

The expected time until death of the last survivor is given by;

(12)

where and  are the expected time until death for  and  respectively and  is the expected time until first death of the component lives.

2.2.4. Insurance for Multiple Life Statuses

For a continuous whole life insurance for joint lives, the APV using a constant force model is given by;

(13)

where is the constant force of mortality for  and  is the constant force of mortality for  and  is the force of interest.

For a continuous whole life insurance for the last survivor, the APV using a constant force model is given by;

(14)

2.2.5. The Annuities for Multiple Life Statuses

The Actuarial Present Value (APV) for Joint Life Annuity is given by;

(15)

where is the constant force of mortality for  and  is the constant force of mortality for  and  is the force of interest.

The APV of the last survivor annuity is given by;

(16)

where and  are the annuities for a continuous whole life insurance of  and , is APV for Joint Life Annuity.

2.2.6. Premium for Multiple Life Statuses

The premium for Joint Life is given by;

(17)

where is the benefit,  is the APV for the continuous whole life insurance for Joint life and  is the APV for Joint Life Annuity.

The premium for Last survivor is given by;

(18)

where is the benefit,  is the APV for the continuous whole life insurance for the last survivor and  is the APV of the last survivor annuity.

3. Results and Discussion

From figure 1, the mortality rate was seen as an increasing function of age. There was a smaller mortality rate from age 1 to age 49 but with an increasing mortality rate from age 50 upwards indicating that mortality rate is an increasing function of age. From age 80, the mortality rate raises to 1 in probability indicating that the chances of death is higher as one approaches the terminal age and as such requires higher premium payments. The expected time until death was seen as a decreasing function of age. At age 1, the expected time until death was almost the same as the terminal age. Also the expected time until death was almost zero at age 80 upwards since it was approaching the terminal age. The insurance payment gradually increases with age, thus from age 1 there was insurance payment which was lesser than payments at age 65 upwards. Also as ones age increases, the premium payments also increases. The annuity payments were seen as a decreasing function of age, meaning once the age goes up, the regular payments reduces. Thus the premium at age 1 to age 10 were quite lesser than premiums thereafter. It could be seen that from age 12 upwards the premium was increasing with age and that approaching the terminal age the premium was almost closer to the benefit. This indicates that, in pricing life insurance age plays a crucial role in determining the premium payments for the insurance coverage in that the higher ones age, the higher the premium and vice versa when other risk factors are held constant.

Figure 1. A graph of Mortality, Expected time until death, insurance, annuity and premium for Single Life.

Table 1, shows the estimates for Single Life Status. The constant force of mortality, from age 0 to 4 was constant at 0.010 rate per death, age 5 to 13 have 0.011 rate per death indicating a 0.001 increase from the previous cohort (age 0 to 4). Age 14 to 19 have 0.012 rate per death with age 20 to 25 having 0.013 rate per death. Also, age 26 to 31 have 0.014 rate per death. Age 32 to 35 have 0.015 rate per death with age 36 to 39 having 0.016 rate per death. A mortality rate difference of 0.001 existed for the cohorts (age 40 to 42, age 43 to 45, age 46 to 48, age 49 to 51, age 52 to 53, age 54 to 55, age 56 to 57 and age 58 to 59) respectively from the previous cohort (age 36 to 39) indicated that the mortality rate for each cohort was the same. From age 60 to 99, the mortality rate kept increasing until it was 1.000 at age 99. At the terminal age, there is an assumption that no person reaches that age and thus there will not be any mortality at that age. All these indicated that according to De Moivres Law, mortality is an increasing function of age. The expected time until death of each individual age was not the same, thus decreasing from time 100 years for age 0 to time 0 years for age 100. This indicates that, considering the age of an individual with all other perils held constant, the expected time until death was a decreasing function of age. The Actuarial Present Value (APV) for the individual lives was also not the same and thus shows an increasing function of age (from 0.198 for age 0 to 0.952 for age 99) since the pricing of life insurance takes into consideration the age of the individual. The annuity payments are decreasing function of age thus 0.991 for age 0 to 0.955 for age 99. For an individual to receive a benefit of GH¢ 1000 with an interest rate of 5%, then premium payment tends to be an increasing function of age, from GH¢ 200.373 for age 0 to GH¢ 997.625 for age 99. But these premium estimates will still be increasing with different interest rates and benefits when all other perils aside attain age are held constant. This shows that, no matter the interest and benefits, an insured for a life policy will have a high premium to pay when the age is high and vice versa. Also insurers of life products must consider the age insured to be able to apply the required premium payments.

Table 1. Estimates for Single Life Actuarial Functions.

Age
x Premium (P)
0 0.010 100.000 0.198 0.991 200.373
1 0.010 98.010 0.200 0.990 202.338
2 0.010 96.040 0.202 0.990 204.340
3 0.010 94.090 0.204 0.990 206.379
4 0.010 92.160 0.206 0.990 208.457
5 0.011 90.250 0.208 0.990 210.574
6 0.011 88.360 0.211 0.990 212.731
7 0.011 86.490 0.213 0.990 214.930
8 0.011 84.640 0.215 0.990 217.172
9 0.011 82.810 0.217 0.990 219.457
10 0.011 81.000 0.219 0.990 221.787
11 0.011 79.210 0.222 0.989 224.164
12 0.011 77.440 0.224 0.989 226.588
13 0.011 75.690 0.227 0.989 229.060
14 0.012 73.960 0.229 0.989 231.583
15 0.012 72.250 0.232 0.989 234.156
16 0.012 70.560 0.234 0.989 236.783
17 0.012 68.890 0.237 0.989 239.464
18 0.012 67.240 0.239 0.989 242.200
19 0.012 65.610 0.242 0.988 244.994
20 0.013 64.000 0.245 0.988 247.847
21 0.013 62.410 0.248 0.988 250.760
22 0.013 60.840 0.251 0.988 253.736
23 0.013 59.290 0.254 0.988 256.775
24 0.013 57.760 0.257 0.988 259.881
25 0.013 56.250 0.260 0.988 263.054
26 0.014 54.760 0.263 0.987 266.297
27 0.014 53.290 0.266 0.987 269.612
28 0.014 51.840 0.269 0.987 273.001
29 0.014 50.410 0.273 0.987 276.465
30 0.014 49.000 0.276 0.987 280.008
31 0.014 47.610 0.280 0.987 283.632
32 0.015 46.240 0.283 0.987 287.339
33 0.015 44.890 0.287 0.986 291.131
34 0.015 43.560 0.291 0.986 295.011
35 0.015 42.250 0.295 0.986 298.983
36 0.016 40.960 0.299 0.986 303.047
37 0.016 39.690 0.303 0.986 307.208
38 0.016 38.440 0.307 0.985 311.469
39 0.016 37.210 0.311 0.985 315.832
40 0.017 36.000 0.315 0.985 320.300
41 0.017 34.810 0.320 0.985 324.877
42 0.017 33.640 0.324 0.985 329.567
43 0.018 32.490 0.329 0.984 334.373
44 0.018 31.360 0.334 0.984 339.297
45 0.018 30.250 0.339 0.984 344.346
46 0.019 29.160 0.344 0.984 349.521
47 0.019 28.090 0.349 0.983 354.828
48 0.019 27.040 0.354 0.983 360.270
49 0.020 26.010 0.360 0.983 365.852
50 0.020 25.000 0.365 0.983 371.579
51 0.020 24.010 0.371 0.982 377.455
52 0.021 23.040 0.377 0.982 383.485
53 0.021 22.090 0.383 0.982 389.674
54 0.022 21.160 0.389 0.981 396.027
55 0.022 20.250 0.395 0.981 402.551
56 0.023 19.360 0.401 0.981 409.250
57 0.023 18.490 0.408 0.981 416.130
58 0.024 17.640 0.415 0.980 423.198
59 0.024 16.810 0.422 0.980 430.460
60 0.025 16.000 0.429 0.980 437.923
61 0.026 15.210 0.436 0.979 445.593
62 0.026 14.440 0.444 0.979 453.477
63 0.027 13.690 0.452 0.978 461.584
64 0.028 12.960 0.460 0.978 469.920
65 0.029 12.250 0.468 0.978 478.494
66 0.029 11.560 0.476 0.977 487.314
67 0.030 10.890 0.485 0.977 496.388
68 0.031 10.240 0.494 0.976 505.726
69 0.032 9.610 0.503 0.976 515.337
70 0.033 9.000 0.512 0.976 525.231
71 0.034 8.410 0.522 0.975 535.418
72 0.036 7.840 0.532 0.975 545.908
73 0.037 7.290 0.542 0.974 556.712
74 0.038 6.760 0.553 0.974 567.842
75 0.040 6.250 0.564 0.973 579.310
76 0.042 5.760 0.575 0.973 591.127
77 0.043 5.290 0.586 0.972 603.308
78 0.045 4.840 0.598 0.972 615.865
79 0.048 4.410 0.611 0.971 628.812
80 0.050 4.000 0.623 0.970 642.165
81 0.053 3.610 0.636 0.970 655.937
82 0.056 3.240 0.649 0.969 670.146
83 0.059 2.890 0.663 0.968 684.807
84 0.063 2.560 0.677 0.968 699.937
85 0.067 2.250 0.692 0.967 715.556
86 0.071 1.960 0.707 0.966 731.681
87 0.077 1.690 0.723 0.966 748.332
88 0.083 1.440 0.739 0.965 765.529
89 0.091 1.210 0.755 0.964 783.295
90 0.100 1.000 0.772 0.963 801.650
91 0.111 0.810 0.790 0.962 820.619
92 0.125 0.640 0.808 0.962 840.226
93 0.143 0.490 0.827 0.961 860.497
94 0.167 0.360 0.846 0.96 881.457
95 0.200 0.250 0.866 0.959 903.134
96 0.250 0.160 0.886 0.958 925.559
97 0.333 0.090 0.908 0.957 948.761
98 0.500 0.040 0.930 0.956 972.771
99 1.000 0.010 0.952 0.955 997.625
100 0.000

Also from figure 2, the mortality rate was seen as increasing function of age for the joint life status. From age 1 to 49, the mortality rate was constant until age 50 when it kept increasing all through to the terminal age. The expected time until death for both joint life (ExpectedTDJL) and last survivor (ExpectedTDLS) are seen as a decreasing function of age. But the expected time until death for the last survivor status was higher than the joint life status, and that the expected time until death does not hit zero for the last survivor as it does for the joint life. The insurance payable for both the joint life (InsuranceJL) and last survivor (InsuranceLS) were all increasing function of age but the joint life has a higher insurance payment than the last survivor. The annuity payment was seen as a decreasing function of age with the last survivor (AnnuityLS) having a higher annuity payment than the joint life (AnnuityJL). Nevertheless, the premium payment was seen an increasing function of age. The premium for joint life (PremiumJL) was higher than that of the last survivor (PremiumLS). This indicates that, in considering two lives for life policy, the ages of the two matters in that the age of either life can easily influence the premium payment especially in the case of joint life statuses.

Figure 2. A graph of Mortality Rate, Expected time until death (Joint Life and Last Survivor), Insurance (Joint life and Last survivor), Annuity (Joint life and Last survivor) and Premium (Joint life and Last survivor).

Table 2. Estimates for Multiple Life Actuarial Functions.

Age
x Premium (P) Premium (P)
0 0.020 49.749 148.261 0.287 0.048 14.265 19.040 20.101 2.520
1 0.020 49.249 144.801 0.289 0.049 14.224 19.025 20.305 2.562
2 0.021 48.749 141.381 0.291 0.050 14.182 19.010 20.513 2.605
3 0.021 48.249 138.001 0.293 0.050 14.139 18.994 20.726 2.649
4 0.021 47.749 134.661 0.295 0.051 14.096 18.978 20.943 2.694
5 0.021 47.249 131.361 0.297 0.052 14.052 18.961 21.165 2.740
6 0.021 46.749 128.101 0.300 0.053 14.007 18.944 21.391 2.787
7 0.022 46.249 124.881 0.302 0.054 13.962 18.927 21.622 2.836
8 0.022 45.749 121.701 0.304 0.055 13.916 18.909 21.859 2.886
9 0.022 45.249 118.561 0.307 0.055 13.870 18.890 22.100 2.937
10 0.022 44.749 115.461 0.309 0.056 13.822 18.872 22.347 2.990
11 0.023 44.249 112.401 0.311 0.057 13.774 18.852 22.600 3.044
12 0.023 43.749 109.381 0.314 0.058 13.725 18.833 22.858 3.099
13 0.023 43.249 106.401 0.316 0.059 13.676 18.812 23.122 3.157
14 0.023 42.749 103.461 0.319 0.060 13.625 18.792 23.393 3.215
15 0.024 42.249 100.561 0.321 0.061 13.574 18.770 23.669 3.276
16 0.024 41.749 97.701 0.324 0.063 13.522 18.748 23.953 3.338
17 0.024 41.248 94.882 0.327 0.064 13.469 18.726 24.243 3.403
18 0.025 40.748 92.102 0.329 0.065 13.415 18.703 24.541 3.469
19 0.025 40.248 89.362 0.332 0.066 13.361 18.679 24.846 3.537
20 0.025 39.748 86.662 0.335 0.067 13.305 18.654 25.158 3.607
21 0.025 39.248 84.002 0.338 0.069 13.249 18.629 25.479 3.679
22 0.026 38.748 81.382 0.340 0.070 13.191 18.603 25.808 3.754
23 0.026 38.248 78.802 0.343 0.071 13.133 18.577 26.145 3.831
24 0.026 37.748 76.262 0.346 0.073 13.073 18.549 26.491 3.910
25 0.027 37.248 73.762 0.349 0.074 13.013 18.521 26.847 3.992
26 0.027 36.748 71.302 0.352 0.075 12.951 18.492 27.212 4.077
27 0.028 36.248 68.882 0.356 0.077 12.889 18.462 27.588 4.164
28 0.028 35.748 66.502 0.359 0.078 12.825 18.432 27.973 4.254
29 0.028 35.248 64.162 0.362 0.080 12.760 18.400 28.370 4.348
30 0.029 34.748 61.862 0.365 0.082 12.694 18.367 28.778 4.444
31 0.029 34.248 59.602 0.369 0.083 12.626 18.334 29.199 4.544
32 0.030 33.748 57.382 0.372 0.085 12.558 18.299 29.631 4.648
33 0.030 33.248 55.202 0.376 0.087 12.488 18.263 30.077 4.755
34 0.031 32.748 53.062 0.379 0.089 12.417 18.226 30.536 4.866
35 0.031 32.248 50.962 0.383 0.091 12.344 18.188 31.010 4.981
36 0.031 31.748 48.902 0.386 0.093 12.270 18.149 31.498 5.101
37 0.032 31.248 46.882 0.390 0.095 12.195 18.108 32.002 5.225
38 0.033 30.748 44.902 0.394 0.097 12.118 18.066 32.522 5.353
39 0.033 30.248 42.962 0.398 0.099 12.039 18.022 33.060 5.487
40 0.034 29.748 41.062 0.402 0.101 11.959 17.977 33.616 5.626
41 0.034 29.248 39.202 0.406 0.103 11.878 17.931 34.191 22.649
42 0.035 28.748 37.382 0.410 0.106 11.795 17.882 34.785 22.943
43 0.035 28.248 35.602 0.415 0.108 11.709 17.833 35.401 23.245
44 0.036 27.748 33.862 0.419 0.111 11.623 17.781 36.039 23.557
45 0.037 27.248 32.162 0.423 0.114 11.534 17.727 36.700 23.879
46 0.037 26.748 30.502 0.428 0.116 11.443 17.672 37.386 24.210
47 0.038 26.248 28.882 0.432 0.119 11.351 17.614 38.099 24.552
48 0.039 25.748 27.302 0.437 0.122 11.256 17.554 38.839 24.905
49 0.040 25.248 25.762 0.442 0.125 11.160 17.492 39.608 25.269
50 0.040 24.747 24.263 0.447 0.129 11.061 17.428 40.408 25.646
51 0.041 24.247 22.803 0.452 0.132 10.960 17.361 41.241 26.036
52 0.042 23.747 21.383 0.457 0.135 10.857 17.291 42.110 26.440
53 0.043 23.247 20.003 0.462 0.139 10.751 17.218 43.016 26.858
54 0.044 22.747 18.663 0.468 0.143 10.643 17.143 43.961 27.292
55 0.045 22.247 17.363 0.473 0.147 10.532 17.064 44.949 27.742
56 0.046 21.747 16.103 0.479 0.151 10.419 16.982 45.983 28.210
57 0.047 21.247 14.883 0.485 0.155 10.302 16.897 47.065 28.697
58 0.048 20.747 13.703 0.491 0.160 10.183 16.808 48.200 29.203
59 0.049 20.247 12.563 0.497 0.164 10.061 16.715 49.390 29.730
60 0.051 19.747 11.463 0.503 0.169 9.936 16.617 50.641 30.281
61 0.052 19.247 10.403 0.510 0.174 9.808 16.516 51.957 30.855
62 0.053 18.747 9.383 0.516 0.180 9.677 16.409 53.343 31.456
63 0.055 18.247 8.403 0.523 0.185 9.542 16.298 54.805 32.085
64 0.056 17.746 7.464 0.530 0.191 9.403 16.181 56.349 32.744
65 0.058 17.246 6.564 0.537 0.197 9.261 16.059 57.983 33.437
66 0.060 16.746 5.704 0.544 0.203 9.115 15.931 59.715 34.165
67 0.062 16.246 4.884 0.552 0.210 8.964 15.796 61.553 34.931
68 0.064 15.746 4.104 0.560 0.217 8.810 15.655 63.508 35.740
69 0.066 15.246 3.364 0.567 0.225 8.651 15.506 65.591 36.596
70 0.068 14.746 2.664 0.576 0.233 8.488 15.349 67.816 37.502
71 0.070 14.246 2.004 0.584 0.241 8.320 15.184 70.197 38.463
72 0.073 13.745 1.385 0.593 0.250 8.147 15.009 72.751 39.487
73 0.075 13.245 0.805 0.602 0.259 7.968 14.825 75.499 40.578
74 0.078 12.745 0.265 0.611 0.268 7.784 14.631 78.462 41.745
75 0.082 12.245 0.235 0.620 0.279 7.595 14.425 81.667 42.998
76 0.085 11.745 0.295 0.630 0.290 7.399 14.207 85.145 44.345
77 0.089 11.244 0.114 0.640 0.301 7.198 13.976 88.933 45.800
78 0.093 10.744 0.494 0.651 0.313 6.989 13.731 93.074 47.378
79 0.098 10.244 0.834 0.661 0.327 6.774 13.470 97.619 49.095
80 0.103 9.744 0.134 0.672 0.340 6.552 13.192 102.632 50.972
81 0.108 9.243 0.393 0.684 0.355 6.322 12.896 108.187 53.035
82 0.114 8.743 0.613 0.696 0.371 6.083 12.579 114.379 29.495
83 0.121 8.242 0.792 0.708 0.388 5.837 12.241 121.324 31.692
84 0.129 7.742 0.932 0.721 0.406 5.581 11.879 129.167 34.183
85 0.138 7.241 0.031 0.734 0.425 5.316 11.490 138.095 37.030
86 0.148 6.741 0.091 0.748 0.446 5.042 11.073 148.352 40.314
87 0.160 6.240 0.110 0.762 0.469 4.756 10.623 160.256 44.138
88 0.174 5.739 0.089 0.777 0.493 4.459 10.137 174.242 48.645
89 0.191 5.238 0.028 0.792 0.519 4.151 9.612 190.909 54.031
90 0.211 4.737 0.927 0.809 0.548 3.830 9.044 211.111 60.573
91 0.236 4.235 0.785 0.825 0.579 3.495 8.426 236.111 68.680
92 0.268 3.733 0.603 0.843 0.612 3.146 7.753 267.857 78.976
93 0.310 3.231 0.381 0.861 0.649 2.781 7.019 309.524 92.468
94 0.367 2.727 0.117 0.880 0.689 2.400 6.215 366.667 110.891
95 0.450 2.222 0.812 0.900 0.733 2.000 5.333 450.000 137.500
96 0.583 1.714 0.464 0.921 0.782 1.579 4.363 583.333 179.196
97 0.833 1.200 0.070 0.943 0.835 1.132 3.295 833.333 253.508
98 1.500 1.667 0.617 0.968 0.894 0.645 2.125 1500.000 420.499
99 1.000 1.000 0.990 0.952 0.000 0.952 20.000 1000.000 0.000
100 0.000 0.000 0.000 20.000 0.000 0.000

Table 2, shows the estimates for the joint life and last survivor statuses. For multiple life statuses in this case two lives were considered at a time. The ages were grouped as follows: ages 0 and 1, ages 1 and 2, ages 2 and 3,…, ages 99 and 100. The mortality for joint life started from 0.020 for ages 0 and 1 and same for ages 1 and 2 but increases from there to 1.000 for ages 99 and 100. This indicated that, the mortality for joint life was an increasing function of age. The expected time until first death of the component lives (joint life status) was a decreasing function of age, from 50 years for ages 0 and 1, 49 years for ages 1 and 2 to 1 year for ages 99 and 100. This indicated that the expected time until death of the first component lives which can be either of the two depends on the age of any of the two when all other perils are held constant. The expected time until death of the last component lives (last survivor status) was also a decreasing function of age, that was from 148 years for ages 0 and 1, 145 years for ages 1 and 2 to approximately 1 year for ages 99 and 100. The expected time until death for last component lives has much surviving time than the expected time until death of the first component lives because in the joint life case, it fails only on the first death of one of the component lives. In this case the probability of death was higher than that of the last survivor where both component lives has to die for the status to fail and by the time the two components might have died the time survived would also have increased. The insurance payable immediately on the death of the first component lives (joint life) was an increasing function of age. Thus, from 0.287 for ages 0 and 1, 0.289 for ages 1 and 2 to 0.952 for ages 99 and 100. The insurance payable immediately on the death of the last component lives was also an increasing function of age, that is from 0.048 for ages 0 and 1, 0.049 for ages 1 and 2 to 0.894 for ages 98 and 99. This indicates that a unit insurance payable immediately on the death of first component lives was greater than that of the last survivor. This was because of the variation in the expected time until death of the joint life been much smaller than the last survivor and as such insurance payments on lives with shorter time until death paying more than those with more time until death. The annuity payments until the first death of the component lives was a decreasing function of age, that is from 14.265 for ages 0 and 1, 14.224 for ages 1 and 2 to 0.952 for ages 98 and 99. But age 100 had a higher annuity payment of 20.000 than all the component lives because it was considered to the terminal age. Also the annuity payments until the death of the last component lives was a decreasing function of age, that is from 19.040 for ages 0 and 1, 19.025 for ages 1 and 2 to 2.125 for ages 98 and 99 but ages 99 and 100 had annuity payments of 20.000 since there was an inclusion of the terminal age. In the same way, the annuity payments for the joint life was smaller than the last survivor because the annuity was paid until the first death of one of the component lives and since the expected time until first death was also smaller compared with last survivor. It was also realized that, annuity for the last survivor was continually paid until the last death of the component lives and in this case will make the payments accrual more than that of the joint life. For one to receive a benefit of GH¢ 1000 at 5% force of interest, the premium to be paid for both the joint life and last survivor was an increasing function of age and that the premium for the joint life was greater than the last survivor since the expected time for the last survivor was greater than the joint life when all other perils are held constant. Again the benefit and force of interest when changed will yield an increasing premium with age when all other perils are held constant.

4. Conclusion

This paper compared single life status and multiple life statuses using multiple and single life functions. The analysis revealed that, using age as the main risk evaluation, for single life status and multiple life statuses, their expected time until death are decreasing functions of age. It was realized also that, the premiums for both single life status and multiple life statuses were increasing with age, however the premium for single life was higher than multiple life statuses. In the case of the multiple life statuses, it was revealed that, premium for joint life was greater than the last survivor and that, a change in the interest rate or force of interest and the benefit did not change the trend in premium payments. Also no matter the interest and benefits, an insured for a life policy will have a high premium to pay when the age is high and vice versa and insurers of life products must consider the age of the insured to be able to apply the required premium payments. In considering two lives for life policy coverage, the ages of the two matters in that the age of either life can easily influence the premium payment especially in the case of joint life status in which the premium payments are higher than the last survivor status. Therefore, in pricing these insurances, the age of one is essential in determining the premium.


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