Analysis of the Variable Life Insurance Based on LogNormal Distribution
Shiqi Dong^{1}^{, *}, Shan Pang^{2}
School of Mathematics and Statistics, Central China Normal University, Wuhan, China
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To cite this article:
Shiqi Dong, Shan Pang. Analysis of the Variable Life Insurance Based on LogNormal Distribution. International Journal of Statistical Distributions and Applications. Vol. 1, No. 1, 2015, pp. 511. doi: 10.11648/j.ijsd.20150101.12
Abstract: Fixed rate, premiums and insurance coverage for policyholders and insurance companies in traditional life insurance have increased certain risks. For this reason, we consider studying variable life insurance. The biggest difference between the two insurance is that whether the actual death benefit of volatility is changeable. This paper studied the change of the premium when the premium changes in proportion to the death benefit and when it is fixed. And, it put forward a way to pay the death benefit, named "pay off increasing amount insurance". Finally, this paper simulated the mean and variance of the death benefit using Monte Carlo method, and also compared the advantage and disadvantages of each approach.
Keywords: Variable Life Insurance, The Actual Death Benefit, Change in Proportion, Fixed Premium, Pay off Increasing Amount Insurance, Monte Carlo Method
1. Introduction
Since the reform and opening up, Chinese insurance industry is rapidly becoming one of the fastest growing sectors of the national economy. With the gradual speeding up of the development of the insurance industry and the competition becoming increasingly fierce, the flaws of traditional life insurance are increasingly conspicuous. For policyholders, it is not conducive to change premiums based on their economic situation, and besides, they cannot gain benefits in economic growth. For life insurance companies, it will lead to increased risk if the scheduled interest rate is too high, or else it will be less attractive to clients. Unstable interest rates and fixed insurance coverage increase the risk of earnings, also lead to more people to terminate their traditional life insurance contracts.
In today's rapid economic development, traditional life insurance which is limited by its defects will gradually lose the market. In order to meet the demands of policyholders, and make clients benefited from economic growth while they obtain life insurance protection, Chinese life insurance companies should learn from foreign experience, develop variable life insurance product and research new pricing model of variable life insurance product.
The variable insurance was built in a paper by Duncan (1952). Since 1969, a flurry of activity on variable life insurance has appeared. It is the paper by Fraser, Miller, and Sternhell (1969), and its extensive discussion that be the basic reference. What is more, a less formal introduction and some numerical illustrations is provided by Miller (1971). For more discussions and additional information on variable annuities, see Bowers et al. (1997).
2. The Basic Introduction and Characteristics of Variable Life Insurance
The difference between variable life insurance and traditional life insurance is that the insured amount of policies is variable on the premise of the minimum amount, and this change depends on the benefits of separate accounts which policyholder choose to invest in. The investmentoriented policy is shown in figure 1.
In addition to the fixed minimum death benefit stipulated by the policy, the death benefit of variable life insurance also includes investment income from investment accounts, which is alterable. Insurance company opens a single account or multiple discrete subaccounts which have different risks and different benefits. After deducting running expenses and the cost of death in premiums of policyholders, the insurance company will put the remaining costs into accounts, and establish funds with different investment direction and different risks for the insured. The policyholders could choose a fund or several funds, and insurance company will manage the funds by itself or commission professional company to manage them. The final benefits will return to the policyholders. The higher benefits of investment will lead to the higher cash value of the policy, and the insured amount would be higher too; on the contrary, lower investment income is, lower cash value of the policy and lower insured amount are.
Thus, the investment income of policyholders is exclusive to them, and they bear the responsibility of investment risk alone. For insurers, they need to bear only the risks that caused by mortality and expense, which is to say, the variable life insurance avoids the risk effectively which is caused by inflation and interest rate variation.
3. The Actuarial Study of Variable Life Insurance Death Benefit
The biggest difference between variable life insurance and traditional life insurance is the variability of death benefits. Therefore, the research on death benefit is the key to actuarial studies of variable life insurance. The actual death benefit is determined by the cumulative value of the policy, but not less than the scheduled minimum death benefit.
For example, consider discrete whole life insurance. Assume that the insured person age , the probability of survival for a year is , the insured amount (the lowest death benefit) isRMB, and the actuarial present value of the insured’s permanent life insurance isat age. indicates actuarial present value of permanent life insurance ofterm when the insured ages. The liability reserve funds on the end of term is, the net annual premium is , and the expected rate of return on investment of the period is , the actual rate of return on investment is .
There are two ways to determine the premium: fixed premium and premium changes in the same proportion with death benefit. We calculate the actual death benefit in both cases respectively:
3.1. Premiums Change in the Same Proportion with Death Benefits
Assume that (,) is the death benefit of term . Therefore, the liability reserve funds in the beginning of term is .
Accumulating to the end of the term at the actual interest rate, the asset share becomes , then
(3.1)
According to the recurrence formula of liability reserve funds, we have
(3.2)
Divide(2.1)by(2.2), and we can get
(3.3)
When , we have . That is to say, even if , the actual rate of return on , investment of term , fall back to the expected rate, the actual death benefit level will remain at last year's level without decreasing, which means the actual death benefit is relatively stable (see Duncan, Robert M, 1952).
3.2. The Premiums Are Fixed
Assume liability reserve funds change in the same proportion with death benefit, the premium is invariant, constant for , and the liability reserve funds at the end of term is , and the premium charged in the beginning of term is , which accumulated to the end of the term at the rate of , then we have
(3.4)
Divide(2.4)by(2.5), we can get
(3.5)
It is obvious that . If , then we can get . That means as long as fall back to the expected rate, the actual death benefit level will drop below the level of the previous period (see Fraser, John C., Walter N.Miller, and Charles M. Sternhell, 1969, pp 175243).
Therefore, the actual death benefit is affected by the actual investment return rate, and it is unstable, so we need to optimize the method of how to determine the actual death benefit when premium is fixed. Writers put forward a way named "pay off increasing amount insurance".
3.3. Pay off Increasing Amount Insurance
The bonus generated then (the asset share of the policy at the end of the term mines liability reserves at the end of term ) is the premium of wholesale payment. Let be the death benefit of term , and the death benefit which is formed only by the bonus be .
Therefore, the liability reserve in the beginning of term is
.
The asset share accumulated to the end of the period at the rate of is
,
and the death benefit of term is . It follows that
(3.6)
Yet
(3.7)
Dividing(2.6)by(2.7)gives
(3.8)
According to the formula of liability reserves,
(3.9)
Substituting(2.9)for(2.8)gives
(3.10)
Hence,
(3.11)
when . That means rather than fall, the actual death benefit will remain at the level of last year even though the actual rate of return on investment drops to. If and only if the actual rate of return on investment is lower than the assumed one, the actual death benefit will decrease, but it is never lower than the scheduled lowest death benefit (see Miller, Walter N, 1971).
4. The Comparison Between Three Methods of Actual Death Benefit
Now there are three approaches to calculate the actual death benefit. The first two are under the condition that the premium changes in proportion to the death benefit and it is fixed respectively. The last one is named "pay off increasing amount insurance".
Consider a variable whole life insurance of unit premium, the insured is 20 years old when sign the insurance contract. Use China Life Insurance Mortality Table (20002003), CL1.
Suppose that the ratio of two adjacent periods’ rate is a lognormal random variable, that is
.
Set , then the confidence interval is .
Yet ,
according to the principle of , we obtain
,
i.e. ,
which means almost always falls into interval (0.04,0.06). Therefore, the assumption is reasonable.
Simulate 10000 times the three methods, and the results is shown in Table 1.
1st method  2ed method  3rd method  
 1.4970  1.8163  1.5353 
 0.0231  0.0395  0.0062 
We can see from Table 1 that the first approach has smallest mean, and it is relatively stable; the second approach has biggest mean, but it is the most unstable; yet last approach, which is based on the former approach, is the most stable even if its mean is lower than the previous one.
Appendices
Appendix one: China Life Insurance Mortality Table (20002003), CL1
China Life Insurance Mortality Table (20002003), CL1  
age  mortality  Survival number  death toll  Life Expectancy  Survival personyear  







0  0.000722  1,000,000  722  76.7  999,639  76,712,704 
1  0.000603  999,278  603  75.8  998,977  75,713,065 
2  0.000499  998,675  498  74.8  998,426  74,714,088 
3  0.000416  998,177  415  73.9  997,969  73,715,662 
4  0.000358  997,762  357  72.9  997,583  72,717,692 
5  0.000323  997,405  322  71.9  997,244  71,720,109 
6  0.000309  997,082  308  70.9  996,928  70,722,865 
7  0.000308  996,774  307  70.0  996,621  69,725,937 
8  0.000311  996,467  310  69.0  996,312  68,729,316 
9  0.000312  996,157  311  68.0  996,002  67,733,004 
10  0.000312  995,847  311  67.0  995,691  66,737,001 
11  0.000312  995,536  311  66.0  995,381  65,741,310 
12  0.000313  995,225  312  65.1  995,070  64,745,929 
13  0.000320  994,914  318  64.1  994,755  63,750,860 
14  0.000336  994,595  334  63.1  994,428  62,756,105 
15  0.000364  994,261  362  62.1  994,080  61,761,677 
16  0.000404  993,899  402  61.1  993,699  60,767,596 
17  0.000455  993,498  452  60.2  993,272  59,773,898 
18  0.000513  993,046  509  59.2  992,791  58,780,626 
19  0.000572  992,536  568  58.2  992,253  57,787,835 
20  0.000621  991,969  616  57.3  991,661  56,795,582 
21  0.000661  991,353  655  56.3  991,025  55,803,922 
22  0.000692  990,697  686  55.3  990,355  54,812,897 
23  0.000716  990,012  709  54.4  989,657  53,822,542 
24  0.000738  989,303  730  53.4  988,938  52,832,885 
25  0.000759  988,573  750  52.4  988,198  51,843,947 
26  0.000779  987,823  770  51.5  987,438  50,855,749 
27  0.000795  987,053  785  50.5  986,661  49,868,311 
28  0.000815  986,268  804  49.6  985,866  48,881,651 
29  0.000842  985,464  830  48.6  985,050  47,895,784 
30  0.000881  984,635  867  47.6  984,201  46,910,735 
31  0.000932  983,767  917  46.7  983,309  45,926,534 
32  0.000994  982,850  977  45.7  982,362  44,943,225 
33  0.001055  981,873  1,036  44.8  981,356  43,960,863 
34  0.001121  980,838  1,100  43.8  980,288  42,979,507 
35  0.001194  979,738  1,170  42.9  979,153  41,999,220 
36  0.001275  978,568  1,248  41.9  977,944  41,020,066 
37  0.001367  977,321  1,336  41.0  976,653  40,042,122 
38  0.001472  975,985  1,437  40.0  975,266  39,065,469 
39  0.001589  974,548  1,549  39.1  973,774  38,090,203 
40  0.001715  972,999  1,669  38.1  972,165  37,116,430 
41  0.001845  971,331  1,792  37.2  970,435  36,144,265 
42  0.001978  969,539  1,918  36.3  968,580  35,173,830 
43  0.002113  967,621  2,045  35.3  966,599  34,205,250 
44  0.002255  965,576  2,177  34.4  964,488  33,238,652 
45  0.002413  963,399  2,325  33.5  962,237  32,274,164 
46  0.002595  961,074  2,494  32.6  959,827  31,311,928 
47  0.002805  958,580  2,689  31.7  957,236  30,352,100 
48  0.003042  955,891  2,908  30.8  954,437  29,394,865 
49  0.003299  952,984  3,144  29.8  951,412  28,440,427 
50  0.003570  949,840  3,391  28.9  948,144  27,489,016 
51  0.003847  946,449  3,641  28.0  944,628  26,540,871 
52  0.004132  942,808  3,896  27.1  940,860  25,596,243 
53  0.004434  938,912  4,163  26.3  936,830  24,655,383 
54  0.004778  934,749  4,466  25.4  932,516  23,718,553 
55  0.005203  930,283  4,840  24.5  927,863  22,786,037 
56  0.005744  925,442  5,316  23.6  922,785  21,858,174 
57  0.006427  920,127  5,914  22.8  917,170  20,935,390 
58  0.007260  914,213  6,637  21.9  910,894  20,018,220 
59  0.008229  907,576  7,468  21.1  903,842  19,107,326 
60  0.009313  900,107  8,383  20.2  895,916  18,203,484 
61  0.010490  891,725  9,354  19.4  887,048  17,307,568 
62  0.011747  882,371  10,365  18.6  877,188  16,420,520 
63  0.013091  872,005  11,415  17.8  866,298  15,543,332 
64  0.014542  860,590  12,515  17.1  854,333  14,677,035 
65  0.016134  848,075  13,683  16.3  841,234  13,822,702 
66  0.017905  834,392  14,940  15.6  826,922  12,981,468 
67  0.019886  819,453  16,296  14.8  811,305  12,154,546 
68  0.022103  803,157  17,752  14.1  794,281  11,343,241 
69  0.024571  785,405  19,298  13.4  775,756  10,548,960 
70  0.027309  766,107  20,922  12.8  755,646  9,773,205 
71  0.030340  745,185  22,609  12.1  733,881  9,017,559 
72  0.033684  722,576  24,339  11.5  710,406  8,283,678 
73  0.037371  698,237  26,094  10.8  685,190  7,573,272 
74  0.041430  672,143  27,847  10.2  658,220  6,888,082 
75  0.045902  644,296  29,574  9.7  629,509  6,229,863 
76  0.050829  614,722  31,246  9.1  599,099  5,600,354 
77  0.056262  583,476  32,828  8.6  567,062  5,001,255 
78  0.062257  550,648  34,282  8.1  533,508  4,434,193 
79  0.068871  516,367  35,563  7.6  498,585  3,900,685 
80  0.076187  480,804  36,631  7.1  462,488  3,402,100 
81  0.084224  444,173  37,410  6.6  425,468  2,939,611 
82  0.093071  406,763  37,858  6.2  387,834  2,514,143 
83  0.102800  368,905  37,923  5.8  349,943  2,126,309 
84  0.113489  330,982  37,563  5.4  312,200  1,776,366 
85  0.125221  293,419  36,742  5.0  275,048  1,464,166 
86  0.138080  256,677  35,442  4.6  238,956  1,189,118 
87  0.152157  221,235  33,662  4.3  204,404  950,162 
88  0.167543  187,572  31,426  4.0  171,859  745,759 
89  0.184333  156,146  28,783  3.7  141,754  573,899 
90  0.202621  127,363  25,806  3.4  114,460  432,145 
91  0.222500  101,557  22,596  3.1  90,258  317,685 
92  0.244059  78,960  19,271  2.9  69,325  227,427 
93  0.267383  59,689  15,960  2.6  51,709  158,102 
94  0.292544  43,729  12,793  2.4  37,333  106,392 
95  0.319604  30,937  9,887  2.2  25,993  69,059 
96  0.348606  21,049  7,338  2.0  17,380  43,067 
97  0.379572  13,711  5,204  1.9  11,109  25,686 
98  0.412495  8,507  3,509  1.7  6,752  14,577 
99  0.447334  4,998  2,236  1.6  3,880  7,825 
100  0.484010  2,762  1,337  1.4  2,094  3,945 
101  0.522397  1,425  745  1.3  1,053  1,851 
102  0.562317  681  383  1.2  489  798 
103  0.603539  298  180  1.0  208  309 
104  0.645770  118  76  0.9  80  101 
105  1.000000  42  42  0.5  21  21 
Appendix two: The project used in the paper
function result=baoe(x,i,M,SI,n)
%b is the matrix of the death benefit, i is the expected rate, M and SI are actual rate and variance
%read the data
W=xlsread('D:\survival.xlsx');
p=W(:,3);
A=xlsread('D:\A.xls');
%simulating the n insureds
u=rand(n,1);
X=zeros(n,1);
for j=1:n
k=0;
m=p(x+k,1);
while u(j,1)<m
k=k+1;
m=m*p(x+k,1);
end
X(j,1)=k;
end
%actual rate
Q=normrnd(M,SI,max(X),1);
Q=exp(Q);
I=zeros(max(X),1);
%I is the vector of actual rates
I(1,1)=0.04;
for j=1:max(X)1
I(j+1,1)=Q(j+1,1)*(I(j,1)+1)1;
end
%generate the present value of nyear fixed life insurance
A1=zeros(max(X),1);
q=1p;
for k=1:max(X)
A1(k,1)=q(x+k1,1)/(1+i);
end
%generate net premiums P
P=zeros(max(X),1);
for j=1:max(X)
P(j,1)=i*A(j)/((1+i)* (1A(j)));
end
%generate liability reserve V
V=zeros(max(X)1,1);
for j=1:max(X)1
V(j,1)=A(j+1,1)*(1(P(1,1)/P(j+1,1)))+0.002;
end
V=[0;V];
%b1 change in proportion
b1=ones(n,1);
for j=1:n
b11=1;
for k=1:X(j,1)1
b11=b11*(1+I(k,1))/(1+i);
end
b1(j,1)=b11;
end
%b2 fixed premium
b2=ones(n,1);
for j=1:n
b21=1;
for k=1:X(j,1)1
b21=b21*(V(k,1)+(P(1,1)/b21A1(k,1)))*(1+I(k,1))/((1+i)*V(k,1)+P(1,1)A1(k,1));
end
b2(j,1)=b21;
end
%b3 pay off increasing amount insurance
b3=ones(n,1);
for j=1:n
b31=1;
for k=1:X(j,1)1
b31=(b31(P(1,1)/P(k+1,1)))*(1+I(k,1))/(1+i)+(P(1,1)/P(k+1,1));
end
b3(j,1)=b31;
end
b=[b1,b2,b3];
m=mean(b);
v=var(b);
result=[m;v];
References