International Journal of Statistical Distributions and Applications
Volume 1, Issue 1, September 2015, Pages: 5-11

Analysis of the Variable Life Insurance Based on Log-Normal Distribution

Shiqi Dong1, *, Shan Pang2

School of Mathematics and Statistics, Central China Normal University, Wuhan, China

(Shiqi Dong)
(Shan Pang)

Shiqi Dong, Shan Pang. Analysis of the Variable Life Insurance Based on Log-Normal Distribution. International Journal of Statistical Distributions and Applications. Vol. 1, No. 1, 2015, pp. 5-11. 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 investment-oriented 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 sub-accounts 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.

Figure 1. The investment-oriented policy.

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 175-243).

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 (2000-2003), CL1.

Suppose that the ratio of two adjacent periods’ rate is a log-normal 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.

Table 1. The mean and variance of each approach after simulating 10000 times.

 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 (2000-2003), CL1

 China Life Insurance Mortality Table (2000-2003), CL1 age mortality Survival number death toll Life Expectancy Survival person-year 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

p=W(:,3);

%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 n-year fixed life insurance

A1=zeros(max(X),1);

q=1-p;

for k=1:max(X)

A1(k,1)=q(x+k-1,1)/(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)* (1-A(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=ones(n,1);

for j=1:n

b21=1;

for k=1:X(j,1)-1

b21=b21*(V(k,1)+(P(1,1)/b21-A1(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

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2. Duncan, R. M. (1952). A retirement system granting unit annuities and investing in equities. Transactions of the Society of Actuaries, 4(9): 317-344.
3. Fraser, J. C., Miller, W. N., & Sternhell, C. M. (1969). Analysis of basic actuarial theory for fixed premium variable benefit life insurance. publisher not identified.
4. Miller, W. N. (1971). Variable Life Insurance Product Design.Journal of Risk and Insurance,527-542.
5. China Association of Actuaries. (2010). Life insurance actuarial, 1-155. China Financial and Economic Publishing House.
6. Li Xianping. (2010). Foundations of Probability Theory, 3rd edition(In Chinese). Higher Education Press.
7. Kenneth Black, Harold D. Skipper. (1994). Life and Health Insurance, 13th Edition. Prentice Hall Press.
8. Del Moral P, Doucet A, Jasra A. (2006). Sequential Monte Carlo samplers. Journal of theRoyal Statistical Society, Series B, 68(3): 411-436.
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11. Liu Jiazi, Jia Ke. (2010). The comparison of Coverage of variable life insurance in two different actuarial methods. Journal of Insurance Professional College (Bimonthly), 48-54.

 Contents 1. 2. 3. 3.1. 3.2. 3.3. 4.
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