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
To cite this article:
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
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.
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
According to the recurrence formula of liability reserve funds, we have
Divide(2.1)by(2.2), and we can get
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
Divide(2.4)by(2.5), we can get
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
According to the formula of liability reserves,
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 .
according to the principle of , we obtain
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|
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.
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|
Appendix two: The project used in the paper
%b is the matrix of the death benefit, i is the expected rate, M and SI are actual rate and variance
%read the data
%simulating the n insureds
%I is the vector of actual rates
%generate the present value of n-year fixed life insurance
%generate net premiums P
%generate liability reserve V
%b1 change in proportion
%b2 fixed premium
%b3 pay off increasing amount insurance