گزینه تبدیل در بیمه عمر

This paper introduces an option that has been provided by life insurance companies extensively but has not been discussed in much in the literature; the conversion option. By constructing a valuation model, we first confirm that the conversion option may have positive values. We further find that the value of this option highly depends on the difference of the expected and actual mortality pattern after the insured individual converts his/her policy. Meanwhile, considering the general trend of mortality improvement, we incorporate this trend by applying the Lee–Carter model, hoping to provide a reasonable and fair valuation of the conversion option.

Various prestigious papers have discussed valuation of embedded option in life insurance policies. However, most literature on embedded options in life and pension insurance has so far concentrated on the bonus payment option with a number of different models that have been developed, in order to cover different elements of the life and pension insurance contract (Nordahl, 2008). Meanwhile, there have also been papers discussing valuation of surrender options embedded either in traditional life insurance or equity-linked policies (e.g. Bacinello, 2003, Bacinello, 2005, Bauer et al., 2006, Costabile et al., 2008, Grosen and Jørgensen, 1997 and Grosen and Jørgensen, 2000). Steffensen (2002), defined intervention options as the policyholder’s surrender and paid-up policy options. The paper deals with the simplest form of intervention, which is optimal stopping, so that the valuation of intervention options is based on optimal intervention strategies, giving rise to arbitrage free values. Nordahl (2008) introduced an exchange option that is available in Norway. The paper compares the value of the exchange option with the value of the corresponding American surrender option and with combinations of the two options. This paper focuses on the conversion option, an option that has been provided by life insurance companies extensively but has not been discussed much in the literature. The conversion option is a feature that allows the insured individual to convert his/her original policy into another type of policy before the original policy expires. This type of feature is most commonly seen in term insurance policies. It allows the policyholders to convert his/her original term insurance policy to whole life insurance before the initial policy is due. Policyholders are often encouraged to convert their policies even when their health situation declines. This is reasonable for the insurance company since it may be much less expensive to convert the policy instead of issuing a new one. However, it should be noted that the future cash flow after the conversion might change since the insured individual may not remain in the standard class as he/she did when applying for the initial policy. Thus, we intend to show that the conversion option may be valuable for the insured individual when he/she has the option to convert his/her term insurance policy into a whole life insurance policy. Term insurance provides insurance coverage for a term of one or more years. It generally does not build up cash value so that when the term is due and the insured individual is still alive, the contract terminates and no payments would be made. Whole life insurance instead provides coverage as long as the insured individual is alive and the premium is paid. However, the premium could be several times higher than the amount that the insured individual has to pay for the coverage of term insurance. Generally, for each premium dollar paid by the insured, term insurance offers the largest insurance protection.1 Most term insurance policies are renewable. In practice, insurance companies usually provide a guarantee renewal clause so that the individual is sure to obtain insurance after the initial policy is due. However, evidence of insurability is required and the premium rate is adjusted based on the insured person’s age and physical situation. If the individual’s health declines, it may become difficult for him/her to qualify for the new term insurance policy at standard rates. Even though there is a disadvantage in buying term life insurance, these types of products still remain popular in the market because of the significantly smaller amount of premium compared with those of whole life insurance products. Fig. 1 shows the product market share of US individual life insurance from 1990 to 2004. The figure shows that in the past four to six years, not only variable life product lost a share to universal life, but traditional whole life insurance also lost a share to term insurance. A more recent survey showed that, of new individual life insurance policies purchased in 2006, 41%, or 4 million, were term insurance, totaling $1.3 trillion, or 71%, of the individual life face amount issued.2Suppose that an individual buys a term insurance policy. When the policy is due and he/she is still alive, he/she may wish to continue the coverage. However, if his/her health situation declines, he/she may be charged a higher premium rate or even be rejected by the insurance company. In this case, another option for the individual would be to convert into a whole life insurance policy any time before the original term insurance is due. The main advantage for the individual to convert his/her policy is that the insurance companies generally will not require any evidence of insurability for the conversion and will calculate the new premium according to the age at the issue of the original contract. The payments for converting a policy are different from applying for a renewal. When the policyholder applies for policy conversion, the insurance company converts the policy retrospectively. Based on this concept, at the time of conversion, the individual has to pay the difference of cash value between the original term insurance and converted whole life insurance policy; after conversion, the premium that the insurance company charges will then be set according to the converted policy.3 Besides the pay up of the cash value and the additional premium that will be charged thereafter, there is usually no additional cost for the policyholder to activate this option. To evaluate the conversion option, we first use a static model to show that the conversion option may be valuable when the decline in health situation increases the mortality rate and thus should not be ignored by the insurance company. Under the static mortality model, our simulation results show that, for a 10-year term insurance policy with insured amount US $10,000, the value of the conversion option ranges from $0.34 to $10.30, depending on the initial age insured, the probability that the individual’s health will decline, the impact on mortality when health declines and our assumptions. Next, we take mortality improvement into consideration by applying the Lee–Carter method when deriving mortality. The approach is similar to that of Denuit et al., 2007. As expected, when there is a general trend of mortality improvement, the value of the conversion option will be lower, ranging from $0.13 to $9.95. The remainder of this paper is organized as follows: In Section 2, we set up the valuation model using traditional actuarial notation to illustrate that the conversion option may be valuable. When deriving the stochastic mortality, the Lee–Carter model was applied. Section 3 presents our simulation results and Section 4 concludes the paper.

The literature has identified various options embedded in life insurance contracts and provides thorough analysis and valuation for these options. This paper contributes to the literature by introducing another option that has been provided by life insurance companies extensively but has not been discussed much in the literature, which is the conversion option. By constructing a valuation model, we first confirm that the conversion option may have positive values. In such cases, life insurance companies may lose money from under pricing the actual cost of their life insurance policies. We then evaluate the conversion option using a static model. We find that the value of this option highly depends on the difference of the expected and actual mortality pattern after the insured converts his/her policy. In our model, we illustrate this difference by setting a parameter aa, which is a level impact on mortality for all ages. Furthermore, since there is a general pattern of mortality improvement, we incorporate this trend by applying the Lee–Carter model, hoping to provide a reasonable and fair valuation of the conversion option. In the valuation process, the probability, θx,nθx,n, which describes the tendency that the health status of the insured individual may decline in the future, has been a relatively important factor. In this paper, we derived the probability rates by referring to a data set of life insurance applicants. It may be interesting if we are able to look into more details of how policyholders retain their health status after they buy a life insurance policy. Furthermore, even though our valuation results show that the conversion option may have a positive value, not all policyholders choose to convert their policies. A more thorough discussion into individuals’ behavior regarding their successive insurance purchase decision may be of interest for future studies.