國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
12
1. Introduction
In 2003, the increasing whole life insurance product was sold prosperously in Taiwan. The increasing whole life insurance product was developed in early stage in the history of Taiwan life insurance industry. The kind with death benefit compounded by high increasing rate stood out in the market with popularity. Since the product was designed against the currency inflation in increasing its death benefit by compound increasing rate, a lot of consumers brought this kind of compound increasing rate whole life insurance products at the time.
From the point of view of the life insurance companies, the kind of compound increasing rate whole life insurance products can catch the eye of consumers through the selling technique of salespersons. The life insurance company may quickly collect the premium income that makes income statement look great. The most important thing is that the company may take advantage of the collecting cash and financial leverage technique to optimize their investment strategy. For a life insurance sells an insurance product is not just a product sold, the main reason for them is to increase its premium income that let them obtain the right on the usage of money from the insurance customers. This is not a news from the life insurance industry and easily to understand why the life insurance companies try very hard in product design in order to attract customers. A branch company1 of global life insurance group in Taiwan developed and sold first this kind of compound increasing rate whole life insurance products in 2003. As they wish, the growth of premium income of the life insurance company climbs sharply.
1 The Aegon Life Insurance Company in Taiwan was the primary company that developed the first product of the kind of compound increasing rate whole life insurance products in 2003
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
13
At the same time, there are two to three life insurance companies selling the kind of compound increasing rate whole life insurance products in the market. On the other hand, most companies in the market state in a cautious way though the product may carry in a lot of premium income. They did not follow the trend to sell the kind of compound increasing rate whole life insurance products because they did not know the risk very well with the kind of the products. They cannot explain why both of the payoff for the buyers and the commission for the salespersons of products are better than other kinds of insurance products and then who is the loser in this business.
Most people would ask “What kind of risk of the product is exposed to?” “How much is the risk?” or “Is the risk underestimated?” Unfortunately, the actuaries of those companies selling the compound increasing rate whole life insurance hardly know the risk and don’t know how to manage such kinds of life insurance products.
In 2004, Taiwan Insurance Institute initiates a project on the study of this kind of increasing whole life products. They took an example of this kind of compound increasing rate whole life insurance products in analysis that may reveal some exotic phenomenon of the product. The background information is in Table 1 and the cash flow testing of the product is shown in the Figure 1. The researchers at the time discovered that the severity of insolvency existed in the kind of products.
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
14
Table 1 Background Information for Cash Flow Testing of a Popular Compound Increasing Rate Whole Life Insurance Product
Age of insured 35
Gender Male
Face amount 200,000
The initial value of force of interest rate (δ) 2%
Death benefit 200,000 compounded by 3% and 6%
Benefit period Whole life
Method of paying premium 20-year period installment premium
Investment Return 5%
We can see the cash flows of the product are downward sharply on the latter age of the policy year. That is to say that the risk is exposed to the stage of older age of the policyholder. If the policy lasts longer, the risk of this policy is getting severer.
This is unusual to a life insurance product. Normally, the risk of a life insurance Figure 1 Cash flows of the Popular Increasing Whole Life Insurance, with δ =
2% and γ = 3% during the premium payment period and 6%
afterwards.
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
15
product happens in the early policy years of the life insurance contracts. That is why the underwriting of life insurance is very important in order to prevent the early payout of death benefit of life insurance. But the risk profile of the kind of compound increasing rate life insurance seems not like that of a typical life insurance product. It is getting curious for all the actuaries in solving the puzzle in the life insurance industry.
First of all, we focus on what the risk for the kind of insurance product is. If it is not exposed to the mortality risk, then we want to find out what risk it should be and how much it is exposed to. And, moreover, we try to prove that the increasing rate is important element in related to the risk navigation and try to come out a strategy in management the risk for the kind of the product in the different alternatives.
In this article we find out the mystery of this kind of compound increasing rate whole life insurance product. It is the product with compound increasing rate higher than preset interest rate of the product. We can see it is reasonable as it corresponds to the theoretical results that we shows in the section 2 “Theoretical Development”:
when the (δ – γ) < 0, the product is exposed to the longevity risk. It shows that the company did not evaluate the product properly so as to collect not enough premiums and had to depend on a good return in the investment to cover the shortage in cash, the insolvency of the product would happen in decades later. And the illustration is presented under the condition that the company did not distribute the earnings during the whole policy year. We can see the kind of products without properly evaluated that lay the company in dangerous financial situation. We know the risk of the kind of the product and we can go further to deliver an optimal strategy in the product design according to its risk appetite.
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
16
The remainder of this article is organized as follows. In “Literature Review”, we study most papers of the subjects of risk mitigation and risk management in the previous researches. In the section “Theoretical Development”, we deduce and elaborate our theoretical development for a natural hedging strategy using the Laplace transform that we generate a specified life insurance. In the next section “Numerical Analysis” we use an existing model with U.S. mortality experience and demonstrate how to implement our proposed strategies to assess the natural hedging effect with numerical examples in the section. We extend our analysis to the cases of annuity product in the next section, and then we conclude in the last section.