4. Numerical Illustrations
4.1 The optimal strategy with γ=δ
國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
34
changes of mortality rate and will be equal to the realized reserve. As the realized reserve remains the same as the expected reserve with the changes of mortality rate, the risk is immunized within the policy.
4. Numerical Illustrations
We provide two strategies of risk mitigation: the optimal strategy and the secondary strategy in product design whether is appropriate to the condition of insurance providers. The optimal strategy can minimize the risk emerged from insurance products. The secondary strategy is used when the optimal strategy is not available or for other risk managements purpose.
4.1 The optimal strategy with γ=δ
To optimize the strategy of natural hedging in product design, we apply our theoretical derivation to keep the γ in consistent with the force of interest rate δ as much as possible. For the timing respective to the most update information of the force of interest rate δ, in determining the force of amount γ, we can separate into two different timing categories in product design, one is pre-determinate and the other is post-determinate. The pre-determined category is that we design the product and determine the force of amount before we get the information of the in-time interest rate in real time. Thus, the post-determined category is the product design in determining the force of amount γ, which is determined immediately after we know correspondent the interest rate in real time. We also provide the effects according to stochastic interest rate model and stochastic mortality rate model.
4.1.1 The Pre-determined Category
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
35
The typical product in this category is increasing whole life insurance with a general increasing amount on death benefit.
The basic assumptions are set up in Table 2. We assume that the face amount is US$100,000 for the specified increasing whole life insurance and the premium is paid by single premium. Assume that the company only sells the specified increasing whole life insurance product with given value of the force of interest rate δ. The natural hedging strategy depends on the product design of the force of amount γ. The γ can be any given value in a reasonable risk mitigation strategy.
Table 2 Basic Assumptions for the New Form of the Whole Life Insurance Product
Age of insured 25, 45
Gender Male
Face amount 100,000
The initial value of force of interest rate (δ) 4%
Death benefit 100,000 compounded by γ(t)
Benefit period Whole life
Method of paying premium Single premium
We first investigate the case with the value of γ equal to the value of δ. Our pricing mortality rate is based on the LC model. We take the 5th policy year as an example to examine the mortality rate risk which is the difference of the realized reserve and the expected reserve. The realized reserve is evaluated by 20% up shock or 20% down shock of the mortality rate. Keeping the setting of δ and γ to satisfy the equation (δ – γ) = 0 in all time, we illustrate three different effects of the designed product as the following situations.
With γ = δ = constant number and fixed in all time
Product design 1: traditional increasing whole life insurance.
‧
compound at γ continuously and it is indicated as the following equationFt = F0 exp(γt),
where F0 is the face amount, t is the policy year. The outcome is shown in the Table 3.
Table 3 The Liability at the End of the 5th Policy Year of Illustrated Insurance Product for Different Mortality Bases
γ(=
4%)
= δ(=4%)
analysis. Thus, the mortality rate risk of the specified product is none since the risk exposure does not increase or decrease caused by the 20% shock at the 5th policy year.The specified whole life insurance product appears no mortality risk or longevity risk in all ages.
The product design 1 is a traditional increasing whole life insurance. If the interest rate keeps the same as the force of amount γ, the product fulfills the criterion δ= γ and there is no mortality rate risk at all. While in reality, the interest rate is not
fixed along with the whole policy years, the product may be exposed to risks.
4.1.2 The Post-determined Category
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
37
Product Design 2: The type of product is an interest rate variable whole life insurance.
In order to keep the criterion of γ(t) = δ(t) during the same time period, t =1, 2…etc. We decide the γ(t) immediately after we obtain a new δ(t) in the market each time. We let γ(t) as close to δ(t) as possible. The δ(t) is piecewise continuously along with time t and so is the γ(t).We assume δ(1) = δ, γ(1) = δ(1) for the first policy year. From the second policy year on, we declare a new interest rate at the beginning of each policy year that generates a new force of interest rate δ(t). Let γ(t)
= γt = δ(t) for all t. We obtain the indication of the death benefit for this kind of policies is
0 ( 0 )
Ft F (s)
t
xp ds
e
.The interest rate is up-and-down at each policy year. We assume the realized interest rate for the 2nd to 5th policy year, then the values of the force of interest rate for the first five policy years are as shown in Table 4.
Table 4 The Realized Values of the Force of Interest Rate for the First Five Policy Years.
Policy Year 1 2 3 4 5
Interest rate A δ=4% δ(2)=4.25% δ(3)=4.50% δ(4)=4.50% δ(5)=4.75%
Interest rate B δ=4% δ(2)=3.75% δ(3)=3.75% δ(4)=3.50% δ(5)=3%
The outcome of product design 2 is shown in Table 5. We can see that the values of reserve are not changed by the shock of mortality rate in Table 5. The mortality risk and longevity risk indicated in column (4) and column (5), respectively, are shown no risk by the changes of mortality rate because the total mortality rate risk
‧
is immunized within the policy.
Table 5 The Liability at the End of the 5th Policy Year of Product Design 2 for Different Mortality Bases benefit (i.e. dividend or increment of death benefit) that depends on the declaration of interest rate each policy year. The interest rate variable life insurance in the United States is one of the kinds in this product group. Our product design is based on the interest rate variable life insurance and keeping the criterion of γ(t)=δ(t). The difference from that the interest rate variable insurance product provides dividend to policyholders, the product 2 provides the increment of death benefit. Each increment of death benefit in our product should be the same as the dividend of interest rate variable insurance product generated by the declared interest rate. As long as the product meets the criterion of γ(t)=δ(t), whether the extra benefit is called either the increment of death benefit in our design or the dividend in the content of interest rate variable life insurance, it does not matter the achievement of risk mitigation. In our product design here, we take the extra death benefit increased by
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
39
each declared interest rate and provide no dividend. As to fulfill the criteria, the design of interest rate variable life insurance should take death benefit and dividends into account on assessment of risk. Since the interest rate is declared in related to market interest rate, the type of product declines the threat of interest rate risk comparing to the product with fixed interest rate.
Product Design 3: The interest rate variable increasing whole life insurance.
The idea of this product is similar to Product design 2 that the decision of γ(t) is soon after the newest δ(t) we can get in the market. Let γ(t) = γ0 +Δγt= δ(t), during the same time period, t =1, 2…etc. Set the initial value of the force of amount γ0 = 2% < δ = 4% at time 0. Let δ(1) = δ, γ(1) = γ0 +Δγ1 = δ(1) in the first policy year.
From the 2nd policy year on, declare a new interest rate in each following policy year that generate a new force of interest rate δ(t). As γ(t) = γ0 + Δγt = δ(t), the death benefit is
0 0
0 ( ) 0 ( )
Ft F F (s)
exp t exp t ds
.The product is a combination of product 1 and product 2. The product contains a fixed increment death benefit that makes this part of product looks like a traditional increasing whole life and a variant increment death benefit that makes this part of product seems like an interest rate variable life insurance. The variant increment death benefit is like product design 2 determined by the declaration of interest rate.
We assume the realized interest rate for the 2nd to 5th policy year. The values of the force of interest rate for the first five policy years are as shown in Table 6.
‧
Table 6 The Realized Values of the Force of Interest Rate for the First Five Policy Years of reserve are not changed by the shock of mortality rate in Table 7. The mortality risk and longevity risk indicated in column (4) and column (5), respectively, are shown no risk by the changes of mortality rate because the total mortality rate risk is immunized within the policy.
Table 7 The Liability at the End of the 5th Policy Year of Product 3 for Different Mortality Bases