Practice and Conclusion

國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

42

longevity risk and the longevity risk is less than the mortality risk. The product in Panel B is exposed to mortality risk only. When the mortality rate is changed by 20% shock, the expected reserve of both products is changed. With increase of reserve as indicated in column 4 with respect to 20% up shock of mortality rate, product in Panel A is exposed to risk of 3% ~ 4% more than that in basis mortality rate. And the product in Panel B is exposed to that of 6 ~ 8% more than that in basis mortality rate by the changes of mortality rate. The secondary strategy is to diminish the risk within the policy by creating longevity risk in a life insurance product to lower the exposure of mortality risk and the strategy helps us to design such products upon the demand of risk exposure.

5. Practice and Conclusion

The evolution of the kind of compound increasing rate whole life insurances may generally be divided into two stages. The first stage is selling those products that the force of amount (i.e. increasing rate of the death benefit, γ) is much bigger than the force of interest rate (i.e. δ) of the policy from 2004~2009. And the second stage is selling the products that the force of amount (i.e. γ) is mostly equal to the force of interest rate (i.e. δ) of the policy from 2009 till now.

In 2009, the Financial Supervisory Authority of Taiwan mandated that increasing rate designed in the product is forbidden higher than the interest rate of the product.

Thus, the life companies did not present the increasing whole life product with overly high increasing rate in death benefit ever since. Furthermore, the companies have known the compound increasing whole life insurance product better than before.

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

43

From 2009 until now is the second stage of the product, almost every insurance company in the market designs and sells such compound increasing whole life insurance product. In the perspective of the insurance companies, the product is quick money collecting product that they have soon accumulated the premium income by selling such compound increasing whole life following the mandate of authority that meets the criterion of (δ – γ) = 0, which is optimal strategy in risk mitigation.

The product is a tool in accumulating richness for the companies, needless in complicated underwriting process but charging the highest penalty in the condition of policy withdrawal.

From the viewpoint of a buyer of the kind of insurance products, the compound increasing whole life insurance product is an insurance product that may resist the currency inflation and is convenient in saving without renewal, like a long-term bond accumulated the coupons until the day of death. The best of all, the death benefit income is tax-free without upper limit at amount of death benefit in Taiwan. The drawback of the product is the penalty charge of the policy withdrawal that is 25% of the policy value and it is way too much in charge for an insurance policy.

After the theoretical development and numerical analysis in the article, we are surely that the compound increasing rate whole life insurance product developing in the first stage is exposed to the longevity risk which is systemic risk along with the lifelong of the population. While in the second stage, the compound increasing rate whole life insurance product is limit in the criterion of (δ ≧ γ) which means the increasing rate should be less than the interest rate of the product. Under the condition of (δ ≧ γ), the compound increasing whole insurance is exposed to the mortality risk that is more controllable than longevity risk.

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

44

Besides that we solve the mystery for the compound increasing rate whole life insurance product, we develop the risk mitigation in product design. We discover that natural hedging strategy through product design is an important step that we can immunize/mitigate the mortality rate risk within a policy. The key point is that we should take the risk rooting in “when to pay” and the risk rooting in “how much to pay” into consideration in risk mitigation technique. We introduce a factor γ, the force of amount (i.e. the variable of increasing rate in death benefit), as a risk factor rooting in “how much to pay” in the product design.

We utilize the γ to design a death benefit protection life insurance that the mortality rate risk is immunized. We deduce the optimal natural hedging strategy within a policy with the criterion of settings on γ as to let γ = δ. When the γ is equal to δ, the specified product appears no risk with the changes of mortality rate. The mortality rate risk is immunized in such a kind of products with death benefit protection in all ages. When the γ is not equal to δ, the strategy is used to diminish the risk rather than to immunize the risk. If δ > γ, the death benefit protection product is exposed to mortality risk rather than longevity risk with the changes of mortality rate and the life insurance product with settings of γ is less exposed to mortality risk than the one without settings of γ.

Following the optimal strategy in the product design, the insurance product has a lot more possibility in engaging to financial instrument without considering the mortality rate risk. Our finding can also provide a further research or re-design of previous studies that ignore the existence of a mortality rate risk in an insurance product valuation or risk management consideration.

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

45

Essay II :

The Uncertainty to Optimal Strategy

in Life Insurance Product design

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

46

1. Introduction

The dynamic pattern of mortality risk is focusing on a sudden mortality rate shock. Those risks are not only a risk to the product providers but also a costly problem to economic society. The technique of risk mitigation in risks is an important subject for insurance providers and supervisors.

Comparing to the mortality risk, longevity risk catches more attention owing to its persistent trend. And, many studies start with the longevity risk. Transferring longevity risk externally with the financial vehicles of capital market is the earliest suggestion in literatures to solve the problem. Blake and Burrows (2001) propose a solution accordingly through capital market by issuing survivor bonds to mitigate the longevity risk exposed to living benefit providers. Other prior studies also provide mitigation solutions through capital market, including survivor bonds (Denuit, Devolder, and Goderniaux, 2007), survivor swaps (e.g. Dowd, Blake, Cairns, and Dawson, 2006), mortality swaps (Lin and Cox, 2007), mortality securitization (e.g.

Dowd, 2003; Lin and Cox, 2005; Cairns, Blake, and Dowd, 2006a; Blake, Cairns, and Dowd, 2006; Cox, Lin, and Wang, 2006; Blake, Cairns, and Dowd, 2006; Blake, Cairns, Dowd, and MacMinn, 2006). These studies suggest mitigating longevity risk by transferring the risk to the investors of capital market. Transferring the risk externally may be a possible way but may not diminish the primary risk rooted in the policy. These methods are also involved with uncertainty in market environment and transaction cost to the providers.

Among the strategies of mitigating longevity risk, natural hedging is a choice strategy in insurance companies internally. Companies have advantage to diminish the threat of longevity risk through selling life insurance products. Cox and Lin

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

47

(2007) create a product portfolio with a life insurance and an annuity product. They show the existence of natural hedging effect between life insurance and annuity product. They also propose a mitigating strategy of issuing mortality swaps for insurers and pension providers as they don’t think the internal hedging is practical in existing business.

Wang et al. (2010) find that the natural hedging strategy is applicable in insurance industry. They propose an immunization model to achieve the optimal life insurance–annuity product mix ratio to hedge against longevity risks. They apply the hedging effect to the liability of product mix and utilize mortality durations and convexities to present the optimal ratio by the impact of mortality changes.

Mortality duration/convexity is the most common used technique in determining optimal natural hedging strategy. Most recent studies adopt the duration/convexity matching to the price of life insurance or annuity with mortality changed proportionally or constantly( Tsai and Chung, 2013; Lin and Tsai, 2013).They extend the duration/convexity matching application to determine the weights of two or three products

Taking higher-order moments of the mortality risk distribution into consideration, Tsai, Wang, Tzeng (2010) instead propose Conditional Value-at-Risk Minimization (CVaRM) approach. The model generates a narrower quantile of loss distribution after hedging. They show CVaRM approach is better off on risk reduction in comparison to immunization model (Wang et al., 2010). Unlike previous studies, Wang, Huang, and Hong (2013) employ an experience mortality data set that allows them to build up mortality rate of life insurance and annuity separately. They consider variance and mispricing effects of longevity risk at the same time and address a natural hedging strategy practically.

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

48

To enhance the accuracy of projection of mortality movement, the field of studies addresses the methodology in modeling longevity risk. Many studies involve in stochastic mortality approach (Marceau and Gaillardetz, 1999; Wilkie, Waters, and Yang, 2003; Cairns, Blake, and Dowd, 2006a, b). The model proposed by Lee and Carter (1992) is the most commonly used method in mortality projection. More recently, the CBD model is another widely used model proposed by Cairns et al.

(2006). These models facilitate the forecasting of longevity risk in determining mitigating risk strategy.

In Essay I, we find a way of natural hedging strategy within a policy that we can immunize/mitigate the mortality rate risk through delicate product design. We are able to catch the risk as much as we need in a simply product design and have not to build up a weighted portfolio of life insurance and annuity.

To be close to the reality, we attempt to utilize the optimal strategy for the product design based on the future of uncertainty in this Essay. We build up the stochastic models of mortality rate and interest rate. We use Lee-Carter model on mortality rate and CIR model on interest rate model. The uncertainty of mortality and interest rate may catch by the two most used models in literatures. And, in fact of the numerical analysis of the illustrated product, the optimal strategy of product design keeps its optimism and minimizes the risk of the product through appropriate product design.

The remainder of this article is organized as follows. In the section

“Theoretical Development” that we apply the theoretical development in the Essay I for a natural hedging strategy. In the next section “Models”, we build up a LC mortality model and CTE interest rate model. In the “Numerical Analysis” we use

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

49

the models with U.S. mortality data and to get evidence of risk mitigation that we implement the proposed strategies with numerical examples in the section.

2. Theoretical Development

In Essay I, for a traditional whole life insurance product, the net single premium with 1- unit face amount expressed by the notations of Bowers et al. (1997) can also be expressed in Laplace transform format as:

   

0 ^s

( )

_{s x x s}

( ) ( )

x

b s p b s x s

e ^

^

ds

f  ^

 

   

 ^L

^{, (1)}

where δ is the force of interest, b(s) is the death benefit at time s per 1- unit face amount, fx(s) is the probability density function of the future lifetime random variable S at age x: spx



x+s, in which μx+s is the force of mortality at age of x+s and spx denotes the probability of a person at age x who will survive s years.

Note that b(s)=1 for traditional whole life insurance, which means that the death benefit is fixed at the face amount of the policy. In such a case, equation (1) implies that the net single premium is a function of the force of interest δ with respect to fx(s).

Should mortality rise unexpectedly, the collected premium has insufficient time to accumulate to the death benefit; on the other hand, the policyholders as a whole pay too much if mortality improves more than expected. The product makes the insurer and insured subject to the mortality risk. Can we mitigate such risk through product design?

One design is to make the death benefit an increasing function of death time so that the accumulated value of the premium can match the death benefit no matter how mortality varies. The Laplace transform format sheds light on a possible solution:

‧

The product implied by equation (2) is an increasing whole life insurance policy.

Its death benefit increases continuously at the annual rate of

. The parameter γ

controls how much to pay while δ would reflect the time value of payment.

Appropriate choices on (,δ), such as  δ, can make the present value of the death benefit insensitive to the timing of death that in turn is affected by mortality. Such design can thus mitigate the mortality risk.

We elaborate the above idea by examining the expected reserve of the appropriately calibrated whole life insurance product to see whether it can be immunized from the mortality risk by itself. When a new policy of the calibrated product is sold now (i.e., at time 0) to a customer at age x with face amount 1 (without loss of generality), the expected reserve at time t of the policy is as in equation (3):

0

t s s

t

V

x

 

^

e e e

^{  } s

p

x t_



x t s_{ }

ds

^{. (3)} In Essay I, we know that the derivation of reserves is

 

Then we have the following boundary conditions of the expected reserves:

1. when μx+t+δ–γ> 0 and _tVx

‧

The equation of the boundary can be written as:

 ^

^{x t}  

^{ }  

^t

V

^x

e

^t



 

e

^t

^{  }



^

. (5)

When narrowing down the possibility of (δ – γ) in the four conditions we find out the boundary reserve is equal to the expected reserve if (δ – γ) = 0. When (δ – γ) is equal to 0, the lower/upper bound of the expected reserve is folded together with the upper/lower bound. The feature of the special case with (δ – γ) = 0 turns into a horizontal line as shown in Figure 8. It is the optimal strategy when (δ – γ) = 0.

Figure 8 The boundary reserve of optimal strategy on (μx+t, tVx) plane, with δ

=2.5% and γ=2.5% in the case

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

52

In Essay I we then transform the horizontal axis in Figure 8 to age x+t and display the case of (δ – γ) = 0 on (x+t, tVx) plane as shown in Figure 9. The upper bound is overlapped with the lower bound of the expected reserve. The feature of the boundary reserve is a horizontal line that is the same as in Figure 8. We thus come out an only solution that fulfills the criteria of the four conditions of equation (5) represented the relationship between expected reserve and age.

When we design such a specified product with (δ – γ) =0, it is the optimal strategy that the mortality risk is mitigated totally by the longevity risk for each age in the case.

Thus, the expected reserve is flat to all ages and to the force of mortality of each age.

In the Essay II, we furthermore examine the expected reserve of such products Figure 9 The boundary reserve of optimal strategy on (x+t , tVx) plane , with δ

=2.5% and γ=2.5% in the case

Note that the illustration is using the Makeham model _{9.566 10}⁴ _{+ 5.162 10}⁵ _1.09369^{x t}

x t_ ^{      ,}

which is cited from Melnikov and Romaniuk (2006) and the original data is based on the mortality rates from 1959 to 1999 in American (Pollard, 1973)

Upper bound=Lower bound

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

53

will not be affected by the changes of mortality rate by generated morality rate model and interest rate model. Does the realized reserve remain the same as the expected reserve with the changes of mortality rate by uncertainty?

3. Models

3.1 Model for Mortality Rate

In the last section, we take advantage of an example of mortality model from previous studies to quickly illustrate the graphic figures to the relationship between ages and boundary reserves. In order to close to the reality of mortality nowadays, we update the mortality data and build up a mortality model based on the Lee-Carter mortality model (Lee and Carter 1992), which is one of the most used models in the literatures. We first define the notation as follows.

 μ(x,T) denotes the force of mortality for age x and time T.

 qx(T) denotes the probability that an individual aged x in year T dies before reaching age x+1, where x is an integer number.

 px (T) is the probability that an individual aged x in year T survives one year.

 npx (T) is the probability that an individual aged x in year t survives n years,

And, the relation of px (T) and qx(T) is that px (T) = 1 - qx(T). Furthermore, the relation of px (T) and npx (T) is

npx (T) = px (t) px+1 (T+1) · · ·px+n-1 (T+n-1).

m(x,T) denotes the central death rate, which is defined as a weighted average of

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

54

μ(x + s, T + s) for s ∈ [0, 1). We assume that the force of mortality within age interval

[x, x+1). Thus, we may say that m(x,T) =μ(x,T), and

px (T) = exp [−μ(x, T )] = exp [−m(x,T)].

3.1.1 The Lee-Carter (LC) Mortality Rate Model

Lee and Carter (1992) suggest the following mortality model for the central death rate m(x,T) for an individual aged x at time T.

log m(x,T) = αx + βx kT + εxT, (6)

where the parameter αx describes the average age-specific mortality; kT is the time-varying index representing the general mortality level changes; βx describes the age response to kT and shows the decline in mortality at age x. The term εxT is the model error and is white noise with zero and relatively small variance (R. D. Lee, 2000). When we determine the parameters, we can forecast the age-specific mortality rates by estimated αx , βx, and kT.

First of all, we adopt a random walk with drift process to model the time-varying index kT as following expression.

kT = kT-1 + c + eT with i.i.d. eT ∼ N(0, σe2), (7) where drift term c is normally a negative number, indicating the trend of decline in the mortality rate.

Before we are able to predict the future mortality rate, we first estimate the past value for αx , βx, and kT from mortality data. To better predict the future stochastic mortality, we gauge the parameters in the LC model by fitting historical mortality data.

‧

time-varying index at future time T then follows the dynamics:

0 fitting historical U.S. male mortality data from1961 to 2010 of the HMD data.

3.1.2 The Parameters of LC Model

We use the U.S. male mortality data from 1961 to 2010 and maximum likelihood estimation to calibrate the mortality models (Brouhns et al. 2002b; Cairns et al. 2009).

We apply the age range from 0 to 110. The key parameter estimates for the LC model of representative age 25 and 45 are given in Table 9.

Table 9 The Parameters of the LC model

Gender Age αx βx

kT

c k0 σe

M 25 -6.39977 0.0075668 -1.066826 -30.55546 1.53409 45 -5.36876 0.0111848 -1.066826 -30.78606 1.53409 F 25 -7.39813 0.0102035 -1.063359 -18.48581 1.22860 45 -5.95441 0.0116959 -1.063359 -21.47925 1.22860

7 The web address of the human mortality database is http://www.mortality.org/.

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

56

3.2 Model for Interest Rate

We adopt the short rate modeling framework where the short rate rt is the driving force of interest rate risk. Thus, we consider the Cox–Ingersoll–Ross model (or CIR model) in our analysis.

3.2.1 The Cox–Ingersoll–Ross (CIR) Interest Rate Model

The model describes the dynamics of short rate on the probability space as the follows.

(

_t

)

_t

t

a b dt

t

dr   r   r dW

(10)

where a, b, and σ are the parameters. And Wt is a Wiener process.

The parameter a is speed of adjustment, b is the long-term mean and σ, is volatility. The a(b-rt) is the drift term in the model. It ensures mean reversion of the interest rate towards the long run value b, with speed of adjustment governed by the positive parameter a.

The standard deviation factor,

 r

t, avoids the possibility of negative interest rates under the condition of all positive values of a and b.

3.2.2 The Parameters of CIR Model

We apply the result of parameter estimated from Liu (2013), using interest rate data from the mid-1960s to the early 1990s obtained by various methods⁸ (see Chan

8 For the limitation of data searching, we are not able to reach appropriate resources of the interest rate data. The parameters are so important that we cite the article that provide the result for us ready to use. It will not bias the theoretical derivation due to the model deviation.

‧ 國

立 政 治 大 學

‧

N a tio na

l C h engchi U ni ve rs it y

57

et al. 1992; Duan and Simonato 1999, and references therein). The parameters are as the following Table 10.

Table 10 The Parameters of CIR Model

a b σr

CIR model parameters* 0.25 0.065 0.07

*parameters applied from Liu (2013)

4. Numerical Illustrations for Uncertainty

We use the same strategies of risk mitigation in Essay I: the optimal strategy and

We use the same strategies of risk mitigation in Essay I: the optimal strategy and

在文檔中 死亡風險的自然避險與商品設計 - 政大學術集成 (頁 42-0)