Essay II : The Uncertainty to Optimal Strategy in Life Insurance
2. Theoretical Development for Annuity Products
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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
dsf
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:
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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
sp
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
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The equation of the boundary can be written as:
x t
tV
xe
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
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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 9.566 104 + 5.162 105 1.09369x 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
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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
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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.
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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/.
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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)
tt
a b dt
tdr 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.
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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 the secondary strategy in product design whether it will also meet the conditions of insurance providers according to the situation of uncertainty. We can obtain the optimal strategy can minimize the risk emerged from insurance products in the cases of future conditions. The secondary strategy is used if the optimal strategy is not the considering risk mitigation purpose in the view of overall framework of risk management.
4.1 The optimal strategy with γ(t) =δ(t)
In the case of optimal strategy of natural hedging in product design, we apply our theoretical derivation to keep the γ(t) in seeming consistent with the force of interest rate δ(t) as possible. The pre-determined category and the post-determined category, as we defined in Essay I are the product design in determining the timing of the force of amount γ(t), which is determined immediately before / after we know the realized interest rate in real time. In the essay, we assess the effects in a way of stochastic interest rate model and stochastic mortality rate model.
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4.1.1 The Pre-determined Category
We take the traditional increasing whole life product as our illustrating product in this category with uncertain increasing amount on death benefit that is determined by the realized interest rate.
The basic assumptions are set up in Table 11. We assume that the face amount is US$100,000 for this increasing whole life insurance and the premium is paid by single premium as same as in the Essay I. We also assume that the company only sells this increasing whole life insurance product with the realized value of the force of interest rate δ(t). The natural hedging effect depends on the product design of the force of amount γ(t). The γ(t) can be decided in any value under risk mitigation strategy of the company.
Table 11 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) 6.5%
Death benefit 100,000 compounded by γ(t)
Benefit period Whole life
Method of paying premium Single premium
We try to fit the criteria of optimal strategy of the value of γ(t) close to the value of δ(t) as possible. 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
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setting of δ(t) and γ(t) to minimize the equation (δ – γ) as possible, we illustrate two designed product as in the Essay I.
With the force of interest rate δ obtained by CIR model
Product design 1: traditional increasing whole life insurance.
Let γ(t) = δ(t) during the same time period through the whole policy year. The
Case I: Apply LC model without stochastic process on mortality rate forecasting To capture the risks in reality at the beginning of the policy, we establish an interest rate model of one factor CIR9 model and mortality rate model of LC10 (Lee
& Carter, 1992) model. In this case, we first want to consider the dynamic of interest rate in the whole policy years. Thus, we forecast the mortality rate by LC model without volatility of mortality rate. We remain γ(t) constant in the product design and consider the risk in giving the guarantee of γ(t). We obtain 10000 results with 10,000 simulations of interest rates and guarantee. The product design 1 of c + eT. The data is US mortality rates from 1961 to 2010 obtained on www.mortality.org. The parameter estimates for LC model are: c = -1.131556, σe = 1.26965.
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Table 12 The Liability at the End of the 5th Policy Year of Illustrated Insurance Product with Stochastic Interest Rate Model for Different Mortality Bases
(1) (2) (3) (4)=[(2)-(1)]
/(1)
(5)=[(3)-(1)]
/(1)
Age/
Gender Premium Basis
Mortality
In Table 12, we can see that after the shock of mortality rate, the reserve changed in Panel A is less than 1%. This outcome is consistent with our theoretical strategy that we keep the criterion of (δ – γ) as small as possible. In Table 12, we display that the effect of risk mitigation in Panel A is better off. The strategy of the product design 1 in Panel A is not exactly keeping γ(t) = δ(t), but it still better than the results with γ(t)=0 as in the Panel B. It shows that the risk exposure in the Panel A does not that much increase or decrease caused by the 20% shock at the 5th policy year. This increasing whole life insurance of product design 1 demonstrates much less mortality risk or longevity risk in all ages.
Case II: Apply LC model with stochastic process on mortality rate forecasting
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interest rate model of one factor CIR model and mortality rate model of LC model to simulate both the dynamic of interest rate and mortality in the whole policy years.We realize the interest rate and mortality rate by the models to catch the volatile in the dynamic process. The γ(t) is constant in the product design. We deliver the results with the guarantee of γ(t).All outcomes are obtained with 10,000 simulations. The product design 1 of dynamic interest rate δ(t) with a constant long term mean that is equal to γ(t). We display the results of γ(t) = 6.5% and the comparison panel of γ(t)
= 0% as shown in the Table 13.
Table 13 The Liability at the End of the 5th Policy Year of Illustrated Insurance Product with Stochastic Interest Rate and Stochastic Mortality Rate for Different Mortality Bases
Gender Premium Basis
Mortality
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As we can see in Table 13, after the shock of mortality rate, the reserve changed in Panel A is less than 1%. The result is stick with the criterion of (δ – γ) = 0 as possible. In Table 13, that shows mitigation of risk in Panel A is better than those in Panel B. Even though that the product design 1 in Panel A is not exactly keeping γ(t)
= δ(t) point to point, it still better off the results with γ(t)=0. The risk exposure in the Panel A appears not that much increase or decrease caused by the 20% shock at the 5th policy year. The specified whole life insurance of product design 1 shows much less mortality risk or longevity risk in all ages.
4.1.2 The Post-determined Category
Product Design 2: The type of product is an interest rate variable whole life insurance.
We keep the criterion of γ(t) = δ(t) during the same time period, t =1, 2…etc.
And then we decide the γ(t) immediately after we obtain the most updated δ(t) in the market each time. We let γ(t) almost the same as the δ(t) as possible. The δ(t) is piecewise continuously along with time t and so is the γ(t). Thus, 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 comes out 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 in this kind of policies is
0 ( 0 )
Ft F (s)
t
xp ds
e
The interest rate is volatile during the policy year. We assume that we obtain the realized interest rate for the 2nd to 5th policy year based on the interest rate model.
The values of the force of interest rate for the first five policy years are as shown in Table 14.
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Table 14 The Realized Values of the Force of Interest Rate for the First Five Policy Years 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 15 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
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States is one of the kinds in this product group. Our product design is based on the interest rate variable life insurance and we let it meet the criterion of γ(t)=δ(t). 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 in pre-determined category..
4.2 The secondary strategy with 0< γ (t)< δ (t)
In the consideration of secondary strategy with 0<γ(t)<δ(t), for the company, the purpose of the strategy is not to optimal the risk mitigation. Their objective is only to moderate the mortality rate risk, not to totally immunize the risk.
4.2.1 The Pre-determined Category
We take the compound increasing whole life insurance product as an example and a flat death benefit whole life insurance as a comparison to the strategic product.
And, the difference from the Essay I is that we concern the uncertainty of factors in the model. The outcome of the changes of the mortality rate with 20% shock is in Table 16.
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Table 16 The Liability at the End of the 5th Policy Year of Illustrated Insurance Product with Stochastic Interest Rate and Stochastic Mortality Rate for Different Mortality Bases
Gender Premium Basis
20% Up
Panel A: γ(t) (=3.5%) < Long Term Mean of Interest Rate Model, 6.5%25/M 26,798 31,061 32,539 29,168 4.76% -6.09%
45/M 48,795 51,520 53,670 48,750 4.17% -5.38%
Panel B: γ(t) (=0%) < Long Term Mean of Interest Rate Model, 6.5%
25/M 5,399 6,378 7,133 5,536
11.84% -13.20%45/M 13,788 17,509 19,195 15,570
9.63% -11.07%The product in Panel A with long term mean of δ(t) is 6.5% and that of γ(t) is 3.5%. We compare hedging effect in Panel A to that in Panel B. The product in Panel A is exposed to both the mortality risk and the 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, the product in Panel A is exposed to risk of 4% ~ 5% more than that in basis mortality rate. And the product in Panel B is exposed to that of 9 ~ 12% 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.
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4.2.2 The Applications of Pre-determined Category
The secondary strategy aims at keeping the profit in situation that realized morality rate is less than expected mortality rate. The trend in mortality rate is less and less. Following the trend of population mortality, the insurance companies may earn the profit in selling life insurance product exposed to mortality risk. When the companies do not want to expose too much mortality risk in a flat whole life insurance product and want to keep the risk in a controllable way. They can apply the secondary strategy.
We say that the company has a target profit rate in mortality of a whole life
We say that the company has a target profit rate in mortality of a whole life