3. Theoretical Development
3.2 Formal Development
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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:
changing the parameter of (.) as
0 s ss x x s
s( )
x
e e p
ds ef
x s
L
. (2)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. We notice that x(0) =
0 sp
x
x sds 1
. 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.3.2 Formal Development
We elaborate the above idea that we further investigate the impact of risks on the expected reserve of the appropriately calibrated whole life insurance product. The mortality pressure increases when the policyholder gains his age. We try to find a way that may eliminate the difference of reserves by the impact of ages (with assumption that the mortality rate is a function of age). That also means the calibrated
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policy of the calibrated product is sold now (i.e., at time 0) to a customer at age x with face amount 1, 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)Referring to the derivation in Bowers et al (1997, chapter 4), we may decompose
x
Substituting equations (5) and (6) into (4) yields
( )
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The limits of the two items on the right-hand side of equation (8) are as follows:
( ) ( )
The limit of equation (8) is thus
et x ttVx x t . (11)
The above derivation means that
Rearranging equation (12), we get
t following boundary conditions of the expected reserves:
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The line of boundary reserve is
=
The above equation of the boundary reserve can be written3 as:
x t
tV
xe
t
e
t
. (15)3.2.1 The Analyses on the (μ
x+t,
tV
x) plane
As Figures 2 and 3 display, Equation (15) represent hyperbolas centered at (–δ+γ, eγt) on the (μx+t, tVx) plane. The two asymptotes of the hyperbola are given by μx+t
= –δ+γ and tVx = eγt. The hyperbola with positive (δ – γ) lays in the 1st and 3rd quadrants coordinated with respect to the center while the hyperbola with negative (δ – γ) lays in the 2nd and 4th quadrants as shown by Figure 2 and Figure 3 respectively. The hyperbola in Figure 2 exhibit positive slopes while that in Figure 3
3 Please see the appendix for the detail deducing process
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have negative slopes.
Condition 1 Condition 2
Figure 2 The boundary reserve with positive (δ – γ) on (μx+t, tVx) plane, with δ
=2.5% and γ=2.3% in the case.
Note: We are aware that negative μx+t is not reasonable but retain them to show a complete hyperbola.
Condition 3
Condition 4
Figure 3 The boundary reserve with negative (δ – γ) on (μx+t, tVx) plane, with δ
=2.5% and γ=2.75% in the case.
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In Figure 2, the curve on the down-right of the center matches the criteria of the condition 1. Assuming that μx+t is an increasing function4 of age x, with criterion should be on the down-right in respect to the center of hyperbola. The up-left curve in Figure 2 matches the criteria of the condition 2. With criteria of μx+t+δ–γ< 0 and
tVx
x
≥ 0, note that the condition 2 shown on the down-left part of the hyperbola does not exist in reality since the force of mortality μx+t should not be negative.
Following the same deduction, we identify the hyperbola with negative (δ – γ) in Figure 3, of which the up-right curve represents the upper bound of the expected reserve in the condition 3 and the down-left curve represents the lower bound of the expected reserve in the condition 4. The feature of boundary reserve in condition 3 and 4 is decreasing along with μx+t. The expected reserve may be more/less than or equal to the lower/upper bound of expected reserve in practice.
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 in Figure 2 is folded together with the upper/lower bound of that in Figure 3. The feature of the special case with (δ – γ) = 0 turns into a horizontal line as shown in Figure 4 which is exactly one of the asymptotes of the hyperbola tVx = eγt.
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3.2.2 Further Analyses on the (x+t,
tV
x) plane
Mortality rate risk exists because we evaluate risk according to ages. If insurance companies price life insurance products according to real aging situation, the companies may not have mortality rate risk. In fact, neither can we observe aging of individuals directly, nor can we price the mortality risk by each individual’s aging situation. What we can observe in nature is individual age and mortality rate with respect to age statistically. Then, the force of mortality is quantified to express aging with respect to age in methodology. Since we price insurance product based on ages, not aging, mortality rate risk emerges accordingly. Thus, acquiring the information on (x+t, tVx) plane is important in determining the mortality rate risk of the specified product along with the changes of mortality rate.
Next, we intend to demonstrate the relationship of expected reserve and age by the
feature of the boundary reserve
t x t t x
x t
V e
coordinated on the (x+t , tVx) plane.Figure 4 The boundary reserve with (δ – γ) = 0 on (μx+t, tVx) plane, with δ =2.5%
and γ=2.5% in the case.
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Condition 1:
Upper bound
With the information on the (μx+t, tVx) plane in last section, we can transform the
feature of the boundary reserve
t x t t x
x t
V e
with respect to the force ofmortality into the feature of that with respect to age by assuming that the force of mortality μx+t is a monotonic increasing function of age x+t. We then can illustrate features of the boundary reserve according to age x+t on the (x+t , tVx) plane and explore the relationship of expected reserve and age.
In case of the positive (δ – γ), we transform the feature of the boundary reserve on the (μx+t, tVx) plane in Figure 2 into the feature on the (x+t , tVx) plane as shown in Figure 5. Also, in case of the negative (δ – γ), we transform the feature of the boundary reserve in Figure 3 into the feature on the (x+t , tVx) plane as shown in Figure 6. The negative values of μx+t on the (μx+t, tVx) plane cannot exhibit on the (x+t , tVx) plane for each age is of positive force of mortality.
Figure 5 The boundary reserve with positive (δ – γ) on (x+t, tVx) plane, with δ
=2.5% and γ=2.3% 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)
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We demonstrate the four conditions of equation (13) in corresponding features on the (x+t ,tVx) plane. The feature of the boundary reserve in Figure 5 shows the ages with positive value of μx+t which is the case in condition 1. The case in condition 2 does not show out on the (x+t ,tVx) plane for the value of μx+t is negative. In Figure 6, the feature of boundary reserve on the up-right part is the case in condition 3 that follows the criterion of μx+t+δ – γ>0; while the other on the down-left part is the case in condition 4 that follows the criterion of μx+t+δ – γ<0 on the (x+t , tVx) plane.
The feature of the boundary reserve in Figure 5 appears faster increasing than that in 1st quadrant of Figure 2 along with the horizontal axis. As the horizontal axis of μx+t is changed into that of the age x+t, the horizontal axis is re-scaled evenly at the
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).
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of the elderly age is much higher than that of the young ages.
In Figure 5, the boundary reserve of specified product with positive (δ – γ), appears that the risk rooting in “when to pay” is more than that rooting in “how much to pay”. As for our death benefit product, the risk rooting in “when to pay”
represents mortality risk because when the realized mortality rate rises up, it means that more than expected people of cohorts are paid earlier than expected timing. The insurance companies get loss by the time value of extra death benefit claim due to increase of mortality rate. The risk rooting in “how much to pay” in our design is that we set the increment of death benefit each time that is only provided to the survivors at the time. The risk rooting in “how much to pay” is a form of longevity risk. The mortality risk exposure is more than longevity exposure in Figure 5. In general, most life insurance products are grouped in this type of feature in Figure 5 as the mortality risk is more than the longevity risk.
The feature in Figure 6 is seemingly but not exact hyperbola on the (x+t , tVx) plane. The age of 33 years old is a critical age and the boundary reserve of all the ages younger than age 33 are unbounded. In this case, this means the maximum reserve is unlimited and the company selling this product younger than age 33 may be responsible for more liability than he can afford. The kind of compound increasing rate whole life insurance products may lead the insurance company into dangerous situation. We have to be cautious in selling this product to the younger ages.
The boundary reserve in the case of negative (δ – γ) is decreasing along with age more than 33 years old. When (δ – γ) is negative, we may say that the risk rooting in
“how much to pay” in our design is more than that rooting in “when to pay”. That is to say the longevity risk of the case is more than mortality risk following the similar
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elaboration in the case of positive (δ – γ) above. This type of insurance product is rare. But it ever shortly appeared in Taiwan insurance market. The companies designed an increasing whole life insurance with γ > δ. For instance, the death benefit is based on face amount compounded by 4% annually and annual interest rate is 2.5%.
As for the special case on the (μx+t, tVx) plane in Figure 4, we transform the horizontal axis to age x+t and display the case of (δ – γ)=0 on (x+t, tVx) plane as shown in Figure 7. 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 4. The difference of features between Figure 4 and Figure 7 is the measurement of horizontal axis that cannot alter the shape of horizontal line. The function of the special horizontal line tVx = eγt on (x+t, tVx) plane is the same as that on the (μx+t, tVx) plane since the case is irrelevant to the variable of horizontal axis. In this special case, the feature is not just the boundary reverse but also the expected reserve. We then come out an only solution that fulfills the criteria of the four conditions of equation (13) represented the relationship between expected reserve and age.
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The expected reserve is a horizontal line on (x+t, tVx) plane reveals that the expected reserve is irrelevant to age. We may make use of this property on risk mitigation within product. When we design such a specified product with (δ – γ)=05, the risk rooting in “when to pay” is exactly equal to the risk rooting in “how much to pay”. The mortality risk is mitigated totally by the longevity risk for each age in the case. Furthermore, the expected reserve is level to all ages and to the force of mortality of each age. The expected reserve of the policy will not be affected by the
5 Please note that the indication of the insurance product with criterion of (δ – γ) = 0 is based on that premium is paid completely. That is single premium paid for the product. For an installment premium paid for the product, we may take each time of installment premium as a single premium each time.
With each single premium, we then design its benefit accordingly. Then, combining all installment premiums is also combining their benefit of each single premium of the policy. We may see the installment premium paid of the product as combining a lot of single premium paid products. If we want to have the optimal strategy for the installment premium paid in design of the kind of compound increasing rate whole life insurance product, we only need to optimize in design of the single premium paid for the product.
Figure 7 The boundary reserve with δ – γ = 0 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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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.