Limited-Payment Whole-Life Benefit Decomposition

Why Decompose a Limited-Payment Whole-Life Benefit?

A whole-life policy can pay a death benefit whenever death occurs, even if premiums are payable only during the first few years. The premium-paying period is not a coverage deadline: after premiums stop, the whole-life insurance remains in force. To value its benefit, divide possible death times into two disjoint intervals: death during the premium-paying period, and death after that period.

This division is useful because the first part has the timing of a term insurance benefit, while the second part is a deferred whole-life benefit. The components add to the value of the whole-life benefit, provided they use the same benefit amount, interest basis, and mortality model.

Definition: Let $T_x$ be the future lifetime of a life aged $x$, and suppose a unit death benefit is paid immediately at death. For a premium-paying period of $n$ years, the benefit during that period is paid if $0<T_x\le n$. The deferred benefit is paid if $T_x>n$. Denote their expected present values by $C_n$ and $D_n$, respectively. A limited-payment whole-life policy provides both benefits; premiums stopping at time $n$ do not end coverage.

Let the force of interest be constant at $\delta$, and let ${}_tp_x=P(T_x>t)$ be the probability of survival for $t$ years. With benefit amount $B$, the present value at issue is $B e^{-\delta T_x}$ if death occurs at $T_x$. The benefit in force at every possible death time is therefore the whole-life benefit, split at time $n$.

\bar A_x=C_n+D_n,\qquad C_n=E\!\left[e^{-\delta T_x}\mathbf{1}_{\{T_x\le n\}}\right],\qquad D_n=E\!\left[e^{-\delta T_x}\mathbf{1}_{\{T_x>n\}}\right].

Here $\bar A_x$, $C_n$, and $D_n$ are values per unit of benefit. For amount $B$, multiply each by $B$. The decomposition concerns the death benefit; a limited premium-paying period does not mean that the policy pays a benefit only during that period.

Integral Form and Deferred Value

If the future lifetime has density $f_{T_x}(t)$, then the present-value contributions can be written as integrals over the two time intervals. The density satisfies $f_{T_x}(t)={}_tp_x\mu_{x+t}$, where $\mu_{x+t}$ is the force of mortality at attained age $x+t$.

C_n=\int_0^n e^{-\delta t}{}_tp_x\mu_{x+t}\,dt, \qquad D_n=\int_n^\infty e^{-\delta t}{}_tp_x\mu_{x+t}\,dt.

The deferred component has a useful factorization. To receive its benefit, the insured must first survive to time $n$. Conditional on survival to $n$, the remaining lifetime has the distribution of the future lifetime of a life then aged $x+n$, under the same mortality model. Discounting the value at time $n$ back to issue gives

D_n=e^{-\delta n}{}_np_x\,\bar A_{x+n}.

This expression assumes the benefit is paid immediately at death and the interest rate is constant over the period. The value $\bar A_{x+n}$ is the continuous whole-life value per unit of benefit at age $x+n$. The factor $e^{-\delta n}{}_np_x$ combines discounting to time $n$ with the probability of being alive then. It is not just a discount factor.

Derivation: In the deferred integral, set $t=n+s$, where $s$ is time after the deferral point. Conditional survival gives ${}_{n+s}p_x={}_np_x\,{}_sp_{x+n}$, and the mortality force at attained age $x+n+s$ is $\mu_{x+n+s}$. Thus

\begin{aligned} D_n &=\int_n^\infty e^{-\delta t}{}_tp_x\mu_{x+t}\,dt\\ &=e^{-\delta n}{}_np_x \int_0^\infty e^{-\delta s}{}_sp_{x+n}\mu_{x+n+s}\,ds\\ &=e^{-\delta n}{}_np_x\,\bar A_{x+n}. \end{aligned}

The final integral is the continuous whole-life insurance value at age $x+n$. The derivation also explains why the deferred value decreases when survival to the deferral point becomes less likely, even if the post-deferral whole-life value stays the same.

Constant Force of Mortality

Under a constant force of mortality $\mu>0$, survival is ${}_tp_x=e^{-\mu t}$, and the future lifetime density is $f_{T_x}(t)=\mu e^{-\mu t}$. With constant force of interest $\delta\ge 0$, the whole-life value is $\mu/(\delta+\mu)$. Integrating over the two segments gives

C_n=\frac{\mu}{\delta+\mu}\left(1-e^{-(\delta+\mu)n}\right), \qquad D_n=\frac{\mu}{\delta+\mu}e^{-(\delta+\mu)n}.

The sum is $\mu/(\delta+\mu)$, as required. In the deferred expression, the exponent includes both mortality and interest because surviving to $n$ has probability $e^{-\mu n}$, and discounting to $n$ contributes $e^{-\delta n}$.

Worked Example: Splitting a Ten-Year Limited-Payment Benefit

A hypothetical continuous whole-life policy has a death benefit of $100,000. Premiums are payable for 10 years. Assume a constant force of mortality $\mu=0.04$ and a constant force of interest $\delta=0.06$. Find the expected present value of the benefit paid during the 10-year payment period and the value of the benefit deferred until afterward.

Method. Use the constant-force decomposition per unit of benefit, then multiply by $100,000. The common mistake to avoid is treating the end of the premium period as the end of insurance coverage.

First, $\delta+\mu=0.06+0.04=0.10$, and $\mu/(\delta+\mu)=0.04/0.10=0.40$. Also, $e^{-(\delta+\mu)n}=e^{-0.10(10)}=e^{-1}=0.3678794$.

The during-period value per unit is

C_{10}=0.40(1-0.3678794)=0.40(0.6321206)=0.2528482.

The deferred value per unit is

D_{10}=0.40(0.3678794)=0.1471518.

Multiplying by $100,000 gives $25,284.82 for the benefit during the payment period and $14,715.18 for the benefit afterward, rounded to the nearest cent. The check is $25,284.82+$14,715.18=$40,000.00. The whole-life value is also $100,000(0.40)=$40,000.00, so the two parts reconcile.

When Mortality Changes After the Payment Period

The decomposition does not require constant mortality. If the force of mortality changes with time, calculate the during-period integral using mortality up to $n$, then value the deferred part using survival to $n$ and the whole-life value at age $x+n$. In particular, do not extend the pre-$n$ mortality force into the deferred period unless the model says it remains in force.

Worked Example: A Change in Mortality at the End of Premium Payments

A hypothetical policy pays one unit at death whenever death occurs. Premiums are payable for 5 years. Assume constant force of interest $\delta=0.03$. The force of mortality is $\mu=0.02$ for the first 5 years and $\mu=0.05$ thereafter. Find the two benefit values per unit.

Method. Integrate using the first mortality force for the during-period benefit. For the deferred part, first calculate the probability and discount factor for reaching year 5, then use the whole-life value under the post-year-5 force. This avoids incorrectly using the first mortality force after year 5.

During the first 5 years, $\delta+\mu=0.03+0.02=0.05$. Therefore

C_5=\frac{0.02}{0.05}\left(1-e^{-0.05(5)}\right) =0.40(1-e^{-0.25}) =0.40(1-0.7788008) =0.0884797.

The probability of surviving to year 5 is $e^{-0.02(5)}=e^{-0.10}$, and the discount factor to year 5 is $e^{-0.03(5)}=e^{-0.15}$. Their product is $e^{-0.25}=0.7788008$. After year 5, the whole-life value per unit is $0.05/(0.03+0.05)=0.625$. Hence

D_5=e^{-0.25}(0.625)=0.7788008(0.625)=0.4867505.

The total is $0.0884797+0.4867505=0.5752302$ per unit. As a check, direct valuation uses the first-period value plus the discounted-survival factor times the post-period whole-life value, which gives the same sum. The components are rounded to seven decimal places here; unrounded values add to the unrounded total.

Using the Decomposition as a Design Check

The identity can also answer questions about how long premiums must be payable to achieve a specified share of the benefit value during or after that period. Under constant forces, the fraction of the whole-life value deferred beyond $n$ is $e^{-(\delta+\mu)n}$. This fraction is not simply the probability of survival: it also reflects discounting.

Worked Example: Choosing a Payment Period From a Deferred-Value Target

A hypothetical whole-life policy has a death benefit of $200,000. Assume constant force of mortality $\mu=0.04$ and constant force of interest $\delta=0.06$. Determine the premium-paying period $n$ for which exactly 25% of the whole-life benefit value is deferred until after premiums stop. Then find the two component values.

Method. Set the deferred component equal to 25% of the whole-life value and solve for $n$. The common mistake to avoid is solving with survival probability alone; the deferred value also includes interest discounting.

The whole-life value per unit is $0.04/(0.04+0.06)=0.40$. The deferred fraction is $e^{-(\delta+\mu)n}=0.25$, so

e^{-0.10n}=0.25,\qquad -0.10n=\ln(0.25),\qquad n=\frac{-\ln(0.25)}{0.10}=13.8629\text{ years}.

Thus the deferred value per unit is $0.25(0.40)=0.10$, and the during-period value per unit is $(1-0.25)(0.40)=0.30$. For a $200,000 benefit, the deferred value is $20,000 and the during-period value is $60,000. Their sum, $80,000, equals $200,000 times the whole-life value per unit, $0.40$. The period is rounded to four decimal places; the dollar amounts are exact under the stated target.

Common Mistakes

Ending coverage when premiums stop. A limited-payment whole-life policy continues to provide its death benefit after the payment period. It is the premiums, not the coverage, that are limited.
Omitting survival from the deferred value. The deferred component must include ${}_np_x$, as well as discounting to time $n$. Under constant mortality, the combined factor is $e^{-(\delta+\mu)n}$.
Using the wrong mortality force after $n$. If mortality changes at the end of the payment period, the deferred whole-life value must use the post-$n$ mortality model.
Adding values with inconsistent benefit amounts or bases. Both components must use the same death benefit and valuation assumptions as the whole-life value they decompose.

Key takeaway: A limited-payment whole-life death benefit splits into a term component for deaths during the payment period and a deferred whole-life component for deaths afterward. The deferred value includes both survival to the deferral time and discounting back to issue; the two components must sum to the whole-life benefit value.

Check Your Understanding

For each problem, identify the during-period and deferred components before checking the answer.

A continuous whole-life policy has a unit benefit, premiums payable for 10 years, $\mu=0.03$, and $\delta=0.02$, both constant. Find each component value. (Answer: during-period $0.2361$; deferred $0.3639$; total $0.6000$.)
A hypothetical $80,000 policy has constant $\mu=0.05$, constant $\delta=0.05$, and a 4-year premium-paying period. Find the two benefit values in dollars. (Answer: during-period $13,187.20; deferred $26,812.80; total $40,000.00.)
A unit-benefit policy has $\delta=0.02$, with $\mu=0.01$ for the first 3 years and $\mu=0.04$ thereafter. Premiums stop at year 3. Find each component value to four decimal places. (Answer: during-period $0.0287$; deferred $0.6093$; total $0.6380$.)
A unit-benefit policy has constant $\mu=0.04$ and $\delta=0.01$. Premiums are payable for 6 years. Find the during-period and deferred values, and check their sum against the whole-life value. (Answer: during-period $0.2073$; deferred $0.5927$; total $0.8000$.)
Under constant forces $\mu=0.02$ and $\delta=0.03$, what premium-paying period makes the deferred component equal to 40% of the whole-life value? (Answer: $n=-\ln(0.40)/0.05=18.3258$ years.)

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