Limited-Payment Versus Limited-Term Coverage

Two Different Meanings of “Limited”

The word limited can describe either how long premiums are paid or how long insurance protection lasts. These are different contract features. A whole-life policy with premiums limited to a fixed period continues to provide its death benefit after that period, while term insurance provides a benefit only if death occurs during the stated term.

This distinction matters when comparing benefit present values. The premium schedule does not determine when the insurance benefit is payable. In this tutorial, benefit values are evaluated separately from premiums: the questions are when a benefit is paid and what its present value is, not how the policy is funded.

Definition: A limited-payment whole-life policy provides a death benefit whenever death occurs, but its premiums are payable only during a specified period. A limited-term policy provides a death benefit only if death occurs within its specified coverage period; no death benefit is paid for death after that period.

For example, suppose both hypothetical policies have a ten-year period associated with them. Under the limited-payment whole-life policy, premiums stop after ten years but the death benefit remains in force for the insured’s lifetime. Under ten-year term insurance, coverage itself ends after ten years. These contracts can have the same stated period and very different benefit values.

Present Values of Continuous Death Benefits

Let $T_x$ be the future lifetime of a life aged $x$, and let $\mu_{x+t}$ be the force of mortality at duration $t$. The probability of surviving $t$ years is ${}_tp_x$. Assume a constant force of interest $\delta$, so a dollar payable at time $t$ has present value $e^{-\delta t}$ dollars.

Consider a benefit of $B$ dollars paid immediately at death. For whole-life coverage, the benefit is paid at $T_x$. For $n$-year term coverage, it is paid at $T_x$ only when $T_x\leq n$. In a continuous lifetime model, whether the endpoint is written as $T_x<n$ or $T_x\leq n$ does not change the value when death at exactly $n$ has probability zero.

\text{Whole-life benefit EPV} = B\int_0^\infty e^{-\delta t}{}_tp_x\mu_{x+t}\,dt = B\bar{A}_x

\text{\(n\)-year term benefit EPV} = B\int_0^n e^{-\delta t}{}_tp_x\mu_{x+t}\,dt

The integrand combines three factors: survival to time $t$, death during the next infinitesimal interval as represented by the mortality force, and discounting to time 0. The term integral stops at $n$ because deaths after the coverage period do not produce a benefit.

Why Whole Life Equals Term Plus Deferred Whole Life

The whole-life integral can be split at the end of the term period. Its first part is exactly the term insurance value. The remaining part represents benefits paid only after surviving to time $n$.

B\bar{A}_x = B\int_0^n e^{-\delta t}{}_tp_x\mu_{x+t}\,dt + B\int_n^\infty e^{-\delta t}{}_tp_x\mu_{x+t}\,dt

For the second integral, write $t=n+s$. Survival to time $n+s$ factors as survival to $n$, followed by survival for another $s$ years conditional on being alive at $n$. Discounting also factors into discounting to $n$ and then from $n$ to $n+s$. Therefore the second part is the value of a whole-life benefit at age $x+n$, multiplied by the probability of surviving to $n$ and the discount factor to $n$.

B\int_n^\infty e^{-\delta t}{}_tp_x\mu_{x+t}\,dt = B e^{-\delta n}{}_np_x\bar{A}_{x+n}

Theorem: For a level benefit $B$, the whole-life benefit EPV is the sum of the $n$-year term benefit EPV and the EPV of whole-life coverage deferred $n$ years: $$ B\bar{A}_x = B\int_0^n e^{-\delta t}{}_tp_x\mu_{x+t}\,dt + B e^{-\delta n}{}_np_x\bar{A}_{x+n}. $$ The second term is positive when survival to $n$ and the future whole-life benefit value are positive. Thus, under the same mortality, interest, and benefit assumptions, whole-life coverage has a greater benefit EPV than term coverage of duration $n$.

The decomposition is a benefit comparison, not a statement that the premiums are equal. A premium comparison would also need the premium amounts, timing, and payment conditions. In particular, the fact that premiums stop after $n$ years does not remove the deferred whole-life benefit from the first contract.

Closed Forms Under Constant Forces

Suppose the force of mortality is constant at $\mu>0$ at every duration, and the force of interest is constant at $\delta\geq 0$. Then ${}_tp_x=e^{-\mu t}$, and the density of death at time $t$ is $\mu e^{-\mu t}$. Combining this density with discounting gives the factor $\mu e^{-(\mu+\delta)t}$.

\bar{A}_x = \int_0^\infty \mu e^{-(\mu+\delta)t}\,dt = \frac{\mu}{\mu+\delta}

\text{\(n\)-year term benefit EPV} = B\int_0^n \mu e^{-(\mu+\delta)t}\,dt = B\frac{\mu}{\mu+\delta}\left(1-e^{-(\mu+\delta)n}\right)

The difference between the whole-life and term benefit EPVs is the deferred part:

B\bar{A}_x-\text{\(n\)-year term benefit EPV} = B\frac{\mu}{\mu+\delta}e^{-(\mu+\delta)n}

These closed forms require constant mortality and interest forces over the relevant durations. If mortality varies with duration, the integral decomposition remains valid, but the constant-force expressions generally do not.

Worked Example: Comparing Ten-Year Term and Whole-Life Benefits

A hypothetical policy pays $B=\$100{,}000$ immediately at death. The force of mortality is constant at $\mu=0.06$, the force of interest is constant at $\delta=0.04$, and the comparison period is $n=10$ years. Find the benefit EPV for whole-life coverage, for ten-year term coverage, and for the deferred whole-life portion.

Method. Use the constant-force formulas and check that the term and deferred values sum to the whole-life value. This avoids treating a ten-year premium period as if it ended whole-life coverage.

First, $\mu+\delta=0.06+0.04=0.10$. The whole-life EPV is

\$100{,}000\frac{0.06}{0.10} = \$100{,}000(0.6) = \$60{,}000.00.

For the ten-year term benefit, the exponent is $(\mu+\delta)n=(0.10)(10)=1$, and $e^{-1}=0.3678794412$. Thus,

\begin{aligned} \text{Term EPV} &=\$100{,}000(0.6)(1-e^{-1})\\ &=\$60{,}000(1-0.3678794412)\\ &=\$60{,}000(0.6321205588)\\ &=\$37{,}927.23. \end{aligned}

The deferred whole-life portion is

\begin{aligned} \text{Deferred EPV} &=\$100{,}000(0.6)e^{-1}\\ &=\$60{,}000(0.3678794412)\\ &=\$22{,}072.77. \end{aligned}

Check. $\$37{,}927.23+\$22{,}072.77=\$60{,}000.00$, equal to the whole-life value. Dollar amounts are rounded to the nearest cent only at the final step.

When Mortality Changes After the Term

The decomposition does not depend on constant mortality. When the force of mortality changes, the whole-life value can be calculated by valuing the period before $n$ and the period after $n$ separately. The survival and discount factors to time $n$ multiply the conditional whole-life value from that point onward.

Worked Example: A Change in Mortality at the End of the Term

A hypothetical contract pays $\$100{,}000$ immediately at death. The force of interest is $\delta=0.03$. The force of mortality is $\mu=0.02$ for the first five years and $\mu=0.05$ thereafter. Compare five-year term coverage with whole-life coverage.

Method. Integrate over the first five years using $\mu=0.02$, then value the post-five-year whole-life benefit conditional on survival to year five. This avoids applying the later mortality force during the first five years.

During the first five years, $\mu+\delta=0.02+0.03=0.05$, so the five-year term factor is

\begin{aligned} \int_0^5 0.02e^{-0.05t}\,dt &=\frac{0.02}{0.05}(1-e^{-0.25})\\ &=0.4(1-0.7788007831)\\ &=0.4(0.2211992169)\\ &=0.0884796868. \end{aligned}

Therefore, the five-year term EPV is $ \$100{,}000(0.0884796868)=\$8{,}847.97$. The probability-and-discount factor to year five is $e^{-(0.02+0.03)5}=e^{-0.25}=0.7788007831$.

After year five, the whole-life value conditional on survival is $\mu/(\mu+\delta)=0.05/(0.05+0.03)=0.625$. The deferred whole-life EPV at issue is therefore

\begin{aligned} \$100{,}000(0.7788007831)(0.625) &=\$100{,}000(0.4867504894)\\ &=\$48{,}675.05. \end{aligned}

Adding the two parts gives the whole-life benefit EPV: $\$8{,}847.97+\$48{,}675.05=\$57{,}523.02$. This exceeds the term value by exactly the deferred portion. Dollar amounts are rounded to the nearest cent only at the final step.

Common Exam Trap and Practical Checks

The most common error is reading “premiums payable for $n$ years” as “coverage lasts $n$ years.” Those statements describe different features. When the benefit is whole life, the death-benefit integral extends to infinity even if premiums are payable only for a limited period. When the benefit is term insurance, the benefit integral ends at the term limit.

Worked Example: Correcting a Misread Coverage Period

A hypothetical contract has a death benefit of $\$50{,}000$ for life, with premiums payable for at most eight years. Assume constant forces $\mu=0.03$ and $\delta=0.02$. A candidate uses the eight-year term formula and reports the contract’s benefit EPV. Identify the error and calculate the correct benefit EPV.

Method. Classify the benefit duration before selecting an integral. The wording says the benefit is for life; only the premium period is limited. The error to avoid is truncating the benefit at year eight.

The correct whole-life benefit factor is

\frac{\mu}{\mu+\delta} = \frac{0.03}{0.03+0.02} = 0.6.

Thus the correct EPV is $\$50{,}000(0.6)=\$30{,}000.00$. The mistaken eight-year term calculation would be

\begin{aligned} \$50{,}000(0.6)(1-e^{-0.05(8)}) &=\$30{,}000(1-e^{-0.4})\\ &=\$30{,}000(1-0.6703200460)\\ &=\$9{,}890.40. \end{aligned}

That smaller value is appropriate for eight-year term coverage, not for the stated whole-life benefit. The difference, $\$30{,}000.00-\$9{,}890.40=\$20{,}109.60$, is the value of the benefit after year eight. Dollar amounts are rounded to the nearest cent only at the final step.

Before calculating, identify whether the specified period applies to premiums or benefits. Then state the benefit timing, interest assumption, mortality assumption, and endpoint of the valuation integral. These checks help prevent using a term value for a whole-life benefit or a whole-life value for a benefit that expires.

Key takeaway: A limited-payment whole-life policy has whole-life benefit duration and a limited premium duration. An $n$-year term policy has limited benefit duration. Their benefit EPVs differ by the value of whole-life coverage deferred until after year $n$.

Check Your Understanding

For each problem, distinguish the benefit duration from any premium-payment period before calculating.

A hypothetical $\$50{,}000$ benefit is payable at death during four-year term coverage. Assume constant $\mu=0.04$ and $\delta=0.06$. Find its benefit EPV and the whole-life EPV under the same assumptions. (Answer: term EPV $\$6{,}593.60$; whole-life EPV $\$20{,}000.00$)
A hypothetical $\$120{,}000$ benefit is payable at death. Compare ten-year term with whole-life coverage under constant $\mu=0.02$ and $\delta=0.03$. (Answer: term EPV $\$18{,}886.53$; whole-life EPV $\$48{,}000.00$; deferred portion $\$29{,}113.47$)
A hypothetical $\$80{,}000$ benefit is payable at death, and the mortality and interest forces are both $0.05$. Find the six-year term EPV and the deferred whole-life portion. (Answer: term EPV $\$18{,}047.53$; deferred portion $\$21{,}952.47$)
A hypothetical $\$100{,}000$ benefit is payable at death. The force of interest is $0.02$; the force of mortality is $0.01$ for the first three years and $0.04$ thereafter. Find the three-year term EPV and the whole-life EPV. (Answer: term EPV $\$2{,}868.96$; whole-life EPV $\$63{,}797.71$)
A policy has premiums payable for at most six years, but its stated death benefit remains payable for life. Under constant forces $\mu=0.04$ and $\delta=0.06$, should its benefit integral end at six years or infinity, and what is the EPV of a $\$75{,}000$ benefit? (Answer: infinity; $\$30{,}000.00$)

What You'll Learn