Limited-Payment Life Insurance Premiums

Why Limit the Premium-Payment Period?

A whole-life policy can remain in force for the insured’s lifetime even when premiums are payable only for a fixed number of years. For example, a hypothetical contract might provide a death benefit whenever death occurs but require annual premiums only during the first 10 years. The end of the premium-payment period does not end the coverage.

To calculate the net level premium, distinguish the benefit from the premium stream. The benefit is valued over the whole lifetime; the premiums are valued only over the stated payment period and only for times when the insured is alive to pay them. In this tutorial, premiums are annual and payable in advance, beginning at issue. The death benefit is paid at the end of the year of death. Expenses are excluded, so the premiums are net premiums.

Definition: An $n$-payment limited-payment whole-life policy provides a whole-life death benefit while requiring at most $n$ annual premiums. Premiums are payable at times $0,1,\ldots,n-1$, provided the insured is alive at each payment time. If death occurs, no later premiums are paid; the death benefit remains payable under the whole-life contract.

Present Values in the Equivalence Equation

Let $i$ be the effective annual interest rate and $v=1/(1+i)$. Let ${}_kp_x$ be the probability that a life aged $x$ survives $k$ years, with ${}_0p_x=1$. For annual discrete whole-life insurance with benefit amount $B$, the benefit is paid at time $k+1$ if death occurs during policy year $k+1$.

Definition: The present value of a unit whole-life insurance benefit is $A_x$. The present value of a unit annual premium payable in advance for at most $n$ years, while the insured is alive, is the $n$-year temporary life annuity-due value $\ddot{a}_{x:\overline{n}|}$. In annual discrete notation, $$ A_x=\sum_{k=0}^{\infty}v^{k+1}{}_kp_xq_{x+k}, \qquad \ddot{a}_{x:\overline{n}|}=\sum_{k=0}^{n-1}v^k{}_kp_x. $$

Here $q_{x+k}$ is the probability that a life aged $x+k$ dies within the following year. The first annuity term equals 1 because the first premium is paid at issue with certainty. Each later term is discounted and weighted by the probability the insured survives to that premium date.

Under the equivalence principle, the expected present value of premiums equals the expected present value of benefits. If $P$ is the annual net level premium, the two sides are $P\ddot{a}_{x:\overline{n}|}$ and $BA_x$, respectively. Therefore:

P\ddot{a}_{x:\overline{n}|}=BA_x, \qquad P=\frac{BA_x}{\ddot{a}_{x:\overline{n}|}}.

Why the Premium Formula Holds

The formula follows by adding the expected present values of the individual payments. A premium of amount $P$ is due at time $k$ only if the insured survives to that time. Its expected present value is $Pv^k{}_kp_x$. Summing over the $n$ possible premium dates gives:

\operatorname{EPV}(\text{premiums}) =\sum_{k=0}^{n-1}Pv^k{}_kp_x =P\sum_{k=0}^{n-1}v^k{}_kp_x =P\ddot{a}_{x:\overline{n}|}.

For the benefit, death in policy year $k+1$ has probability ${}_kp_xq_{x+k}$, and the benefit is discounted $k+1$ years. Thus the expected present value of the benefit is:

\operatorname{EPV}(\text{benefit}) =\sum_{k=0}^{\infty}Bv^{k+1}{}_kp_xq_{x+k} =BA_x.

Equating these present values gives the limited-payment premium equation. It assumes premiums are paid at the beginning of each year while the insured is alive, and that the benefit is paid at the end of the year of death. If premium timing or benefit timing differs, the present-value expressions must be changed accordingly.

Worked Calculations Under Constant Mortality

The following examples use a constant annual mortality model: the probability of death in each year is $q$, conditional on survival to that year, and the probability of surviving one year is $p=1-q$. This assumption applies at every attained age in these examples. Then ${}_kp_x=p^k$, and the whole-life insurance value is a geometric series:

A_x=\sum_{k=0}^{\infty}v^{k+1}p^kq =\frac{qv}{1-pv}, \quad\text{provided }pv<1.

Worked Example: Three Annual Premiums for Whole-Life Coverage

A hypothetical policy provides a \$120,000 whole-life benefit paid at the end of the year of death. Premiums of level amount $P$ are payable annually in advance for at most three years, while the insured is alive. Assume $q=0.20$ each year and $i=0.04$. Find the net annual premium.

Method. Calculate the whole-life benefit value and the three-payment temporary life annuity-due value, then use the equivalence principle. This method avoids treating the three-year premium period as a three-year coverage period.

Step 1: Interest and survival factors. The one-year survival probability is $p=1-q=0.80$, and $v=1/1.04=25/26$. Hence $pv=(4/5)(25/26)=10/13$.

Step 2: Whole-life benefit value. Using the constant mortality formula, $A_x=(qv)/(1-pv)$. Thus $qv=(1/5)(25/26)=5/26$, while $1-pv=1-10/13=3/13=6/26$. Therefore $A_x=(5/26)/(6/26)=5/6$. The expected present value of the benefit is $\$120{,}000(5/6)=\$100{,}000$.

Step 3: Premium value. The three premiums are due at times 0, 1, and 2 if the insured is alive at the relevant time: $$ \ddot{a}_{x:\overline{3}|} =1+\frac{10}{13}+\left(\frac{10}{13}\right)^2 =\frac{169+130+100}{169} =\frac{399}{169}. $$

Step 4: Solve and check. The equivalence principle gives $P(399/169)=100{,}000$, so $P=100{,}000(169/399)=16{,}900{,}000/399=\$42{,}355.8897\ldots$. Rounded to the nearest cent, the annual net premium is \$42,355.89. Using the unrounded premium, $P(399/169)=\$100{,}000$, which matches the benefit present value.

Worked Example: Comparing Two- and Five-Payment Policies

A hypothetical whole-life policy has a \$90,000 benefit, paid at the end of the year of death. Compare the annual net premiums if premiums are payable annually in advance for at most two years versus at most five years. Assume $q=0.20$ each year and $i=0.04$, as in the constant mortality model.

Method. The benefit present value is the same for both contracts. Calculate a separate temporary life annuity-due value for each payment period. This avoids confusing the number of premiums with the duration of coverage.

Step 1: Benefit present value. As calculated from the stated assumptions, $A_x=5/6$. Therefore the benefit present value is $\$90{,}000(5/6)=\$75{,}000$.

Step 2: Two-payment premium. With $pv=10/13$, $\ddot{a}_{x:\overline{2}|}=1+10/13=23/13$. Thus $P_2=75{,}000/(23/13)=975{,}000/23=\$42{,}391.3043\ldots$, or \$42,391.30 to the nearest cent.

Step 3: Five-payment premium. The annuity-due value is $$ \ddot{a}_{x:\overline{5}|} =1+\frac{10}{13}+\frac{100}{169}+\frac{1000}{2197}+\frac{10000}{28561} =\frac{90{,}431}{28{,}561}. $$ Therefore $P_5=75{,}000(28{,}561/90{,}431) =2{,}142{,}075{,}000/90{,}431 =\$23{,}687.3970\ldots$, or \$23,687.40 to the nearest cent.

Step 4: Check and interpret. Each premium multiplied by its corresponding annuity-due value equals the same \$75,000 benefit present value. The five-payment premium is lower because it can be collected at more payment dates, but the policy still provides whole-life coverage in both cases.

Worked Example: Four Payments at a Different Mortality and Interest Rate

A hypothetical whole-life contract pays \$60,000 at the end of the year of death. Level annual premiums are payable in advance for at most four years while the insured is alive. Assume $q=0.10$ each year and $i=0.05$. Calculate the annual net premium.

Method. Use the geometric-series formula for $A_x$ and the four-term temporary life annuity-due sum. The common trap avoided here is using an interest factor from one rate with mortality probabilities from another model.

Step 1: Model factors. Here $p=0.90$ and $v=1/1.05=20/21$, so $pv=(9/10)(20/21)=6/7$.

Step 2: Whole-life benefit value. We have $qv=(1/10)(20/21)=2/21$ and $1-pv=1/7$. Therefore $A_x=(2/21)/(1/7)=2/3$. The expected present value of the benefit is $\$60{,}000(2/3)=\$40{,}000$.

Step 3: Four-premium value. The premium dates are 0, 1, 2, and 3, so $$ \ddot{a}_{x:\overline{4}|} =1+\frac67+\frac{36}{49}+\frac{216}{343} =\frac{343+294+252+216}{343} =\frac{1105}{343}. $$

Step 4: Solve and check. The equivalence equation is $P(1105/343)=40{,}000$. Hence $P=40{,}000(343/1105)=13{,}720{,}000/1105 =2{,}744{,}000/221=\$12{,}416.2896\ldots$. Rounded to the nearest cent, $P=$\$12,416.29. The unrounded premium times $1105/343$ is exactly \$40,000, as required.

Common Mistake: Using an Annuity-Certain Factor

The premium stream is not an annuity certain when premiums stop on death. A three-payment annuity-certain factor at $i=0.04$ would be $1+v+v^2=1+25/26+625/676=1951/676$. That factor assumes all three payments are made, including payments after the insured has died.

For the first worked example, incorrectly using that factor would give $P=100{,}000/(1951/676)$, rather than the correct $P=100{,}000/(399/169)=\$42{,}355.89$. The correct denominator weights each later premium by the probability the insured is alive when it is due. Also avoid the opposite mistake: stopping premiums after year $n$ does not stop whole-life coverage. The benefit value remains $BA_x$, not a term-insurance value over $n$ years.

Key takeaway: For an annual net level premium payable in advance for at most $n$ years on whole-life coverage, equate $P\ddot{a}_{x:\overline{n}|}$ to $BA_x$. The annuity-due term reflects the limited, survival-contingent premium stream; $A_x$ reflects the full lifetime benefit.

Check Your Understanding

For each problem, use annual premiums payable in advance while the insured is alive and a benefit paid at the end of the year of death. Round monetary answers to the nearest cent.

A hypothetical whole-life policy has benefit \$100,000, annual $q=0.10$, and $i=0.05$. Premiums are payable for at most three years. Find the net annual premium. (Answer: \$25,721.78)
For a whole-life benefit of \$72,000, assume constant annual $q=0.20$ and $i=0.04$. Premiums are payable for at most two years. Find the annual net premium. (Answer: \$33,913.04)
Assume $q=0.20$ and $i=0.04$. A whole-life policy has annual net premium \$25,000 payable for at most three years. Use the equivalence principle to find the death benefit. (Answer: \$70,828.40)
A \$120,000 whole-life benefit is priced with constant annual $q=0.20$ and $i=0.04$. If premiums are payable for at most four years, find the annual net premium. (Answer: \$35,509.94)
Under constant annual $q=0.20$ and $i=0.04$, compare annual net premiums for a \$90,000 whole-life benefit when premiums are payable for at most two years and at most five years. (Answer: Two payments: \$42,391.30; five payments: \$23,687.40)

What You'll Learn