Premiums with Different Benefit and Payment Periods

Why the Two Periods Must Be Kept Separate

A policy can provide protection for longer than the period in which premiums are paid. For example, a hypothetical whole life policy might require premiums for only a fixed number of years, while its death benefit remains in force for life. Premium payments can also begin after a specified interval. In each case, the benefit period and the premium-paying period describe different parts of the contract.

The net level premium is found by equating the expected present value of benefits to the expected present value of premiums. The key practical decision is therefore not just which benefit factor to use, but also which annuity factor matches the actual premium dates and the condition that premiums are payable only while the insured is alive.

This treatment is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus. The examples use annual premiums payable in advance and benefits payable at the end of the year of death unless stated otherwise.

Definition: Suppose a life aged $x$ receives premiums of amount $P$ at times $0,1,\ldots,r-1$, provided the life is then alive. The expected present value of those premiums is $P\ddot{a}_{x:\overline{r}|}$, where $$ \ddot{a}_{x:\overline{r}|}=\sum_{j=0}^{r-1}v^j\,{}_jp_x. $$ This is an $r$-payment temporary life annuity-due factor. It includes the premium at time 0, which is payable at issue, and excludes a payment at time $r$.

Here $i$ is the effective annual interest rate, $v=1/(1+i)$, and ${}_jp_x$ is the probability that a life aged $x$ survives $j$ years. In particular, ${}_0p_x=1$. If premiums are deferred for $s$ years and then payable at times $s,s+1,\ldots,s+r-1$, the factor is

{}_{s|}\ddot{a}_{x:\overline{r}|} =\sum_{j=s}^{s+r-1}v^j\,{}_jp_x =v^s\,{}_sp_x\,\ddot{a}_{x+s:\overline{r}|}.

The deferred factor reflects both discounting to the issue date and the probability of surviving to the first premium date. It does not include premiums during the deferral period.

Building the Net Premium Equation

For a term insurance benefit $B$ payable at the end of the year of death, with coverage for $n$ years, the expected present value of benefits is

B A_{x:\overline{n}|} = B\sum_{k=1}^{n}v^k\,{}_{k-1}p_xq_{x+k-1}.

The probability of a claim in year $k$ is ${}_{k-1}p_xq_{x+k-1}$: the insured must first survive $k-1$ years and then die in the next year. If level premiums are payable at times $0$ through $r-1$, conditional on being alive on each due date, the equivalence principle gives

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

The coverage duration $n$ determines which deaths can generate benefits. The premium duration $r$ determines which premium dates appear in the denominator. They are not interchangeable.

Why the Premium Factor Has This Form

Each possible premium payment contributes its amount multiplied by the probability it is paid and the discount factor to its due date. A premium due at time $j$ is paid only if the insured survives to time $j$, an event with probability ${}_jp_x$. Its expected present value is therefore $P v^j{}_jp_x$. Adding the contributions for every due date gives the temporary annuity-due factor.

For premiums deferred $s$ years, the same reasoning starts at time $s$, not at time 0. Factoring out the discount and survival terms to that first payment date gives

\sum_{j=s}^{s+r-1}v^j{}_jp_x = v^s{}_sp_x \sum_{h=0}^{r-1}v^h{}_hp_{x+s} = v^s{}_sp_x\ddot{a}_{x+s:\overline{r}|}.

In the middle expression, $h=j-s$ measures time after the first due date. Conditional on survival to age $x+s$, the later survival probability is ${}_hp_{x+s}$. This is why a deferred premium stream requires both the deferral factor and the temporary annuity factor.

Worked Examples

Worked Example: Shorter Premium Period Than Term Coverage

A hypothetical four-year term policy pays $10,000 at the end of the year of death. The annual effective interest rate is $i=5\%$. Assume an independent annual mortality model with $q_{x+k}=0.10$ in every year, so $p_{x+k}=0.90$. Level premiums are due at times 0 and 1 while the insured is alive. Find the net annual premium.

Method. Calculate the four-year benefit present value, then divide by the two-payment temporary life annuity-due factor. The benefit period is four years, but there are only two premium dates.

First, $v=1/1.05=20/21$. Let $r=vp_{x+k}=(20/21)(0.90)=6/7$. The probability of death in year $k$ is $0.10(0.90)^{k-1}$, so after discounting, the year-$k$ probability-weight is $0.10v r^{k-1}$. Thus

A_{x:\overline{4}|} =0.10v(1+r+r^2+r^3) =\frac{2}{21}\left(1+\frac67+\frac{36}{49}+\frac{216}{343}\right) =\frac{2210}{7203} \approx 0.3068166.

The expected present value of benefits is $10{,}000(0.3068166)\approx \$3{,}068.17$. The premium factor is

\ddot{a}_{x:\overline{2}|} =1+v\,{}_1p_x =1+\frac67 =\frac{13}{7} \approx 1.8571429.

Therefore,

P=\frac{10{,}000(2210/7203)}{13/7} \approx \$1{,}652.09.

The calculation uses an end-of-year benefit and premiums at times 0 and 1. As a check, multiplying the premium by its factor gives approximately $\$1{,}652.09(1.8571429)=\$3{,}068.17$, matching the benefit present value. The common mistake avoided here is using a four-payment premium factor merely because coverage lasts four years.

Worked Example: Premiums Deferred Within the Coverage Period

A hypothetical four-year term policy pays $10,000 at the end of the year of death. It covers deaths in years 1 through 4. The annual effective interest rate is $5\%$, and the annual mortality probability is $q=0.10$ in every year. Premiums are waived for the first two years; if the insured is alive, level premiums are due at times 2 and 3. Find the net annual premium.

Method. The benefit present value is the same four-year term value calculated from the stated mortality and interest assumptions: $\$10{,}000(2210/7203)\approx \$3{,}068.17$. The premiums, however, are deferred. With $v p=6/7$, the factor for payments at times 2 and 3 is

{}_{2|}\ddot{a}_{x:\overline{2}|} =v^2{}_2p_x+v^3{}_3p_x =\left(\frac67\right)^2+\left(\frac67\right)^3 =\frac{468}{343} \approx 1.3644315.

The first term is the time-2 payment: it is discounted for two years and paid only if the insured survives two years. The second term applies the same reasoning to the time-3 payment. Hence

P=\frac{10{,}000(2210/7203)}{468/343} \approx \$2{,}248.68.

The check is $P(1.3644315)\approx \$3{,}068.17$, equal to the expected present value of benefits. The premium is higher than in the first example because the same benefit value is funded by fewer expected present-value units of premium. A common error is to use $\ddot{a}_{x:\overline{2}|}$, which would incorrectly place the premiums at times 0 and 1.

Worked Example: Limited-Payment Whole Life Insurance

A hypothetical whole life policy pays $50,000 at the end of the year of death. The annual effective interest rate is $5\%$. Assume $q_{x+k}=0.10$ for every year, with $p_{x+k}=0.90$. Premiums of amount $P$ are due at times 0, 1, and 2 while the insured is alive; no premiums are due later. Find $P$.

Method. The benefit is whole life, so sum the discounted death probabilities over all possible years. The premium stream is a three-payment temporary life annuity-due.

As before, $v=20/21$ and $vp=6/7$. The whole life insurance factor is a geometric series:

A_x =\sum_{k=1}^{\infty}v^k(0.90)^{k-1}(0.10) =\frac{0.10v}{1-0.90v} =\frac{2/21}{1-6/7} =\frac23.

The expected present value of benefits is $\$50{,}000(2/3)=\$33{,}333.33$. The three-payment premium factor is

\ddot{a}_{x:\overline{3}|} =1+\frac67+\frac{36}{49} =\frac{127}{49} \approx 2.5918367.

Thus

P=\frac{50{,}000(2/3)}{127/49} =\frac{4{,}900{,}000}{381} \approx \$12{,}860.89.

Checking, $\$12{,}860.89(2.5918367)\approx \$33{,}333.33$. The benefit remains payable for life, but that does not make the premium factor a whole life annuity: only three premiums are due. The calculation avoids using an annuity factor that continues premium payments after the stated payment period.

Timing Checks and Common Traps

Before calculating, write down the premium dates and benefit dates. This simple step prevents a frequent error: assigning a payment at time $r$ when an $r$-payment annuity-due stream actually ends at time $r-1$. It also makes the survival condition visible. A premium at time $j$ is multiplied by ${}_jp_x$, not by the probability of surviving $j+1$ years.

For example, if premiums are due at times 0 and 1, the factor is $1+v{}_1p_x$. It is not $1+v{}_1p_x+v^2{}_2p_x$. The latter has three payments, at times 0, 1, and 2. Likewise, if premiums start at time $s$, the first premium's probability of payment is ${}_sp_x$, not ${}_{s+1}p_x$.

Key takeaway: Calculate the benefit present value from the coverage period and benefit timing. Calculate the premium present value from the actual premium dates and survival condition. The net level premium is the benefit present value divided by the matching premium annuity factor.

Check Your Understanding

For each hypothetical contract, identify the appropriate premium timing and calculate the net annual premium. Premiums are payable only if the insured is alive at the due date.

A three-year term policy pays $20,000 at the end of the year of death. The effective annual interest rate is $0\%$, and $q=0.05$ each year. Premiums are due at times 0 and 1. (Answer: $1,462.82)
A four-year term policy pays $15,000 at the end of the year of death. The effective annual interest rate is $0\%$, and $q=0.10$ each year. Premiums are due at times 0, 1, 2, and 3. (Answer: $1,500.00)
A three-year term policy pays $30,000 at the end of the year of death. The effective annual interest rate is $5\%$, and $q=0.10$ each year. Premiums are due at times 0 and 1. (Answer: $3,987.44)
A four-year term policy pays $10,000 at the end of the year of death. The effective annual interest rate is $0\%$, and $q=0.20$ each year. Premiums are deferred until times 1 and 2. (Answer: $4,100.00)

What You'll Learn