Net Premiums for Varying Benefit Amounts

Why Varying Benefits Need a Year-by-Year Calculation

Some hypothetical life insurance contracts pay different amounts depending on when death occurs. For example, a contract might pay a smaller amount in its first policy year and larger amounts in later years. A single benefit amount cannot represent this design: the expected present value must account for the benefit payable in each year and the probability that the insured survives long enough to reach that year.

The calculation uses the same net-premium principle as level-benefit insurance: the expected present value of premiums equals the expected present value of benefits, under the stated interest and mortality assumptions. The difference is that the benefit present value is a sum of year-specific terms. A net premium does not include expenses or profit.

Definition: Suppose an $n$-year discrete insurance pays benefit $B_k$ at the end of policy year $k$ if death occurs during that year, for $k=1,\ldots,n$. Let $q_{x+k-1}$ be the conditional probability of death during policy year $k$, given survival to its start. The probability of death in year $k$, conditional on being alive at issue, is ${}_{k-1}p_xq_{x+k-1}$. With effective annual interest rate $i$ and $v=1/(1+i)$, the expected present value of benefits is

\operatorname{EPV}(\text{benefits}) =\sum_{k=1}^{n} v^k B_k\,{}_{k-1}p_xq_{x+k-1}.

This formula assumes benefits are paid at the end of the policy year of death, interest is effective annually, and mortality probabilities are conditional on survival to the start of the relevant year. If a contract pays at another time, such as immediately at death, its discounting must reflect that timing. The formula also assumes no benefit is paid if the insured survives the term.

Why the Benefit Present-Value Formula Holds

The possible claim events are mutually exclusive: death occurs in year 1, year 2, and so on, or no claim occurs during the term. For death in year $k$, the insured must first survive $k-1$ years, with probability ${}_{k-1}p_x$, and then die in year $k$, with conditional probability $q_{x+k-1}$. The claim amount $B_k$ is paid at time $k$, so its present value in that event is $v^kB_k$. The expectation of the claim present value is therefore the sum of these present values multiplied by their event probabilities:

\sum_{k=1}^{n} (v^kB_k)P(\text{death in year }k) = \sum_{k=1}^{n}v^kB_k\,{}_{k-1}p_xq_{x+k-1}.

This is an application of the expectation rule for mutually exclusive outcomes. The survival factor is essential in every year after the first; a conditional death probability by itself is not the probability of a claim in that year from issue.

Worked Example: Net Single Premium for a Three-Year Benefit Schedule

A hypothetical three-year term policy pays $B_1=\$10{,}000$, $B_2=\$20{,}000$, or $B_3=\$30{,}000$ at the end of the year of death. The effective annual interest rate is $i=5\%$. Conditional death probabilities are $q_x=0.10$, $q_{x+1}=0.12$, and $q_{x+2}=0.15$. Find the net single premium.

Method. Calculate the unconditional-from-issue probability of death in each policy year, multiply by that year’s benefit and discount factor, and add the expected present values.

First, the survival probabilities at the starts of years 1, 2, and 3 are $1$, $0.90$, and $0.90(1-0.12)=0.792$. Thus the probabilities of death in the respective years are $0.10$, $0.90(0.12)=0.108$, and $0.792(0.15)=0.1188$.

The discount factors are $v=1/1.05$, $v^2=1/1.05^2$, and $v^3=1/1.05^3$. The three benefit present-value contributions are

\begin{aligned} \text{Year 1: }&\$10{,}000(0.10)v=\$952.380952\ldots,\\ \text{Year 2: }&\$20{,}000(0.108)v^2=\$1{,}959.183673\ldots,\\ \text{Year 3: }&\$30{,}000(0.1188)v^3=\$3{,}078.716991\ldots. \end{aligned}

Adding the unrounded contributions gives $\$5{,}990.281616\ldots$. The net single premium is therefore \$5,990.28, rounded to the nearest cent. The result is checked by summing the three mutually exclusive claim-year contributions; no contribution is included for survival beyond the three-year term.

From Expected Benefits to a Level Net Premium

If premiums are paid annually in advance while the insured is alive, a premium at time $j$ is payable only if the insured survives to that time. For an $n$-year term with annual premium $P$, the expected present value of premiums is

P\sum_{j=0}^{n-1}v^j\,{}_jp_x.

The $j=0$ term is $P$, paid at issue with certainty. For $j\geq 1$, the premium at time $j$ is paid only if the insured survives $j$ years. Equating this premium value to the expected present value of the varying benefits gives

P= \frac{\displaystyle\sum_{k=1}^{n}v^kB_k\,{}_{k-1}p_xq_{x+k-1}} {\displaystyle\sum_{j=0}^{n-1}v^j\,{}_jp_x}.

This level-premium formula assumes premiums are annual, payable in advance while the insured is alive, and cease at death or at the end of the term. If the contract has a different premium schedule, the denominator must be changed to the expected present value of that schedule. The benefit schedule and premium schedule are separate parts of the calculation.

Worked Example: Level Net Premium with Changing Benefits

A hypothetical three-year policy pays $\$10{,}000$, $\$15{,}000$, or $\$20{,}000$ at the end of the year of death in years 1, 2, or 3. The effective annual interest rate is $5\%$. The conditional death probabilities are $0.10$, $0.20$, and $0.25$, respectively. Annual premiums are payable in advance while the insured is alive, beginning at issue. Calculate the annual net premium.

Method. Find the benefit present value by year, then divide by the expected present value of the premium stream.

Death in year 1 has probability $0.10$. Death in year 2 has probability $0.90(0.20)=0.18$. Survival to the start of year 3 is $0.90(0.80)=0.72$, so death in year 3 has probability $0.72(0.25)=0.18$. The benefit present value is

\begin{aligned} \operatorname{EPV}(\text{benefits}) &=\$10{,}000(0.10)v+\$15{,}000(0.18)v^2+\$20{,}000(0.18)v^3\\ &=\frac{\$1{,}000}{1.05} +\frac{\$2{,}700}{1.05^2} +\frac{\$3{,}600}{1.05^3}\\ &=\$6{,}511.175790\ldots. \end{aligned}

Premiums can be paid at time 0, at time 1 if alive, and at time 2 if alive. The expected present value of a unit annual premium is

1+0.90v+0.72v^2 =1+\frac{6}{7}+\frac{32}{49} =\frac{123}{49}.

Consequently,

P=\frac{\$6{,}511.175790\ldots}{123/49} =\$2{,}593.883\ldots.

The annual net premium is \$2,593.88, rounded to the nearest cent. As a check, multiplying the unrounded premium by the unit-premium present value $123/49$ recovers the benefit present value. This method avoids treating all three premium payments as certain: the later payments depend on survival.

Benefit Schedule, Mortality Schedule, and Timing

A reliable calculation keeps three schedules distinct: the benefit $B_k$, the conditional death probability $q_{x+k-1}$, and the discount factor $v^k$. Their product, together with survival to the start of the year, is the expected present value contribution for year $k$. A change in any one schedule affects the price, but it should not be substituted for either of the others.

Item for policy year $k$	Meaning	Role in the benefit present value
$B_k$	Benefit payable if death occurs in year $k$	Claim amount in that year
${}_{k-1}p_x$	Probability of surviving to the start of year $k$	Accounts for reaching that year
$q_{x+k-1}$	Conditional probability of death during year $k$	Gives death probability conditional on reaching the year
$v^k$	Present-value factor for payment at the end of year $k$	Discounts the claim to issue

Worked Example: Increasing Benefits and Year-Specific Death Probabilities

A hypothetical four-year term policy pays $\$5{,}000$, $\$10{,}000$, $\$15{,}000$, and $\$20{,}000$ at the end of the year of death. The effective annual interest rate is $4\%$. The conditional death probabilities by year are $0.05$, $0.10$, $0.15$, and $0.20$. Find the net single premium.

Method. Build the survival probability forward one year at a time, then calculate each discounted claim contribution.

The probabilities of death in years 1 through 4 are, respectively, $0.05$, $0.95(0.10)=0.095$, $0.95(0.90)(0.15)=0.12825$, and $0.95(0.90)(0.85)(0.20)=0.14535$. Using $v=1/1.04$, the benefit contributions are

\begin{aligned} \text{Year 1: }&\$5{,}000(0.05)v=\$240.384615\ldots,\\ \text{Year 2: }&\$10{,}000(0.095)v^2=\$878.328402\ldots,\\ \text{Year 3: }&\$15{,}000(0.12825)v^3=\$1{,}710.206745\ldots,\\ \text{Year 4: }&\$20{,}000(0.14535)v^4=\$2{,}484.915783\ldots. \end{aligned}

The sum of the unrounded contributions is $\$5{,}313.835545\ldots$, so the net single premium is \$5,313.84, rounded to the nearest cent. The death probabilities add to $0.41865$; the remaining probability, $1-0.41865=0.58135$, is the probability of surviving the full term and receiving no term benefit. This provides a probability check on the year-by-year mortality calculation.

Common Exam Trap: Using a Conditional Rate as a Claim Probability

A frequent error is to multiply a later-year benefit by that year’s conditional death probability alone. In the first worked example, the incorrect year-3 calculation would be

\$30{,}000(0.15)v^3=\$3{,}887.27\text{ (approximately)}.

This treats the insured as certain to be alive at the start of year 3. The insured must survive the first two years, with probability $0.90(0.88)=0.792$. The correct year-3 contribution is

\$30{,}000(0.792)(0.15)v^3 =\$3{,}078.716991\ldots.

The conditional rate $q_{x+2}$ applies only to those who reach the start of year 3. Always multiply it by the probability of reaching that year before applying the benefit and discount factor.

Key takeaway: For each policy year, multiply the benefit payable in that year by the probability of surviving to its start, the conditional probability of death during it, and the appropriate discount factor. For a level net premium, divide the resulting benefit present value by the expected present value of the stated premium stream.

Check Your Understanding

For each problem, use the stated payment timing and assumptions. Answers are given at the end of each item for checking.

A two-year term policy pays $\$4{,}000$ for death in year 1 and $\$9{,}000$ for death in year 2. The effective annual interest rate is $0\%$, and the conditional death probabilities are $0.10$ and $0.20$. Find the net single premium. (Answer: $\$2{,}020.00$)
A two-year policy pays $\$6{,}000$ or $\$12{,}000$ at the end of the year of death in years 1 and 2. The effective annual interest rate is $5\%$, and the conditional death probabilities are $0.10$ and $0.20$. Find the net single premium. (Answer: $\$2{,}530.61$)
Use the policy assumptions in Question 2, but let premiums be level and payable at times 0 and 1 while alive. Find the annual net premium. (Answer: $\$1{,}362.64$)
A three-year policy pays $\$5{,}000$, $\$10{,}000$, or $\$15{,}000$ at the end of the year of death. The effective annual interest rate is $0\%$, and the conditional death probabilities are $0.10$, $0.10$, and $0.20$. Find the net single premium. (Answer: $\$3{,}830.00$)
A two-year policy pays $\$8{,}000$ in year 1 and $\$16{,}000$ in year 2. The effective annual interest rate is $4\%$, and the conditional death probabilities are $0.05$ and $0.10$. Find the net single premium. (Answer: $\$2{,}170.12$)

Item for policy year \(k\)	Meaning	Role in the benefit present value
\(B_k\)	Benefit payable if death occurs in year \(k\)	Claim amount in that year
\({}_{k-1}p_x\)	Probability of surviving to the start of year \(k\)	Accounts for reaching that year
\(q_{x+k-1}\)	Conditional probability of death during year \(k\)	Gives death probability conditional on reaching the year
\(v^k\)	Present-value factor for payment at the end of year \(k\)	Discounts the claim to issue

What You'll Learn