Term Insurance With a Non-Level Benefit Schedule

Why a Year-by-Year Benefit Schedule Matters

A term policy need not pay the same amount whenever death occurs. A hypothetical policy might pay one benefit if death occurs during the first policy year and a different benefit if it occurs during a later year. To value it, pair each possible death-year benefit with the probability of death in that year and the present value of that payment.

This tutorial assumes a discrete benefit paid at the end of the policy year of death. The insured is age $x$ at issue, and the term lasts $n$ years. The benefit for death in policy year $k$ is $B_k$, for $k=1,\ldots,n$. The interest rate is an effective annual rate $i$, with $v=1/(1+i)$. Benefits are paid only if death occurs during the term.

Definition: The expected present value (EPV) of a term insurance benefit schedule $B_1,\ldots,B_n$ is the probability-weighted average of the present value of the benefit paid. If $D_k$ is the probability that death occurs in policy year $k$, then $$ \operatorname{EPV}=\sum_{k=1}^{n} B_k v^k D_k. $$ Here $D_k$ is an unconditional probability: it is the probability, measured at issue, that death occurs in that particular policy year.

The year index matters twice. The benefit $B_k$ must match death in year $k$, and a payment at the end of that year is discounted for $k$ years. A benefit paid at the end of year 3 is discounted by $v^3$, not $v^2$. The formula does not apply unchanged to a benefit paid at the moment of death or at the start of a policy year, because those payment times have different present values.

Building the Death-Year Probabilities

The earlier material on term insurance distinguishes the one-year mortality rate at age $x+k-1$, denoted $q_{x+k-1}$, from the probability of surviving to the start of year $k$. For a life aged $x$ at issue, the probability of death in policy year $k$ is

D_k={}_{k-1}p_x q_{x+k-1}.

This multiplication is needed when the inputs are conditional one-year mortality rates: the person must first survive the preceding $k-1$ years and then die during year $k$. In particular, $D_1=q_x$, and for $k>1$, ${}_{k-1}p_x$ is the product of the preceding one-year survival probabilities. If the problem instead supplies unconditional death-year probabilities directly, use those as $D_k$; do not multiply them by survival again.

Theorem: For an $n$-year discrete term policy with benefit $B_k$ paid at the end of policy year $k$ if death occurs in that year, and with effective annual interest rate $i$, $$ \operatorname{EPV}=\sum_{k=1}^{n} B_k v^k {}_{k-1}p_x q_{x+k-1}, \qquad v=\frac{1}{1+i}. $$ This form uses conditional mortality rates. If unconditional probabilities $D_k$ are given, replace ${}_{k-1}p_xq_{x+k-1}$ with $D_k$.

The formula follows by splitting the policy’s present-value random variable into mutually exclusive death-year outcomes. On death in year $k$, its value at issue is $B_kv^k$; if death does not occur during the term, its value is zero. Taking the expected value over these outcomes gives the sum below.

E[\text{present value}] =\sum_{k=1}^{n}(B_kv^k)P(\text{death in year }k) +0\cdot P(\text{survival through year }n) =\sum_{k=1}^{n}B_kv^kD_k.

This argument also explains why the benefit amount must be inside the year-by-year sum. A single level-benefit term-insurance factor cannot be applied to a non-level schedule unless the schedule is first represented as separate benefit components.

Worked Examples

Worked Example: Conditional Mortality Rates and Changing Benefits

A hypothetical three-year term policy pays $10,000 for death in year 1, $6,000 for death in year 2, and $14,000 for death in year 3. The effective annual interest rate is $i=0.05$. The conditional mortality rates are $q_x=0.10$, $q_{x+1}=0.20$, and $q_{x+2}=0.25$. Find the EPV, rounded to the nearest cent.

Method. First calculate the unconditional death probability for each year, then multiply each year’s benefit by that probability and its year-specific discount factor.

The annual survival probabilities are $p_x=1-0.10=0.90$, $p_{x+1}=1-0.20=0.80$, and $p_{x+2}=1-0.25=0.75$. Thus the death-year probabilities are:

D_1=0.10,\qquad D_2=0.90(0.20)=0.18,\qquad D_3=0.90(0.80)(0.25)=0.18.

The discount factors are $v=1/1.05$, $v^2=1/1.05^2$, and $v^3=1/1.05^3$. The contributions to the EPV are:

\begin{aligned} \text{Year 1: }&10{,}000(0.10)/1.05=952.3810,\\ \text{Year 2: }&6{,}000(0.18)/1.05^2=979.5918,\\ \text{Year 3: }&14{,}000(0.18)/1.05^3=2{,}176.8710. \end{aligned}

Adding the unrounded contributions gives $952.380952\ldots+979.591837\ldots+2{,}176.870964\ldots=4{,}108.843753\ldots$. The EPV, rounded to the nearest cent, is $4,108.84.

Check. The probability of surviving all three years is $0.90(0.80)(0.75)=0.54$, and $0.10+0.18+0.18+0.54=1$. The death-year probabilities are consistent. This method avoids the mistake of using $q_{x+1}$ alone as the probability of a year-2 death.

Worked Example: A Supplied Death-Year Probability Schedule

A hypothetical four-year policy pays $1,100 for death in year 1, $5,500 in year 2, and $5,000 in either year 3 or year 4. The effective annual interest rate is $i=0.10$. The probabilities of death in years 1 through 4, measured at issue, are $0.10$, $0.11$, $0.1331$, and $0.14641$. Find the EPV to the nearest cent and check that the probabilities are possible.

Method. The probabilities given are already unconditional, so use them directly. The probability of surviving through year 4 is the probability left over after the four death-year probabilities are added:

0.10+0.11+0.1331+0.14641=0.48951,\qquad P(\text{survive through year 4})=1-0.48951=0.51049.

These probabilities are nonnegative and, with the survival probability, sum to 1. The discount factors are $v=1/1.1$, $v^2=1/1.21$, $v^3=1/1.331$, and $v^4=1/1.4641$. The year-by-year present-value contributions are:

\begin{aligned} \text{Year 1: }&1{,}100(0.10)/1.1=100,\\ \text{Year 2: }&5{,}500(0.11)/1.21=500,\\ \text{Year 3: }&5{,}000(0.1331)/1.331=500,\\ \text{Year 4: }&5{,}000(0.14641)/1.4641=500. \end{aligned}

The EPV is $100+500+500+500=$ $1,600.00. No further survival factors belong in the calculation: each supplied probability already describes death in its stated year from the issue date.

Worked Example: Recovering Conditional Rates Before Valuation

A hypothetical three-year term policy pays $5,000, $8,000, and $10,000 for death in years 1, 2, and 3, respectively. The conditional mortality rates are $q_x=0.10$, $q_{x+1}=0.20$, and $q_{x+2}=0.25$. The effective annual interest rate is $i=0$. Find the EPV and the probability of surviving the term.

Method. A zero interest rate means $v=1$, so discounting does not change the benefit amounts. It does not remove the need to calculate the probability of reaching each policy year.

The death-year probabilities are $D_1=0.10$, $D_2=0.90(0.20)=0.18$, and $D_3=0.90(0.80)(0.25)=0.18$. Their weighted benefits are:

\begin{aligned} \text{Year 1 contribution: }&5{,}000(0.10)=500,\\ \text{Year 2 contribution: }&8{,}000(0.18)=1{,}440,\\ \text{Year 3 contribution: }&10{,}000(0.18)=1{,}800. \end{aligned}

Therefore, the EPV is $500+1{,}440+1{,}800=$ $3,740.00. The probability of surviving all three years is $0.90(0.80)(0.75)=0.54$. As a check, the probability of death during the term is $0.10+0.18+0.18=0.46$, and $0.46+0.54=1$.

This example isolates the mortality calculation: even when there is no discounting, a year-3 death benefit is weighted by the probability of surviving the first two years and then dying in year 3.

Common Mistakes and Practical Checks

The most frequent error is mixing up a conditional one-year mortality rate and an unconditional death-year probability. The conditional rate $q_{x+1}$ answers, “What is the probability of dying during the next year, given survival to age $x+1$?” It is not, by itself, the probability of dying in policy year 2 from issue.

For example, with $q_x=0.10$ and $q_{x+1}=0.20$, the incorrect year-2 contribution for an $8,000 benefit at $i=0$ would be $8{,}000(0.20)=\$1{,}600$. This omits the condition that the insured must reach year 2. The correct calculation is $8{,}000(0.90)(0.20)=\$1{,}440$. If instead the problem states that the unconditional probability of death in year 2 is $0.18$, use $8{,}000(0.18)$ directly.

Match the benefit to the death year. Use $B_k$ only with the probability and discount factor for year $k$.
Discount to the payment time. An end-of-year-$k$ payment uses $v^k$, even though death occurs during that policy year.
Do not double-count survival. Apply ${}_{k-1}p_x$ when starting from conditional mortality rates, but not when $D_k$ is already supplied.
Check probability totals. The death-year probabilities over the term cannot sum to more than 1; the remainder is the probability of survival through the term.

A reliable calculation table has one row per policy year and separate columns for the benefit, death-year probability, discount factor, and present-value contribution. This layout makes it easier to detect a shifted benefit, a missing survival factor, or a discount exponent that does not match the payment time. The result is specific to the stated benefit schedule, mortality assumptions, interest rate, and payment timing.

Check Your Understanding

For each problem, use the stated payment timing and round money to the nearest cent where needed.

A two-year policy pays $4,000 for death in year 1 and $9,000 for death in year 2. The conditional rates are $q_x=0.10$ and $q_{x+1}=0.20$, and $i=0$. Find the EPV. (Answer: $2,020.00)
A three-year policy pays $2,000, $4,000, and $5,000 by death year. Its unconditional death-year probabilities are $0.10$, $0.15$, and $0.20$, and $i=0.05$. Find the EPV. (Answer: $1,598.53)
A two-year policy pays $3,000 in year 1 and $7,000 in year 2. The conditional rates are $q_x=0.20$ and $q_{x+1}=0.10$, with $i=0$. Find the EPV. (Answer: $1,160.00)
A three-year policy pays $5,000, $4,000, and $9,000 by death year. The conditional mortality rates are $0.10$, $0.20$, and $0.25$, and $i=0.10$. Find the EPV. (Answer: $2,266.72)
A four-year policy pays $5,000, $8,000, $12,000, and $20,000 by death year. Its unconditional death-year probabilities are $0.08$, $0.12$, $0.10$, and $0.05$, and $i=0.10$. Find the EPV. (Answer: $2,741.62)

What You'll Learn