Decomposing Term Insurance Into Time Segments

Why Decompose a Changing Term Benefit?

A term contract may provide one death benefit during the first few years, a different amount later, and no benefit during a gap. Rather than valuing the entire schedule at once, divide the coverage period at every point where the benefit changes. Each resulting component is a term benefit restricted to one time interval. The contract’s expected present value is the sum of the interval values because only one death time can occur, and the benefit at that time is the amount assigned to its interval.

This approach is useful when benefits increase or decrease by stage, when a contract has a waiting period before coverage begins, or when coverage temporarily ends and later resumes. The key is to keep the benefit amount, covered interval, and payment timing together. In this tutorial, the benefit is paid immediately at death, so its present value depends on the actual time of death within the interval.

Definition: Suppose a life aged $x$ has a death benefit $b(t)$ payable immediately at death time $t$, provided death occurs within the term. A segment on the interval $(s,t]$ has benefit $B$ if the payment is $B$ when death occurs after time $s$ and no later than time $t$, and zero otherwise. A changing-benefit term contract is the sum of its segment benefits on nonoverlapping intervals. If $s=0$, the segment is not deferred; if $s>0$, it is deferred until time $s$.

Valuing One Interval and Adding the Components

Let $T_x$ be the future lifetime of a life aged $x$, and let $\delta$ be a constant force of interest. A dollar paid at time $u$ has present value $e^{-\delta u}$. If the force of mortality at age $x+u$ is $\mu_{x+u}$, the probability density of death at time $u$ is ${}_up_x\mu_{x+u}$. Therefore, a benefit $B$ payable immediately at death only during $(s,t]$ has expected present value

B\int_s^t e^{-\delta u}{}_up_x\mu_{x+u}\,du.

For a contract with $m$ segments, let the $j$th segment have benefit $B_j$ on $(s_j,t_j]$. Its total expected present value is

\sum_{j=1}^{m} B_j\int_{s_j}^{t_j} e^{-\delta u}{}_up_x\mu_{x+u}\,du.

This integral form applies whenever the survival model has the stated density and the interest force is constant. If the force of interest varies with time, the discount factor must instead reflect accumulated interest; if mortality is specified by another model, use that model’s survival probabilities and death-time distribution. Do not apply the constant-force shortcut below unless both forces are constant over the intervals being valued.

Theorem: If the force of mortality is the constant $\mu$ and the force of interest is the constant $\delta$, then ${}_up_x=e^{-\mu u}$, and the expected present value of a benefit $B$ payable immediately at death during $(s,t]$ is $$ B\frac{\mu}{\mu+\delta}\left(e^{-(\mu+\delta)s}-e^{-(\mu+\delta)t}\right). $$ For $\mu+\delta=0$, the integral form applies directly; in the usual life insurance setting, $\mu>0$ and $\delta\geq 0$.

The result follows by substituting the constant-force survival model and mortality density into the integral. This derivation also explains why the exponent contains the sum of the mortality and interest forces: survival to time $u$ contributes $e^{-\mu u}$, and discounting contributes $e^{-\delta u}$.

\begin{aligned} B\int_s^t e^{-\delta u}{}_up_x\mu\,du &=B\int_s^t e^{-\delta u}e^{-\mu u}\mu\,du\\ &=B\mu\int_s^t e^{-(\mu+\delta)u}\,du\\ &=B\frac{\mu}{\mu+\delta} \left(e^{-(\mu+\delta)s}-e^{-(\mu+\delta)t}\right). \end{aligned}

Because the intervals do not overlap, the full schedule can be valued one segment at a time and then summed. Under a continuous lifetime distribution, the probability of death at any one exact boundary time is zero, so assigning a boundary to either adjacent interval does not change the value. The coverage intervals must still be specified consistently so that the schedule does not accidentally leave a gap or count a benefit twice.

Worked Example: Two Consecutive Benefit Levels

A hypothetical policy pays $50,000 immediately at death during years 0 to 1 and $80,000 immediately at death during years 1 to 2. There is no benefit after year 2. Assume a constant force of mortality $\mu=0.04$ and a constant force of interest $\delta=0.01$. Find the expected present value, rounded to the nearest cent.

Method. Split the schedule into $(0,1]$ with benefit $50,000 and $(1,2]$ with benefit $80,000. Here $\mu+\delta=0.05$ and $\mu/(\mu+\delta)=0.04/0.05=0.8$.

For the first interval, the expected present value is

\begin{aligned} 50{,}000(0.8)(1-e^{-0.05}) &=40{,}000(1-0.9512294245)\\ &=40{,}000(0.0487705755)\\ &=\$1{,}950.8230. \end{aligned}

For the second interval, the benefit is deferred to year 1 and covers deaths through year 2:

\begin{aligned} 80{,}000(0.8)(e^{-0.05}-e^{-0.10}) &=64{,}000(0.9512294245-0.9048374180)\\ &=64{,}000(0.0463920065)\\ &=\$2{,}969.0884. \end{aligned}

Adding the components gives $1{,}950.8230+2{,}969.0884=\$4{,}919.9114$, so the expected present value is $4,919.91 to the nearest cent. The method avoids treating both years as though they had the same benefit: each interval is valued using its own amount and endpoints.

Worked Example: A Waiting Period and a Coverage Gap

A hypothetical contract pays $30,000 immediately at death during the first year, pays nothing from year 1 through year 2, and pays $60,000 immediately at death during years 2 to 4. Assume constant forces $\mu=0.02$ and $\delta=0.03$. Calculate the expected present value, rounded to the nearest cent.

Method. The segments are $(0,1]$ and $(2,4]$. The gap $(1,2]$ contributes zero; it is not treated as covered merely because the later benefit begins at year 2. Here $\mu+\delta=0.05$ and $\mu/(\mu+\delta)=0.4$.

For the first segment,

\begin{aligned} 30{,}000(0.4)(1-e^{-0.05}) &=12{,}000(1-0.9512294245)\\ &=12{,}000(0.0487705755)\\ &=\$585.2469. \end{aligned}

For the deferred segment from year 2 to year 4,

\begin{aligned} 60{,}000(0.4)(e^{-0.10}-e^{-0.20}) &=24{,}000(0.9048374180-0.8187307531)\\ &=24{,}000(0.0861066649)\\ &=\$2{,}066.5600. \end{aligned}

The total is $585.2469+2{,}066.5600=\$2{,}651.8069$, which rounds to $2,651.81. This calculation makes the gap explicit: there is no component for deaths in years 1 to 2.

Worked Example: Three Benefit Stages

A hypothetical three-year policy pays $20,000 during years 0 to 1, $40,000 during years 1 to 2, and $60,000 during years 2 to 3, always immediately at death. The force of mortality and force of interest are each constant at $0.05$. Find the total expected present value, rounded to the nearest cent, and verify it by combining the integrals.

Method. The sum of forces is $0.10$, and the ratio $\mu/(\mu+\delta)$ is $0.05/0.10=0.5$. The endpoint exponential values needed are $e^{-0.10}=0.904837418$, $e^{-0.20}=0.818730753$, and $e^{-0.30}=0.740818221$.

The first segment contributes

20{,}000(0.5)(1-e^{-0.10}) =10{,}000(0.095162582) =\$951.6258.

The second segment contributes

40{,}000(0.5)(e^{-0.10}-e^{-0.20}) =20{,}000(0.086106665) =\$1{,}722.1333.

The third segment contributes

60{,}000(0.5)(e^{-0.20}-e^{-0.30}) =30{,}000(0.077912532) =\$2{,}337.3759.

Thus the total is $951.6258+1{,}722.1333+2{,}337.3759=\$5{,}011.1350$, or $5,011.14 to the nearest cent. Combining the integrals before evaluating gives the same result:

\begin{aligned} &10{,}000\int_0^1 e^{-0.10u}\,du +20{,}000\int_1^2 e^{-0.10u}\,du +30{,}000\int_2^3 e^{-0.10u}\,du\\ &\qquad=951.6258+1{,}722.1333+2{,}337.3759 =\$5{,}011.1350. \end{aligned}

The equality checks that the interval-by-interval calculation agrees with the direct piecewise-integral valuation.

Common Timing Error

A frequent mistake is to take the probability of death during a segment and discount the entire benefit at the segment’s starting time. That calculation treats every death in the segment as if payment occurred at its start. For continuous insurance payable immediately at death, payments later in the interval receive more discount.

For example, in the first worked example, the incorrect start-of-interval discounting for the second segment would give

80{,}000e^{-0.01}(e^{-0.04}-e^{-0.08}) \approx \$2{,}983.86,

whereas the correct value is $80{,}000(0.8)(e^{-0.05}-e^{-0.10})=\$2{,}969.09$. The incorrect expression uses the death probability over years 1 to 2 but discounts all those payments at year 1. The correct integral, or its constant-force formula, discounts each possible death payment at its actual time. If the contract instead paid only at the end of each year, a discrete-payment calculation would be needed; that is a different payment timing assumption.

Key takeaway: Mark every time at which the benefit changes, write one interval component for each covered segment, and value each component using the actual death-payment timing. Add the segment values only after checking that their intervals and benefit amounts reproduce the contract schedule.

Check Your Understanding

For each hypothetical contract, use immediate payment at death, the stated constant forces, and round the final expected present value to the nearest cent.

A policy pays $100,000 during years 0 to 1 and $150,000 during years 1 to 2. The forces are $\mu=0.03$ and $\delta=0.02$. Value both segments and add them. (Answer: $7,101.52)
A policy pays $25,000 during years 0 to 2 and $50,000 during years 2 to 4. The forces are $\mu=0.06$ and $\delta=0.04$. Calculate the two interval values and their sum. (Answer: $7,171.36)
A policy pays $40,000 only during years 1 to 3 and $90,000 only during years 4 to 5; there is no benefit in the other periods. The forces are $\mu=0.02$ and $\delta=0.03$. Include the deferral of both covered segments. (Answer: $2,885.82)
A policy pays $10,000 during years 0 to 1, nothing during years 1 to 2, and $30,000 during years 2 to 3. The forces are $\mu=0.08$ and $\delta=0.02$. Value the covered intervals only, then check that the gap contributes no benefit. (Answer: $2,631.20)

What You'll Learn