Deferred Term Insurance Values

Why the Coverage Interval Matters

A deferred term contract pays a death benefit only if death occurs in a specified interval: not during the deferment period, and not after the term ends. Its value therefore depends on the exact times at which coverage applies. A correct mortality model and interest rate cannot compensate for an incorrectly identified interval.

Measure time $t$ from the policy issue date. Let $m$ be the deferment period and $n$ the length of the term after deferment. The covered interval is from time $m$ through time $m+n$. Thus, a deferment of 3 years followed by a 5-year term means coverage from year 3 to year 8 after issue—not from year 3 to year 5.

Definition: A deferred term insurance with level benefit $B$, deferment period $m$, and subsequent term length $n$ pays $B$ at the time of death if death occurs during the covered interval from $m$ to $m+n$. It pays nothing for death before the deferment ends or after the term ends. Time is measured from issue throughout.

For a continuously distributed future lifetime, the probability of death at either exact endpoint is zero, so whether the interval is written with open or closed endpoints does not affect its expected present value. The formulas below assume a continuous lifetime model with no probability mass at a fixed time. They also assume a constant force of interest $\delta$, so a payment at time $t$ has present value $Be^{-\delta t}$.

Direct Valuation Over the Covered Interval

Let ${}_tp_x$ be the probability that a life aged $x$ survives for $t$ years, and let $\mu_{x+t}$ be the force of mortality at attained age $x+t$. Under a continuous lifetime model, the density of death at time $t$ is ${}_tp_x\mu_{x+t}$. The expected present value is the discounted benefit multiplied by the death density and integrated only over times when the contract pays.

\operatorname{EPV} = B\int_m^{m+n}e^{-\delta t}\,{}_tp_x\mu_{x+t}\,dt.

This direct integral is the general valuation method for a level benefit when mortality may vary with age or duration. The integral must include the probability of reaching each time $t$, including the mortality experience before coverage starts. If mortality is specified only during the covered interval, the probability of surviving to the beginning of that interval must also be supplied before a numerical value can be found.

Constant Force of Mortality

Suppose the force of mortality is a constant $\mu$ from issue through the end of coverage, and the force of interest is the constant $\delta$. Then ${}_tp_x=e^{-\mu t}$, and the death density at time $t$ is $\mu e^{-\mu t}$. Substituting these expressions into the direct integral gives a convenient closed form.

\begin{aligned} \operatorname{EPV} &=B\int_m^{m+n}e^{-\delta t}\mu e^{-\mu t}\,dt\\ &=B\mu\int_m^{m+n}e^{-(\mu+\delta)t}\,dt\\ &=B\frac{\mu}{\mu+\delta} \left(e^{-(\mu+\delta)m}-e^{-(\mu+\delta)(m+n)}\right). \end{aligned}

The last step follows by integrating the exponential: for $r=\mu+\delta>0$, an antiderivative of $e^{-rt}$ is $-e^{-rt}/r$. Evaluating it at the upper and lower limits produces $e^{-rm}-e^{-r(m+n)}$. The factor $\mu$ in the density then gives $\mu/r$. This also explains why the expression uses the combined rate $\mu+\delta$: mortality reduces the probability of a payment at later times, while interest reduces its present value.

The formula requires a constant mortality force over the entire period from issue to the term boundary, not merely a constant force during the covered term. With piecewise or otherwise varying mortality, use the direct integral and account for the appropriate survival probability in each interval.

Worked Example: Constant Forces Over a Deferred Term

A hypothetical policy pays $50,000 at death during the 4-year term that begins after a 2-year deferment. Assume a constant force of mortality $\mu=0.04$ from issue onward and a constant force of interest $\delta=0.05$. Find the expected present value, rounded to the nearest cent.

Method and interval. Time is measured from issue. The deferment is $m=2$, and the subsequent term is $n=4$, so the covered interval is year 2 to year 6. Set $B=50{,}000$ and $\mu+\delta=0.04+0.05=0.09$.

Substitute into the constant-force formula:

\begin{aligned} \operatorname{EPV} &=50{,}000\frac{0.04}{0.09} \left(e^{-0.09(2)}-e^{-0.09(2+4)}\right)\\ &=50{,}000\left(\frac{4}{9}\right) \left(e^{-0.18}-e^{-0.54}\right). \end{aligned}

Evaluate the exponential terms: $e^{-0.18}\approx0.8352702114$ and $e^{-0.54}\approx0.5827482524$. Their difference is $0.2525219590$. Multiplying by $4/9$ gives $0.1122319818$, and multiplying by $50,000 gives $5,611.5991. Rounded to the nearest cent, the expected present value is $5,611.60.

Check. The result is positive and is less than the face amount of $50,000. The calculation also discounts from issue: the covered interval is years 2 through 6, not a 4-year interval starting at time zero. The common mistake avoided here is treating the term length as the final policy year.

Piecewise Mortality Forces

When the force of mortality changes during the contract, split the valuation integral wherever the force changes. On each segment, use the force for that segment and the survival probability accumulated from issue up to the time being valued. Do not restart survival at a segment boundary as if the life were known to be alive without accounting for the probability of reaching that boundary.

For example, if mortality has one constant force up to time 2 and a different constant force after time 2, and the covered interval crosses year 2, the integral has one part before year 2 and another after year 2. In the second part, survival to year 2 is already reflected in the cumulative survival probability.

Worked Example: A Mortality Change Within the Covered Term

A hypothetical $40,000 benefit is payable at death from year 1 through year 4 after issue. The force of mortality is $0.03$ for $0\le t<2$, and $0.05$ for $t\ge2$. The force of interest is $\delta=0.02$. Find the expected present value, rounded to the nearest cent.

Method and interval. The coverage runs from $t=1$ to $t=4$, and the mortality force changes at $t=2$, so split the integral into $[1,2]$ and $[2,4]$. On the first segment, survival from issue is $e^{-0.03t}$. On the second segment, the cumulative mortality force through time $t$ is $0.03(2)+0.05(t-2)$.

For years 1 to 2, the combined mortality and interest exponent has rate $0.03+0.02=0.05$:

\begin{aligned} I_1 &=\int_1^2 0.03e^{-(0.03+0.02)t}\,dt\\ &=\frac{0.03}{0.05}\left(e^{-0.05}-e^{-0.10}\right)\\ &=0.6(0.951229425-0.904837418)\\ &\approx0.027835204. \end{aligned}

For years 2 to 4, mortality accumulated before year 2 is $0.03(2)=0.06$. Including interest, the exponent at $t=2$ is $0.06+0.02(2)=0.10$, and the combined rate after year 2 is $0.05+0.02=0.07$. Therefore:

\begin{aligned} I_2 &=\int_2^4 0.05e^{-0.06-0.05(t-2)-0.02t}\,dt\\ &=\frac{0.05}{0.07}\left(e^{-0.10}-e^{-0.24}\right)\\ &=\frac{5}{7}(0.904837418-0.786627861)\\ &\approx0.084435398. \end{aligned}

The unit-benefit value is $I_1+I_2\approx0.027835204+0.084435398=0.112270602$. Thus the expected present value is $40{,}000(0.112270602)\approx\$4{,}490.8241$, or $4,490.82 to the nearest cent.

Check. The covered interval is years 1 to 4, and both mortality segments are represented. The second segment begins with the accumulated mortality exponent $0.06$; omitting it would incorrectly treat the life as certain to have survived to year 2. The common mistake avoided is restarting the survival calculation at the force-change point.

Changing a Boundary or Solving for a Term

A boundary change affects only the part of the interval that is added or removed. For example, extending the end of coverage from year $b$ to year $b+1$ adds the value of death benefits during years $b$ to $b+1$; the earlier covered years do not need to be revalued. Similarly, a target expected present value can be used to solve for an unknown term length by isolating the exponential associated with the term boundary.

Worked Example: Incremental Value of Extending Coverage

A hypothetical $100,000 benefit is payable after a 2-year deferment. Under constant forces $\mu=0.03$ and $\delta=0.05$, calculate the additional expected present value from extending the coverage endpoint from year 7 to year 8 after issue. Round to the nearest cent.

Method and interval. The extension adds only the interval from year 7 to year 8. Here $\mu+\delta=0.08$, so the value of that added interval is

\begin{aligned} \Delta\operatorname{EPV} &=100{,}000\frac{0.03}{0.08} \left(e^{-0.08(7)}-e^{-0.08(8)}\right)\\ &=100{,}000(0.375)\left(e^{-0.56}-e^{-0.64}\right). \end{aligned}

Using $e^{-0.56}\approx0.571209063$ and $e^{-0.64}\approx0.527292424$, the difference is $0.043916639$. Multiplication by $0.375$ gives $0.016468740$; multiplication by $100,000 gives $1,646.87396. The additional expected present value is therefore $1,646.87.

Check. The answer values only the added year, so it is smaller than the value of the full contract. The common mistake avoided is valuing the entire deferred term again instead of isolating the new coverage interval.

Worked Example: Solving for the Subsequent Term Length

A hypothetical $60,000 benefit is payable at death after a 4-year deferment and during the following term. Assume constant forces $\mu=0.04$ and $\delta=0.03$. Find the subsequent term length $n$ that gives an expected present value of $6,000, rounded to three decimal places.

Method and interval. The covered interval is year 4 to year $4+n$, and $\mu+\delta=0.07$. The target unit-benefit value is $6{,}000/60{,}000=0.1$. Set the formula equal to this target:

\begin{aligned} 0.1 &=\frac{0.04}{0.07} \left(e^{-0.07(4)}-e^{-0.07(4+n)}\right),\\ e^{-0.07(4+n)} &=e^{-0.28}-\frac{0.1}{0.04/0.07}\\ &=e^{-0.28}-0.175\\ &\approx0.755783741-0.175\\ &=0.580783741. \end{aligned}

Taking natural logarithms gives $-0.07(4+n)=\ln(0.580783741)$, so

n=\frac{-\ln(0.580783741)}{0.07}-4\approx3.76253\text{ years}.

Rounded to three decimal places, the subsequent term length is 3.763 years. Substituting the unrounded value into the original expression gives unit value $0.1$, and hence an expected present value of $60{,}000(0.1)=\$6{,}000$. Using the rounded term length causes only a small rounding difference.

Check. The solved quantity is the term after the deferment, not the total time from issue. The common mistake avoided is subtracting 4 years from the target value or treating the 4-year deferment as part of the unknown term.

Common Checks and Exam Traps

Deferred term calculations are especially sensitive to time indexing. Before evaluating an integral or formula, write the covered interval in years measured from issue. Then identify every mortality-force change and every interest assumption. These checks help catch errors before arithmetic begins.

Confusing deferment with term length. A deferment of $m$ followed by a term of length $n$ ends at $m+n$, not at $n$.
Discounting from the start of coverage. Unless the problem asks for a value at time $m$, payments are discounted from issue. The factor for a payment at time $t$ is $e^{-\delta t}$.
Using a constant-force formula outside its assumptions. The closed form above requires the same constant $\mu$ from issue through the endpoint. With changing mortality, integrate piece by piece.
Dropping survival before coverage. A death benefit is paid only if the insured is alive at the start of coverage. In a direct integral, this is included in ${}_tp_x$ or the appropriate cumulative survival expression.
Valuing too much when extending coverage. The incremental value is the integral over the added interval only.

Key takeaway: Define $m$ and $n$ from issue, so the deferred term covers years $m$ through $m+n$. Value only deaths in that interval, discount from issue, and use the direct integral whenever the mortality force is not constant over the full period.

Check Your Understanding

For each problem, measure time from issue and use a constant force of interest as stated.

A hypothetical $30,000 benefit is payable from year 2 through year 5. If $\mu=0.02$ throughout and $\delta=0.03$, calculate the expected present value to the nearest cent. (Answer: $1,512.44)
A hypothetical $20,000 benefit is payable from year 1 through year 3. Mortality has force $0.02$ before year 1 and $0.04$ from year 1 onward; $\delta=0.01$. Calculate the expected present value to the nearest cent. (Answer: $1,477.60)
Under constant forces $\mu=0.025$ and $\delta=0.035$, a hypothetical $80,000 deferred term contract has its endpoint extended from year 5 to year 6. Find the increase in expected present value to the nearest cent. (Answer: $1,438.06)
A hypothetical $25,000 benefit is payable from year 1 through year 4. Mortality has force $0.02$ before year 2 and $0.04$ from year 2 onward; $\delta=0.01$. Split the valuation at year 2 and calculate the expected present value to the nearest cent. (Answer: $2,270.43)

What You'll Learn