How Coverage Duration Changes a Term APV

Why Coverage Duration Matters

A level-benefit term policy pays only if death occurs during its stated coverage period. Extending that period adds a possible payment in the newly covered year, while leaving the earlier covered years unchanged. This makes it possible to compare two durations directly: the difference in their expected present values (APVs) is the discounted expected payment from the added year alone.

That comparison is useful when evaluating a sequence of coverage durations or checking a term-insurance calculation. The key is to account for two things in the added year: the insured must survive to the start of that year, and death must occur during it. The benefit is then discounted from the end of that year.

Definition: Consider a hypothetical discrete $n$-year term insurance on a life aged $x$, with level benefit $B$ paid at the end of the year of death if death occurs within the $n$-year coverage period. Let $i$ be the effective annual interest rate and $v=1/(1+i)$. The APV is $$ B A^1_{x:\overline{n}|} = B\sum_{k=0}^{n-1}v^{k+1}{}_kp_xq_{x+k}. $$ Here, ${}_kp_x$ is the probability of surviving $k$ years from age $x$, and $q_{x+k}$ is the probability, conditional on being alive at age $x+k$, of dying during the following year. The formula assumes annual benefit payments at the end of the year of death and a constant effective annual interest rate.

The first term in the sum is the probability of dying in the first year, discounted for one year. The next term is the probability of surviving one year and then dying in the second year, discounted for two years. Each successive term follows the same pattern. This death-year partition is the basis for understanding how duration changes the APV.

The Added-Year Formula

Compare an $n$-year policy with an otherwise identical $(n+1)$-year policy. The first $n$ death-year contributions are exactly the same in both APVs. The longer policy adds the possibility of payment in year $n+1$, which requires survival through $n$ years followed by death in the next year.

Theorem: For a level benefit $B$, the increase in APV when coverage is extended from $n$ years to $n+1$ years is $$ B\left(A^1_{x:\overline{n+1}|}-A^1_{x:\overline{n}|}\right) = Bv^{n+1}{}_np_xq_{x+n}. $$ The expression is the added-year contribution, not the APV of the entire longer policy.

This formula also gives a quick check on calculations: if the inputs are nonnegative, extending coverage cannot reduce the APV. The increase is strictly positive when $B>0$, $v>0$, and the probability of surviving to year $n+1$ and dying in that year is positive. If that death probability is zero, the extension adds no APV.

Why the Difference Is One Death-Year Term

Write the APV for each duration using the death-year sum. The shorter duration includes the contributions for $k=0,\ldots,n-1$, while the longer duration includes those same contributions and one more term, for $k=n$. Subtracting cancels every shared term.

\begin{aligned} A^1_{x:\overline{n+1}|} &=\sum_{k=0}^{n-1}v^{k+1}{}_kp_xq_{x+k} +v^{n+1}{}_np_xq_{x+n},\\ A^1_{x:\overline{n}|} &=\sum_{k=0}^{n-1}v^{k+1}{}_kp_xq_{x+k},\\ A^1_{x:\overline{n+1}|}-A^1_{x:\overline{n}|} &=v^{n+1}{}_np_xq_{x+n}. \end{aligned}

Multiplying by $B$ gives the added-year formula for a level benefit. This reasoning does not require mortality rates to be constant across ages. It requires only that the probabilities describe the same life, starting age, and mortality basis in both contracts being compared.

Identify the added policy year.
Extending $n$-year coverage to $n+1$ years adds year $n+1$, not year $n$.

Find the probability of payment.
Use ${}_np_xq_{x+n}$: survival through the first $n$ years, then death in the next year.

Discount and scale by the benefit.
Multiply the probability by $v^{n+1}$ and by $B$, if the requested result is in dollars.

Worked Examples

Worked Example: Extending Two-Year Coverage by One Year

A hypothetical policy pays $100,000 at the end of the year of death. The effective annual interest rate is $i=5.2632\%$, so use $v=0.95$. The one-year death probabilities are $q_x=0.02$, $q_{x+1}=0.03$, and $q_{x+2}=0.04$. Calculate the APVs for two- and three-year coverage and the increase from extending coverage.

Method. Calculate each death-year contribution, then use the added-year formula to check the difference. The probability of surviving one year is $p_x=1-q_x=0.98$. The probability of surviving two years is ${}_2p_x=0.98(1-0.03)=0.9506$.

For the first year, the death probability from the original starting age is $0.02$. Its discounted contribution per dollar of benefit is $$ 0.95(0.02)=0.019. $$ The dollar contribution is $100{,}000(0.019)=\$1{,}900.00$.

For the second year, the probability of surviving the first year and then dying is $0.98(0.03)=0.0294$. The discounted contribution per dollar is $$ 0.95^2(0.0294)=0.9025(0.0294)=0.0265335. $$ Thus, the two-year APV is $100{,}000(0.019+0.0265335)=\$4{,}553.35$.

For year three, the probability of surviving two years and then dying is $0.9506(0.04)=0.038024$. Its discounted contribution per dollar is $$ 0.95^3(0.038024)=0.857375(0.038024)=0.032600827. $$ The three-year APV is $100{,}000(0.019+0.0265335+0.032600827)=\$7{,}813.43$, rounded to the nearest cent.

The increase is $\$7{,}813.43-\$4{,}553.35=\$3{,}260.08$. The added-year formula gives the same result: $100{,}000(0.95^3)(0.9506)(0.04)=\$3{,}260.08$, after rounding to the nearest cent. This check confirms that the extra contribution is exactly the third-year death contribution.

Common mistake avoided: Using $q_{x+2}$ by itself ignores the requirement that the insured survive the first two years. The correct probability from age $x$ is ${}_2p_xq_{x+2}=0.9506(0.04)$.

Worked Example: Building APVs One Year at a Time

For a hypothetical life aged $x$, the effective annual interest rate is $i=11.1111\%$, so $v=0.90$. The one-year death probabilities for the first four policy years are $0.01, 0.02, 0.03,$ and $0.04$. Find the unit-benefit APVs for one- through four-year term insurance and the increase at each extension.

Method. Each extension adds $v^{n+1}{}_np_xq_{x+n}$. The successive survival probabilities are ${}_0p_x=1$, ${}_1p_x=0.99$, ${}_2p_x=0.99(0.98)=0.9702$, and ${}_3p_x=0.9702(0.97)=0.941094$.

The first-year contribution is $0.90(0.01)=0.009$, so the one-year unit-benefit APV is $0.009$.

The second-year contribution is $0.90^2(0.99)(0.02)=0.81(0.0198)=0.016038$. Therefore the two-year APV is $0.009+0.016038=0.025038$.

The third-year contribution is $0.90^3(0.9702)(0.03)=0.729(0.029106)=0.021218274$. The three-year APV is $0.025038+0.021218274=0.046256274$.

The fourth-year contribution is $0.90^4(0.941094)(0.04)=0.6561(0.03764376)=0.024698070936$. The four-year APV is $0.046256274+0.024698070936=0.070954344936$.

The resulting unit-benefit APVs are $0.009$, $0.025038$, $0.046256274$, and $0.070954344936$. The successive increases are $0.009$, $0.016038$, $0.021218274$, and $0.024698070936$. Each increase is nonnegative, as required. The increases themselves need not always rise with duration; their sizes depend on the mortality and discount factors in the added year.

Common mistake avoided: Adding each new contribution to the preceding contribution instead of to the cumulative APV would report only the latest year’s value. The APV at each duration is the sum of all contributions through that duration.

Worked Example: Extending Coverage Using Life-Table Counts

A hypothetical life table gives $\ell_x=80{,}000$, $\ell_{x+1}=78{,}000$, $\ell_{x+2}=75{,}000$, $\ell_{x+3}=70{,}000$, and $\ell_{x+4}=67{,}200$. A policy pays $250,000 at the end of the year of death. The effective annual interest rate is $i=4.1667\%$, so $v=0.96$. Calculate the APV for three-year coverage, the added contribution from year four, and the four-year APV.

Method. First express each year’s deaths as a fraction of the original cohort, $\ell_x$. This gives the probability of death in that year from the policy’s starting age. Then discount each probability by the appropriate power of $v$.

The numbers dying in the first three years are $80{,}000-78{,}000=2{,}000$, $78{,}000-75{,}000=3{,}000$, and $75{,}000-70{,}000=5{,}000$. As fractions of the original cohort, these are $2{,}000/80{,}000=0.025$, $3{,}000/80{,}000=0.0375$, and $5{,}000/80{,}000=0.0625$.

The three discounted contributions per dollar of benefit are $$ 0.96(0.025)=0.024,\qquad 0.96^2(0.0375)=0.03456,\qquad 0.96^3(0.0625)=0.055296. $$ Their sum is $0.024+0.03456+0.055296=0.113856$. The three-year APV is therefore $250{,}000(0.113856)=\$28{,}464.00$.

In year four, $70{,}000-67{,}200=2{,}800$ lives die. The probability from the original cohort is $2{,}800/80{,}000=0.035$. The year-four contribution per dollar is $$ 0.96^4(0.035)=0.84934656(0.035)=0.0297271296. $$ The added dollar amount is $250{,}000(0.0297271296)=\$7{,}431.78$, rounded to the nearest cent. Adding this to the three-year APV gives a four-year APV of $\$28{,}464.00+\$7{,}431.78=\$35{,}895.78$.

Common mistake avoided: The year-four deaths must be divided by the original cohort, $\ell_x$, to obtain their probability from the policy’s starting age. Dividing by $\ell_{x+3}$ gives the conditional one-year death probability at age $x+3$; that conditional probability must also be multiplied by the probability of surviving to age $x+3$.

Common Duration and Indexing Traps

The added-year calculation is simple once the policy-year index is clear. In year $n+1$, the insured must survive $n$ complete years, so the survival factor is ${}_np_x$. Death occurs during the following year, so the conditional death probability is $q_{x+n}$. Since the benefit is paid at the end of that year, the discount factor is $v^{n+1}$.

A common incorrect calculation for extending a two-year policy to three years is $v^2{}_2p_xq_{x+2}$. That uses the correct survival and death probabilities but discounts the benefit for only two years. The correction is $v^3{}_2p_xq_{x+2}$, because the added payment occurs at the end of policy year three. Another error is to discount by $v^3$ but omit ${}_2p_x$, treating $q_{x+2}$ as if it were measured from the original starting age.

For a level benefit and a consistent interest and mortality basis, the longer-duration APV is the shorter-duration APV plus a nonnegative term. This conclusion is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus: compare nested coverage periods by identifying the payment events that differ, rather than recalculating or altering the shared years.

Check Your Understanding

For each problem, identify the contribution from the added coverage before combining it with any earlier APV.

A unit-benefit policy has $v=0.95$, $q_x=0.04$, and $q_{x+1}=0.05$. What is the increase in APV from one-year to two-year coverage? (Answer: $0.04332$)
A hypothetical policy pays $50,000 at the end of the year of death. Let $v=0.90$, $q_x=0.02$, and $q_{x+1}=0.03$. How much does extending one-year coverage to two years add to the APV? (Answer: $1,190.70)
A unit-benefit term policy has $v=0.80$ and successive one-year death probabilities $0.01$, $0.02$, and $0.03$. Calculate the three-year APV. (Answer: $0.035574272$)
A hypothetical life table has $\ell_x=50{,}000$, $\ell_{x+2}=44{,}000$, and $\ell_{x+3}=42{,}000$. For a $100,000 benefit and $v=0.97$, find the increase in APV when coverage is extended from two to three years. (Answer: $3,650.69)
A unit-benefit two-year APV is $0.08$. For the next year, let $v=0.90$, ${}_2p_x=0.94$, and $q_{x+2}=0.05$. Find the three-year APV. (Answer: $0.114263$)

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