Deferred Term Insurance from Deferred Whole-Life Values

Why a Difference of Whole-Life Values Gives Deferred Term Insurance

A deferred term contract pays only if death occurs in a specified future interval. For an $n$-year deferral followed by $m$ years of coverage, the covered policy years are $n+1$ through $n+m$. One way to value that interval is to add the discounted death benefits in those years. Another is to value all benefits after year $n$, then subtract the benefits after year $n+m$. The second approach expresses deferred term insurance as a difference between two deferred whole-life values.

The distinction between the deferral period and the coverage period matters. The contract does not pay for death during the first $n$ years, and it does not pay for death after the following $m$ years. In this tutorial, all benefits are payable at the end of the year of death, and values are calculated at issue.

Definition: Assume a unit benefit is payable at the end of the year of death, with effective annual interest rate $i$ and $v=(1+i)^{-1}$. The $n$-year deferred whole-life insurance value is $$ {}_{n|}A_x=\sum_{k=n+1}^{\infty}v^k\,{}_{{k-1}}p_xq_{x+k-1}. $$ The $n$-year deferred, $m$-year term insurance value, denoted ${}_{n|m}A_x$, is $$ {}_{n|m}A_x=\sum_{k=n+1}^{n+m}v^k\,{}_{{k-1}}p_xq_{x+k-1}. $$ Here ${}_{k-1}p_xq_{x+k-1}$ is the probability that a life aged $x$ survives $k-1$ years and then dies in the next year.

These definitions describe a discrete insurance benefit. They do not describe a benefit payable immediately on death, a continuous insurance value, or a contract that pays a maturity benefit. Those benefit timings and contract features require their own valuations.

The Difference Identity

The deferred whole-life value ${}_{n|}A_x$ includes death benefits in every year after the deferral period: years $n+1,n+2,\ldots$. The value ${}_{n+m|}A_x$ includes the tail beginning in year $n+m+1$. Subtracting removes that tail and leaves exactly the $m$ covered years.

Theorem: For a unit benefit payable at the end of the year of death, $$ {}_{n|m}A_x={}_{n|}A_x-{}_{n+m|}A_x. $$ The identity assumes the same interest and mortality basis for both deferred whole-life values and the deferred term value.

The result follows directly from the sums. Both deferred whole-life values use the same terms from year $n+m+1$ onward, so those terms cancel. What remains is the sum for years $n+1$ through $n+m$.

\begin{aligned} {}_{n|}A_x-{}_{n+m|}A_x &=\sum_{k=n+1}^{\infty}v^k\,{}_{{k-1}}p_xq_{x+k-1} -\sum_{k=n+m+1}^{\infty}v^k\,{}_{{k-1}}p_xq_{x+k-1}\\ &=\sum_{k=n+1}^{n+m}v^k\,{}_{{k-1}}p_xq_{x+k-1}\\ &={}_{n|m}A_x. \end{aligned}

The finite-sum result is also useful as a check: the difference of two deferred whole-life values must equal the present value of the death benefits in the covered years, not the probability of surviving to the end of the term. The identity requires no constant mortality assumption. It applies with age-varying mortality provided the same mortality basis is used throughout.

Valuation Method

Mark the covered years.
An $n$-year deferral followed by $m$ years of term coverage covers years $n+1$ through $n+m$.

Identify both deferred whole-life values.
Use ${}_{n|}A_x$ for the tail after year $n$ and ${}_{n+m|}A_x$ for the tail after year $n+m$.

Subtract the tails.
Calculate ${}_{n|}A_x-{}_{n+m|}A_x$, retaining full precision during the calculation.

Check the covered interval.
If needed, sum $v^k\,{}_{{k-1}}p_xq_{x+k-1}$ for $k=n+1,\ldots,n+m$. The direct sum should agree.

Worked Calculations

Worked Example: Two Years of Deferred Coverage

A hypothetical policyholder aged $x$ is insured for a unit benefit payable at the end of the year of death. The effective annual interest rate is $i=25\%$, so $v=1/1.25=0.8$. Assume $q_x=0.20$, $q_{x+1}=0.25$, $q_{x+2}=0.50$, and $q_{x+3}=1$. Find the value of a two-year deferred, two-year term insurance, and verify it directly.

Method. Subtract the deferred whole-life value beginning after year $4$ from the deferred whole-life value beginning after year $2$. The covered years are $3$ and $4$.

First, the probability of surviving two years is ${}_2p_x=(1-0.20)(1-0.25)=0.80(0.75)=0.60$. At age $x+2$, the conditional death probabilities are $0.50$ in the first year and $0.50$ in the second year. Thus $A_{x+2}=0.50(0.8)+0.50(0.8^2)=0.40+0.32=0.72$.

The two-year deferred whole-life value is therefore ${}_{2|}A_x=v^2{}_2p_xA_{x+2}=0.64(0.60)(0.72)=0.27648$. By year $4$, survival is impossible because $q_{x+3}=1$, so ${}_{4|}A_x=0$. The difference gives ${}_{2|2}A_x=0.27648-0=0.27648$.

Direct check. The unconditional probability of death in year $3$ is $0.60(0.50)=0.30$. The unconditional probability of death in year $4$ is $0.60(0.50)(1)=0.30$. Discounting each end-of-year benefit gives $0.30(0.8^3)+0.30(0.8^4)=0.30(0.512)+0.30(0.4096)=0.1536+0.12288=0.27648$. The check agrees. The common mistake avoided is using the conditional mortality at age $x+2$ as though it were already an unconditional probability from age $x$; the factor ${}_2p_x$ is needed.

Worked Example: Subtracting the Correct Deferred Tail

A hypothetical insurer values a unit benefit payable at the end of the year of death using $i=25\%$, so $v=0.8$. The mortality assumptions are $q_x=0.10$, $q_{x+1}=0.20$, $q_{x+2}=0.25$, $q_{x+3}=0.50$, and $q_{x+4}=1$. Find the value of a one-year deferred, two-year term insurance using deferred whole-life values.

Method. The covered years are $2$ and $3$. Calculate ${}_{1|}A_x$ and ${}_{3|}A_x$, then subtract. The unconditional death probabilities in years $2$ through $5$ are, respectively, $0.90(0.20)=0.18$, $0.90(0.80)(0.25)=0.18$, $0.90(0.80)(0.75)(0.50)=0.27$, and $0.90(0.80)(0.75)(0.50)(1)=0.27$.

The first deferred whole-life value includes all four of these years: ${}_{1|}A_x=0.18(0.8^2)+0.18(0.8^3)+0.27(0.8^4)+0.27(0.8^5)$. The individual terms are $0.1152$, $0.09216$, $0.110592$, and $0.0884736$, so ${}_{1|}A_x=0.4064256$.

The second deferred whole-life value begins after year $3$, so it includes only years $4$ and $5$: ${}_{3|}A_x=0.27(0.8^4)+0.27(0.8^5)=0.110592+0.0884736=0.1990656$. Therefore ${}_{1|2}A_x=0.4064256-0.1990656=0.20736$.

Direct check. The covered benefits are in years $2$ and $3$, with present values $0.18(0.64)=0.1152$ and $0.18(0.512)=0.09216$. Their sum is $0.20736$, as required. This check also confirms that the subtraction removes years $4$ and later, rather than removing year $3$.

Worked Example: A Longer Deferred Term With Changing Mortality

A hypothetical contract pays a unit benefit at the end of the year of death. Use an effective annual interest rate of $25\%$, so $v=0.8$. Assume $q_x=0.20$, $q_{x+1}=0.10$, $q_{x+2}=0.20$, $q_{x+3}=0.25$, $q_{x+4}=0.50$, and $q_{x+5}=1$. Find the value of a two-year deferred, three-year term insurance by taking a difference of deferred whole-life values.

Method. The covered years are $3,4,5$. First calculate the survival probability to age $x+2$: ${}_2p_x=0.80(0.90)=0.72$. At age $x+2$, the conditional probabilities of death in the next four years are $0.20$, $0.80(0.25)=0.20$, $0.80(0.75)(0.50)=0.30$, and $0.30(1)=0.30$. Hence $A_{x+2}=0.20(0.8)+0.20(0.8^2)+0.30(0.8^3)+0.30(0.8^4)=0.16+0.128+0.1536+0.12288=0.56448$.

Thus ${}_{2|}A_x=v^2{}_2p_xA_{x+2}=0.64(0.72)(0.56448)=0.260112384$. To calculate ${}_{5|}A_x$, note that survival through five years is ${}_5p_x=0.80(0.90)(0.80)(0.75)(0.50)=0.216$. At age $x+5$, death in the next year is certain, so $A_{x+5}=0.8$. It follows that ${}_{5|}A_x=v^5{}_5p_xA_{x+5}=0.32768(0.216)(0.8)=0.056623104$.

Subtracting gives ${}_{2|3}A_x=0.260112384-0.056623104=0.20348928$.

Direct check. The unconditional death probabilities in years $3,4,5$ are $0.72(0.20)=0.144$, $0.72(0.80)(0.25)=0.144$, and $0.72(0.80)(0.75)(0.50)=0.216$. Their discounted values are $0.144(0.512)=0.073728$, $0.144(0.4096)=0.0589824$, and $0.216(0.32768)=0.07077888$. The sum is $0.20348928$, matching the difference. The common mistake avoided is omitting the year-5 discount factor: a year-$k$ end-of-year benefit is discounted by $v^k$, not $v^{k-1}$.

Common Errors and When the Identity Applies

The identity is a convenient valuation tool, but it is reliable only when the contract definition and the values being subtracted agree. Before calculating, identify the covered policy years and make sure that the benefit amount, timing, interest basis, and mortality basis match in both deferred whole-life values.

Using $n$ as the term length. $n$ is the waiting period before coverage starts. The coverage length is $m$, so the subtraction is ${}_{n|}A_x-{}_{n+m|}A_x$.
Subtracting at the wrong endpoint. The second value starts after year $n+m$, not after year $n+m-1$. Otherwise one covered year is removed.
Discounting from the wrong date. For a benefit paid at the end of policy year $k$, the discount factor from issue is $v^k$.
Mixing benefit conventions. A continuous whole-life value cannot be subtracted from a discrete deferred whole-life value to obtain the discrete term value.
Assuming constant mortality is necessary. It is not. The cancellation argument works with varying annual death probabilities as long as the same mortality basis is used for both values.

This relationship is useful when deferred whole-life values are available from a table or have already been calculated. It also provides a strong internal check: a deferred term value must be nonnegative and cannot exceed the corresponding deferred whole-life value, since it covers only some of that whole-life value’s possible death years. The identity and its checks are aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Check Your Understanding

For each hypothetical contract, use end-of-year death benefits and the stated effective annual interest rate. Check that the deferred period and term length identify the correct covered years.

Let $i=25\%$, $q_x=0.20$, $q_{x+1}=0.50$, and $q_{x+2}=1$. Find the value of a one-year deferred, two-year term insurance. (Answer: $0.4608$)
Let $i=25\%$, $q_x=0.10$, $q_{x+1}=0.20$, $q_{x+2}=0.50$, and $q_{x+3}=1$. Find the value of a two-year deferred, one-year term insurance. (Answer: $0.18432$)
Let $i=25\%$, $q_x=0.25$, $q_{x+1}=0.20$, $q_{x+2}=0.40$, and $q_{x+3}=1$. Find the value of a one-year deferred, two-year term insurance by summing the covered death benefits. (Answer: $0.21888$)
Let $i=25\%$ and $q_x=0.10$, $q_{x+1}=0.20$, $q_{x+2}=0.25$, $q_{x+3}=0.50$, and $q_{x+4}=1$. Find the value of a one-year deferred, two-year term insurance. (Answer: $0.20736$)
Let $i=25\%$ and $q_x=0.20$, $q_{x+1}=0.10$, $q_{x+2}=0.20$, $q_{x+3}=0.25$, $q_{x+4}=0.50$, and $q_{x+5}=1$. Find the value of a two-year deferred, three-year term insurance. (Answer: $0.20348928$)

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