Assemble Deferred and Limited-Payment Insurance Values

Why Separate Deferral, Coverage, and Payment Periods?

A life insurance contract can have several time periods that sound similar but describe different things. A deferral period specifies how long the policy must be in force before a death benefit can be paid. A benefit period specifies which deaths are covered. A premium-payment period specifies when premiums are due. These periods need not be equal.

For example, a deferred whole-life policy pays only if death occurs after a stated waiting period, but then covers death for the rest of life. A limited-payment whole-life policy also covers death for life, but requires premiums for only a specified number of years. The benefit value is determined by the covered death events and their payment times; the premium value is determined by the premium cash-flow schedule.

Throughout this tutorial, insurance benefits are discrete and paid at the end of the year of death. Mortality probabilities are conditional on being alive at the start of each year, and premiums described as paid at the beginning of a year are paid only if the insured is alive then. Unless a problem states otherwise, ignore expenses, lapses, and other policy features.

Definition: An $m$-year deferred whole-life insurance pays a benefit at the end of the year of death if death occurs after the insured has survived $m$ full years. A deferred $n$-year term insurance with an $m$-year deferral pays only for deaths in years $m+1,\ldots,m+n$. A limited-payment whole-life policy covers death throughout life but has premiums due for only a specified limited period.

Values from the Covered Death Years

Let $i$ be the effective annual interest rate and $v=1/(1+i)$. The probability of death in year $k$, measured from issue at age $x$, is ${}_{k-1}p_xq_{x+k-1}$: the insured must survive the first $k-1$ years and then die during year $k$. A unit benefit paid at the end of that year has present value $v^k$ if it is paid.

Consequently, the expected present value of an $m$-year deferred, $n$-year term insurance with unit benefit is the sum of the discounted death probabilities for precisely the covered years. This sum makes the benefit period explicit and prevents a deferral from being mistaken for an extra period of coverage.

\text{EPV}=\sum_{k=m+1}^{m+n}v^k\,{}_{k-1}p_xq_{x+k-1}.

For deferred whole-life insurance, the sum continues over all possible death years after the deferral period. Equivalently, condition on survival to the end of year $m$. Given survival, the remaining benefit has the whole-life value at age $x+m$, and its value at issue must be discounted $m$ years.

Theorem: Under the discrete end-of-year-of-death benefit convention, the value of an $m$-year deferred whole-life insurance with unit benefit is $$ {}_{m|}A_x=v^m\,{}_{m}p_xA_{x+m}. $$ The factor ${}_{m}p_x$ accounts for the chance that the benefit becomes eligible; $v^m$ discounts the value from the end of the deferral period to issue.

Why the Deferred Whole-Life Formula Holds

The formula follows by partitioning the possible benefit payments according to the year of death. Death during the first $m$ years produces no payment. For any later year $k$, the probability of death in year $k$ from issue is the probability of surviving $m$ years multiplied by the conditional probability of death in year $k-m$ from age $x+m$.

{}_{k-1}p_xq_{x+k-1} ={}_{m}p_x\,{}_{k-m-1}p_{x+m}q_{x+k-1}, \qquad k>m.

The issue-date present value for that death year is therefore $v^k=v^m v^{k-m}$. Summing over all $k>m$ gives

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

This derivation also shows why multiplying by the discount factor alone is insufficient: it would value a benefit as though survival through the deferral period were certain. The survival factor is essential.

Worked Example: Deferred Whole-Life Insurance with Constant Mortality

A hypothetical policy pays $1 at the end of the year of death, provided death occurs after a two-year deferral. The annual effective interest rate is $i=0.25$. At every age, the annual mortality probability is $q=0.25$, so $p=0.75$. Find the expected present value at issue.

Method. First find the whole-life value at the starting age, then apply the deferral survival and discount factors. This avoids treating survival through the waiting period as certain.

Here $v=1/1.25=0.8$. Because mortality is the same at every age, the whole-life value satisfies the geometric sum

A_x=\sum_{k=1}^{\infty}v^kp^{k-1}q =\frac{vq}{1-vp} =\frac{(0.8)(0.25)}{1-(0.8)(0.75)} =\frac{0.20}{0.40}=0.5.

The probability of surviving two years is ${}_{2}p_x=(0.75)^2=0.5625$, and the two-year discount factor is $v^2=(0.8)^2=0.64$. Thus

{}_{2|}A_x=(0.64)(0.5625)(0.5)=0.18.

A direct check gives the same result. The discounted probability of death in year 3 is $(0.8)^3(0.75)^2(0.25)=0.072$; each later year's term is $0.6$ times the preceding term. The sum is $0.072/(1-0.6)=0.18$. The two methods agree.

Deferred Term Insurance: Stop at the End of Coverage

A deferred term benefit has a finite set of eligible death years. It is not enough to value a whole-life benefit and then remove the first few years: the benefit also ends after the term. Use the death-year sum directly, or calculate each eligible year and add the values.

Worked Example: Two-Year Deferral Followed by Three Years of Coverage

A hypothetical contract pays $10,000 at the end of the year of death only if death occurs in years 3, 4, or 5 after issue. The annual effective interest rate is $i=0.25$. Each year's mortality probabilities are $q=0.25$ and $p=0.75$. Find the expected present value at issue.

Method. Enumerate only the covered death years. Here $v=0.8$; the death probability in year $k$ is $p^{k-1}q$, and the issue-date discount factor is $v^k$.

For year 3, the death probability is $(0.75)^2(0.25)=0.140625$, and the discounted unit-benefit value is $(0.8)^3(0.140625)=0.072$.

For year 4, the death probability is $(0.75)^3(0.25)=0.10546875$, and the discounted unit-benefit value is $(0.8)^4(0.10546875)=0.0432$.

For year 5, the death probability is $(0.75)^4(0.25)=0.0791015625$, and the discounted unit-benefit value is $(0.8)^5(0.0791015625)=0.02592$.

Adding the three covered years gives a unit-benefit value of $0.072+0.0432+0.02592=0.14112$. Scaling by the $10,000 benefit, the expected present value is $10{,}000(0.14112)=\$1{,}411.20$. A check is that the year-4 term is $0.6$ times the year-3 term, and the year-5 term is $0.6$ times the year-4 term, as $vp=(0.8)(0.75)=0.6$. The three-term total is consistent with that progression.

Limited-Payment Whole-Life Insurance

In a limited-payment whole-life contract, the benefit is still payable on covered death throughout the insured's life. Limiting the years in which premiums are due does not limit the death years that generate the benefit value. Therefore, the expected present value of the benefit remains the whole-life insurance value $A_x$, multiplied by the benefit amount, under the same mortality and interest assumptions.

The premium cash flows have a separate value. If a level premium is due at the beginning of each of the first $r$ years while the insured is alive, its expected present-value factor is

\sum_{j=0}^{r-1}v^j\,{}_{j}p_x.

The $j=0$ term is 1 because the first premium is due immediately at issue. Each later term is discounted $j$ years and multiplied by the probability of being alive when that premium is due. This premium factor is not an insurance value: it describes a different cash-flow stream.

Worked Example: Lifelong Benefit with Three Annual Premiums

A hypothetical policy pays $100,000 at the end of the year of death for death at any future time. Premiums of $20,000 are due at the beginning of each of the first three years, while the insured is alive. Assume $i=0.25$, $q=0.25$, and $p=0.75$ every year. Calculate the expected present value of the benefit and of the scheduled premiums.

Method. Value the lifelong benefit over all death years, then value the three-premium stream only over its stated payment period. This avoids truncating the insurance benefit at the last premium year.

From the geometric sum in the first example, $A_x=0.5$. The benefit expected present value is therefore $100{,}000(0.5)=\$50{,}000$.

The premium dates are time 0, time 1 if alive, and time 2 if alive. With $v=0.8$, the premium factor is

$$ 1+v p+v^2p^2 =1+(0.8)(0.75)+(0.8)^2(0.75)^2 =1+0.60+0.36=1.96. $$

Thus the expected present value of the scheduled premiums is $20{,}000(1.96)=\$39{,}200$. This calculation values two distinct streams: the benefit remains lifelong, while premiums stop after the third scheduled payment. The arithmetic checks because the three premium contributions are $20,000, $12,000, and $7,200, which sum to $39,200.

Common Exam Trap: Confusing the Periods

A frequent error is to use one duration in place of another. In a deferred term contract, the deferral period identifies the years with no benefit; it does not itself tell you how long coverage lasts. In limited-payment whole life, the premium period does not define the benefit period.

Wrong for deferred whole life: use $v^m A_{x+m}$ and omit ${}_{m}p_x$. This assumes survival to the end of the deferral is certain.
Correction: use $v^m{}_{m}p_xA_{x+m}$; both discounting and survival are required.
Wrong for deferred term: include death years $m+1$ onward without stopping at $m+n$. That values a deferred whole-life benefit instead of a deferred term benefit.
Wrong for limited-payment whole life: stop the benefit sum when premium payments stop. That changes the product to term insurance, although the benefit is specified as lifelong.

Before calculating, write down three separate facts: the first eligible death year, the last eligible death year (if any), and the dates premiums are due. Then assign survival and discount factors to each cash flow using its own timing.

Key takeaway: Deferred insurance values are built from covered death years, with survival to the deferred period and discounting to issue both accounted for. Limited-payment whole life limits premium duration, not benefit duration.

Check Your Understanding

For each problem, identify the relevant benefit or premium period before calculating.

A unit-benefit deferred whole-life policy has $i=1/9$ and constant annual $q=0.20$. Find its value with a one-year deferral. (Answer: $81/175$, approximately $0.4629$.)
A unit benefit is paid only for deaths in years 2 and 3. Assume $i=1/9$, $p=0.80$, and $q=0.20$ each year. Find its expected present value at issue. (Answer: $0.222912$.)
Premiums are due at the beginning of years 1 and 2 only while alive. If $v=0.95$ and $p=0.90$, find the expected present-value factor for this two-payment schedule. (Answer: $1+0.95(0.90)=1.855$.)
A unit benefit is paid only for death in year 3. The annual mortality probabilities in years 1, 2, and 3 are $0.10$, $0.20$, and $0.30$, respectively, and $i=1/9$. Find the issue-date expected present value. (Answer: $(0.9)^3(0.9)(0.8)(0.3)=0.157464$.)
A $50,000 lifelong death benefit is paired with three beginning-of-year premiums of $12,000, payable only while alive. Assume $i=1/9$ and constant $q=0.20$. Find the benefit expected present value and the premium expected present value. (Answer: benefit $=\$32{,}142.86$; premium $=\$26{,}860.80$.)

What You'll Learn