Worked Values for Deferred and Endowment Contracts

Reading the Contract Before Calculating

A deferred insurance contract has a waiting period before death benefits become payable. An endowment contract can also promise a benefit at a specified date if the insured is then alive. To value either contract, identify the event that triggers each payment, the payment date, and the survival required for the payment. The word “deferred” changes the covered years; it does not by itself specify when money is paid.

Throughout this tutorial, assume a discrete model with annual effective interest rate $i$, discount factor $v=1/(1+i)$, and annual survival probabilities ${}_tp_x$. Deaths during policy year $k$, the interval from time $k-1$ to time $k$, are paid at the end of that year. Unless stated otherwise, benefits are paid only when their contract conditions are met.

Definition: An $m$-year deferred, $n$-year term insurance pays a death benefit if death occurs during one of policy years $m+1,\ldots,m+n$. A pure endowment payable at time $r$ pays only if the insured survives to time $r$. An $n$-year temporary life annuity-due pays at times $0,1,\ldots,n-1$, provided the insured is alive at each payment time.

For a death benefit of $B$, the payment for death in year $k$ has probability ${}_{k-1}p_x q_{x+k-1}$ and is discounted by $v^k$. A pure endowment of $C$ payable at time $r$ has present value $C v^r {}_rp_x$. An annuity-due payment at time $j$ has present value only when the insured is alive then, so its expected present value is the payment amount multiplied by $v^j {}_jp_x$.

Deferred Term Insurance

The deferred period requires survival through the first $m$ years, but coverage applies only to deaths in the next $n$ years. The direct calculation sums over the covered years. Equivalently, one can condition on survival to the start of coverage and value an $n$-year term insurance at that point:

\operatorname{EPV} = B\sum_{k=m+1}^{m+n} v^k\,{}_{k-1}p_x q_{x+k-1} = v^m\,{}_mp_x \left( B\sum_{j=1}^{n}v^j\,{}_jp_{x+m-1}q_{x+m+j-1} \right).

In the parenthesized expression, the year-$j$ death probability after reaching age $x+m$ is ${}_{j-1}p_{x+m}q_{x+m+j-1}$. Thus, written with that survival index, the expression is:

\operatorname{EPV} = v^m\,{}_mp_x \left( B\sum_{j=1}^{n}v^j\,{}_{j-1}p_{x+m}q_{x+m+j-1} \right).

The second form separates the value of reaching the start of coverage, $v^m{}_mp_x$, from the value of the insurance after that point. The benefit is not paid merely because the insured survives the deferral period; survival is a condition for entering the covered period.

Worked Example: A Deferred Two-Year Death Benefit

A hypothetical policy pays $800 at the end of the year of death if death occurs during policy years 2 or 3. There is no payment for death during year 1. Assume $i=25\%$ effective annually and a constant annual survival probability $p=0.80$, so $q=0.20$. Find the actuarial present value.

Method. Value each covered death year separately. Here $v=1/1.25=0.80$. Death in year 2 requires survival through year 1, so its probability is $0.80(0.20)=0.16$. Its discounted expected benefit is $800(0.16)(0.80^2)=800(0.1024)=\$81.92$.

Death in year 3 requires survival through two years, so its probability is $0.80^2(0.20)=0.128$. Its discounted expected benefit is $800(0.128)(0.80^3)=800(0.065536)=\$52.4288$.

Adding the two covered years gives $81.92+52.4288=\$134.3488$, or $134.35 to the nearest cent. The result is checked by confirming that only deaths in years 2 and 3 were included and that each benefit was discounted to its actual payment time.

Common mistake avoided: Applying an additional survival factor after the death probability counts survival twice. For example, the year-2 death probability is $0.80(0.20)$, not $0.80^2(0.20)$; the second $0.80$ belongs in the discount factor $v^2$, not in the death probability.

Pure Endowments and Temporary Annuities

A pure endowment has a different trigger from death insurance: it pays at maturity only if the insured survives to that date. If its amount is $C$ and maturity is time $r$, its value is $C v^r{}_rp_x$. The discount factor and survival probability both run to the maturity date.

A temporary annuity-due pays at the beginning of each year while the insured is alive. For level payment $R$ over $n$ possible payment dates, its value is:

R\sum_{j=0}^{n-1}v^j\,{}_jp_x.

The term for $j=0$ is $R$: the payment at time 0 is immediate and requires no prior survival. A payment at time 1 requires survival for one year and is discounted once; a payment at time 2 requires survival for two years and is discounted twice. If the contract instead pays at the end of each year while alive, the payment dates and corresponding survival events differ, so this annuity-due formula does not apply unchanged.

Worked Example: Combining Deferred Insurance, an Endowment, and an Annuity

A hypothetical contract provides a $500 death benefit for deaths in policy years 2 or 3, a $1,000 pure endowment at time 3 if the insured survives, and a $120 annuity-due payment at times 0, 1, and 2 if alive at each date. Assume $i=25\%$, constant annual survival probability $p=0.80$, and $q=0.20$. Find the total actuarial present value.

Method. Value each benefit stream separately and add the results. Present-value linearity permits this addition even though the benefits have different payment conditions.

For the deferred death benefit, the two probabilities and discount factors are those from the covered years: the year-2 contribution is $500(0.16)(0.80^2)=\$51.20$, and the year-3 contribution is $500(0.128)(0.80^3)=\$32.768$. The insurance value is $51.20+32.768=\$83.968$.

For the pure endowment, survival to time 3 has probability $0.80^3=0.512$, and the discount factor is $v^3=0.80^3=0.512$. Its value is $1{,}000(0.512)(0.512)=\$262.144$.

For the annuity-due, the time-0 payment has value $120$. The time-1 payment has expected present value $120(0.80)(0.80)=\$76.80$. The time-2 payment has expected present value $120(0.80^2)(0.80^2)=120(0.4096)=\$49.152$. Thus the annuity value is $120+76.80+49.152=\$245.952$.

The total is $83.968+262.144+245.952=\$592.064$, or $592.06 to the nearest cent. This total includes a death payment in year 3 only on death during that year; the pure endowment is paid at time 3 only on survival to that time.

Common mistake avoided: Do not give the pure endowment to someone who dies during year 3, and do not require survival for the annuity payment at time 0. Each payment has its own event condition.

An Insurance-Annuity Check

A useful check connects term insurance, a temporary annuity-due, and a pure endowment over the same $n$-year period. Let $A^{1}_{x:\overline{n}|}$ denote the actuarial present value of a unit death benefit paid at the end of the year of death during the next $n$ years. Let $\ddot{a}_{x:\overline{n}|}$ denote the value of a unit temporary life annuity-due over those same $n$ payment dates. The corresponding pure endowment value is $v^n{}_np_x$.

Identity: For annual effective interest $i$, with $d=i/(1+i)=1-v$, these three values satisfy $A^{1}_{x:\overline{n}|}+d\ddot{a}_{x:\overline{n}|}+v^n{}_np_x=1$. The identity assumes death benefits are paid at the end of the year of death and annuity-due payments are made at times $0,\ldots,n-1$ while alive.

The pure endowment term is essential: a term insurance does not pay if the insured survives the full term. The identity follows by pairing the annuity and insurance terms year by year. Write ${}_jp_x$ for survival through $j$ years, with ${}_0p_x=1$. For year $j+1$, the annuity contribution multiplied by $d$ and the insurance contribution are:

d v^j\,{}_jp_x+v^{j+1}\,{}_jp_x q_{x+j} = v^j\,{}_jp_x\bigl(d+v q_{x+j}\bigr) = v^j\,{}_jp_x-v^{j+1}\,{}_{j+1}p_x.

The last equality uses $d=1-v$ and ${}_{j+1}p_x={}_jp_x(1-q_{x+j})$. Summing for $j=0,\ldots,n-1$ makes the intermediate terms cancel, leaving $1-v^n{}_np_x$. Adding the pure endowment value $v^n{}_np_x$ gives 1, which proves the identity.

Worked Example: Using the Identity for Deferred Coverage

A hypothetical policy pays $1,000 at the end of the year of death for deaths in years 2 or 3. Assume $i=10\%$ effective annually and constant annual survival probability $p=0.90$, so $q=0.10$. Use the insurance-annuity identity to find the value, then check it directly.

Method. First value a two-year term insurance beginning at time 1, and then multiply by the probability-discount factor for reaching that point. Here $v=1/1.1=10/11$, $d=1/11$, and $vp=9/11$.

For the two-year term at the start of coverage, the unit temporary annuity-due value is $1+vp=1+9/11=20/11=220/121$. The two-year pure endowment value is $(vp)^2=81/121$. Therefore the unit term insurance value is $1-(1/11)(220/121)-81/121$. Since $(1/11)(220/121)=20/121$, this is $1-20/121-81/121=20/121$.

To reach the start of coverage, the insured must survive one year and the value must be discounted one year, giving $vp=9/11$. The deferred unit insurance value is therefore $(9/11)(20/121)=180/1331$. For a $1,000 benefit, the value is $1{,}000(180/1331)=\$135.2367$, or $135.24 to the nearest cent.

Direct check. The year-2 contribution is $1{,}000(0.90)(0.10)(10/11)^2=\$74.3802$. The year-3 contribution is $1{,}000(0.90^2)(0.10)(10/11)^3=\$60.8565$. Their sum is $\$135.2367$, agreeing with the identity-based value before rounding.

Common mistake avoided: Using only $1-d\ddot{a}_{x:\overline{n}|}$ for the term insurance omits the pure endowment. The correct term value is $1-d\ddot{a}_{x:\overline{n}|}-v^n{}_np_x$.

Practical Checks and Common Traps

Before accepting a result, compare every payment date with the probability and discount factor used for it. For a death benefit at the end of year $k$, the event is survival through $k-1$ years followed by death during year $k$, and the discount exponent is $k$. For a pure endowment at time $r$, both the survival period and discount exponent are $r$. For an annuity-due, the first payment occurs at time 0, not at the end of year 1.

Starting coverage too early. An $m$-year deferred contract does not cover deaths during the first $m$ years.
Discounting to the wrong date. A death benefit paid at the end of year $k$ uses $v^k$, even if coverage began later.
Confusing the pure endowment with a death benefit. The pure endowment is payable only on survival to maturity.
Forgetting the immediate annuity-due payment. If the insured is alive at issue, the payment at time 0 has no discount or survival factor.
Using an identity with mismatched terms. The insurance, annuity, and pure endowment must share the same term and timing conventions.

Key takeaway: Write down each benefit’s trigger and payment time before calculating. Deferred insurance, pure endowments, and temporary annuities can then be valued separately and combined, with the identity providing a useful check when all three terms match.

Check Your Understanding

For each problem, state the payment conditions before calculating. Round money to the nearest cent.

A hypothetical policy pays $500 at the end of year 2 if death occurs during that year; death in year 1 is not covered. Assume $i=10\%$ and constant annual survival probability $p=0.90$. Find the actuarial present value. (Answer: $37.19)
A hypothetical pure endowment pays $2,000 at time 4 if the insured survives. Assume $i=5\%$ and constant annual survival probability $p=0.95$. Find its actuarial present value. (Answer: $1,340.19)
A hypothetical temporary life annuity-due pays $300 at times 0, 1, and 2 while the insured is alive. Assume $i=25\%$ and constant annual survival probability $p=0.80$. Find its actuarial present value. (Answer: $614.88)
A hypothetical $1,000 death benefit covers policy year 3 only, with no benefit for earlier deaths. Assume $i=10\%$ and constant annual survival probability $p=0.90$. Find its actuarial present value. (Answer: $60.86)
A hypothetical contract pays $400 at the end of the year of death for deaths in years 2 or 3, and $600 at time 3 if the insured survives. Assume $i=25\%$ and constant annual survival probability $p=0.80$. Find the combined actuarial present value. (Answer: $224.46)

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