Calculate Reserves for Deferred Benefits with Finite Premium Terms

Why Deferral and Premium Term Must Be Tracked Separately

A deferred life insurance contract has two distinct time features. The deferral period determines when the insurance benefit first becomes payable; the premium term determines when premiums stop. Those periods need not be equal. A policy may still require premiums during the deferral period, or it may become paid-up before its benefit coverage begins.

To value the contract at a duration, first identify the benefit payable after that date, conditional on survival to the valuation time. Then subtract the value of any premiums still due. The result depends on whether the valuation date is before, at, or after the end of deferral, and whether it is before or after the last scheduled premium.

Definition: Consider a discrete deferred whole life insurance contract issued at age $x$. Let $m$ be the number of years of deferral and $n$ the number of annual premiums. A benefit of $B$ is payable at the end of the year of death only if the insured survives the first $m$ years and subsequently dies while the contract remains in force. Premiums of $P$ are payable at times $0,1,\ldots,n-1$, provided the insured is alive at each payment time. A reserve at duration $k$ is taken immediately before any premium scheduled at time $k$.

Under this definition, a death during the first $m$ years produces no benefit. At duration $m$, the deferral is complete and the benefit is whole life insurance on the life then aged $x+m$. The reserve convention matters: before a premium at time $k$, that premium is included among future premiums; after the premium, it is not.

Present Value of the Deferred Benefit

Use the standard discrete life insurance notation: $A_y$ is the expected present value of a unit whole life benefit payable at the end of the year of death for a life aged $y$, and ${}_tp_y$ is the probability that a life aged $y$ survives $t$ years. The effective annual interest rate is $i$, with $v=1/(1+i)$. These values are calculated on the same mortality and interest basis.

Theorem: At duration $k$, conditional on survival to that duration, the expected present value of a unit deferred benefit is $$ \begin{cases} v^{m-k}{}_{m-k}p_{x+k}A_{x+m}, & k<m,\\ A_{x+k}, & k\ge m. \end{cases} $$ For benefit amount $B$, multiply the appropriate expression by $B$.

The first line applies before the benefit becomes available. It requires both surviving the remaining $m-k$ years of deferral and then dying at some later time. The second line applies once the deferral has ended: the insured is then covered for whole life insurance from the attained age. The formula does not describe a contract that returns premiums or pays another benefit if death occurs during deferral; such a benefit would need its own present value.

Derivation by Conditioning on Survival

Suppose $k<m$. From duration $k$, no benefit is possible during the next $m-k$ years. The insured must survive that interval, whose probability is ${}_{m-k}p_{x+k}$. If survival occurs, the life is aged $x+m$, and the expected present value at that time of a unit whole life benefit is $A_{x+m}$. Discounting that value back $m-k$ years gives the first line of the theorem.

\begin{aligned} \text{EPV at duration }k &= {}_{m-k}p_{x+k}\,v^{m-k}A_{x+m}. \end{aligned}

This conditioning argument explains why both a survival probability and a discount factor appear. Neither can be omitted: survival determines whether coverage is reached, while discounting converts the later value to the valuation date.

Net Premium and Prospective Reserve

For a net premium, apply the equivalence principle: at issue, the expected present value of premiums equals the expected present value of benefits. With annual premiums payable in advance for at most $n$ years while alive, the premium value is $P\ddot{a}_{x:\overline{n}|}$, where this notation denotes the $n$-year temporary life annuity-due value. Therefore, for $m$ years of deferral,

P=\frac{Bv^m\,{}_mp_x A_{x+m}}{\ddot{a}_{x:\overline{n}|}}.

At duration $k$, the prospective reserve is the present value of future benefits less the present value of future premiums, conditional on survival to $k$. Before the premium at duration $k$, the number of scheduled premium dates remaining is $n-k$ if $k<n$. If $k\ge n$, no future premiums remain.

{}_kV_x = B \begin{cases} v^{m-k}{}_{m-k}p_{x+k}A_{x+m}, & k<m,\\ A_{x+k}, & k\ge m \end{cases} - P \begin{cases} \ddot{a}_{x+k:\overline{n-k}|}, & k<n,\\ 0, & k\ge n. \end{cases}

The annuity term includes the premium due immediately at duration $k$, consistent with the stated before-premium reserve convention. If the reserve is instead required immediately after that premium, its value is different by the value, at that time, of the premium just paid. Do not use the displayed formula without checking the valuation instant.

Locate the duration.
Compare $k$ with the deferral endpoint $m$ and the last premium date $n-1$.

Value the benefit.
Before the end of deferral, include survival and discounting to age $x+m$. At or after the endpoint, use whole life insurance at the attained age.

Value premiums still due.
Before a scheduled premium, include it. After the premium term, the future premium value is zero.

Subtract and check.
Subtract the future premium value from the future benefit value and verify the timing at each boundary.

Worked Calculations

The following examples use hypothetical contracts and explicitly state their mortality and interest assumptions. For convenient arithmetic, the examples use a constant annual mortality probability: $q_y=0.20$ at every age, so $p_y=0.80$, together with an effective annual interest rate $i=0.25$, so $v=0.80$. Deaths are assumed to occur within annual intervals, with benefits paid at the end of the year of death.

Worked Example: Find the Net Premium for a Deferred Contract

A hypothetical contract pays $B=\$10{,}000$ at the end of the year of death, but only if the insured survives the first $m=2$ years. Premiums are level and payable annually in advance for $n=4$ years while alive. Find the net annual premium.

Method. Use equivalence at issue. Under the constant mortality and interest assumptions, the one-year survival-discount factor is $vp=0.80(0.80)=0.64$. The whole life insurance value at any age is $$ A_y=\sum_{j=0}^{\infty}(vp)^j vq =\frac{vq}{1-vp} =\frac{0.80(0.20)}{1-0.64} =\frac{0.16}{0.36} =\frac{4}{9}. $$

The expected present value of the benefit per dollar at issue is $$ v^2{}_2p_x A_{x+2} =(0.64)^2\left(\frac{4}{9}\right) =0.4096\left(\frac{4}{9}\right) =0.182044444\ldots. $$

Thus, for the $\$10{,}000$ benefit, the expected present value is $$ \$10{,}000(0.182044444\ldots)=\$1{,}820.444444\ldots. $$ The four-premium annuity-due value is $$ \ddot{a}_{x:\overline{4}|} =1+0.64+(0.64)^2+(0.64)^3 =1+0.64+0.4096+0.262144 =2.311744. $$

Equating benefit and premium values gives $$ P=\frac{\$1{,}820.444444\ldots}{2.311744} =\$787.476660\ldots. $$ Rounded to the nearest cent, the net annual premium is $\$787.48$.

Check. Using the unrounded premium, $P(2.311744)=\$1{,}820.444444\ldots$, equal to the benefit present value. A common mistake is to omit the two-year survival probability or its discount factor; the benefit cannot be valued as an immediate whole life benefit at issue.

Worked Example: Calculate Reserves Before, At, and After Deferral

Use the contract and basis from the preceding example. Find the reserve immediately before the scheduled premium at durations $k=1,2,3,4$. Use the unrounded premium $P=\$787.476660\ldots$.

At duration 1, before the end of deferral. One year of deferral remains. The benefit value is $$ \$10{,}000(0.64)\left(\frac{4}{9}\right) =\$2{,}844.444444\ldots. $$ There are $4-1=3$ premiums remaining, including the one due at duration 1. Their value is $$ P(1+0.64+0.4096) =\$787.476660\ldots(2.0496) =\$1{,}614.012163\ldots. $$ Therefore, $$ {}_1V_x=\$2{,}844.444444\ldots-\$1{,}614.012163\ldots =\$1{,}230.432281\ldots, $$ or $\$1{,}230.43$ to the nearest cent.

At duration 2, when deferral ends. The insured is now covered for whole life insurance, so the benefit value is $$ \$10{,}000\left(\frac{4}{9}\right)=\$4{,}444.444444\ldots. $$ Two premiums remain, at durations 2 and 3. Their value is $$ P(1+0.64)=\$787.476660\ldots(1.64)=\$1{,}291.461723\ldots. $$ Thus ${}_2V_x=\$4{,}444.444444\ldots-\$1{,}291.461723\ldots=\$3{,}152.982721\ldots$, or $\$3{,}152.98$.

At duration 3, after deferral but during the premium term. The benefit value remains $\$4{,}444.444444\ldots$. Only the premium due at duration 3 remains, so $$ {}_3V_x=\$4{,}444.444444\ldots-\$787.476660\ldots =\$3{,}656.967784\ldots, $$ or $\$3{,}656.97$.

At duration 4, after the premium term. No premiums remain. The reserve is the whole life benefit value: $$ {}_4V_x=\$4{,}444.444444\ldots, $$ or $\$4{,}444.44$.

Check. At duration 4 the premium term has ended, so subtracting another premium would be incorrect. The reserve rises from duration 3 to 4 because the final premium is no longer a future outgo; the benefit value is unchanged on this constant mortality basis.

Worked Example: Premiums End Before the Benefit Begins

A hypothetical policy pays $\$5{,}000$ after the year of death, but only after survival through a four-year deferral period. It charges level annual premiums in advance for two years only. Use $q_y=0.20$ at every age and $i=0.25$. Calculate the net premium and the reserve immediately before the premium at duration 1, then find the reserve at duration 2.

Net premium at issue. The benefit present value is $$ \$5{,}000(0.64)^4\left(\frac{4}{9}\right) =\$5{,}000(0.16777216)\left(\frac{4}{9}\right) =\$372.827022\ldots. $$ The two-premium annuity-due value is $1+0.64=1.64$. Hence $$ P=\frac{\$372.827022\ldots}{1.64} =\$227.333550\ldots, $$ or $\$227.33$ to the nearest cent.

Reserve at duration 1. Three years of deferral remain, so the benefit value is $$ \$5{,}000(0.64)^3\left(\frac{4}{9}\right) =\$582.542222\ldots. $$ Only one premium remains, the premium due at duration 1. Therefore, $$ {}_1V_x=\$582.542222\ldots-\$227.333550\ldots =\$355.208672\ldots, $$ or $\$355.21$.

Reserve at duration 2. The premium term has ended, although two years of deferral remain. No future premiums are subtracted. The benefit value is $$ {}_2V_x=\$5{,}000(0.64)^2\left(\frac{4}{9}\right) =\$910.222222\ldots, $$ or $\$910.22$.

Check. At duration 2 the insured has not yet reached the benefit coverage period, but the contract is already paid-up. The reserve is positive and consists entirely of the deferred benefit value. The mistake to avoid is treating the end of premiums as if it ended the deferral period as well.

Common Timing Errors and Practical Checks

Using the wrong endpoint. The benefit changes from deferred to whole life insurance at duration $m$, while premiums stop according to $n$. These are independent dates.
Forgetting survival during deferral. Before duration $m$, the benefit value must include ${}_{m-k}p_{x+k}$, as well as $v^{m-k}$.
Including or omitting the wrong premium. This tutorial values reserves immediately before a scheduled premium. The premium at duration $k$ is included only when $k<n$.
Applying whole life value too early. $A_{x+k}$ alone is appropriate only once the deferral period has ended. Before then, value the remaining deferral first.
Rounding the premium too soon. Keep the full premium precision in reserve calculations, then round the final dollar amount to cents.

A useful check is to evaluate the formula at each boundary. At $k=m$, the deferred-benefit expression becomes $A_{x+m}$, matching the post-deferral expression. At $k=n$, the future premium value becomes zero. At issue, a net premium set by equivalence gives a zero prospective reserve immediately before the first premium.

Key takeaway: A deferred benefit and a finite premium term create two separate timelines. Value the benefit according to the deferral endpoint, value premiums according to the premium endpoint and valuation instant, and subtract the latter from the former.

Check Your Understanding

For each question, assume annual discrete death benefits paid at the end of the year of death, premiums payable in advance while alive, and reserves measured immediately before any premium due at that duration. Unless stated otherwise, use constant annual mortality $q_y=0.20$, $p_y=0.80$, and effective annual interest $i=0.25$, so $v=0.80$ and $vp=0.64$.

A hypothetical contract provides a $\$5{,}000$ benefit after a one-year deferral. What is the benefit present value at issue? (Answer: $\$1{,}422.22$)
A contract provides a $\$9{,}000$ benefit after a one-year deferral and charges premiums for three years. Find the level net annual premium. (Answer: $\$1{,}249.02$)
For the contract in Question 2, find the reserve at duration 2, immediately before the scheduled premium. (Answer: $\$2{,}750.98$)
A contract provides a $\$6{,}000$ benefit after a four-year deferral and charges premiums for two years. Find the level net annual premium. (Answer: $\$272.80$)
A contract provides a $\$1{,}000$ benefit after a three-year deferral and charges premiums for two years. Find its reserve at duration 2, when the premium term has ended but one year of deferral remains. (Answer: $\$284.44$)

What You'll Learn