Annuity-Due Reserves at Premium Dates

Why the Valuation Moment Matters

A whole-life annuity-due pays at the beginning of each year, provided the annuitant is alive on the payment date. At a policy anniversary, a payment may therefore be due at the very moment a reserve is being valued. Whether that payment is included depends on whether the valuation is made immediately before or immediately after the payment.

This timing distinction is especially important at a premium date. A date on the premium schedule does not, by itself, determine whether an annuity benefit is payable then: the benefit schedule must also be checked. When both a benefit and a premium are due, their cash flows belong in the value only if they have not yet been settled.

Definition: Consider a policy in force at age $x$. It pays 1 at times $0,1,2,\ldots$ as long as the annuitant is alive at each payment time. The prospective value immediately before the payment at time 0 is the whole-life annuity-due value $\ddot{a}_x$. With effective annual interest rate $i$, discount factor $v=1/(1+i)$, and survival probability ${}_jp_x$, it is $\ddot{a}_x=\sum_{j=0}^{\infty}v^j{}_jp_x$, where ${}_0p_x=1$.

The term for $j=0$ is the payment due immediately. The next term is the expected present value of the payment one year later, and each later term accounts for both survival to that payment date and discounting. For an annual benefit amount $B$, the value is $B\ddot{a}_x$. These expressions assume annual payments in advance, payments only when the annuitant is alive, and a specified interest and mortality model. They do not describe an annuity with a different payment frequency or a guarantee independent of survival.

Before and After the Due Payment

Suppose the policy has reached duration $k$, and the annuitant is alive. The attained age is $x+k$. If the reserve is evaluated immediately before the scheduled payment, the payment at that moment is included. For a level benefit $B$, the benefit value is $B\ddot{a}_{x+k}$. If the reserve is evaluated immediately after that payment has been made, the payment is no longer a future liability; the remaining payments begin one year later, and their value is $Ba_{x+k}$.

For a whole-life annuity, the annuity-immediate value $a_{x+k}$ includes payments at the ends of future years while the annuitant survives. Thus the distinction is exactly one payment of amount $B$ at the valuation date:

\ddot{a}_{x+k}=1+a_{x+k}, \qquad B\ddot{a}_{x+k}=B+Ba_{x+k}.

This comparison concerns the annuity benefit alone. A net prospective reserve also accounts for future premiums, if any. Its value depends on whether a premium is due at the valuation moment and whether it has already been paid.

Worked Example: Constant Annual Survival Probability

A hypothetical whole-life annuity pays $1,000 annually in advance while the annuitant is alive. At every age, the probability of surviving one year is $0.90$, and the effective annual interest rate is $5\%$. Calculate the benefit reserve at a duration anniversary immediately before the payment due at that anniversary, and immediately after it.

Method and assumptions. The mortality assumption is a constant one-year survival probability $p=0.90$, so ${}_jp_x=(0.90)^j$. Interest is annual at $i=0.05$, giving $v=1/1.05=20/21$. Survival probabilities and interest are treated as deterministic model inputs; the reserve is conditional on the annuitant being alive at the valuation date.

Immediately before the due payment, the unit annuity-due value is a geometric series. Its ratio is $vp=(20/21)(0.90)=6/7$:

\ddot{a}_{x+k} =\sum_{j=0}^{\infty}(vp)^j =\frac{1}{1-6/7} =7.

The benefit reserve before the payment is therefore $1{,}000(7)=\$7{,}000.00$. Immediately after the payment, the current $1,000 has been paid, so the remaining value is the annuity-immediate value:

a_{x+k}=\ddot{a}_{x+k}-1=7-1=6, \qquad 1{,}000(6)=\$6{,}000.00.

Check. The before-payment value exceeds the after-payment value by exactly the payment made, $\$7{,}000-\$6{,}000=\$1{,}000$. The common mistake this avoids is using $a_{x+k}=6$ before the payment, which would omit the $1,000 immediately due.

Why the One-Year Recursion Holds

The annuity-due value can be evaluated from its first payment and the value of the remaining stream. At age $x$, the payment of 1 is due now. The payments from one year onward have value $ \ddot{a}_{x+1}$ at time 1 if the annuitant survives to that time. Discounting that conditional future value by $v$, and multiplying by the probability $p_x$ of surviving the year, gives the continuation value at time 0.

\ddot{a}_x=1+vp_x\ddot{a}_{x+1}.

The recursion follows directly by separating the $j=0$ term from the defining sum. For every $j\geq 1$, survival to year $j$ can be written as $p_x\,{}_{j-1}p_{x+1}$. Factoring out $vp_x$ from all those terms leaves the annuity-due sum at age $x+1$:

\begin{aligned} \ddot{a}_x &=1+\sum_{j=1}^{\infty}v^j{}_jp_x\\ &=1+vp_x\sum_{r=0}^{\infty}v^r{}_rp_{x+1}\\ &=1+vp_x\ddot{a}_{x+1}. \end{aligned}

This relation requires the usual conditional survival probabilities to be consistent across successive years, and it uses the same annual effective interest rate for the one-year discount. It applies to the value just before a payment is due. Immediately after that payment, the current 1 is no longer part of the liability.

Worked Example: Survival Probabilities Vary by Year

A hypothetical annuity pays $2,500 at the beginning of each year while the annuitant is alive. At the valuation date immediately before a payment, the one-year survival probabilities are $0.95$ for the first year, $0.90$ for the second year, and $0.80$ for the third year and every year thereafter. The effective annual interest rate is $25\%$. Find the reserve immediately before the due payment and immediately after it.

Method and assumptions. The mortality model specifies survival for each future year conditional on being alive at the start of that year. Thus the probabilities of surviving to years 0, 1, 2, and 3 are $1$, $0.95$, $0.95(0.90)=0.855$, and $0.855(0.80)=0.684$. From year 3 onward, each additional year multiplies survival by $0.80$. The discount factor is $v=1/1.25=0.80$.

The first three terms of the unit annuity-due value, for payment dates 0, 1, and 2, are:

1,\qquad (0.80)(0.95)=0.76,\qquad (0.80)^2(0.855)=0.5472.

The year-3 payment term is $(0.80)^3(0.684)=0.350208$. Each later term is $0.80(0.80)=0.64$ times the preceding term, since both discounting and conditional survival contribute a factor of $0.80$. The tail starting at year 3 is therefore a geometric series:

\frac{0.350208}{1-0.64}=0.9728.

The full unit annuity-due value is $1+0.76+0.5472+0.9728=3.28$. Hence the reserve immediately before the payment is $2{,}500(3.28)=\$8{,}200.00$. Immediately after the payment, the value is $2{,}500(3.28-1)=\$5{,}700.00$.

Check. The difference is $\$8{,}200-\$5{,}700=\$2{,}500$, exactly one scheduled payment. The common mistake avoided is treating the year-3 survival probability as $0.80$ rather than $0.684$: the annuitant must also survive the first two years to receive the year-3 payment.

Including Premiums Due at the Same Duration

For a net prospective reserve, subtract the expected present value of future premiums from the expected present value of future benefits. Consider a hypothetical contract where, while the annuitant is alive, both the annual benefit $B$ and annual premium $P$ are payable at the beginning of each year for life. Immediately before the payments at duration $k$, both cash flows are still outstanding. If the same survival and interest assumptions apply to each stream, the net reserve is:

{}_kV_x =B\ddot{a}_{x+k}-P\ddot{a}_{x+k} =(B-P)\ddot{a}_{x+k}.

If valuation occurs immediately after the benefit and premium have both been settled, neither current cash flow remains in the prospective value. The remaining net value is $(B-P)a_{x+k}$. If only one of the two payments is due, only that cash flow is included at the valuation moment. Always identify the cash flows due and whether they have been settled before applying a formula.

Worked Example: Net Value Before and After a Premium Date

A hypothetical contract pays an annuity benefit of $1,200 at the beginning of each year while the annuitant is alive. It also requires a premium of $200 at the same times, for as long as the annuitant is alive. The one-year survival probability is $0.90$ at every age, and the effective annual interest rate is $5\%$. Find the net prospective reserve at a duration premium date immediately before both payments, and immediately after both payments.

Method and assumptions. This is a net prospective calculation: value future benefits and subtract future premiums. Both streams have the same annual due timing and survival condition. From the geometric-series calculation, $v=20/21$, $vp=6/7$, and $\ddot{a}_{x+k}=1/(1-6/7)=7$.

Immediately before settlement, the benefit value is $1{,}200(7)=\$8{,}400.00$, and the premium value is $200(7)=\$1{,}400.00$. Subtracting gives a net reserve of $\$8{,}400-\$1{,}400=\$7{,}000.00$, or equivalently $(1{,}200-200)(7)=\$7{,}000.00$.

Immediately after settlement, the current benefit and premium have both been paid. The unit annuity-immediate value for the remaining stream is $a_{x+k}=7-1=6$. The net reserve is then $(1{,}200-200)(6)=\$6{,}000.00$.

Check. The net reserve falls by the net cash flow at the date, $\$1{,}200-\$200=\$1{,}000$: $\$7{,}000-\$1{,}000=\$6{,}000$. The common mistake avoided is subtracting only future premiums while also omitting the premium due now, or including that premium after it has already been paid.

Common Timing Errors

The central exam skill is to establish the reserve’s timing before evaluating survival or discounting. A compact check is to write down the first payment date still included in each cash-flow stream.

Using the wrong annuity value. Immediately before an annual due benefit, use the due value, including the current payment. Immediately after it, the remaining stream is valued as an annuity-immediate.
Assuming every premium date is a benefit date. Read the benefit schedule and premium schedule separately. A premium date determines a benefit payment only when the contract specifies that both are due then.
Ignoring whether the annuitant is alive. A payment conditional on survival is included only when the annuitant is alive at its payment date. For future payments, the appropriate survival probability must multiply each discounted amount.
Mixing payment amounts with unit values. The notation $\ddot{a}_{x+k}$ is the value per unit payment. Multiply by the benefit amount, and subtract the premium value when calculating a net reserve.
Using a constant-survival formula without its assumption. The geometric-series shortcut requires a constant annual survival probability at the relevant ages. With varying probabilities, sum the appropriately weighted terms or use the one-year recursion.

These checks are aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus: reserve calculations depend not only on mortality and interest, but also on the precise ordering of valuation and scheduled cash flows.

Key takeaway: At a duration immediately before a whole-life annuity-due payment, include that payment and value the benefit stream with $\ddot{a}_{x+k}$. Immediately after the payment, the remaining stream has value $a_{x+k}=\ddot{a}_{x+k}-1$. For a net reserve, apply the same timing discipline to premiums.

Check Your Understanding

For each question, state whether the valuation is before or after any due cash flow, then calculate the requested value.

A hypothetical annuity pays $5,000 annually in advance while the annuitant is alive. The annual survival probability is $0.96$ at every age and the effective annual interest rate is $4\%$. Find the benefit reserve immediately before a due payment. (Answer: $65,000.00)
An annuity pays $1,200 annually in advance. The first one-year survival probability is $0.90$, and the probability is $0.80$ in every subsequent year. The effective annual interest rate is $25\%$. Find its benefit reserve immediately before the payment due at the valuation date. (Answer: $3,600.00)
A contract pays a $900 annuity benefit and receives a $150 premium at the beginning of each year while the annuitant is alive. At a premium date, the unit annuity-due value is 8. Find the net reserve immediately before both payments. (Answer: $6,000.00)
A benefit of $2,000 is payable annually in advance. At a duration date, the annuitant is alive, the unit annuity-due value is 4.6, and the due payment has just been made. Find the benefit reserve immediately after the payment. (Answer: $7,200.00)
At age $x+1$, the unit annuity-due value is 5. The one-year survival probability from age $x$ is $0.90$, and the effective annual interest rate is $5\%$. Use the recursion to find the unit annuity-due value at age $x$, immediately before its due payment. (Answer: 5.2857)

What You'll Learn