Life Annuity-Immediate Reserve Recursions

Why Timing Matters in an Annuity Reserve

A life annuity-immediate pays at the end of each year, provided the annuitant is then alive. Its reserve must therefore reflect two events over the coming year: the annuitant may survive and receive the next payment, and a survivor may continue to be entitled to later payments. If death occurs before the payment date, neither that payment nor the later annuity payments are made.

That timing determines the reserve recursion. The next payment is not payable with certainty, and the reserve at the end of the year applies only if the annuitant survives to that time. The recursion is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Definition: Consider a level life annuity-immediate with annual payment $b$, where $b>0$. At time $k$, immediately after the payment scheduled for that time (if any), let $V_k$ be the expected present value at time $k$ of the future annuity payments, conditional on the annuitant being alive at time $k$. There are no future premiums in this definition. For a whole-life annuity, payments continue while the annuitant is alive; for a temporary annuity, they stop after a specified final payment date.

The annual effective interest rate is $i$, and $v=1/(1+i)$. Conditional on being alive at age $x+k$, the probability of surviving one more year is ${}_1p_{x+k}$, and the probability of dying during that year is ${}_1q_{x+k}=1-{}_1p_{x+k}$. These probabilities describe survival over the year, not survival to the payment date given some earlier event.

At time $k$, the payment at time $k+1$ is $b$ if the annuitant survives to that date. Immediately after receiving it, a surviving annuitant has a further reserve $V_{k+1}$. If the annuitant dies before time $k+1$, both amounts are zero under this annuity contract. These cash flows give the recursion.

Deriving the Annual Recursion

Theorem: For a life annuity-immediate with annual payment $b$, no future premiums, and reserve measured immediately after the payment at time $k$, the annual reserve recursion is

V_k=v\,{}_1p_{x+k}\bigl(b+V_{k+1}\bigr) =\frac{{}_1p_{x+k}}{1+i}\bigl(b+V_{k+1}\bigr).

The formula applies when the annuity payment and the continuation reserve are both contingent on survival to time $k+1$. For a temporary annuity, set the reserve after its final scheduled payment equal to zero.

To justify the formula, condition on survival through the coming year. With probability ${}_1p_{x+k}$, the annuitant reaches time $k+1$, and the insurer pays $b$ then. At that point, the future liability is $V_{k+1}$. The value of those amounts at time $k+1$ is $b+V_{k+1}$; discounting one year and weighting by the survival probability gives the value at time $k$. In the death outcome, the annuity makes no payment and has no continuing liability, so that outcome contributes zero.

Solving for the next reserve is also useful:

V_{k+1}=\frac{(1+i)V_k}{{}_1p_{x+k}}-b, \qquad {}_1p_{x+k}>0.

This rearrangement is appropriate when $V_k$, the interest rate, the annual payment, and the one-year survival probability are known. It does not turn a reserve into a guaranteed amount payable at time $k+1$: $V_{k+1}$ remains conditional on survival.

Fix the reserve time.
Use a value immediately after the payment at time $k$, so the next possible payment is at time $k+1$.

List the survival-contingent amounts.
If alive at time $k+1$, the annuitant receives $b$, and the continuing liability is $V_{k+1}$.

Take the conditional expected value and discount.
Multiply the total time-$k+1$ amount by ${}_1p_{x+k}$, then by $v$.

Apply the endpoint condition if needed.
For a temporary annuity, the reserve immediately after its last scheduled payment is $0$.

For a whole-life annuity-immediate paying 1 per year, the same reasoning gives the familiar relation $a_{x+k}=v\,{}_1p_{x+k}(1+a_{x+k+1})$. The term $1$ represents the next year-end payment; the annuity factor at the next age represents subsequent payments. For a level payment $b$, multiplying the unit-payment values by $b$ gives the corresponding reserves.

Worked Example: Calculate the Reserve from the Next-Year Reserve

A hypothetical whole-life annuity-immediate pays $12,000 at each year-end while the annuitant is alive. At a particular duration, the annual effective interest rate is 5%, the one-year death probability is 0.04, and the reserve immediately after the next payment is $30,000. Find the reserve immediately after the current payment.

Method. Apply the annual recursion, conditioning both the next payment and the continuation reserve on survival. The one-year survival probability is ${}_1p_{x+k}=1-0.04=0.96$.

Substitute $b=12{,}000$, $V_{k+1}=30{,}000$, and $i=0.05$:

V_k=\frac{0.96}{1.05}(12{,}000+30{,}000) =\frac{0.96(42{,}000)}{1.05} =\frac{40{,}320}{1.05} =38{,}400.

The reserve is $38,400.00. As a check, accumulating this reserve for one year gives $1.05(38{,}400)=40{,}320$, equal to the survival-weighted amount $0.96(12{,}000+30{,}000)$. The common mistake is to use the death probability 0.04 as the multiplier; the payment and continuation occur in the survival outcome, so the multiplier is 0.96.

Temporary Annuities and the Final Payment

A temporary life annuity has a fixed final payment date. If its last payment is scheduled at time $n$, then immediately after that payment there are no future benefits. Its endpoint reserve is therefore $V_n=0$. The recursion is then applied backward from that endpoint. This condition is important: before the final payment, a survivor may still be owed that final amount.

For example, if only one payment remains, then $V_{n-1}=v\,{}_1p_{x+n-1}b$. There is no continuation reserve after that payment. Setting $V_{n-1}=0$ instead would incorrectly omit the final possible payment.

Worked Example: Work Backward Through a Two-Year Temporary Annuity

A hypothetical two-year life annuity-immediate pays $1,000 at the end of each year if the annuitant is alive. The annual effective interest rate is 25%. The one-year survival probabilities are 0.8 from issue to time 1 and 0.5 from time 1 to time 2. Calculate the reserves immediately after issue and immediately after the first payment, using issue as time 0.

Method. Start with the reserve after the final scheduled payment: $V_2=0$. Then apply the recursion backward. The stated survival probabilities are conditional one-year probabilities at their respective durations.

At time 1, there is one payment remaining, so

V_1=\frac{0.5}{1.25}(1{,}000+0) =0.4(1{,}000) =400.

At time 0, survival to time 1 has probability 0.8. If the annuitant survives, the $1,000 payment is made and the continuing reserve is $400. Therefore,

V_0=\frac{0.8}{1.25}(1{,}000+400) =0.64(1{,}400) =896.

The reserves are $896.00 at time 0 and $400.00 at time 1. Check by valuing the two payments directly at time 0. The first has probability $0.8$, and the second has probability $0.8(0.5)=0.4$:

\frac{0.8(1{,}000)}{1.25} +\frac{0.4(1{,}000)}{1.25^2} =640+256 =896.

The direct value matches the recursion. A common error is to use 0.5 as the probability of the second payment from time 0; it is conditional on survival to time 1. From time 0, the probability of reaching time 2 is $0.8(0.5)=0.4$.

Recovering Survival or a Missing Reserve

The recursion can be rearranged to find an unknown quantity when adjacent reserves are known. For example, if $V_k$, $V_{k+1}$, $b$, and $i$ are given, then

{}_1p_{x+k}=\frac{(1+i)V_k}{b+V_{k+1}}.

This calculation is meaningful only when $b+V_{k+1}>0$, and a valid probability must lie between 0 and 1. A result outside that range is a signal to check the data, reserve timing, or model assumptions. Likewise, the backward rearrangement for $V_{k+1}$ requires a positive survival probability.

Worked Example: Infer the One-Year Survival Probability

A hypothetical life annuity-immediate pays $1,000 annually. At a duration with annual effective interest 25%, its reserve immediately after the current payment is $3,840, and its reserve immediately after the next payment is $5,000. Infer the one-year survival and death probabilities implied by these reserves.

Method. Use the rearranged recursion and then check that the inferred survival probability is valid. The timing convention is unchanged: both the next payment and next reserve arise only if the annuitant survives the year.

{}_1p_{x+k} =\frac{(1+i)V_k}{b+V_{k+1}} =\frac{1.25(3{,}840)}{1{,}000+5{,}000} =\frac{4{,}800}{6{,}000} =0.8.

Thus the implied one-year death probability is ${}_1q_{x+k}=1-0.8=0.2$. The probability is between 0 and 1, as required. Check the original relation: $0.8(1{,}000+5{,}000)/1.25=4{,}800/1.25=3{,}840$. The common mistake is to divide by the continuation reserve alone; the next-year amount contingent on survival is $b+V_{k+1}$, not just $V_{k+1}$.

Common Timing and Conditioning Errors

The recursion is compact, but it is easy to misapply when the reserve date is not stated carefully. In particular, the life annuity-immediate payment is made at the end of the year, not at its beginning. The following checks help keep the model consistent.

Issue	Correct treatment	Typical error
Next payment	Include $b$ at time $k+1$ only on survival.	Treat the payment as certain, or pay it on death during the year.
Continuation reserve	Include $V_{k+1}$ only if alive at time $k+1$.	Add $V_{k+1}$ without the survival condition.
Discounting	Discount the time-$k+1$ amounts by $v=1/(1+i)$.	Multiply by $1+i$ when calculating the current reserve.
Temporary endpoint	Set the reserve after the final scheduled payment to zero.	Set the reserve before that payment to zero and omit the final possible benefit.

For instance, the incorrect expression $V_k=v\,{}_1p_{x+k}b+V_{k+1}$ discounts and survival-weights the next payment but leaves the continuation reserve at its future value. The correct expression discounts and survival-weights both amounts: $V_k=v\,{}_1p_{x+k}(b+V_{k+1})$. Always identify the time at which each amount is valued before combining it with another amount.

Key takeaway: For a life annuity-immediate, the next year-end payment and the reserve after that payment are both contingent on survival. With reserve measured just after the current payment, combine those amounts, multiply by the one-year survival probability, and discount one year. For a temporary annuity, work backward from a zero reserve immediately after the final payment.

Check Your Understanding

For each problem, keep the reserve timing explicit and check the result in the recursion.

A hypothetical whole-life annuity-immediate pays $2,000 annually. The annual effective interest rate is 4%, the one-year death probability is 0.05, and $V_{k+1}=$30,000$. Find $V_k$. (Answer: $29,230.77)
A hypothetical life annuity-immediate pays $1,000 annually. At a duration, $i=5\%$, ${}_1p_{x+k}=0.8$, and $V_k=$8,000$. Find $V_{k+1}$. (Answer: $9,500.00)
A hypothetical two-year temporary annuity-immediate pays $1,000 annually. The annual effective interest rate is 25%, and the conditional one-year survival probabilities are 0.8 and 0.5. Find the reserve at time 0. (Answer: $896.00)
A hypothetical annuity-immediate pays $3,000 annually. At a particular duration, $i=5\%$, ${}_1q_{x+k}=0.03$, and $V_{k+1}=$20,000$. Find $V_k$. (Answer: $21,247.62)
A hypothetical life annuity-immediate pays $1,000 annually. At a particular duration, $i=8\%$, $V_k=$6,000$, and $V_{k+1}=$5,500$. Infer the one-year survival probability and death probability. (Answer: ${}_1p_{x+k}=0.9969$, ${}_1q_{x+k}=0.0031$, to four decimal places)

Issue	Correct treatment	Typical error
Next payment	Include \(b\) at time \(k+1\) only on survival.	Treat the payment as certain, or pay it on death during the year.
Continuation reserve	Include \(V_{k+1}\) only if alive at time \(k+1\).	Add \(V_{k+1}\) without the survival condition.
Discounting	Discount the time-\(k+1\) amounts by \(v=1/(1+i)\).	Multiply by \(1+i\) when calculating the current reserve.
Temporary endpoint	Set the reserve after the final scheduled payment to zero.	Set the reserve before that payment to zero and omit the final possible benefit.

What You'll Learn