Life Annuity-Due Reserve Recursions

Why Payment Timing Matters

A life annuity-due pays at the beginning of each year, provided the annuitant is alive then. That first payment is made immediately at the valuation date; later payments depend on survival through one or more years. A reserve recursion must keep these two facts separate. In particular, the payment due now is not discounted and is not contingent on surviving the year that follows.

This tutorial considers annual payments that stop on death, with no death benefit or refund. The interest rate is an effective annual rate, and mortality is described by one-year survival probabilities. The resulting recursion is a liability reserve: the expected present value of future scheduled payments, conditional on the annuitant being alive at the valuation time.

Definition: Consider an annuity issued at age \(x\), with payment \(b_k\) at time \(k\) if the annuitant is alive then. Let \({}_kV_x\) denote the reserve at time \(k\), immediately before the payment scheduled at that time. For a fixed effective annual interest rate \(i\), write \(v=1/(1+i)\). The one-year survival probability from age \(x+k\) is \({}_1p_{x+k}\). The reserve includes the payment due immediately at time \(k\).

With this convention, \({}_kV_x\) includes \(b_k\) with certainty, conditional on being alive at time \(k\). Payments after time \(k\) are included only to the extent that the annuitant survives to their due dates. For a level annuity, \(b_k=b\) throughout the payment period. The same recursion also applies when scheduled amounts vary by year.

Deriving the Annual Recursion

After the payment at time \(k\) has been made, the reserve for the remaining payments is \({}_kV_x-b_k\). One year later, if the annuitant is alive, the reserve immediately before the next payment is \({}_{k+1}V_x\). If the annuitant dies during the year, there are no further payments and no remaining liability. Discounting the expected reserve at the end of the year gives:

{}_kV_x-b_k = v\,{}_1p_{x+k}\,{}_{k+1}V_x.

Adding the current payment to both sides gives the annuity-due recursion:

Theorem: For an annual life annuity-due with payment \(b_k\) at time \(k\), payable only if alive at that time, and with no further benefit after death,

{}_kV_x=b_k+v\,{}_1p_{x+k}\,{}_{k+1}V_x.

The payment \(b_k\) is not discounted or multiplied by a survival probability: the reserve is defined conditional on being alive immediately before that payment.

The derivation also gives a useful accumulation interpretation. The assets remaining after the payment at time \(k\) accumulate for one year. Their value at time \(k+1\), before the next payment, must equal \({}_{k+1}V_x\) if the annuitant survives, and zero if the annuitant dies. Thus, in expectation,

({}_kV_x-b_k)(1+i) = {}_1p_{x+k}\,{}_{k+1}V_x.

This is the same relation rearranged. It depends on the stated timing: payments occur at the beginning of each year, the annual interest rate applies over the following year, and death ends the payment obligation. It would not apply unchanged if a payment were made at year-end, if payments continued after death, or if there were a refund or death benefit.

Finite Annuities and the Terminal Reserve

Suppose the final scheduled payment is at time \(n-1\), so the annuity has at most \(n\) payments, at times \(0,1,\ldots,n-1\). Immediately before that last payment, no later payments remain. Therefore the terminal reserve is \({}_{n-1}V_x=b_{n-1}\). Equivalently, if the contract is treated as expired just before time \(n\), its reserve then is zero: \({}_nV_x=0\), and the recursion at time \(n-1\) gives the last payment alone.

Set the payment timeline.
List the payment due at each time and establish whether the reserve is measured before or after that payment.

Set the endpoint.
For a finite annuity, use zero reserve after the last payment, or use the last payment as the reserve immediately before it.

Work backward.
At each year, add the payment due now to the discounted, survival-weighted reserve one year later.

Worked Example: A Three-Payment Level Annuity-Due

A hypothetical annuity pays \(1{,}000\) dollars at times \(0\), \(1\), and \(2\), if the annuitant is alive at each payment time. The effective annual interest rate is \(i=5\%\). The one-year survival probabilities from ages \(x\) and \(x+1\) are \({}_1p_x=0.90\) and \({}_1p_{x+1}=0.80\). Find the reserve at issue and at time \(1\), each immediately before its scheduled payment.

Method. Start with the final payment and apply the annuity-due recursion backward. There is no reserve after the payment at time \(2\), so immediately before that last payment, \({}_2V_x=1{,}000\).

The discount factor is \(v=1/1.05=0.95238095\). At time \(1\),

{}_1V_x = 1{,}000+\frac{0.80}{1.05}(1{,}000) = 1{,}000+761.9047619 = 1{,}761.9047619.

At time \(0\),

{}_0V_x = 1{,}000+\frac{0.90}{1.05}(1{,}761.9047619) = 1{,}000+1{,}510.2040816 = 2{,}510.2040816.

The reserve at issue is \(2{,}510.20\) dollars, and the time-1 reserve is \(1{,}761.90\) dollars, rounded to the nearest cent. As a check, the direct prospective value at issue is \(1{,}000+(0.90/1.05)(1{,}000)+(0.90)(0.80)/(1.05)^2(1{,}000)=2{,}510.2040816\), agreeing with the recursion. A common mistake is to discount or survival-weight the time-0 payment; neither applies to the payment due immediately.

Varying Payments and Direct Prospective Checks

The recursion does not require level payments. If the payment changes by duration, use the amount due at each time in that year’s equation. A direct prospective calculation is also a useful check: each future payment is multiplied by the probability of surviving to its due date and discounted from that date. The probability of surviving from time \(k\) to time \(j\), for \(j>k\), is the product of the intervening one-year survival probabilities.

Worked Example: A Schedule of Increasing and Decreasing Payments

A hypothetical three-payment annuity-due pays \(500\) dollars at time \(0\), \(700\) dollars at time \(1\), and \(900\) dollars at time \(2\), conditional on being alive at each payment date. The effective annual interest rate is \(10\%\), and the one-year survival probabilities are \(0.80\) from age \(x\) and \(0.50\) from age \(x+1\). Calculate the reserve at issue and verify it directly.

At time \(2\), immediately before the final payment, \({}_2V_x=900\). At time \(1\), the payment due is \(700\), and the next payment is due only if the annuitant survives the year:

{}_1V_x = 700+\frac{0.50}{1.10}(900) = 700+409.0909091 = 1{,}109.0909091.

At time \(0\),

{}_0V_x = 500+\frac{0.80}{1.10}(1{,}109.0909091) = 500+806.6115702 = 1{,}306.6115702.

Thus the reserve at issue is \(1{,}306.61\) dollars, rounded to the nearest cent. For a direct check, the expected present value is

500+\frac{0.80(700)}{1.10} +\frac{0.80(0.50)(900)}{(1.10)^2} = 500+509.0909091+297.5206612 = 1{,}306.6115703.

The tiny difference in the last decimal is due to displayed rounding. The check confirms that the time-0 payment is certain conditional on being alive at issue, while the payments at times \(1\) and \(2\) are contingent on survival to those dates. The method avoids treating all three payments as though they shared the same survival probability.

Reserves Measured Immediately After a Payment

Some calculations describe the reserve immediately after a payment rather than immediately before it. Let \(R_k\) be the reserve at time \(k\), immediately after the payment at that time. If the next payment is \(b_{k+1}\), then the relation to the before-payment convention is \({}_kV_x=b_k+R_k\). The post-payment recursion is

R_k = v\,{}_1p_{x+k}\bigl(b_{k+1}+R_{k+1}\bigr).

Here both the next payment and the reserve remaining after it are contingent on surviving to time \(k+1\). This formula is not a competing result; it is the same recursion with the reserve measured on the other side of the current payment.

Worked Example: Converting to a Post-Payment Reserve

Immediately after a hypothetical payment of \(250\) dollars at time \(0\), an annuity has two future scheduled payments: \(200\) dollars at time \(1\) and \(300\) dollars at time \(2\), each payable only if the annuitant is alive. The effective annual interest rate is \(10\%\), and the one-year survival probability in each year is \(0.80\). Find the reserve immediately after the time-0 payment.

Method. Work backward using post-payment reserves. Immediately after the time-2 payment, the reserve is zero. Thus, immediately after the time-1 payment, the only possible remaining payment is the \(300\)-dollar payment at time \(2\):

R_1 = \frac{0.80}{1.10}(300) = 218.1818182.

At time \(0\), the next payment is \(200\) dollars at time \(1\). Both that payment and the reserve after it are payable only if the annuitant survives the year:

R_0 = \frac{0.80}{1.10}(200+218.1818182) = 304.1322314.

The reserve immediately after the time-0 payment is therefore \(304.13\) dollars, rounded to the nearest cent. A direct check gives \(200(0.80/1.10)+300(0.80)^2/(1.10)^2=145.4545455+158.6776860=304.1322315\), with the difference again due only to displayed rounding. The already-paid \(250\) dollars is not included in this post-payment reserve.

Common Timing Errors

The most reliable way to avoid mistakes is to state the reserve convention before writing an equation. For the before-payment convention used in the main recursion, \({}_kV_x\) includes \(b_k\). Only the reserve after that payment is carried forward one year, and that future reserve is discounted and survival-weighted.

Discounting the immediate payment. This incorrectly treats a payment due now as if it were due one year later. The term \(b_k\) appears without a factor of \(v\).
Applying the next year’s survival probability to the current payment. The reserve is conditional on being alive at time \(k\), so the payment at time \(k\) is included in full.
Using a before-payment formula with a post-payment reserve. Confirm whether the amount at time \(k+1\) includes the payment due at time \(k+1\). If it does not, use the post-payment version consistently.
Setting the final pre-payment reserve to zero. If a final payment is still scheduled at that time, the reserve immediately before it equals that payment. Zero applies immediately after it, or at a time when no payment remains.

Key takeaway: For a life annuity-due valued immediately before its time-\(k\) payment, add that payment in full, then add the next reserve multiplied by the one-year discount factor and the probability of surviving to the next payment date: \({}_kV_x=b_k+v\,{}_1p_{x+k}\,{}_{k+1}V_x\).

Check Your Understanding

For each problem, use the stated reserve timing and round money to the nearest cent.

A two-payment annuity-due pays \(400\) dollars at times \(0\) and \(1\), conditional on being alive at each date. If \(i=5\%\) and \({}_1p_x=0.75\), find the reserve at issue immediately before the time-0 payment. (Answer: \(685.71\) dollars.)
A three-payment annuity-due pays \(100\) dollars at times \(0\), \(1\), and \(2\). The interest rate is \(i=0\%\), and the one-year survival probabilities from ages \(x\) and \(x+1\) are \(0.90\) and \(0.80\). Find the reserve immediately before the time-0 payment. (Answer: \(262.00\) dollars.)
Immediately after the time-0 payment, a two-payment annuity has payments of \(250\) dollars at time \(1\) and \(400\) dollars at time \(2\). The effective annual interest rate is \(8\%\), and each one-year survival probability is \(0.90\). Find the post-payment reserve at time \(0\). (Answer: \(486.11\) dollars.)
A three-payment annuity-due pays \(600\), \(500\), and \(300\) dollars at times \(0\), \(1\), and \(2\), respectively. The effective annual interest rate is \(5\%\), with one-year survival probabilities \(0.80\) and \(0.70\) for the first and second years. Find the reserve at time \(0\), immediately before its payment. (Answer: \(1{,}133.33\) dollars.)
A four-payment level annuity-due pays \(100\) dollars at times \(0\), \(1\), \(2\), and \(3\). The effective annual interest rate is \(10\%\), and the one-year survival probability is \(0.50\) in every year. Find the reserve at issue immediately before the first payment. (Answer: \(175.51\) dollars.)

What You'll Learn