Why the Remaining Term and Payment Timing Matter
A temporary life annuity pays only while the annuitant is alive and only during a stated finite term. Its reserve therefore depends on two distinct questions: which scheduled payments remain, and when the next payment is due. A payment made at the valuation time is included if the reserve is calculated immediately before that payment, but excluded if it is calculated immediately after.
The prospective approach values the expected future benefits and premiums from the valuation time onward. The life-contingent payments must be weighted by the probability that the annuitant survives to each payment date. This treatment is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.
Let \(i\) be the effective annual interest rate and \(v=1/(1+i)\). Write \(B_j\) for the benefit and \(P_j\) for the premium scheduled \(j\) years after the valuation time. If both are payable only when the annuitant is alive, the value of their net cash flow at that date is weighted by \({}_jp_{x+k}\), the probability of surviving \(j\) years from age \(x+k\).
For an annuity-immediate, annual payments are at the ends of policy years. If the reserve is immediately after the cash flows at time \(k\), only dates \(k+1,\ldots,n\) remain. For an annuity-due, payments are at the beginnings of policy years. Immediately before a scheduled payment at time \(k\), that payment is still future from the reserve’s perspective and must be included; immediately after it, it must be excluded.
| Convention and valuation instant | First future payment date | Survival factor for that payment |
|---|---|---|
| Annuity-immediate, immediately after time \(k\) cash flows | Time \(k+1\) | \({}_1p_{x+k}\) |
| Annuity-due, immediately before time \(k\) cash flows | Time \(k\) | \({}_0p_{x+k}=1\) |
| Annuity-due, immediately after time \(k\) cash flows | Time \(k+1\) | \({}_1p_{x+k}\) |
Prospective Reserve Formula
Suppose the valuation is at time \(k\), while the annuitant is alive. Let \(J\) be the set of future payment offsets, measured in years from that instant. An offset of zero is included only when the valuation is immediately before a payment due at time \(k\). The general discrete formula is
This form applies when benefits and premiums are both payable on the same survival-contingent dates. When their dates differ, value the future benefits and premiums separately, using their own payment dates and survival factors:
For a level net payment \(C=B-P\) at each remaining payment date, the formula reduces to \(C\) times the sum of the discount-and-survival factors for those dates. The finite term is essential: payments after the contract’s expiry are not part of either sum. If the annuitant is alive at the end of the term and no payment is scheduled then, the reserve immediately after expiry is zero.
Why the Formula Works
A scheduled payment at offset \(j\) is made only if the annuitant survives to that date. Let \(I_j\) equal 1 if the annuitant is alive at the payment date and 0 otherwise. The present value at time \(k\) of a net payment \(B_j-P_j\) is \(v^j I_j(B_j-P_j)\). Conditional on being alive at time \(k\), the probability that \(I_j=1\) is \({}_jp_{x+k}\). Taking the conditional expectation therefore gives
Adding the expected present values over the scheduled future dates gives the prospective reserve. The sum stops at the contract’s final eligible payment date because the indicator for any later payment is not part of the benefit design. This derivation also explains why the survival probability is conditional on being alive at the valuation time, rather than a probability measured from issue.
State whether it is immediately before or after any cash flow scheduled at that time.
Exclude past payments and dates after the temporary annuity’s term ends.
Discount by \(v^j\), multiply by \({}_jp_{x+k}\), and sum the resulting values.
Worked Examples
Worked Example: Annuity-Immediate Reserve After a Payment
A hypothetical four-year annuity-immediate pays \$1,500 at times 1, 2, 3, and 4, provided the annuitant is alive. A premium of \$400 is due at each of those same dates, also only if alive. The annual survival probability is \(p=0.8\) at every age, and the effective annual interest rate is \(i=25\%\), so \(v=1/1.25=0.8\). Find the reserve at time 2 immediately after both scheduled cash flows.
Method. Since the time-2 benefit and premium have already been paid, the remaining dates are times 3 and 4. These occur one and two years after valuation, so the net payment at each is \(1{,}500-400=\$1{,}100\).
At offset \(j=1\), the discount-and-survival factor is \(v\,{}_1p_{x+2}=0.8(0.8)=0.64\). At offset \(j=2\), the two-year survival probability is \(0.8^2=0.64\), and the factor is \(v^2\,{}_2p_{x+2}=0.8^2(0.8^2)=0.4096\).
Therefore,
The reserve is \$1,154.56. As a check, both remaining net payments are positive, so the reserve is positive; the time-2 cash flows do not appear in the calculation. The common mistake avoided here is adding a time-2 payment after it has already occurred.
Worked Example: Annuity-Due Reserve Immediately Before a Payment
A hypothetical five-year annuity-due pays \$1,000 at times 0, 1, 2, 3, and 4 if the annuitant is alive. A \$250 premium is due on the same dates, subject to the same survival condition. Assume an effective annual interest rate of \(i=10\%\), so \(v=1/1.10=10/11\), and a constant annual survival probability of \(p=0.9\). Find the reserve at time 2 immediately before the scheduled time-2 benefit and premium.
Method. Since the reserve is immediately before the time-2 cash flows, the payment at time 2 is still due. The remaining payment dates are 2, 3, and 4, corresponding to offsets \(j=0,1,2\). The net amount at each date is \(1{,}000-250=\$750\).
The offset-zero factor is \(v^0\,{}_0p_{x+2}=1\). At offset 1 the factor is \((10/11)(0.9)=9/11\). At offset 2 it is \((10/11)^2(0.9^2)=(9/11)^2=81/121\). Hence
The reserve is \$1,865.70, rounded to the nearest cent. The offset-zero factor is 1 because the annuitant is alive at the valuation instant and the payment is due immediately. If the reserve were instead calculated immediately after those time-2 cash flows, the offset-zero term would be removed.
Worked Example: Varying Payments and Age-Specific Survival
A hypothetical five-year annuity-immediate has its first two payments already made. Immediately after the time-2 cash flows, its remaining benefits at times 3, 4, and 5 are \$1,500, \$1,700, and \$1,900. The premiums at those dates are each \$500. The effective annual interest rate is \(i=5\%\), so \(v=1/1.05\). From the attained age at time 2, the one-year survival probabilities for successive years are \(0.9\), \(0.8\), and \(0.75\). Find the reserve.
Method. The remaining net payments are \$1,000, \$1,200, and \$1,400. Because survival is age-specific, the cumulative probabilities of reaching offsets 1, 2, and 3 are \(0.9\), \(0.9(0.8)=0.72\), and \(0.9(0.8)(0.75)=0.54\).
The corresponding discounted survival factors are
Thus the three present values are \(1{,}000(6/7)=\$857.142857\ldots\), \(1{,}200(32/49)=\$783.673469\ldots\), and \(1{,}400(160/343)=\$653.061224\ldots\). Adding before rounding gives
The reserve is \$2,293.88, rounded to the nearest cent. The check is that each payment is discounted and weighted by survival to its own date; the three survival probabilities are cumulative, not just the one-year probabilities.
Common Timing Errors
The most consequential error is to use the right annuity factors for the wrong valuation instant. Suppose the time-2 cash flows in the first example have just occurred. An incorrect calculation might include an additional net payment at offset zero:
That calculation treats the payment at time 2 as still future. The correct calculation starts at offset 1, so the reserve is \(1{,}100(0.64+0.4096)=\$1{,}154.56\). By contrast, when the reserve is immediately before a due payment, the offset-zero term belongs in the prospective value.
- Do not count elapsed cash flows. Write down the valuation instant and explicitly list the future dates before calculating.
- Do not value beyond the temporary term. The reserve includes only eligible payment dates remaining in the contract.
- Do not use unconditional survival from issue. At time \(k\), use survival conditional on being alive then, beginning from age \(x+k\).
- Do not confuse due with immediate timing. The label describes scheduled payment timing; the valuation instant determines whether the next scheduled cash flow is still future.
Check Your Understanding
For each problem, identify the remaining payment offsets before evaluating the reserve. Round dollar answers to the nearest cent.
- A hypothetical three-year annuity-immediate pays \$900 and charges a \$300 premium at times 1, 2, and 3, with both cash flows contingent on survival. At time 1 immediately after the cash flows, the annual survival probability is \(0.8\) and the effective annual interest rate is \(0\%\). Find the reserve. (Answer: \$864.00)
- A hypothetical four-year annuity-due pays \$1,000 and charges a \$400 premium at times 0, 1, 2, and 3. At time 1 immediately before the time-1 cash flows, the annual survival probability is \(0.75\) and the effective annual interest rate is \(25\%\). Find the reserve. (Answer: \$1,176.00)
- A hypothetical four-year annuity-immediate has level benefits of \$2,000 and level premiums of \$500 at each payment date. At time 2 immediately after the cash flows, the effective annual interest rate is \(4\%\), and the successive annual survival probabilities from the attained age are \(0.95\) and \(0.90\). Find the reserve. (Answer: \$2,555.94)
- A hypothetical four-year annuity-due is valued at time 2 immediately before that date’s cash flows. Its remaining benefits are \$800 at time 2 and \$1,000 at time 3; premiums of \$200 are due on both dates. The effective annual interest rate is \(5\%\), and the one-year survival probability from age \(x+2\) is \(0.9\). Find the reserve. (Answer: \$1,285.71)