Introduction
The present value of a life-contingent contract depends on more than its benefit amount. A payment made at the end of a year of death has a different timing condition from an annuity payment made only if the insured is alive at the end of that year. A pure endowment, in turn, pays at a specified future date only if the insured survives to that date. To value any of these cash flows, identify when it is paid and what must happen for it to be paid.
This tutorial develops a cash-flow method for annual discrete benefits and payments. It uses the effective annual interest rate \(i\), the discount factor \(v=1/(1+i)\), and survival probabilities such as \({}_kp_x\). The focus is on setting up and evaluating present values, not on deriving identities that relate one type of contract to another.
Match Each Payment to Its Date and Condition
Let \(T_x\) be the future lifetime of a life aged \(x\), measured in years from the valuation date. The probability that the life survives at least \(k\) years is \({}_kp_x=P(T_x>k)\). Its complement is \({}_kq_x=P(T_x\leq k)\). In a life table, \(\ell_x\) is the number of lives at age \(x\); for example, when the table’s assumptions apply, \({}_kp_x=\ell_{x+k}/\ell_x\).
For annual insurance paid at the end of the policy year of death, define the probability of death during policy year \(k\) as \({}_{k-1|}q_x=P(k-1<T_x\leq k)\). Under the usual annual life-table convention, it can be calculated as \({}_{k-1}p_xq_{x+k-1}\), where \(q_{x+k-1}\) is the one-year death probability at age \(x+k-1\). This uses the convention that death at the boundary \(k\) belongs to the year ending at \(k\). If a lifetime model can assign positive probability to an integer lifetime, its boundary convention must be stated and used consistently; one must not silently equate this interval event with an integer-valued curtate lifetime event.
| Cash flow | Payment date | Condition for payment | Probability used |
|---|---|---|---|
| Annual death benefit in policy year \(k\) | End of year \(k\) | Death during year \(k\) | \({}_{k-1|}q_x\) |
| Annuity-immediate payment at \(k\) | End of year \(k\) | Alive at time \(k\) | \({}_kp_x\) |
| Annuity-due payment at \(k\) | Time \(k\) | Alive at time \(k\) | \({}_kp_x\) |
| Pure endowment at \(n\) | End of year \(n\) | Alive at time \(n\) | \({}_np_x\) |
An annuity-immediate payment stream with payment \(c_k\) at each year-end \(k=1,\ldots,n\), payable only if the life is then alive, has APV \(\sum_{k=1}^n c_kv^k{}_kp_x\). An annuity-due instead pays at times \(k=0,\ldots,n-1\); if the payment at time \(k\) requires survival to that time, its APV is \(\sum_{k=0}^{n-1}c_kv^k{}_kp_x\), with \({}_0p_x=1\). In particular, a payment at time zero is not discounted and is not conditional on future survival.
A pure endowment with maturity amount \(B\) at time \(n\), payable only if the life is alive then, has APV \(Bv^n{}_np_x\). An \(n\)-year term insurance with benefit \(B\) paid at the end of the policy year of death has APV \(B\sum_{k=1}^n v^k{}_{k-1|}q_x\). The benefit is discounted to the end of the year in which it is paid, not to the time of death within that year.
Why the Present-Value Sums Work
For each possible payment, its discounted value is a random variable: it equals the discounted amount if the payment condition is met and zero otherwise. Its expected value is the discounted amount multiplied by the probability that the condition is met. Linearity of expectation then permits the separate expected values to be added, whether or not the payment events are independent.
A payment at time \(k\) receives discount factor \(v^k\). Do not substitute the time of death if the contract specifies an end-of-year payment.
For a survival-contingent payment at \(k\), use \({}_kp_x\). For a death benefit in policy year \(k\), use the probability of death in that year, \({}_{k-1|}q_x\).
Multiply the benefit by \(v^k\) and by the probability that it is paid. Sum the contributions over all payment dates.
Worked Example: Two-Year Term Insurance
A hypothetical two-year term policy pays $10,000 at the end of the policy year of death, if death occurs within two years. The effective annual interest rate is 5%. Conditional on being alive at the start of a year, the probabilities of surviving years 1 and 2 are 0.90 and 0.80, respectively. Find the APV. State the method and avoid treating the conditional survival probabilities as cumulative probabilities.
Method. Find the probability of death in each policy year, then discount each year-end benefit to time zero.
Year 1. The probability of death during year 1 is \(1-0.90=0.10\). Its contribution is \(10{,}000(0.10)v\), where \(v=1/1.05=0.95238095\). Thus the contribution is \(1{,}000/1.05=\$952.38095\).
Year 2. The probability of surviving year 1 and then dying during year 2 is \(0.90(1-0.80)=0.18\). Its contribution is \(10{,}000(0.18)v^2=1{,}800/(1.05)^2=1{,}800/1.1025=\$1{,}632.65306\).
Total and check. The APV is \(\$952.38095+\$1{,}632.65306=\$2{,}585.03401\), or $2,585.03 to the nearest cent. The year-2 death probability is 0.18, not 0.20: the year-2 conditional death probability must be multiplied by the probability of reaching year 2.
Survival-Contingent Annuities and Pure Endowments
For an annuity, the relevant probability is survival to each payment date. If the given survival probabilities are conditional from one year to the next, multiply them to obtain the cumulative probability of reaching a later payment date. For instance, if the successive conditional survival probabilities are \(p_1,p_2,p_3\), then the probabilities of being alive at times 1, 2, and 3 are \(p_1\), \(p_1p_2\), and \(p_1p_2p_3\). They are not \(p_1,p_2,p_3\).
Worked Example: Annuity-Immediate with Varying Payments
A hypothetical life annuity pays $500, $750, and $1,000 at times 1, 2, and 3, respectively, provided the annuitant is alive at each payment date. The effective annual interest rate is 25%. Conditional on survival to the start of each year, the probabilities of surviving years 1, 2, and 3 are 0.90, 0.80, and 0.75. Find the APV. The main trap is using the last two conditional probabilities as though they were probabilities of survival from time zero.
Method. First calculate cumulative survival probabilities, then multiply each payment by the appropriate survival probability and discount factor. Here \(v=1/1.25=0.8\).
At time 1, survival probability is \(0.90\), so the contribution is \(500(0.90)(0.8)=\$360.00\).
At time 2, survival probability is \(0.90(0.80)=0.72\), so the contribution is \(750(0.72)(0.8^2)=750(0.72)(0.64)=\$345.60\).
At time 3, survival probability is \(0.90(0.80)(0.75)=0.54\), so the contribution is \(1{,}000(0.54)(0.8^3)=1{,}000(0.54)(0.512)=\$276.48\).
Total and check. The APV is \(\$360.00+\$345.60+\$276.48=\$982.08\). The cumulative survival probabilities decrease from 0.90 to 0.72 to 0.54, as they must; using 0.80 and 0.75 directly for years 2 and 3 would overstate the chance of receiving those payments.
Worked Example: Combining a Death Benefit and a Pure Endowment
A hypothetical two-year contract pays $4,000 at the end of year 1 if the insured dies during year 1, and pays $10,000 at the end of year 2 if the insured is alive then. The effective annual interest rate is 25%. The conditional probabilities of surviving years 1 and 2 are 0.90 and 0.80. Find the APV of these benefits. The trap is to use the death probability for the maturity benefit or the maturity survival probability for the first-year death benefit.
Method. Value each benefit separately using its own payment condition and date, then add the expected present values.
The probability of death during year 1 is \(1-0.90=0.10\). The death-benefit contribution is \(4{,}000(0.10)v=400(0.8)=\$320.00\).
The probability of being alive at time 2 is \(0.90(0.80)=0.72\). The pure-endowment contribution is \(10{,}000(0.72)v^2=7{,}200(0.64)=\$4{,}608.00\).
Total and check. The APV is \(\$320.00+\$4{,}608.00=\$4{,}928.00\). The year-1 death event and survival to year 2 are mutually exclusive outcomes, and each benefit has been discounted to its stated payment date.
Common Timing and Probability Traps
A frequent error is to confuse the probability of death during a particular year with the probability of being dead by the end of that year. For a benefit paid at the end of year \(k\) only if death occurs during that year, use the probability of surviving to the start of year \(k\) and then dying in the year. The cumulative probability \({}_kq_x\) includes deaths in earlier years and is not the correct weight for that single year’s benefit.
Another common error is to apply a survival probability at the wrong date. For an annuity-immediate payment at time \(k\), use survival to time \(k\), not to time \(k-1\). For an annuity-due payment at time \(k\), use survival to time \(k\); the payment at time zero is certain under the usual assumption that the contract is in force at the valuation date. For a pure endowment, survival is required through the maturity date.
- Wrong: Treat successive conditional survival probabilities as cumulative. Correction: Multiply them to get the probability of reaching each payment date.
- Wrong: Discount an end-of-year death benefit to the exact time of death. Correction: Use \(v^k\) when the contract pays at the end of policy year \(k\).
- Wrong: Use the probability of death by year \(k\) for a benefit payable only for death in year \(k\). Correction: Use the probability of death in that specific year.
- Wrong: Discount a time-zero annuity-due payment. Correction: Its discount factor is \(v^0=1\).
These cash-flow distinctions are a foundation for insurance and annuity valuation aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus. A reliable setup states the effective annual interest rate, writes each payment date, and pairs each payment with the probability of its own condition before doing arithmetic.
Check Your Understanding
For each problem, identify the payment dates and conditions before calculating. Round dollar answers to the nearest cent.
- A two-year annuity-immediate pays $300 at each year-end if the annuitant is alive. The effective annual interest rate is 5%, and conditional survival probabilities for years 1 and 2 are 0.90 and 0.80. Find the APV. (Answer: $453.06)
- A two-year term policy pays $5,000 at the end of the policy year of death. The effective annual interest rate is 10%, and conditional survival probabilities for years 1 and 2 are 0.80 and 0.75. Find the APV. (Answer: $1,735.54)
- A pure endowment pays $12,000 at time 3 if the insured is alive then. The effective annual interest rate is 5%, and conditional survival probabilities for years 1, 2, and 3 are 0.95, 0.90, and 0.80. Find the APV. (Answer: $7,090.38)
- An annuity-due pays $250 at times 0, 1, and 2 if the annuitant is alive at the payment time. The effective annual interest rate is 4%, and conditional survival probabilities for years 1 and 2 are 0.80 and 0.75. Find the APV. (Answer: $580.99)
- A hypothetical two-year contract pays $8,000 at the end of year 1 if death occurs in year 1, $8,000 at the end of year 2 if death occurs in year 2, and an additional $2,000 at time 2 if the insured is alive then. The effective annual interest rate is 25%, and conditional survival probabilities for years 1 and 2 are 0.90 and 0.50. Find the APV. (Answer: $3,520.00)