Valuing Benefits at Death or Maturity
An equity-linked policy may pay a benefit if the insured dies during the term, or a maturity benefit if the insured survives to the end. The benefit amounts can depend on projected fund values and may differ from year to year. To value these payments, combine the amount payable on each event with the probability of that event and the discount factor for its payment time.
The key is to distinguish death during a particular year from survival through the entire term. A maturity benefit is payable only on the latter event; it is not an additional payment in a year in which a death benefit is paid. The calculations below use a discrete annual model, with all benefits paid at the end of the year of death or at the end of the term.
Assume that the valuation uses an effective annual interest rate \(i\), with \(v=1/(1+i)\). The mortality rates are conditional on survival to the start of each year. Assume also that the policy remains in force to death or maturity, and that the benefit amounts are specified on the valuation basis. In an equity-linked setting, fund projections may determine those amounts; for the mortality valuation here, treat the projected amounts as fixed for the scenario being valued.
This expression allows the death benefit to vary by year. If it is level, set \(D_k=D\) for each \(k\). The formula does not apply unchanged if benefits are payable at a different time, if the contract can terminate through another event, or if the benefit amount is itself random under the valuation being performed. Those situations require the event probabilities, cash-flow times, or benefit distribution to be represented accordingly.
Why the Event-Weighted Formula Holds
The present value is a random amount: it depends on whether death occurs in a particular year or the life survives to maturity. These possibilities are mutually exclusive. Under the stated assumptions, they also exhaust the possible outcomes for a policy that remains in force until death or maturity. The expected present value is therefore the sum of each event's present value multiplied by its probability.
For death in year \(k\), the life must survive the first \(k-1\) years and then die during year \(k\). The event probability is \({}_{k-1}p_xq_{x+k-1}\).
The death benefit \(D_k\) is paid at time \(k\), so its present value at issue is \(v^kD_k\). The maturity benefit's present value is \(v^nM\).
Multiply each present value by its event probability, then add across the mutually exclusive death-year events and survival to maturity.
The survival probabilities follow by multiplying conditional probabilities. In particular, \({}_kp_x={}_{k-1}p_xp_{x+k-1}\). Thus \({}_{k-1}p_xq_{x+k-1}\) is the probability of surviving to the start of year \(k\) and dying in that year. The event probabilities sum to one because \(\sum_{k=1}^{n}{}_{k-1}p_xq_{x+k-1}+{}_np_x=1\). This identity is also a useful check on the probability calculations.
Worked Valuations
Worked Example: Two-Year Benefit with a Separate Maturity Payment
A hypothetical two-year policy pays $1,100 at the end of the year of death. If the insured survives two years, it pays $1,331 at maturity. The annual effective interest rate is 10%. Conditional death probabilities are 0.10 in year 1 and 0.20 in year 2. Find the expected present value at issue, to the nearest cent.
Method. First calculate the mutually exclusive event probabilities. Death in year 1 has probability \(0.10\). Death in year 2 requires survival through year 1, so its probability is \(0.90(0.20)=0.18\). Survival to maturity has probability \(0.90(0.80)=0.72\). They sum to \(0.10+0.18+0.72=1.00\).
The discount factor is \(v=1/1.10\). The year-1 death benefit has present value $1,100/1.10 = $1,000 before weighting by its event probability. Its contribution to the expected present value is \(0.10(1{,}000)=\$100.00\).
The year-2 death benefit has present value \(1{,}100/(1.10)^2=\$909.0909\ldots\). Its weighted contribution is \(0.18(909.0909\ldots)=\$163.6363\ldots\). The maturity benefit has present value \(1{,}331/(1.10)^2=\$1{,}100\), giving a weighted contribution of \(0.72(1{,}100)=\$792.00\).
Adding the contributions gives \(100+163.6363\ldots+792=\$1{,}055.6363\ldots\), or $1,055.64 to the nearest cent. This is checked by weighting each year's payment only by the event that triggers it; death in year 2 and survival to maturity are distinct outcomes.
Common mistake avoided: using 0.20 as the probability of death in year 2 ignores that the insured must first survive year 1. The unconditional probability is \(0.90(0.20)=0.18\).
Worked Example: Different Death Benefits by Year
A hypothetical two-year contract pays $10,000 at the end of year 1 if death occurs in that year, and $12,000 at the end of year 2 if death occurs in year 2. It pays $12,000 at maturity if the insured survives. The annual effective interest rate is 4%, and the conditional death probabilities are 0.05 and 0.10 in years 1 and 2, respectively. Find the expected present value to the nearest cent.
Method. The year-1 death probability is \(0.05\). The year-2 death probability is \(0.95(0.10)=0.095\). The maturity survival probability is \(0.95(0.90)=0.855\). These probabilities sum to \(0.05+0.095+0.855=1.000\).
Use \(v=1/1.04\), so the discount factors for years 1 and 2 are \(1/1.04\) and \(1/(1.04)^2=1/1.0816\). The year-1 contribution is \(10{,}000(0.05)/1.04=\$480.7692\ldots\).
The year-2 death contribution is \(12{,}000(0.095)/1.0816=1{,}140/1.0816=\$1{,}053.9941\ldots\). The maturity contribution is \(12{,}000(0.855)/1.0816=10{,}260/1.0816=\$9{,}485.9467\ldots\).
Therefore the expected present value is \(480.7692\ldots+1{,}053.9941\ldots+9{,}485.9467\ldots=\$11{,}020.7100\ldots\), or $11,020.71 to the nearest cent.
As a check, the year-2 death and maturity payments have the same amount, and their event probabilities sum to \(0.095+0.855=0.95\). Their combined present-value contribution is therefore \(12{,}000(0.95)/(1.04)^2\), equal to the sum of the two separate contributions above.
Common mistake avoided: the two year-2 events must both be included, but neither is discounted for one year. Both payments occur at time 2 and use \(v^2\).
Worked Example: Three-Year Contract with Changing Benefits
A hypothetical three-year policy pays $8,000 at the end of the year of death in year 1, $10,000 in year 2, or $12,000 in year 3. If the insured survives to maturity, it pays $15,000. The annual effective interest rate is 5%. Conditional death probabilities are 0.10, 0.20, and 0.25 in years 1, 2, and 3. Find the expected present value to the nearest cent.
Method. Death in year 1 has probability \(0.10\). Death in year 2 has probability \(0.90(0.20)=0.18\). Death in year 3 requires survival through the first two years, so its probability is \(0.90(0.80)(0.25)=0.18\). Survival to maturity is \(0.90(0.80)(0.75)=0.54\). The probabilities sum to \(0.10+0.18+0.18+0.54=1.00\).
For year 1, the weighted present value is \(8{,}000(0.10)/1.05=\$761.9048\ldots\). For year 2, it is \(10{,}000(0.18)/(1.05)^2=\$1{,}632.6531\ldots\). For year 3 death, it is \(12{,}000(0.18)/(1.05)^3=\$1{,}865.8890\ldots\).
The maturity contribution is \(15{,}000(0.54)/(1.05)^3=\$6{,}997.0845\ldots\). Thus the total is \(761.9048\ldots+1{,}632.6531\ldots+1{,}865.8890\ldots+6{,}997.0845\ldots=\$11{,}257.5314\ldots\), or $11,257.53 to the nearest cent.
The event probabilities sum to one, and each benefit is discounted to its own payment year. These checks help detect both a missing maturity event and an incorrect discount exponent.
Common mistake avoided: applying the year-3 conditional death probability 0.25 directly as the probability of death in year 3. The unconditional probability is \(0.90(0.80)(0.25)=0.18\), because survival through the first two years is required.
Interpreting the Valuation and Avoiding Traps
For an equity-linked benefit, the amounts \(D_k\) and \(M\) may be obtained from projected fund values and contract terms. Once those amounts are specified on a particular deterministic valuation basis, the mortality valuation weights each amount by the probability of its triggering event and discounts it to the valuation date. If the benefit is a fund value plus a guaranteed top-up, use the full contractual amount payable on the event, not just one component.
This expected present value is a valuation of benefits under the assumptions stated; it is not automatically a reserve or a measure of profit. Those quantities require additional cash flows or valuation definitions. The present tutorial isolates the death-or-maturity benefit cash flows.
| Check | Correct treatment | Typical error |
|---|---|---|
| Death in year \(k\) | Use \({}_{k-1}p_xq_{x+k-1}\) | Use the conditional death probability alone |
| Payment in year \(k\) | Discount by \(v^k\) | Discount every payment by the same factor |
| Maturity benefit | Weight by \({}_np_x\) | Add it for policies that ended in death |
| Probability check | Death-event probabilities plus survival to maturity equal 1 | Omit an event or count an outcome twice |
A useful order of work is to calculate the event probabilities first, record the relevant benefit and payment year for each event, then calculate and sum the weighted present values. Keeping these steps separate makes the calculation auditable and makes errors in mortality, benefit amount, or discounting easier to locate. This treatment is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.
Check Your Understanding
For each problem, identify the death and survival events before calculating the present value. Round monetary answers to the nearest cent.
- A one-year hypothetical policy pays $6,000 at the end of the year if death occurs, or $7,000 at maturity if the insured survives. The annual effective interest rate is 5%, and the conditional death probability is 0.08. Find the expected present value. (Answer: $6,590.48)
- A two-year contract pays $5,000 on death in year 1, $7,000 on death in year 2, or $8,000 at maturity. The annual effective interest rate is 4%, with conditional death probabilities 0.10 and 0.20. Find the expected present value. (Answer: $6,971.15)
- A three-year contract pays $1,000 at the end of the year of death in any year, or $2,000 at maturity if the insured survives. The annual effective interest rate is 10%, and each year's conditional death probability is 0.10. Find the expected present value. (Answer: $1,321.56)
- A two-year policy pays $9,000 on death in year 1, $11,000 on death in year 2, or $14,000 at maturity. The annual effective interest rate is 5%, and the conditional death probabilities are 0.20 and 0.25. Find the expected present value. (Answer: $11,328.80)
- A two-year hypothetical fund-linked contract has projected death benefits of $8,000 in year 1 and $9,000 in year 2, plus a maturity benefit of $11,000. The annual effective interest rate is 5%, and the conditional death probabilities are 0.10 and 0.20. Treat the projected amounts as fixed for this valuation. Find the expected present value. (Answer: $9,414.97)