Why Death-Benefit Timing Matters
In an equity-linked policy, the policy fund can change as unit prices move and charges are deducted. A death claim may be based on the fund at the date of death, a fixed sum assured, or the greater of those two amounts. The contract also specifies when the claim is paid. These are separate questions: the amount of the benefit and the date the benefit is paid.
This tutorial focuses on calculating the contractual death benefit once the death time and relevant fund value are known. It does not calculate the probability of death or the present value of future claims. The distinctions are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.
The notation \(F(t-)\) emphasizes that the claim is assessed using the fund before the death benefit is paid and the policy terminates. A contract might instead specify a particular valuation date, such as the next monthly policy date. In that case, use the fund value required by that wording, not an assumed value at the instant of death.
Separate the Valuation Time from the Payment Time
Suppose death occurs during a policy year and the contract says that the death benefit is payable at the end of that year. The payment date is later than the death date, but this does not by itself mean that the benefit amount changes. If the contract fixes the amount using the fund at death, the amount is determined at death and paid later. Do not continue accumulating the fund after death unless the contract explicitly defines the benefit using a later fund value or provides for interest on the claim.
For a fund-only benefit, the key question is therefore: which fund value does the contract use? For a fixed sum assured, the fund calculation does not change the benefit amount unless the contract includes the fund as an additional payment. For a greater-of benefit, compare the specified fund value and sum assured, then pay the larger amount.
Worked Example: Fund Benefit with a Delayed Claim Payment
A hypothetical policy has a fund of $20,000 at the start of a period. It earns 2% per quarter, with no premiums or charges during the period. The contract pays the fund value immediately before death, and the benefit is paid at the end of the policy year. The insured dies after three quarters. Calculate the death benefit and identify its payment time.
Method. Project the fund only to the death time, because the contract defines the amount using the fund immediately before death. The quarterly accumulation recurrence is \(F_{j+1}=1.02F_j\), where \(F_j\) is the fund after \(j\) quarters.
Thus \(F(t-)=\$21{,}224.16\), so the fund-only death benefit is $21,224.16. The claim is paid at the end of the policy year, three months after death. Under the stated contract wording, that delay changes the payment date, not the amount.
Check. The cumulative factor is \(1.02^3=1.061208\), and \(20{,}000(1.061208)=\$21{,}224.16\), matching the step-by-step projection. A common error is to apply another quarter of growth and report $21,648.64. That would use a later fund value, contrary to the stated death-date valuation basis.
Calculating the Three Common Benefit Forms
For each benefit form, first identify the relevant fund value and any deductions made before the claim is assessed. Then apply the contract’s benefit definition. A fund-only benefit uses the specified fund value; a fixed-sum-assured benefit uses the stated amount; and a greater-of benefit compares both. A greater-of benefit is not the sum of the fund and the sum assured.
Fund only: \(B(t)=F(t-)\).
Fixed sum assured: \(B(t)=S\).
Greater of fund and sum assured: \(B(t)=\max(F(t-),S)\).
The greater-of rule follows directly from its two cases. If \(F(t-)\geq S\), the larger of the two amounts is \(F(t-)\), so \(B(t)=F(t-)\). If \(F(t-)<S\), the larger amount is \(S\), so \(B(t)=S\). The equality case gives the same benefit whichever amount is selected. This case check is also a useful way to avoid accidentally adding the two amounts.
Worked Example: Fixed Sum Assured Despite a Different Fund Value
A hypothetical policy provides a fixed death benefit of $75,000. The fund is $68,400 immediately before the claim. The insured dies in month 9, and the contract pays the claim at the end of month 10. There is no additional fund payment and no interest added to the claim. Calculate the amount and payment time.
Method. Apply the fixed-sum-assured definition. The fund value is not part of the benefit under the stated contract.
The death benefit is $75,000, payable at the end of month 10. The fund value of $68,400 does not reduce the amount and is not added to it.
Check. The contract specifies one fixed payment, $75,000, at the stated claim date. A common mistake is to calculate \(\$75{,}000+\$68{,}400=\$143{,}400\), or to substitute the fund for the fixed amount. Neither is consistent with this benefit definition.
Worked Example: Greater-of Benefit after Unit Cancellations
A hypothetical policy begins a quarter with 1,000 units. At each quarter-end, the unit price is first set and then 5 units are cancelled for charges. The prices at the first three quarter-ends are $25, $26, and $27. There are no other premiums, charges, or unit changes. Death occurs immediately after the third quarter-end charge. The contract pays the greater of the fund at death and a sum assured of $26,000, at the end of the policy year. Calculate the fund, benefit, and payment time.
Method. At each quarter-end, apply the price change and then cancel the specified units. Use the resulting units and price at the death time to calculate the fund.
After the first charge, the account has \(1{,}000-5=995\) units. Its value is \(995\times \$25=\$24{,}875\).
After the second charge, the account has \(995-5=990\) units. Its value is \(990\times \$26=\$25{,}740\).
After the third charge, the account has \(990-5=985\) units. Its value at death is \(985\times \$27=\$26{,}595\).
Compare the fund with the sum assured: \(\$26{,}595>\$26{,}000\). Therefore, the greater-of death benefit is $26,595, paid at the end of the policy year.
Check. The unit count falls by \(5\times 3=15\), from 1,000 to 985, and the final fund is \(985\times 27=\$26{,}595\). A common mistake is to add the fund and sum assured and report $52,595. The contract requires the larger amount, not their sum. Another timing error would be to deduct a further charge after death; the stated final charge has already been applied immediately before death.
A Reliable Calculation Sequence
A consistent sequence helps prevent errors when the fund changes near the claim date. Confirm what the policy says about valuation and payment before performing arithmetic. Then calculate the fund at the required time, apply the benefit formula, and report the amount separately from the date it is payable.
Identify the death time, the fund valuation time specified by the contract, and the claim payment time. Do not assume these are the same.
Apply investment changes and charges up to the specified valuation time. Keep track of whether a charge is applied before or after a unit-price change.
Use the fund, the fixed sum assured, or the greater of the two, as the contract requires.
Report when the claim is payable separately. Add post-death accumulation or claim interest only if the contract provides for it.
Common Timing and Benefit Traps
- Using the payment date’s fund automatically. A later payment date does not necessarily change a benefit fixed using the fund at death.
- Accumulating the fund after death without authority. Once the death-date benefit is fixed, further fund growth is not part of that amount unless the contract says otherwise.
- Adding a fixed sum assured to the fund. A greater-of benefit selects one amount. It does not pay both amounts.
- Ignoring charges around the valuation time. A charge applied immediately before death may change the fund used in the benefit calculation. Follow the stated ordering.
- Using a fund value from the wrong date. Some contracts specify a policy date or another valuation convention rather than the exact instant of death. Use the contractual basis supplied in the problem.
These distinctions matter because two policies with identical investment performance can produce different death payments if their benefit definitions or valuation rules differ. For calculation questions, write down the benefit form and timing basis before substituting numbers. This keeps the fund projection separate from the rule that turns that fund into a claim amount.
Check Your Understanding
For each problem, state the death benefit amount and, where given, its payment time. Assume no unmentioned additions, deductions, or claim interest.
- A fund-only policy starts with $40,000 and earns 1.5% per quarter for four quarters. Death occurs immediately after the fourth quarter’s accumulation, and the benefit is payable immediately. What is the benefit? (Answer: $42,454.54.)
- A policy pays a fixed sum assured of $90,000. Its fund is $87,250 at death in month 5, and the claim is payable at the end of month 6. What amount is payable? (Answer: $90,000 at the end of month 6.)
- A greater-of policy uses the fund immediately before death, which is $31,680, and has a sum assured of $30,000. What is the death benefit? (Answer: $31,680.)
- A policy begins with 800 units. At each of two quarter-ends, 4 units are cancelled after the unit price is set. The prices are $12 and $13.50. Death occurs immediately after the second cancellation. The benefit is the greater of the fund and $11,000, payable at year-end. What is the benefit? (Answer: $11,000 at year-end.)
- A fund-only contract fixes the benefit using the fund at death. That fund is $46,000, and the claim is paid six months later. The fund would have grown by 3% over those six months if the policy had continued, but the contract provides no post-death accumulation or claim interest. What amount is paid? (Answer: $46,000.)