Three Amounts That Must Not Be Confused
In an equity-linked policy, investment performance can change the value attributed to the policy, but it does not necessarily change every amount in the contract. A sound projection keeps three quantities separate: the policyholder fund, the insurer’s cash flows, and the contractual benefit. The distinction is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.
The fund is the value of the policy’s investment account at a specified time. Insurer cash flows include premiums received, amounts allocated to the fund, charges, expenses and benefits paid. The contractual benefit is the amount the insurer is obliged to pay when a stated event occurs. It may equal the fund, or it may include a guarantee that requires the insurer to pay more.
A fund value may depend directly on investment performance. A fixed guaranteed minimum does not vary with performance, although the top-up needed to meet it does. Premiums and contractual charges may also be specified independently of investment performance, while their effect on the fund depends on when they are deducted.
A Fund Projection Requires a Timing Convention
For a simple annual projection, let \(F_{t-1}\) be the fund at the start of year \(t\), \(P_t\) the gross premium paid at the start of the year, and \(\alpha_t\) the proportion of that premium allocated to the fund. The allocation charge is therefore \((1-\alpha_t)P_t\). Assume the allocated premium is added immediately, the fund earns a deterministic annual return \(r_t\), and a fixed fee \(c_t\) is deducted at year-end. Then the closing fund is
This is a stipulated projection convention, not a universal description of all contracts. If a premium is paid later, a charge is deducted at a different time, or the return is applied to a different balance, the formula must be changed to match those terms. In these examples, returns are deterministic scenarios; no probability distribution or mortality assumption is used to project the account. When a claim is included, it is assumed to occur at the stated time.
The formula follows from the sequence of cash flows. At the start of the year, the account contains \(F_{t-1}+\alpha_tP_t\). Applying the return multiplies this amount by \(1+r_t\). Deducting the year-end fee then gives the closing value. In particular, the fee does not earn the year’s return under this timing convention.
Worked Example: Projecting Two Fund Values and a Death Benefit
A hypothetical policy starts with no fund. In year 1, the gross premium is $1,000, 90% is allocated to the fund, the deterministic return is 4%, and a $20 fee is deducted at year-end. In year 2, the gross premium is $750, 80% is allocated, the deterministic return is 3%, and the year-end fee is $15. A $1,700 death benefit is payable if death occurs at the end of year 2. Calculate each year’s allocation charge and closing fund, then find the required top-up. No mortality probability is required: assume death occurs at the stated time.
Method. Apply the stated annual recurrence in each year, using the first year’s closing fund as the second year’s opening fund.
Year 1 allocation. The amount allocated is \(0.90(1{,}000)=\$900\). The allocation charge is \(1{,}000-900=\$100\). The opening fund is zero, so the closing fund is
Year 2 allocation. The amount allocated is \(0.80(750)=\$600\), and the allocation charge is \(750-600=\$150\). The amount in the fund before investment growth is \(916+600=\$1{,}516\). Thus,
The benefit is $1,700, so the top-up is \(1{,}700-1{,}546.48=\$153.52\). The insurer pays $1,700 in total: $1,546.48 is met by releasing the fund, and $153.52 is the additional amount required to meet the guarantee.
Check. The two allocation charges are $100 and $150, and the year 2 calculation starts with \(916+600=1{,}516\). The fund plus top-up equals the stated benefit: \(1{,}546.48+153.52=1{,}700\). The common error is to use the gross year 2 premium of $750 as the fund addition; only $600 is allocated.
Benefits, Fund Release and Guarantee Cost
If a benefit \(B\) is payable when the fund is \(F\), and the contract permits the fund to be applied toward the benefit, the additional amount required from the insurer is
The maximum is necessary because a benefit below the fund does not create a positive top-up. The treatment of any excess fund in that case depends on the contract; it should not be assumed to be an insurer gain without checking the terms.
For a benefit defined as the greater of the fund and a guaranteed amount \(G\), the contractual payment is \(B=\max(F,G)\). The top-up can then be written as
This identity shows why the guaranteed amount and the top-up are different quantities. The guarantee \(G\) is fixed by the contract, but the top-up changes with the fund. Strong investment performance may eliminate the top-up; weak performance may increase it. Neither the fund nor the top-up alone describes all insurer cash flows, which may also include premiums, charges, expenses and other obligations.
Worked Example: Separating the Fund from a Guaranteed Payment
A hypothetical policy has an opening fund of $1,100. At the start of the year, the policyholder pays a gross premium of $600. The contract allocates 90% of it to the fund, earns a deterministic 5% for the year, and deducts a $25 fee at year-end. If death occurs at year-end, the benefit is the greater of the fund and $1,800. Find the closing fund, the allocation charge, the benefit and the top-up. Assume the death occurs at year-end; no mortality probability is used.
Method. First project the fund under the specified timing convention. Then compare that fund with the guaranteed amount.
The amount allocated is \(0.90(600)=\$540\), so the allocation charge is \(600-540=\$60\). Before investment growth, the fund is \(1{,}100+540=\$1{,}640\). After the 5% return it is \(1{,}640(1.05)=\$1{,}722\). After the fee, the closing fund is \(1{,}722-25=\$1{,}697\).
The insurer’s benefit payment is $1,800, not $103. The $103 is only the portion in excess of the fund released toward the payment. The allocation charge is $60, but it should not be confused with either the benefit or the guarantee top-up.
Check. The allocated premium plus opening fund is $1,640; 5% growth produces $1,722, and the $25 fee gives $1,697. Adding the $103 top-up to that fund gives the $1,800 benefit. A common mistake is to apply the 5% return to the fund and then omit the year-end fee.
Insurer Cash Flows Are Not the Policy Fund
The insurer’s cash-flow view records transactions from the insurer’s perspective. For example, a gross premium is received by the insurer, but only the allocated portion is added to the policy fund under the convention above. The allocation charge is the difference between the gross premium and that allocation; it is not an extra premium received on top of the gross amount. A fee deducted from the fund reduces the policy account and may be a receipt to the insurer under the contract, but a full profit calculation would also need relevant expenses, claim costs and valuation assumptions.
At a claim, it is useful to reconcile the gross contractual payment with the fund applied to it. If the benefit is $2,000 and the fund released is $1,955.60, the insurer’s additional payment is $44.40. The gross benefit remains $2,000. Describing the insurer as paying only $44.40 would omit the fund-funded part of the contractual benefit.
Worked Example: Reconciling Premium, Fund and Claim Cash Flows
A hypothetical unit-linked policy has an opening fund of $940. A gross premium of $1,000 is paid at the start of the year, with 95% allocated to the fund. The deterministic annual return is 4%, and a $10 fee is deducted at year-end. A death benefit of $2,000 is payable if death occurs at year-end. Calculate the allocation charge, closing fund, total benefit payment and top-up. No mortality probability is assumed; the claim is taken to occur at year-end.
Method. Separate the gross premium from the amount allocated, project the fund, and then reconcile the fund with the contractual benefit.
The amount allocated is \(0.95(1{,}000)=\$950\), and the allocation charge is \(1{,}000-950=\$50\). The amount before investment growth is \(940+950=\$1{,}890\). Applying the 4% return gives \(1{,}890(1.04)=\$1{,}965.60\). Deducting the fee gives
The stated death benefit is $2,000, so the top-up is \(2{,}000-1{,}955.60=\$44.40\). The insurer’s gross benefit payment is $2,000, comprising $1,955.60 met by the fund and $44.40 of additional insurer payment.
Check. The premium split reconciles: $950 allocated plus $50 charge equals the $1,000 gross premium. The fund projection starts from \(940+950=1{,}890\), not $1,390. The benefit reconciliation also balances: \(1{,}955.60+44.40=2{,}000\). A common mistake is to add the allocated premium incorrectly or to report the top-up as the full claim payment.
What Varies and What Is Guaranteed?
| Quantity | Typical dependence on investment performance | What to check |
|---|---|---|
| Policy fund | Usually varies with fund returns and deductions | Projection date, allocation and charge timing |
| Gross premium | Often contractually specified rather than return-dependent | Payment amount and payment date |
| Allocation charge or fixed fee | May be fixed or defined by a rate; not necessarily return-dependent | Whether it is deducted from premium or fund |
| Guaranteed benefit | Does not change with fund performance if stated as a fixed amount | Event that triggers it and any conditions |
| Guarantee top-up | Varies with the fund relative to the guarantee | Benefit less fund, floored at zero where appropriate |
These labels are not substitutes for contract terms. A benefit described as “fund value” varies with the fund, while a benefit with a guaranteed floor includes a fixed contractual minimum. The insurer’s exposure to that floor depends on the shortfall at the payment date, not just on the guaranteed amount in isolation.
Common Exam Traps
- Using the gross premium as the fund addition. Add only the allocated premium when the contract deducts an allocation charge first.
- Applying a return to the wrong balance. Confirm whether the premium is invested at the beginning or end of the period and whether fees are deducted before or after growth.
- Calling the top-up the benefit. The top-up is only the amount above the fund; the contractual payment can be larger.
- Treating the guarantee as investment-linked. A fixed guaranteed amount is unchanged by returns, even though the top-up needed to meet it is return-sensitive.
- Calling an allocation charge profit. It is a component of cash-flow accounting, not by itself a complete profit measure.
A reliable solution states the timing convention, projects the fund, identifies the exact contractual benefit and then reconciles the benefit with the fund. Keeping those steps separate prevents an investment result from being mistaken for a claim amount or an insurer cash flow.
Check Your Understanding
For each problem, use the stated timing convention: premium allocated at the start, deterministic return applied to the resulting fund, and fee deducted at year-end. No mortality probability is needed unless a probability is explicitly requested.
- A policy has an opening fund of $500, receives a $400 premium with 90% allocated, earns 2%, and deducts an $8 year-end fee. Find the allocation charge and closing fund. (Answer: Allocation charge $40.00; closing fund $869.20.)
- A fund of $1,200 receives a $500 premium with 80% allocated, earns a deterministic return of -5%, and is charged a $10 year-end fee. The death benefit is the greater of the fund and $1,600. Find the closing fund and top-up if death occurs at year-end. (Answer: Closing fund $1,510.00; benefit $1,600.00; top-up $90.00.)
- A policy starts with $700. In year 1, a $300 premium is 90% allocated, the return is 10%, and the fee is $5. In year 2, a $200 premium is 75% allocated, the return is -4%, and the fee is $8. Find both closing funds and the total allocation charges. (Answer: Year 1 fund $1,062.00; year 2 fund $1,155.52; total allocation charges $80.00.)
- A policy’s closing fund is $2,240. Its maturity benefit is the greater of the fund and 102% of total gross premiums paid, which were $2,100. Find the maturity benefit and guarantee top-up. (Answer: Benefit $2,240.00; top-up $0.00.)
- A policy starts with a $940 fund. A $1,000 premium has 95% allocated, the return is 4%, and a $10 fee is deducted at year-end. The death benefit is $2,000. Find the closing fund, allocation charge, top-up and total benefit payment. (Answer: Closing fund $1,955.60; allocation charge $50.00; top-up $44.40; benefit payment $2,000.00.)