Introduction
A universal life policy has an account value that changes as premiums, credited interest, and policy charges are applied. Under a level death benefit, those account movements do not automatically change the stated death benefit. They do, however, change how much of that benefit is exposed to the insurer’s risk.
The key calculation is the net amount at risk: the death benefit less the account value, when the account value is below the benefit and the specified benefit governs. The account value is not added to the level death benefit. It is part of the financial backing for the claim.
Separate the Benefit from the Account Projection
The account value must first be projected using the policy’s transaction order. A premium may be added before interest is credited, and a charge may be deducted after interest. The ordering matters: interest applies to the amount in the account at the time interest is credited. Once the account value at the relevant date is known, compare it with the fixed benefit to calculate net amount at risk.
For a simple annual projection with a net premium \(P\) added before annual effective interest \(i\), followed by a charge \(C\) deducted after interest, the account value at year-end is:
This recurrence applies only to the stated transaction order. If the contract applies a premium, interest credit, or charge at a different time, the projection must reflect that order instead. The net premium is the amount actually allocated to the account, not necessarily the gross premium paid.
Apply the stated premiums, interest credits, and charges in their specified order.
Use the fixed specified amount \(B\) under the level-benefit assumption in the problem.
When \(AV < B\), subtract the projected account value from the benefit: \(B-AV\).
Why Account Growth Reduces Net Amount at Risk
The relationship follows from a simple decomposition of the claim. When the account value is below the fixed benefit, the benefit can be written as the account value plus the remaining amount:
The second term is the portion not covered by the account value and is the insurer’s net amount at risk under this definition. This is a financial decomposition, not a direction to pay the beneficiary \(AV\) plus \(B\). The contractual payment remains \(B\). If the benefit stays fixed and the account value rises by an amount \(\Delta\), then:
Thus, while the account remains below the benefit, a one-dollar increase in account value reduces net amount at risk by one dollar. Premiums and interest can raise the account and reduce exposure; charges can lower the account and increase exposure. The size of the change depends on the account transactions, not on a change to the level benefit.
Worked Example: Projecting the Account Before Calculating Exposure
A hypothetical policy has a level death benefit of $180,000 and an opening account value of $72,000. A net premium of $4,000 is added before interest. The account earns 3% annual effective interest, and a $2,280 charge is deducted after interest. Find the year-end account value and net amount at risk. Assume the account remains below the benefit and no other benefit provision changes the level amount. Mortality is not projected during the year; evaluate the claim as if death occurs at year-end. No probability distribution is needed for this deterministic calculation.
Method. Follow the stated transaction order, then subtract the resulting account value from the fixed benefit.
Step 1: Add the net premium. The amount eligible for interest is $72,000 + $4,000 = $76,000.
Step 2: Credit interest. Interest is 3% of $76,000, or $76,000 × 0.03 = $2,280. The account before the charge is $76,000 + $2,280 = $78,280.
Step 3: Deduct the charge. The year-end account value is $78,280 − $2,280 = $76,000.
Step 4: Calculate net amount at risk. It is $180,000 − $76,000 = $104,000. As a check, $76,000 + $104,000 = $180,000.
The year-end account value is $76,000 and the net amount at risk is $104,000. A common error is to stop at $78,280 and omit the charge; another is to calculate interest on the opening account alone, even though the premium is added first.
Worked Comparisons: Fixed Benefit, Changing Exposure
The level benefit and net amount at risk answer different questions. The benefit is the contractual death claim under the assumptions of the calculation. Net amount at risk measures the part of that benefit not represented by the account value. Therefore, when a problem asks for both, calculate and label them separately.
Worked Example: A Premium and Charge Change the Net Amount at Risk
A hypothetical policy has a $150,000 level death benefit and an opening account value of $94,000. A net premium of $1,500 is added before interest. The annual effective credited rate is 2%, and a fixed charge of $425 is deducted after interest. Calculate the ending account value and net amount at risk. Assume the account remains below the benefit and the death claim is evaluated at year-end; no mortality distribution is used.
Method. Project the account in the contractual order and compare its year-end value with the unchanged benefit.
Step 1: Add the premium. $94,000 + $1,500 = $95,500.
Step 2: Calculate interest. $95,500 × 0.02 = $1,910. The account before the charge is $95,500 + $1,910 = $97,410.
Step 3: Deduct the charge. The ending account value is $97,410 − $425 = $96,985.
Step 4: Calculate net amount at risk. $150,000 − $96,985 = $53,015. The check is $96,985 + $53,015 = $150,000.
The death benefit remains $150,000; the ending account value is $96,985; and the net amount at risk is $53,015. The common mistake is to add the account value to the benefit and report a claim of $246,985. Under the stated level-benefit assumption, the claim is $150,000, not the benefit plus the account.
Worked Example: Comparing Two Crediting Rates
A hypothetical policy has a $200,000 level death benefit, an opening account value of $118,000, and a net premium of $5,000 added before annual interest. A fixed annual charge of $1,800 is deducted after interest. Compare year-end account value and net amount at risk under annual effective rates of 4% and 2%. Assume the account stays below the benefit and death is evaluated at year-end. The account projection is deterministic, so no probability distribution or mortality model is required.
Method. Keep all cash flows and the benefit fixed; change only the credited rate.
Step 1: Determine the amount before interest. In both cases, $118,000 + $5,000 = $123,000.
Step 2: Calculate the 4% case. Interest is $123,000 × 0.04 = $4,920. After interest the account is $127,920; after the $1,800 charge it is $126,120. Net amount at risk is $200,000 − $126,120 = $73,880.
Step 3: Calculate the 2% case. Interest is $123,000 × 0.02 = $2,460. After interest the account is $125,460; after the charge it is $123,660. Net amount at risk is $200,000 − $123,660 = $76,340.
Step 4: Check the comparison. The account value is $126,120 − $123,660 = $2,460 higher at 4%. Net amount at risk is $76,340 − $73,880 = $2,460 lower at 4%, as expected for the same fixed benefit.
The 4% case has account value $126,120 and net amount at risk $73,880; the 2% case has account value $123,660 and net amount at risk $76,340. The common mistake is to assume a higher credited rate changes the level death benefit. Here it changes the account value and thus the net amount at risk, not the stated benefit.
Interpreting the Result and Avoiding Traps
For a level-benefit calculation, the beneficiary’s contractual death payment and the insurer’s net amount at risk should not be confused. If death occurs at the valuation point in these examples, the policy pays the stated benefit, subject to the assumptions given. Net amount at risk is the benefit less the account value; it is not the amount paid to the beneficiary.
The formula \(B-AV\) is being used here only when the level specified benefit governs and the account value is below it. If a contract’s benefit provisions alter the payable amount, or if the account value is not below the benefit, the assumptions behind this calculation must be revisited. Do not extend the subtraction formula beyond its stated conditions.
A mortality charge may be based on net amount at risk when the contract defines it that way. For example, if a monthly rate is quoted per $1,000 of net amount at risk, first calculate the applicable net amount at risk and then apply the rate using the correct units. The amount of a mortality charge is a policy transaction; it is not itself the death benefit.
- Do not ignore timing. A premium added before interest earns interest in that period; a charge deducted after interest does not reduce the amount on which that period’s interest was calculated.
- Do not add account value to a level benefit. The account supports the benefit; it is not automatically an additional claim payment.
- Do not hold net amount at risk fixed. Even with a constant benefit, account movements change the difference between benefit and account value.
- Do not use a negative difference without checking the contract. The formula here is conditional on the account being below the governing benefit.
The account-value and net-amount-at-risk distinction is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus. In any calculation, state the benefit assumption, follow the transaction order, and show the subtraction that produces the exposure.
Check Your Understanding
For each problem, follow the stated timing and report money to the nearest cent where needed.
- A hypothetical policy has a level death benefit of $125,000 and an account value of $48,000 at the valuation date. The account is below the benefit and no other provision changes the amount. Find the net amount at risk. (Answer: $77,000)
- A policy has a $140,000 level benefit and an opening account value of $86,000. A net premium of $2,000 is added before interest; the annual effective rate is 3%; a $900 charge is deducted after interest. Find the ending account value and net amount at risk. Assume death is evaluated at year-end and the account stays below the benefit. (Answer: Account value $89,740; net amount at risk $50,260)
- A policy has a $210,000 level benefit and an opening account value of $132,000. A net premium of $4,000 is added before annual effective interest of 2%, and a $1,500 charge is deducted after interest. Find the ending account value and net amount at risk. (Answer: Account value $137,220; net amount at risk $72,780)
- A level benefit is $180,000. The account value rises from $101,400 to $108,650, with no change in the governing benefit. Calculate the net amount at risk at each account value and the change in net amount at risk. (Answer: $78,600 initially; $71,350 afterward; decrease of $7,250)
- A hypothetical policy has a level benefit of $160,000 and an account value of $95,500 at the start of a month. The account is below the benefit. A monthly mortality charge is specified as $0.80 per $1,000 of net amount at risk at that time. Calculate the net amount at risk and the charge. (Answer: Net amount at risk $64,500; charge $51.60)