Why Net Amount at Risk Depends on the Benefit Definition
A universal life policy can have an account value and a death benefit that are related in different ways. The insurer’s exposure is therefore not determined by the account value or the stated face amount alone. It depends on the death benefit actually payable and on how much of that payment is supported by the policy account.
The net amount at risk (NAR) measures the death benefit in excess of the account value at the time of the claim. Calculating it requires three decisions: identify the benefit option, apply any contractual minimum benefit, and compare the resulting benefit with the account value at the same instant. A death benefit floor can change the NAR even when the account value and stated amount are unchanged.
Translate the Benefit Option into a Payable Amount
The phrase “specified amount” does not by itself tell you whether the account value is included in the death benefit. Write the benefit formula before subtracting the account value. In a simplified specified-amount design, the benefit before any floor is the specified amount \(F_t\). In an account-value-plus-specified-amount design, that benefit is \(A_t+F_t\). These are distinct definitions, not two ways of calculating the same benefit.
A contract may also impose a floor. For example, the payable benefit might be the greater of a stated amount and the account value, or the greater of the benefit-option formula and a fixed minimum death benefit. Apply the floor to the gross death benefit first. Then subtract the account value once. Applying the floor directly to the NAR, or subtracting the account value before determining which death benefit is payable, can produce the wrong result.
| Benefit design before any additional floor | Gross benefit before any additional floor | Initial NAR calculation |
|---|---|---|
| Specified amount only | \(F_t\) | \(F_t-A_t\) |
| Account value plus specified amount | \(A_t+F_t\) | \(F_t\) |
| Specified amount with an account-value minimum | \(\max(F_t,A_t)\) | \(\max(F_t-A_t,0)\) |
The table gives the calculation before any other contract-specific minimum. If a separate fixed-dollar floor applies, use the actual benefit after that floor instead. Do not assume that every policy uses one of these simplified designs: the contract’s definition of the death benefit governs.
Worked Example: Specified Amount with an Account-Value Floor
A hypothetical policy has a specified-amount benefit of $150,000 and pays at least the account value. At the start of the year, its account value is $80,000. It receives no premiums and has no charges during the year. The annual effective credited interest rate is 5%. Assume the insured’s one-year death probability is 0.01, and the insured dies at the end of the year. Find the death benefit and NAR at that time.
Method. First project the account value to the claim date, then apply the account-value floor to the specified-amount benefit. No probability distribution is used for the point-in-time amount: the calculation is conditional on death at the year-end. The stated death probability does not multiply the claim amount.
The account value after one year is
The specified amount is $150,000, which exceeds the account value of $84,000. Therefore, the payable death benefit is
The NAR is
Check. The payable benefit is $84,000 of account value plus $66,000 of excess benefit, totaling $150,000. The death probability of 0.01 would matter in an expected-value calculation over the year, but not in this conditional claim amount. A common error is to calculate \(0.01(150{,}000-84{,}000)=660\) and call $660 the NAR; that is a probability-weighted amount, not the NAR at death.
Why the Account Value Is Subtracted
The account value is part of the policy’s financial backing for its death benefit. The insurer’s death benefit exposure above that backing is the excess of the gross payment over the account value. This explains the subtraction in the definition; it is not an adjustment for the chance that death occurs.
Express the benefit option as a gross amount payable before applying any separate floor.
Use the greater of the benefit formula and any applicable contractual minimum, as required by the contract.
Use the account value immediately before the claim at the same date as the death benefit.
Subtract the account value from the payable death benefit. Verify that the result agrees with the stated floor and benefit definition.
In symbols, if \(B_t\) is the benefit-option amount before a separate fixed-dollar minimum \(G_t\), and the contract pays the greater of those two amounts, then
When \(B_t=A_t+F_t\), a fixed minimum may bind even though the account-value-plus-specified-amount formula already includes the account value. It is still the gross benefit that is compared with the fixed minimum. The NAR is calculated only after identifying the greater payable amount.
Worked Example: A Fixed Minimum Benefit Binds
A hypothetical policy has an account-value-plus-specified-amount benefit. At the claim date, the account value is $92,000 and the specified amount is $58,000. The contract also requires a gross death benefit of at least $165,000. The account value already reflects all crediting and charges through the claim date; the annual effective credited rate was 3% during the preceding year. Assume a one-year death probability of 0.02 and condition the calculation on death at the claim date. Find the payable benefit and NAR.
Method. Calculate the benefit-option amount, compare it with the fixed minimum, then subtract the account value. The given account value is already at the claim date, so applying another year of interest would mismatch the valuation time. No mortality distribution is needed for a claim amount conditional on death.
Before the fixed minimum, the account-value-plus-specified-amount benefit is
The fixed minimum is greater, so the payable benefit is
Subtract the account value once:
Check. The $165,000 payment consists of $92,000 of account value and $73,000 of excess. A common wrong calculation is to subtract the account value from the option amount first, obtaining $58,000, and then compare $58,000 with the $165,000 gross-benefit minimum. Those quantities are on different bases: the floor is a minimum gross benefit, not a minimum NAR. The 0.02 mortality probability is not applied to the conditional NAR.
When the Account-Value Floor Controls
If the account value exceeds a specified amount and the contract pays at least the account value, the actual benefit is the account value rather than the lower specified amount. The NAR under this simplified design is then zero. A negative difference from subtracting the account value from the specified amount is a useful signal to check the floor; it is not the positive exposure under a contract that pays at least the account value.
Worked Example: Avoiding a Negative NAR
A hypothetical specified-amount policy has a $105,000 specified amount and pays no less than its account value. Immediately before a claim, the account value is $118,000. The account value is supplied at the claim date and already reflects the policy’s prior transactions; the credited rate for the preceding year was an effective annual 4%. Assume the one-year death probability is 0.015 and that death occurs at the stated claim time. Determine the actual benefit and NAR.
Method. Apply the account-value floor before finding the excess over the account value. The calculation is conditional on a claim at the stated time; the mortality probability does not change the payable amount.
The payable death benefit is
Therefore,
Check. The insurer pays no more than the amount represented by the account value under this benefit definition, so there is no positive benefit above the account value. The tempting calculation \(105{,}000-118{,}000=-13{,}000\) is the signed difference between the stated amount and the account value, but it is not the NAR after applying the contractual floor. This does not imply that every policy has zero exposure whenever its account value exceeds its specified amount; the result depends on the actual benefit wording.
Common Exam Traps and Practical Checks
NAR calculations are short, but small interpretation errors can change the result substantially. Before finalizing an answer, check each of the following:
- Adding the account value when the option does not. A specified-amount benefit is not automatically an account-value-plus-specified-amount benefit.
- Omitting the account value when it is included. For a benefit defined as account value plus a specified amount, subtracting the account value leaves the specified amount, before any separate floor changes the payable benefit.
- Applying a floor to the wrong quantity. A minimum death benefit is a minimum gross payment unless the contract explicitly defines a different floor.
- Using inconsistent dates. Do not compare a start-of-month account value with a death benefit at the end of the month. Use the account value at the claim time, after whatever transactions the contract says occur before the claim.
- Multiplying by the probability of death. NAR is conditional on the death claim being made. A probability-weighted claim amount is a different calculation.
- Accepting a negative exposure without checking the contract. Revisit the benefit floor and distinguish a signed difference from the nonnegative exposure when that distinction applies.
NAR is also relevant because mortality-related charges may be tied to an amount at risk. The precise charge calculation depends on the contract’s definitions and timing conventions; do not infer a charge solely from a NAR unless the required rate and charge basis are specified. Keeping the benefit, account value, date, and floor explicit makes the NAR calculation auditable and aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.
Check Your Understanding
For each hypothetical policy, use the benefit definition and floor stated in the question. The account values are at the claim time unless a projection is requested.
- A specified-amount policy pays the greater of its $132,000 specified amount and its account value. The account value is $47,000. Find the death benefit and NAR. (Answer: death benefit $132,000; NAR $85,000)
- A benefit equals account value plus a $44,000 specified amount. The account value is $76,000, and there is no other floor. Find the NAR. (Answer: $44,000)
- A specified-amount policy has a $96,000 amount and an account-value floor. The account value is $111,000. Find the benefit and positive NAR exposure. (Answer: death benefit $111,000; NAR $0)
- A benefit equals account value plus $52,000, but the contract guarantees a gross death benefit of at least $140,000. At the claim time the account value is $81,000. Find the payable benefit and NAR. (Answer: death benefit $140,000; NAR $59,000)
- An account-value-plus-specified-amount policy has a $63,000 specified amount. Its account value is $90,000 at the start of a year. There are no premiums or charges during the year, and the account earns an effective annual credited rate of 4%. At year-end, death occurs and a fixed minimum gross death benefit of $160,000 applies. Find the year-end account value, actual benefit, and NAR. (Answer: account value $93,600; death benefit $160,000; NAR $66,400)