Why Net Amount at Risk Matters
A universal life death claim may include both the policy account value and an amount the insurer must provide beyond that account. The insurer’s exposure to mortality is tied to this second amount, not necessarily to the full death benefit. Calculating it correctly is important when separating account-funded benefits from the cost of the insurance protection and when interpreting a prospective reserve.
The result depends on the contractual death benefit design and on the account value at the time the claim is measured. For a claim during a year, the account may have changed since the beginning of the year because of credited interest, premiums, charges, or other transactions. The timing convention must therefore be stated before calculating the exposure.
For an annual model, let \(q_t\) be the probability that a policy in force at the beginning of year \(t\) dies during that year, conditional on being in force at its start, and let \(p_t=1-q_t\). If death benefits are paid at year-end, expected net mortality outgo for that year is \(q_tN_{t+1}\), where \(N_{t+1}\) is measured at the claim date. This is an expected amount per policy in force at the start of the year, before discounting. It is not the expected claim conditional on death; that conditional amount is \(N_{t+1}\).
Death Benefit Designs and the NAR
Two common designs illustrate why the policy wording matters. Under a level death benefit design, the death benefit is a specified amount, subject to any contractual minimum or corridor. The account value can therefore reduce the portion of the benefit that is an insurer-funded mortality exposure. Under an increasing design, the benefit is the specified amount plus the account value, so the account value is included in the claim and the excess is the specified amount.
| Design, before any corridor or minimum | Death benefit at claim | Net amount at risk |
|---|---|---|
| Level specified amount \(F\) | \(D=F\) | \(N=F-\mathrm{AV}\) |
| Specified amount plus account value | \(D=F+\mathrm{AV}\) | \(N=F\) |
| Benefit subject to a corridor | \(D=\max(F,c\,\mathrm{AV})\) | \(N=\max(F,c\,\mathrm{AV})-\mathrm{AV}\) |
Here \(F\) is the specified amount and \(c\) is the corridor factor stated in the hypothetical contract. A corridor design may be described as a level benefit subject to a minimum multiple of account value. The maximum in the benefit formula must be evaluated before subtracting the account value. If a contract uses a different definition—for example, a different corridor base or a minimum benefit—the actual contractual formula replaces these examples.
Worked Example: Level Benefit and Claim-Date Account Value
A hypothetical policy has a level death benefit of $100,000. At the beginning of the year its account value is $62,000, which is credited interest at an effective annual rate of 4.00%. Assume no other account transactions during the year. Death claims are paid at year-end, and the conditional probability of death during the year is \(q_t=0.0200\). Find the claim-date NAR and expected net mortality outgo per policy in force at the start of the year.
Method. First accumulate the account value to the claim date. Then subtract that value from the level death benefit and multiply by the annual death probability.
The claim-date account value is $$ \mathrm{AV}_{t+1}=62{,}000(1.04)=64{,}480. $$ The benefit is level, so \(D_{t+1}=100{,}000\). Therefore, $$ N_{t+1}=100{,}000-64{,}480=35{,}520. $$ Expected net mortality outgo per policy in force at the start of the year is $$ q_tN_{t+1}=0.0200(35{,}520)=710.40. $$ As a check, expected gross claims are \(0.0200(100{,}000)=2{,}000.00\), and the expected account-value component of those claims is \(0.0200(64{,}480)=1{,}289.60\). Their difference is \(2{,}000.00-1{,}289.60=710.40\), matching the NAR calculation. The common mistake avoided here is subtracting the beginning-year account value rather than the account value at the stated claim time.
Worked Example: Increasing Benefit
A hypothetical increasing-benefit policy pays a specified amount of $80,000 plus the account value at death. At the beginning of the year, the account value is $50,000 and earns 3.00% effective annual interest, with no other account transactions. The annual conditional probability of death is \(q_t=0.0150\), and the claim is paid at year-end. Calculate the death benefit, NAR, and expected net mortality outgo per policy in force at the beginning of the year.
Method. Calculate the account value at the claim date, add it to the specified amount to obtain the gross benefit, and then subtract that same account value to find the NAR.
The account value at the claim date is $$ \mathrm{AV}_{t+1}=50{,}000(1.03)=51{,}500. $$ The death benefit is $$ D_{t+1}=80{,}000+51{,}500=131{,}500. $$ Thus, $$ N_{t+1}=131{,}500-51{,}500=80{,}000, $$ which equals the specified amount. Expected net mortality outgo is $$ 0.0150(80{,}000)=1{,}200.00. $$ As a check, expected gross claims are \(0.0150(131{,}500)=1{,}972.50\), while the expected account-value portion is \(0.0150(51{,}500)=772.50\). The difference is \(1{,}200.00\). The common mistake avoided is treating the entire increasing death benefit as mortality exposure; its account-value component is not part of the NAR.
Worked Example: Applying a Corridor Before Subtraction
A hypothetical policy pays the greater of a specified amount of $90,000 and 2.5 times the account value at death. The beginning-year account value is $40,000, earns 5.00% effective annual interest, and has no other transactions during the year. The conditional annual death probability is \(q_t=0.0120\), with claims paid at year-end. Find the death benefit, NAR, and expected net mortality outgo per policy in force at the start of the year.
Method. Evaluate both branches of the death benefit formula at the claim-date account value, select the greater amount, and only then subtract the account value.
The claim-date account value is $$ \mathrm{AV}_{t+1}=40{,}000(1.05)=42{,}000. $$ The corridor amount is \(2.5(42{,}000)=105{,}000\), which exceeds the specified amount of $90,000. Therefore, $$ D_{t+1}=\max(90{,}000,105{,}000)=105{,}000, $$ and $$ N_{t+1}=105{,}000-42{,}000=63{,}000. $$ Expected net mortality outgo is $$ 0.0120(63{,}000)=756.00. $$ The result is consistent with the selected corridor branch: its benefit is 2.5 times the account value, so its NAR is \(1.5\) times the account value, or \(1.5(42{,}000)=63{,}000\). The common mistake avoided is subtracting the account value from the specified amount without first checking whether the corridor controls.
Why Expected Mortality Outgo Uses the NAR
For a policy in force at the start of the year, a death occurs with probability \(q_t\). If it occurs, the claim-date amount not represented by the account is \(N_{t+1}\). If no death occurs, there is no death claim in this year. Therefore the random net claim amount is \(N_{t+1}\) on death and zero otherwise. Its expected value is the death probability times the amount paid on death:
This argument assumes a single annual death probability and a claim amount that is fixed at the stated claim-date value for the year. If the account value or benefit varies with the precise time of death, then the claim amount is random conditional on death, and the appropriate expression is \(q_t E[N\mid\text{death during year }t]\). If benefits are paid at a different time, discounting and account accumulation must follow that timing. The formula also does not include expenses, premiums, lapse, or other policy cash flows; those require separate terms in the valuation model.
Net Amount at Risk in a Prospective Reserve
A prospective reserve represents the value of future benefits less the value of future premiums, evaluated on a specified basis. For an account-based policy, it is useful to distinguish the liability associated with the account value from the additional insurance exposure. The following recursion shows precisely how the NAR enters that insurance component in a simplified annual model.
Under these assumptions, the full reserve recursion is the discounted expected value of the death benefit on death and the next reserve on survival. Subtracting the corresponding account-value recursion leaves the expected value of the excess claim on death and the next excess reserve on survival.
where \(v=1/(1+i)\), and the same claim-date account amount is used in either outcome under the stated account-growth assumption. Since \(q_t+p_t=1\), the account recursion simplifies to \(\mathrm{AV}_t=v\mathrm{AV}_{t+1}\). Subtract the second recursion from the first and use \(R_{t+1}=V_{t+1}-\mathrm{AV}_{t+1}\):
This recursion makes the role of NAR explicit: the mortality component is the probability-weighted excess claim, while the survival component is the next excess reserve. The result is not a universal recursion for every universal life contract. If the account earns a rate different from the valuation rate, or if premiums, charges, expenses, lapses, or other cash flows are present, their effects must be included; the simple subtraction above may no longer isolate the reserve difference in this form.
Worked Example: One-Year Excess Reserve
Use the simplified reserve model above. At the next year-end, the NAR on death is $50,000. The annual death probability is \(q_t=0.0200\), the effective annual valuation interest rate is 5.00%, and the excess reserve for a surviving policy at year-end is \(R_{t+1}=\)$1,200. Calculate \(R_t\), the beginning-year excess reserve.
Method. Apply the one-year recursion \(R_t=v[q_tN_{t+1}+p_tR_{t+1}]\), with \(p_t=1-q_t\).
First, \(p_t=1-0.0200=0.9800\). The expected year-end excess amount is $$ 0.0200(50{,}000)+0.9800(1{,}200) =1{,}000+1{,}176 =2{,}176. $$ The discount factor is \(v=1/1.05\). Therefore, $$ R_t=\frac{2{,}176}{1.05}=2{,}072.38095\ldots $$ Rounded to the nearest cent, \(R_t=\)$2,072.38. As a check, the year-end expected amount \(2{,}176\) is greater than the reserve because it is discounted at a positive interest rate. The common mistake avoided is discounting the death component but not the survival component; the recursion discounts both year-end outcomes.
Common Exam Traps
The most frequent error is confusing the death benefit with the NAR. For the increasing-benefit policy in the second example, an incorrect calculation would be $$ 0.0150(131{,}500)=1{,}972.50. $$ That is expected gross death claims, not expected net mortality outgo. The correction is to remove the account-funded amount from each death claim: $$ 0.0150(131{,}500-51{,}500)=0.0150(80{,}000)=1{,}200.00. $$ Other checks help prevent related errors:
- Match the account value and death benefit at the same time point. Do not combine a beginning-year account value with an end-of-year benefit unless the question explicitly requires it.
- For an increasing benefit, subtract the account value included in the benefit; do not add it again to the NAR.
- For a corridor or minimum benefit, apply the contractual maximum or minimum before calculating \(D-\mathrm{AV}\).
- Distinguish an expected amount per policy in force at the start of the year, \(q_tN\), from an amount conditional on death, \(N\).
- Use the stated mortality and interest assumptions. A claim-date amount and a present value are different quantities when the payment is made after the valuation date.
Check Your Understanding
For each problem, use the timing and valuation assumptions stated in the question.
- A hypothetical level-benefit policy has a $120,000 death benefit. Its beginning-year account value is $70,000, earns 4.00% effective annual interest, and has no other transactions. The annual conditional death probability is 0.0100 and claims are paid at year-end. Find the claim-date NAR and expected net mortality outgo per policy in force at the start of the year. (Answer: NAR = $47,200; expected net mortality outgo = $472.00.)
- A hypothetical increasing-benefit policy pays $65,000 plus the account value at death. The account value is $38,000 at the beginning of the year and earns 3.00%; the annual death probability is 0.0200. Find the claim-date death benefit, NAR, and expected net mortality outgo. (Answer: death benefit = $104,140; NAR = $65,000; expected net mortality outgo = $1,300.00.)
- A hypothetical policy pays the greater of $75,000 and twice the account value at death. Its beginning-year account value is $40,000, earns 5.00%, and the annual death probability is 0.0100. Claims are paid at year-end. Find the controlling benefit, NAR, and expected net mortality outgo. (Answer: corridor benefit = $84,000; NAR = $42,000; expected net mortality outgo = $420.00.)
- Under the simplified excess-reserve recursion, the claim-date NAR is $50,000, \(q_t=0.0200\), \(R_{t+1}=\)$1,200, and the effective annual valuation rate is 5.00%. Calculate \(R_t\) to the nearest cent. (Answer: $2,072.38.)
- A hypothetical level-benefit policy has a $90,000 claim-date benefit and a $54,000 claim-date account value. The annual conditional death probability is 0.0125. Calculate expected gross claims, the expected account-value portion of claims, and expected net mortality outgo per policy in force at the start of the year. (Answer: expected gross claims = $1,125.00; expected account-value portion = $675.00; expected net mortality outgo = $450.00.)