Death Benefit Options and Their Valuation Cash Flows

Why the Death Benefit Option Matters

A universal life policy can have a substantial account value and still expose the insurer to a death claim that is larger than that account value. The size of the claim depends on the policy’s death benefit option. For valuation, it is therefore important to distinguish the death benefit paid from the portion of that benefit that is in excess of the policy account value.

This tutorial focuses on translating two common options into cash flows: a level death benefit and a death benefit equal to account value plus a specified amount. The calculations support cash-flow valuation, but they are not by themselves a complete prospective reserve calculation. A complete reserve may also involve premiums, policy charges, expenses, persistency, guarantees, and other contractual features.

Definition: Let $F$ be the specified amount and $A_t$ the policy account value immediately before a death claim at time $t$. The death benefit, denoted $D_t$, is the amount payable on death. The net amount at risk at that time is $N_t=D_t-A_t$: the part of the death benefit in excess of the account value.

The net amount at risk is a useful mortality-related measure, but it is not a second payment to the beneficiary. The beneficiary receives the death benefit $D_t$, not $D_t$ plus $N_t$. In the basic decomposition, the claim consists of the account value plus the net amount at risk.

Basic Formulas and Their Assumptions

The formulas below apply to hypothetical policies with no policy loan, no unpaid premium adjustment, and no separate rider benefit. For the level option, assume the specified amount exceeds the account value at every death time considered, and any corridor minimum is nonbinding. These conditions let us use the simplified level-benefit formula without an adjustment.

Definition: Under a level death benefit option, the death benefit is $D_t=F$. When $A_t\leq F$, the net amount at risk is $N_t=F-A_t$. Under an account-value-plus-specified-amount option, the death benefit is $D_t=F+A_t$, so the net amount at risk is $N_t=F$.

Actual contracts may impose a corridor minimum, commonly requiring the death benefit to be at least a specified multiple of account value. If that minimum binds, the level-option benefit is not simply $F$, and the net amount at risk must be recalculated from the actual death benefit. Contract provisions may also define the timing of account-value updates and benefit calculations differently. Always use the contract’s stated terms.

Why the Net Amount at Risk Decomposes the Claim

For any death benefit $D_t$, define $N_t=D_t-A_t$. Rearranging gives $D_t=A_t+N_t$. This identity explains the two components of the death claim in the basic valuation view: the account-value component and the amount in excess of that value.

$$ D_t=A_t+N_t. $$

For the level option, substituting $D_t=F$ gives $N_t=F-A_t$. For the account-value-plus-specified-amount option, substituting $D_t=F+A_t$ gives $N_t=F$. Thus, under the second option, changes in account value change the total death claim but do not change the net amount at risk, provided the specified amount is unchanged.

The identity also helps prevent a common valuation error: treating the account value as an additional amount payable on top of a benefit that already includes it. The account value is part of the total claim under the increasing option, not an extra payment to add again.

Worked Claims and Net Amounts at Risk

Worked Example: Comparing the Two Options at One Death Time

A hypothetical policy has a specified amount of $F=\$120{,}000$ and an account value of $A_t=\$28{,}000$ immediately before death. Assume there is no binding corridor minimum and no other adjustment. Calculate the death benefit and net amount at risk under each option.

Method. Apply the contract’s benefit formula first, then subtract account value to find the net amount at risk.

Level option. The death benefit is $D_t=F=\$120{,}000$. Therefore, $N_t=D_t-A_t=\$120{,}000-\$28{,}000=\$92{,}000$. The check is $\$28{,}000+\$92{,}000=\$120{,}000$, which equals the claim.

Account-value-plus-specified-amount option. The death benefit is $D_t=F+A_t=\$120{,}000+\$28{,}000=\$148{,}000$. Therefore, $N_t=\$148{,}000-\$28{,}000=\$120{,}000$, equal to the specified amount. The check is $\$28{,}000+\$120{,}000=\$148{,}000$.

Common mistake avoided. Adding the account value a second time to the $\$148{,}000$ increasing-option claim would produce an incorrect $\$176{,}000$ benefit. The stated option formula already includes the account value.

Valuing Death-Contingent Cash Flows

For a discrete annual model, suppose a death claim is paid at the end of the year of death. Let $q_k$ be the probability of death during year $k$, conditional on the policyholder being alive at the start of that year. The probability of surviving to the start of year $k$ is the product of the preceding conditional survival probabilities. Consequently, the unconditional probability of death during year $k$ is that survival probability multiplied by $q_k$.

If $i$ is the effective annual interest rate and $v=1/(1+i)$, the expected present value of death benefits over $n$ years is the sum of each year’s death probability times its benefit, discounted to the valuation date. The same construction applies to net amounts at risk. This is a death-contingent valuation only; it does not include premiums, expenses, or other policy cash flows.

\operatorname{EPV}(\text{death benefits}) =\sum_{k=1}^{n} {}_{k-1}p\,q_k\,D_kv^k, \qquad \operatorname{EPV}(\text{net amounts at risk}) =\sum_{k=1}^{n} {}_{k-1}p\,q_k\,N_kv^k,

Here ${}_{k-1}p$ denotes the probability of surviving from issue to the start of year $k$, with ${}_0p=1$. In a life-contingency setting this survival probability can be represented by ${}_{k-1}p_x$ when the policyholder’s age at issue is $x$ and the mortality model is specified by age. If account values differ by year, use the account value applicable at that year’s death time.

Worked Example: Two-Year Expected Present Values by Option

A hypothetical policy has specified amount $F=\$100{,}000$. The account value at death is $\$20{,}000$ in year 1 and $\$25{,}000$ in year 2. Conditional death probabilities are $q_1=0.0100$ and $q_2=0.0200$, respectively. Assume an effective annual interest rate of $i=5\%$, end-of-year claim payment, and no corridor adjustment. Calculate the expected present value of death benefits and net amounts at risk under each option.

Method. First find the unconditional death probabilities. Year 1 probability is $0.0100$. Survival through year 1 is $1-0.0100=0.9900$, so the year 2 death probability is $0.9900(0.0200)=0.0198$. The discount factors are $v=1/1.05$ and $v^2=1/1.1025$.

Level option. The death benefit is $\$100{,}000$ in both years. The net amounts at risk are $\$100{,}000-\$20{,}000=\$80{,}000$ in year 1 and $\$100{,}000-\$25{,}000=\$75{,}000$ in year 2. Thus,

\begin{aligned} \operatorname{EPV}(D) &=\frac{0.0100(100{,}000)}{1.05} +\frac{0.0198(100{,}000)}{1.1025}\\ &=952.380952+1{,}795.918367 =\$2{,}748.30,\\ \operatorname{EPV}(N) &=\frac{0.0100(80{,}000)}{1.05} +\frac{0.0198(75{,}000)}{1.1025}\\ &=761.904762+1{,}346.938776 =\$2{,}108.84. \end{aligned}

Account-value-plus-specified-amount option. The claims are $\$120{,}000$ in year 1 and $\$125{,}000$ in year 2. The net amount at risk is $\$100{,}000$ in each year. Therefore,

\begin{aligned} \operatorname{EPV}(D) &=\frac{0.0100(120{,}000)}{1.05} +\frac{0.0198(125{,}000)}{1.1025}\\ &=1{,}142.857143+2{,}244.897959 =\$3{,}387.76,\\ \operatorname{EPV}(N) &=\frac{0.0100(100{,}000)}{1.05} +\frac{0.0198(100{,}000)}{1.1025}\\ &=952.380952+1{,}795.918367 =\$2{,}748.30. \end{aligned}

The increasing-option benefit EPV exceeds the level-option benefit EPV by $\$639.46$. This is the present value of the account-value amounts included in claims: $0.0100(20{,}000)/1.05+0.0198(25{,}000)/1.1025=\$639.46$, to the nearest cent. The calculation checks that the claim difference is the account-value component, subject to rounding at the final cent.

Common mistake avoided. The year 2 death probability is not $q_2=0.0200$ unconditionally. It must be multiplied by survival through year 1, giving $0.0198$.

When the Option Changes During the Projection

If the policy’s death benefit option changes, calculate the benefit and net amount at risk separately for each period using the option then in force. The timing of an option change matters: use the contractual option applicable at the death time, not the one that applied at issue or at the start of a different period. The account value must also correspond to that same death time.

Worked Example: A Change from Level to Increasing Benefit

A hypothetical policy has specified amount $F=\$90{,}000$. The level option applies in year 1, and the account-value-plus-specified-amount option applies in year 2 after an option change. Account value at death is $\$30{,}000$ in year 1 and $\$20{,}000$ in year 2. Conditional death probabilities are $q_1=0.0200$ and $q_2=0.0300$. Assume an effective annual interest rate of $10\%$, end-of-year claim payment, and no corridor adjustment. Calculate the two-year expected present values of claims and net amounts at risk.

Method. Apply the level formula in year 1 and the increasing formula in year 2. The unconditional year 2 death probability is $(1-0.0200)(0.0300)=0.0294$. The discount factors are $v=1/1.10$ and $v^2=1/1.21$.

In year 1, the claim is $\$90{,}000$, and the net amount at risk is $\$90{,}000-\$30{,}000=\$60{,}000$. In year 2, the claim is $\$90{,}000+\$20{,}000=\$110{,}000$, and the net amount at risk is $\$90{,}000$.

\begin{aligned} \operatorname{EPV}(D) &=\frac{0.0200(90{,}000)}{1.10} +\frac{0.0294(110{,}000)}{1.21}\\ &=1{,}636.363636+2{,}672.727273 =\$4{,}309.09,\\ \operatorname{EPV}(N) &=\frac{0.0200(60{,}000)}{1.10} +\frac{0.0294(90{,}000)}{1.21}\\ &=1{,}090.909091+2{,}186.776860 =\$3{,}277.69. \end{aligned}

As a check, the present value of the account-value component at death is $0.0200(30{,}000)/1.10+0.0294(20{,}000)/1.21=\$1{,}031.40$. Adding this to the net-amount-at-risk EPV gives $\$4{,}309.09$ to the nearest cent, agreeing with the direct claim calculation.

Common mistake avoided. Applying the increasing formula in year 1 would incorrectly include account value in a period when the level option is still in force. Conversely, using the level formula in year 2 would omit the year 2 account value from the claim.

Valuation Checks and Common Traps

A disciplined cash-flow map keeps each amount tied to its role and timing. For every projected death year, record the option in force, account value at death, total death benefit, net amount at risk, conditional mortality probability, and discount factor. Then confirm that the total benefit and net amount at risk satisfy $D_t=A_t+N_t$.

Do not confuse benefit with risk amount. The beneficiary receives $D_t$; the net amount at risk is the difference $D_t-A_t$.
Do not add the account value twice. Under the increasing option, it is already part of the death benefit.
Use the correct mortality probability. A conditional $q_k$ must be multiplied by the probability of surviving to the start of year $k$ when calculating an unconditional EPV.
Keep option timing consistent. Use the option and account value applicable at the projected death time.
Check contract adjustments. A binding corridor, loan, rider, or other provision may alter the simple formulas.

The calculations here isolate death-contingent claims and their net amounts at risk. In a full valuation, those cash flows are considered alongside the other future policy cash flows under the applicable assumptions. The cash-flow map is the necessary first step: an incorrect claim amount or timing will carry through any later valuation.

Check Your Understanding

For each problem, state the relevant benefit option and use the assumptions provided. Dollar answers are rounded to the nearest cent.

A hypothetical level-option policy has specified amount $\$150{,}000$ and account value $\$42{,}000$ at death. Assume no corridor adjustment. Find the death benefit and net amount at risk. (Answer: death benefit $\$150{,}000$; net amount at risk $\$108{,}000$.)
A hypothetical account-value-plus-specified-amount policy has specified amount $\$75{,}000$ and account value $\$11{,}500$ at death. Find the death benefit and net amount at risk. (Answer: death benefit $\$86{,}500$; net amount at risk $\$75{,}000$.)
A level-option policy has specified amount $\$200{,}000$, account value $\$50{,}000$, one-year conditional death probability $0.0120$, and effective annual interest rate $5\%$. A claim is paid at year end and no corridor adjustment applies. Find the one-year expected present value of the death benefit and of the net amount at risk. (Answer: death-benefit EPV $\$2{,}285.71$; net-amount-at-risk EPV $\$1{,}714.29$.)
An increasing-option policy has specified amount $\$80{,}000$. Account value at death is $\$10{,}000$ in year 1 and $\$14{,}000$ in year 2. Conditional death probabilities are $q_1=0.0100$ and $q_2=0.0200$; the effective annual interest rate is $4\%$, and claims are paid at year end. Find the two-year EPV of the death benefit. (Answer: $\$2{,}586.17$.)
A hypothetical policy has specified amount $\$100{,}000$, account value $\$20{,}000$ at death, one-year death probability $0.0200$, and effective annual interest rate $4\%$. Compare the one-year expected present values of death benefits under the level and increasing options, assuming no corridor adjustment. (Answer: level option $\$1{,}923.08$; increasing option $\$2{,}307.69$; the increasing-option EPV is $\$384.62$ greater.)

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