Equity-Linked Death Benefits Under Volatile Fund Values

Why Death Guarantees Need Path-by-Path Analysis

An equity-linked policy may promise a death benefit equal to the value of the policyholder’s investment units, a stated guaranteed amount, or the greater of the two. When fund values fluctuate, that guarantee can become payable in some market outcomes but not others. The insurer therefore needs to distinguish the ordinary unit-value claim from the additional amount due only when the guarantee exceeds the account.

This tutorial uses a discrete annual model. Each example states when investment returns, charges, and death claims occur. A claim is assumed payable at the end of the year of death. Calculations of claim reserves are simplified present values of expected death benefits; they do not include expenses, premiums, taxes, other policy cash flows, or any prescribed regulatory capital formula.

Definition: Let $A_t$ be the policy account value at the end of year $t$, immediately before a death claim is paid, and let $G$ be the guaranteed death-benefit floor. A unit-value death benefit is $B_t^{U}=A_t$. A guaranteed death benefit is $B_t^{G}=\max(A_t,G)$. The incremental guarantee top-up is $C_t=(G-A_t)^+=\max(G-A_t,0)$, so $B_t^{G}=B_t^{U}+C_t$.

These expressions apply to a contract whose death benefit is exactly the account value or the greater of account value and a fixed floor. A contract with a different benefit rule—for example, a percentage of account value plus a separate face amount—requires that rule to be modelled directly. The account value itself must also reflect the contract’s specified charge and investment mechanics.

Projecting the Account and Weighting Claims

For the examples, the account earns a stated annual fund return $r_t$, then a proportional charge $c_t$ is deducted from the post-return account. Thus, if the beginning-of-year account is $A_{t-1}$, the end-of-year value before a claim is

A_t=A_{t-1}(1+r_t)(1-c_t).

This is a particular timing convention, not a universal account formula. If a contract deducts charges before investment returns, or deducts a fixed dollar amount, the projection must follow that alternative rule.

Suppose the probability of death during year $t$, conditional on surviving to its start, is $q_t$. The probability of surviving through that year is $p_t=1-q_t$. The probability of dying in year $t$, measured from the start of the projection, is the probability of surviving each earlier year and then dying in year $t$. For a deterministic benefit $B_t$, this gives the mortality-weighted expected claim

\Pr(\text{death in year }t)B_t = \left(\prod_{j=1}^{t-1}p_j\right)q_tB_t.

If the fund value is random, the benefit is random too. When death is independent of fund returns, the expected claim in year $t$ is the probability of death in that year multiplied by the expected benefit conditional on reaching the start of the year. The independence assumption matters: if mortality and investment outcomes are associated, the joint distribution must be used instead.

Result: For a one-year policy with death probability $q$, a deterministic end-of-year discount rate $i$, and a death benefit $B$ that may depend on the fund outcome, assume death is independent of that outcome. The present value at the start of the year of expected death claims is $qE[B]/(1+i)$. For a multi-year policy, calculate each year’s probability of death from conditional mortality and prior survival, discount that year’s expected claim, and sum.

Worked Examples

Worked Example: A Two-Year Volatile Fund Path

A hypothetical policy starts with an account of $\$10{,}000$. The annual charge is 2% of the account after investment returns. The fund return is $-20\%$ in year 1 and $25\%$ in year 2. The unit-value death benefit is the account value; the guaranteed version pays the greater of the account value and $\$10{,}000$. Conditional death probabilities are $q_1=0.01$ and $q_2=0.02$. Use an effective annual discount rate of $i=4\%$, assume mortality is independent of fund returns, and calculate the present value of expected claims for each benefit.

Method. Project the account using the stated return-then-charge order. Then apply the death benefit rule separately at each year-end. The probability of death in year 2 is conditional on surviving year 1.

Year 1 account.

A_1=10{,}000(1-0.20)(1-0.02) =10{,}000(0.80)(0.98) =\$7{,}840.

The unit-value benefit is $\$7{,}840$, and the guaranteed benefit is $\max(7{,}840,10{,}000)=\$10{,}000$. The year-1 probability of death is $0.01$, so the expected year-1 claims are $\$78.40$ and $\$100.00$, respectively.

Year 2 account. The year-2 return is applied to the year-1 account, followed by the charge:

A_2=7{,}840(1+0.25)(1-0.02) =7{,}840(1.25)(0.98) =\$9{,}604.

The unit-value benefit is $\$9{,}604$, and the guaranteed benefit is $\$10{,}000$. The probability of surviving year 1 and dying in year 2 is $0.99(0.02)=0.0198$. Thus the expected year-2 claims are $0.0198(9{,}604)=\$190.1592$ for the unit-value benefit and $0.0198(10{,}000)=\$198.00$ for the guaranteed benefit.

Present values. Discount year-1 claims by $1.04$ and year-2 claims by $1.04^2=1.0816$:

\begin{aligned} \operatorname{PV}(U) &=\frac{78.40}{1.04}+\frac{190.1592}{1.0816} =75.3846154+175.8128698 =251.1974852,\\ \operatorname{PV}(G) &=\frac{100}{1.04}+\frac{198}{1.0816} =96.1538462+183.0621302 =279.2159764. \end{aligned}

To the nearest cent, the unit-value claim present value is $\$251.20$ and the guaranteed-claim present value is $\$279.22$. The additional guarantee present value is $\$279.2159764-\$251.1974852=\$28.0184912$, or $\$28.02$ to the nearest cent. This equals the present value of the mortality-weighted top-ups: $0.01(2{,}160)/1.04+0.0198(396)/1.0816$. The calculation is checked by reconciling the guaranteed present value to the unit-value present value plus the incremental guarantee.

Common mistake avoided. Year-2 mortality is not $0.02$ of the original policy count; it is $0.99(0.02)$, because only policies surviving year 1 can produce a year-2 death claim.

Worked Example: Volatile One-Year Fund Outcomes

A hypothetical policy has a beginning account of $\$12{,}000$. At year-end, a 1% charge is deducted after the fund return. With probability 0.60 the return is $-30\%$; with probability 0.40 it is $20\%$. The death-benefit floor is $\$10{,}000$. The probability of death during the year is $q=0.015$, independent of the fund outcome, and the effective annual discount rate is $i=5\%$. Find the present value of expected claims for the unit-value and guaranteed benefits, and the incremental guarantee cost.

Method. First calculate each account outcome, then calculate each benefit in that outcome. Average the benefits across fund outcomes before applying the independent death probability and discounting.

For the $-30\%$ outcome, the account is $12{,}000(0.70)(0.99)=\$8{,}316$. The unit-value benefit is $\$8{,}316$, the guaranteed benefit is $\$10{,}000$, and the top-up is $\$1{,}684$. For the $20\%$ outcome, the account is $12{,}000(1.20)(0.99)=\$14{,}256$. Both benefit forms then pay $\$14{,}256$, so the top-up is zero.

The expected unit-value benefit conditional on the year’s fund outcomes is

E[B^U]=0.60(8{,}316)+0.40(14{,}256) =4{,}989.60+5{,}702.40 =\$10{,}692.

The expected guaranteed benefit and expected top-up are

\begin{aligned} E[B^G]&=0.60(10{,}000)+0.40(14{,}256) =6{,}000+5{,}702.40=\$11{,}702.40,\\ E[C]&=0.60(1{,}684)+0.40(0)=\$1{,}010.40. \end{aligned}

The check $E[B^G]=E[B^U]+E[C]$ holds: $\$10{,}692+\$1{,}010.40=\$11{,}702.40$. Expected claims before discounting are $0.015(10{,}692)=\$160.38$ for the unit-value benefit and $0.015(11{,}702.40)=\$175.536$ for the guaranteed benefit. Therefore,

\operatorname{PV}(U)=\frac{160.38}{1.05}=\$152.742857\ldots,\qquad \operatorname{PV}(G)=\frac{175.536}{1.05}=\$167.177142\ldots.

Rounded to the nearest cent, the present values are $\$152.74$ and $\$167.18$; the incremental guarantee cost is $0.015(1{,}010.40)/1.05=\$14.434285\ldots$, or $\$14.43$. These are simplified claim reserves at issue under the stated assumptions, not a complete policy reserve or a regulatory capital measure.

Common mistake avoided. The mean account is $\$10{,}692$, so applying the floor to that mean would give a top-up of only $\max(10{,}000-10{,}692,0)=0$. That is not the expected top-up: the actual expected top-up is $\$1{,}010.40$, because the floor applies separately to each outcome before averaging.

Worked Example: Mortality Stress and the Guarantee Component

A hypothetical one-year linked policy has two equally likely fund outcomes after all charges: an account value of $\$8{,}000$ or $\$12{,}000$. The death-benefit floor is $\$10{,}000$. The base one-year death probability is 1%, and a mortality stress doubles it to 2%. Assume fund outcomes and mortality are independent, claims are paid at year-end, and $i=4\%$. Calculate the base and stressed present values of the incremental guarantee claims.

Method. The top-up is $\$2{,}000$ in the lower-account outcome and zero in the higher-account outcome. Since the outcomes are equally likely, its expected amount conditional on death is $0.50(2{,}000)+0.50(0)=\$1{,}000$. Multiply this by the relevant death probability and discount one year.

\begin{aligned} \operatorname{PV}(C)_{\text{base}} &=\frac{0.01(1{,}000)}{1.04} =\$9.6153846\ldots \approx \$9.62,\\ \operatorname{PV}(C)_{\text{stress}} &=\frac{0.02(1{,}000)}{1.04} =\$19.2307692\ldots \approx \$19.23. \end{aligned}

The stressed incremental guarantee present value increases by $\$19.2307692-\$9.6153846=\$9.6153846$, or $\$9.62$ to the nearest cent. This isolates the mortality effect on the guarantee component, holding the fund-outcome distribution and interest rate fixed. It is a measure of additional expected claim exposure, not a full solvency capital requirement.

Common mistake avoided. The average account is $\$10{,}000$, but that does not make the expected top-up zero. The floor must be applied to each account outcome first; the average of the resulting top-ups is $\$1{,}000$.

Why the Expected Top-Up Is Calculated Outcome by Outcome

The guarantee top-up is a positive-part function: it is zero when the account exceeds the floor and positive when the account falls below it. Averaging the account first can erase the very low outcomes in which the guarantee costs money. In general, the correct quantity is $E[(G-A)^+]$, not $(G-E[A])^+$.

The mortality-weighting formula follows by conditioning on whether death occurs. Let $D$ be the event of death during the year and let $B$ be the benefit payable if death occurs. The claim is $1_D B$, where $1_D$ is 1 on $D$ and 0 otherwise. If death is independent of the fund outcome determining $B$, then

$$ E[1_D B]=E[1_D]E[B]=qE[B]. $$

Discounting a year-end expected claim at effective annual rate $i$ divides it by $1+i$. If the benefit and death event are not independent, the product $qE[B]$ need not be correct; instead calculate $E[1_D B]$ from their joint outcomes. The same caution applies across multiple years: survival to each year must be reflected before weighting that year’s death claim.

Interpreting Charges, Reserves, and Solvency Exposure

A charge affects a death guarantee through the account value. Under the return-then-charge convention used here, a higher charge lowers the account and can increase the guarantee top-up. That does not mean the charge is irrelevant to the insurer’s overall economics: charges are also insurer cash inflows. The examples compare death-claim liabilities only, so they do not net those inflows against claims.

For a guaranteed benefit, the expected claim can be decomposed into the unit-value component and the incremental top-up:

$$ E[B^G]=E[B^U]+E[C]. $$

The same decomposition applies to mortality-weighted claims and their discounted present values, provided the same mortality, fund, and discount assumptions are used for both components. This makes the guarantee exposure visible: it is not the whole death claim, but the additional liability above the unit-value benefit. For a solvency assessment, the top-up under adverse combinations of mortality and fund outcomes is an important exposure to identify; the simplified expected claim present value calculated here should not be mistaken for a capital requirement.

Key takeaway: Project the account using the contract’s timing rules, apply the death-benefit formula to each relevant fund outcome, and then weight each claim by the probability of death and survival to that year. For a guarantee, calculate the top-up outcome by outcome before averaging or discounting.

Check Your Understanding

For each problem, use the stated account and benefit rules, and round dollar answers to the nearest cent where needed.

A hypothetical one-year policy begins with $\$15{,}000$. The fund return is $-10\%$, followed by a 2% charge on the post-return account. The guarantee is $\$14{,}000$, the death probability is 2%, and $i=4\%$. Find the end-year account and the present value of expected guaranteed death claims. (Answer: Account $\$13{,}230$; present value $\$269.23$.)
A one-year policy has a 25% chance of an end-year account of $\$9{,}000$ and a 75% chance of $\$13{,}000$. The guaranteed floor is $\$11{,}000$, and the death probability is 1%. Find the expected top-up conditional on death and the expected incremental claim per policy. (Answer: Expected top-up $\$500$; expected incremental claim $\$5.00$.)
For a two-year policy, unit-value death benefits are $\$8{,}000$ in year 1 and $\$12{,}000$ in year 2. A guaranteed floor of $\$10{,}000$ applies in both years. Conditional death probabilities are $q_1=0.01$ and $q_2=0.03$; $i=5\%$. Find the present value of the incremental guarantee claims. (Answer: $\$19.05$.)
In a one-year policy, the account is $\$8{,}000$ with probability 0.40 and $\$12{,}000$ with probability 0.60. The guaranteed floor is $\$10{,}000$, and the death probability is 2%. Find the expected top-up conditional on death and the expected incremental claim per policy. (Answer: Expected top-up $\$800$; expected incremental claim $\$16.00$.)
A one-year death guarantee has a top-up of $\$1{,}500$ in a low-fund outcome occurring with probability 0.30, and zero otherwise. Death is independent of fund outcomes, with probability 1.2%; the effective annual discount rate is 3%. Find the present value of expected guarantee claims. (Answer: $\$5.24$.)

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