Writing Equity-Linked Guarantees as Payoff Functions

Why Write Guarantees as Functions?

An equity-linked policy’s fund value can rise or fall with the value of its underlying units. A contract may promise that a specified benefit will not fall below a minimum even if the fund is worth less. To apply that promise correctly, first identify the event—death or maturity—and then compare the fund value with the floor that applies to that event.

A compact function makes both the contractual rule and its boundary cases explicit. It also helps prevent a common error: adding the guarantee to the fund even though the contract promises the greater of the two amounts. The functions below describe gross benefits payable at the event. They do not discount payments or calculate the insurer’s cost of providing the guarantee.

Definition: Let $F\geq 0$ be the policy fund value immediately before a benefit is determined. In the contract design used here, the minimum death benefit is the greater of $F$ and a stated death sum assured $S\geq 0$. The minimum maturity benefit is the greater of $F$ and a stated maturity guarantee floor $G\geq 0$. The resulting gross benefits are $B_D(F)$ at death and $B_M(F)$ at maturity. Unless the contract explicitly says otherwise, $S$ and $G$ are event-specific floors: the death benefit uses $S$, and the maturity benefit uses $G$.

These definitions describe one particular contract structure. Some contracts specify a different death benefit, such as the fund plus an additional amount, or set a different floor. Those terms must be represented by their own rules; the formulas here do not apply automatically to every equity-linked policy. In particular, a maturity floor should not be substituted for the death sum assured merely because both are described as guarantees.

Piecewise Payoff Functions

For a death benefit that is the greater of the fund and the sum assured, compare $F$ with $S$. When the fund is below $S$, the sum assured determines the benefit. When the fund equals or exceeds $S$, the fund determines the benefit. Thus:

B_D(F)=\max(F,S)= \begin{cases} S, & 0\leq F<S,\\ F, & F\geq S. \end{cases}

For maturity, the same comparison is made with the maturity floor $G$, not automatically with $S$:

B_M(F)=\max(F,G)= \begin{cases} G, & 0\leq F<G,\\ F, & F\geq G. \end{cases}

At the boundary, either branch gives the same result: $B_D(S)=S$ and $B_M(G)=G$. It is therefore valid to put equality in either adjacent branch, provided the piecewise definition does not leave the boundary out or assign it two different values.

Key interpretation: Each function selects one amount: the fund value or the floor applicable to the event. A death sum assured and a maturity guarantee floor can differ, so identify the benefit trigger before selecting the comparison.

Why the Piecewise Form Follows

The function $B_D(F)=\max(F,S)$ means, by definition, selecting whichever of $F$ and $S$ is larger. If $F<S$, then $S$ is larger, so $B_D(F)=S$. If $F\geq S$, then $F$ is at least as large as $S$, so $B_D(F)=F$. The two cases cover every possible nonnegative fund value, and they agree when $F=S$.

The same reasoning applies at maturity with $G$ in place of $S$. This argument is useful because it makes the comparison condition visible: the guarantee controls the benefit only on the side of the boundary where the fund is lower. Above the boundary, the benefit equals the fund.

B_M(F)=\max(F,G)= \begin{cases} G, & F<G,\\ F, & F\geq G. \end{cases}

The use of $\max$ also gives an immediate check on a calculation: $B_D(F)$ cannot be less than $S$, and $B_M(F)$ cannot be less than $G$. In either case the benefit is also at least the fund value. If a computed amount is below either relevant amount, the comparison has been applied incorrectly.

Evaluating the Benefits in Practice

A disciplined calculation separates the contractual trigger from the numerical comparison. Use the following sequence:

Identify the event.
Determine whether the amount is payable on death or at maturity. Do not choose the floor before identifying the trigger.

Find the applicable fund value and floor.
Use the fund value specified at the contractual valuation time, then select $S$ for death or $G$ for maturity under the design defined above.

Compare and select.
If $F$ is below the applicable floor, the benefit is the floor. Otherwise, it is the fund value.

Check the result.
Confirm that the selected benefit is at least the applicable floor and at least $F$, and that equality at the boundary is handled consistently.

Worked Example: Death Benefit Below, At, and Above the Sum Assured

A hypothetical policy pays the greater of its fund value and a death sum assured of $S=\$120{,}000$. Consider three possible fund values immediately before the death benefit is determined: $\$94{,}000$, $\$120{,}000$, and $\$137{,}000$. Find the gross death benefit in each case. The event is stipulated to be death in each case; no mortality probability is required for these conditional payoff calculations. No interest rate, discounting, or probability distribution is used.

Method. Apply $B_D(F)=\max(F,S)$, comparing each fund value with the death sum assured.

For $F=\$94{,}000$, $F<S$, so the floor applies: $B_D(\$94{,}000)=\max(\$94{,}000,\$120{,}000)=\$120{,}000$.

For $F=\$120{,}000$, $F=S$, so both amounts agree: $B_D(\$120{,}000)=\max(\$120{,}000,\$120{,}000)=\$120{,}000$.

For $F=\$137{,}000$, $F>S$, so the fund determines the benefit: $B_D(\$137{,}000)=\max(\$137{,}000,\$120{,}000)=\$137{,}000$.

Check. Each benefit is at least $\$120{,}000$ and at least its corresponding fund value. The results are $\$120{,}000$, $\$120{,}000$, and $\$137{,}000$. The common mistake avoided is paying the sum assured in every case, including when the contract requires the greater fund value.

Worked Example: Maturity Benefit Near the Guarantee Floor

A hypothetical equity-linked contract pays the greater of the maturity fund and a guarantee floor of $G=\$96{,}000$. Evaluate the maturity benefit for fund values of $\$91{,}500$, $\$96{,}000$, and $\$103{,}200$. Maturity is stipulated to occur in each case. This is a deterministic payoff calculation, so no mortality assumption, interest rate, or probability distribution is needed.

Method. Use $B_M(F)=\max(F,G)$, with the maturity floor—not a death sum assured—as the comparison amount.

At $F=\$91{,}500$, $F<G$, hence $B_M(\$91{,}500)=\max(\$91{,}500,\$96{,}000)=\$96{,}000$.

At $F=\$96{,}000$, $F=G$, hence $B_M(\$96{,}000)=\max(\$96{,}000,\$96{,}000)=\$96{,}000$.

At $F=\$103{,}200$, $F>G$, hence $B_M(\$103{,}200)=\max(\$103{,}200,\$96{,}000)=\$103{,}200$.

Check. The benefits are $\$96{,}000$, $\$96{,}000$, and $\$103{,}200$, each no smaller than either its fund value or the guarantee floor. The calculation avoids treating the floor as an amount to add to the fund.

Worked Example: Choose the Floor After Determining the Event

A hypothetical contract has death sum assured $S=\$112{,}000$ and maturity guarantee floor $G=\$94{,}000$. At the relevant time, the policy holds 4,800 units priced at $\$21.25$ per unit. Evaluate the gross benefit if the insured dies at that time, and separately evaluate the benefit if the policy instead reaches maturity with the same fund value. For each conditional outcome, the event is stipulated; no mortality assumption or probability distribution is needed. No interest rate or discounting is applied because the requested amounts are payments at the event.

Step 1: Calculate the fund value. $F=4{,}800\times \$21.25=\$102{,}000$.

Step 2: Apply the death rule. The applicable floor is $S=\$112{,}000$. Since $\$102{,}000<\$112{,}000$, $B_D(F)=\max(\$102{,}000,\$112{,}000)=\$112{,}000$.

Step 3: Apply the maturity rule separately. The applicable floor is $G=\$94{,}000$. Since $\$102{,}000>\$94{,}000$, $B_M(F)=\max(\$102{,}000,\$94{,}000)=\$102{,}000$.

Check. The death benefit is $\$112{,}000$, while the maturity benefit is $\$102{,}000$. The same fund value produces different benefits because different contractual floors apply to the two triggers. The common mistake avoided is using the death sum assured as the maturity floor, or vice versa.

Common Mistakes and Contract Checks

The function is only as reliable as the interpretation of the contract terms. In particular, verify whether the quoted sum assured or floor is payable as a minimum, whether it applies to the stated event, and which fund value is used. The examples assume that the stated fund value is already the value to use immediately before determining the benefit. They do not include a separate adjustment for charges, timing, or other benefit terms.

Adding instead of comparing. For a benefit defined as the greater of $F$ and $S$, the formula is $\max(F,S)$, not $F+S$. With $F=\$80{,}000$ and $S=\$100{,}000$, the benefit is $\$100{,}000$, not $\$180{,}000$.
Using the wrong event’s floor. A maturity floor does not become the death sum assured unless the contract says so. Select the benefit rule from the event first.
Paying the floor even when the fund is higher. If $F>S$, a “greater of fund or sum assured” death benefit equals $F$. It is not capped at $S$.
Misreading equality. At $F=S$ or $F=G$, the benefit equals that boundary amount. A piecewise definition must assign the boundary consistently.
Confusing a payoff with a value today. These functions calculate the amount payable at the event. They do not by themselves value that contingent payment at an earlier date.

For a final exam-style check, write down the trigger and the relevant floor beside the fund value before doing any arithmetic. This makes the contractual comparison auditable and reduces the risk of carrying a valid formula into the wrong benefit setting. The distinction is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Check Your Understanding

For each problem, apply the contractual rule as stated. Treat each event and fund value as given; no probability or discounting calculation is required.

A hypothetical policy pays the greater of fund value and a death sum assured of $\$72{,}000$. Find the death benefit when $F=\$68{,}000$. (Answer: $\$72{,}000$)
A maturity benefit is the greater of fund value and $G=\$84{,}000$. Find the benefits for fund values $\$83{,}990$, $\$84{,}000$, and $\$84{,}010$. (Answer: $\$84{,}000$, $\$84{,}000$, and $\$84{,}010$)
A contract has death sum assured $S=\$100{,}000$ and maturity floor $G=\$96{,}000$. For a fund value of $\$92{,}400$, calculate the death benefit and the maturity benefit separately. (Answer: death $\$100{,}000$; maturity $\$96{,}000$)
A policy has 3,750 units priced at $\$24.80$ each, death sum assured $S=\$105{,}000$, and maturity floor $G=\$90{,}000$. Calculate the fund value, then the death benefit and maturity benefit. (Answer: fund $\$93{,}000$; death $\$105{,}000$; maturity $\$93{,}000$)
A contract specifies a death benefit of $\max(F,S)$, where $S=\$115{,}000$, and a maturity benefit of $\max(F,G)$, where $G=\$130{,}000$. If $F=\$121{,}000$, state both benefits and identify which floor controls each comparison. (Answer: death $\$121{,}000$, with the fund controlling; maturity $\$130{,}000$, with the guarantee floor controlling)

What You'll Learn