Capital Requirements for Equity-Linked Guarantees

Why Guarantee Capital Is Not the Account Value

An equity-linked policy’s account value moves with its investment fund, while a guarantee may require the insurer to pay more than that account value. The insurer’s exposure is the difference between the contractual benefit and the amount already represented by the account—not the account value itself. Capital analysis therefore begins by identifying the benefit payable in each scenario and isolating any insurer top-up.

This tutorial uses explicitly defined scenario measures. There is no single capital formula that applies to every regulatory or internal framework: the result depends on the valuation date, specified stresses, eligible resources, and the capital definition being used. The calculations below illustrate how to calculate and aggregate guarantee shortfalls under stated assumptions. They are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Definition: Let $A$ be the policyholder’s fund account value at the benefit date and $G$ the guaranteed minimum benefit. For a benefit equal to the greater of the account value and guarantee, the insurer’s top-up is $\max(G-A,0)$. The signed guarantee gap is $G-A$; it can be negative. The top-up cannot be negative because the contract does not require the insurer to pay less than the account value.

A negative signed gap means that the account value exceeds the guarantee. It does not mean the insurer pays a negative amount or that the policyholder loses the account value. For example, if $G=\$100{,}000$ and $A=\$106{,}000$, then $G-A=-\$6{,}000$, while the insurer’s top-up is $\max(-\$6{,}000,0)=\$0$. Keeping the signed gap and the contractual top-up distinct is essential.

Contractual Shortfalls and Scenario Capital

For a maturity guarantee, suppose the policy pays $\max(A_T,G)$ at time $T$. The account value $A_T$ funds the account portion of the benefit; the insurer’s guarantee cost is the excess, if any, of $G$ over $A_T$. For a death guarantee with death benefit $\max(A_t,F)$, where $F$ is a guaranteed death floor, the insurer’s top-up on death at time $t$ is $\max(F-A_t,0)$.

To make a scenario capital measure precise, this tutorial defines additional capital at a valuation date as the positive excess of the present value of the specified stressed guarantee shortfalls over eligible resources assigned to those shortfalls. This is an illustrative measure, not a universal regulatory definition. If the stress is stated at a future benefit date, discount at the stated effective annual interest rate $i$, using $v=1/(1+i)$, before comparing amounts at the valuation date.

Scenario capital measure: For shortfalls $L_j$ payable at time $t_j$, the present value of stressed guarantee shortfalls is $\sum_j v^{t_j}L_j$. If eligible resources at the valuation date are $R$, additional capital under this stated measure is $\max\left(\sum_j v^{t_j}L_j-R,0\right)$. A different capital framework may use different stresses, valuation assumptions, or resource rules.

The shortfall formula follows directly from the contract. If the policyholder is entitled to $\max(A,G)$, the insurer’s payment beyond the account value is $\max(A,G)-A$. If $G\ge A$, this difference is $G-A$; if $G<A$, it is zero. Thus in either case it is $\max(G-A,0)$. This derivation explains why a fund decline by itself is not the insurer’s guarantee loss: the loss is the part of the guaranteed benefit that the account value does not cover.

\max(A,G)-A=\max(G-A,0)

Worked Example: Maturity Guarantee Under a Fund Stress

A hypothetical one-year policy has a maturity guarantee of $\$100{,}000$ and an account value of $\$102{,}500$ at the valuation date. The fund return over the year has a two-point distribution: $3.4146\%$ with probability $0.5$, or $-20\%$ with probability $0.5$. The annual effective interest rate for discounting is $i=5\%$. The calculation is conditional on survival to maturity, so no mortality weighting is applied. Find the signed gap and insurer top-up in each fund scenario, and the present value of the top-up in the adverse scenario.

Method. Project the account under each return, calculate $G-A_T$, apply the positive-part operation only when calculating the contractual top-up, and discount the adverse top-up to time 0.

In the first outcome, the account value is $A_T=\$102{,}500(1.034146)=\$106{,}000$, to the nearest dollar. The signed gap is $G-A_T=\$100{,}000-\$106{,}000=-\$6{,}000$. The insurer top-up is $\max(-\$6{,}000,0)=\$0$.

In the adverse outcome, $A_T=\$102{,}500(0.80)=\$82{,}000$. The signed gap is $\$100{,}000-\$82{,}000=\$18{,}000$, so the insurer top-up is $\$18{,}000$.

The time-0 present value of this stressed top-up is $\$18{,}000/1.05=\$17{,}142.857\ldots$, or $\$17{,}142.86$ to the nearest cent. This is the present value conditional on the adverse scenario, not a probability-weighted expected cost. The example’s common trap is to call the account decline a guarantee loss; only the $\$18{,}000$ excess of the guarantee over the stressed account is the insurer’s maturity top-up.

Check. In the adverse outcome, the policy pays $\$100{,}000$: $\$82{,}000$ from the account and $\$18{,}000$ from the insurer. The amounts sum to the contractual guarantee.

Death Guarantees: Severity and Probability Are Different

A death guarantee requires attention to both the amount payable on death and the probability of death during the period. The top-up conditional on a death is a severity amount. An expected cost also uses the probability of that event. A scenario capital measure might instead evaluate the full conditional shortfall under a specified death scenario. These quantities answer different questions and should not be substituted for one another.

Worked Example: Death Floor and Expected Top-Up

A hypothetical one-year equity-linked contract pays the greater of its account value or a death floor of $\$120{,}000$ if death occurs during the year. The account value at death is assumed to be $\$106{,}000$, with no fund variation in this calculation. The probability of death during the year is $q=0.04$, and the effective annual interest rate is $i=4\%$. Assume the death benefit is paid at the end of the year. Find the conditional insurer top-up, its present value if death occurs, and the present value of the expected top-up.

Method. First calculate the claim conditional on death. Then, for the expected cost, multiply by the one-year death probability before discounting. The probability assumption is a Bernoulli model: death within the year has probability $0.04$, and survival has probability $0.96$.

The signed gap on death is $\$120{,}000-\$106{,}000=\$14{,}000$. Thus the insurer’s top-up conditional on death is $\max(\$14{,}000,0)=\$14{,}000$. Its conditional present value is $\$14{,}000/1.04=\$13{,}461.538\ldots$, or $\$13{,}461.54$.

The expected top-up payable at year-end is $0.04(\$14{,}000)=\$560$. Its present value is $\$560/1.04=\$538.4615\ldots$, or $\$538.46$. Equivalently, the result is $0.04(\$14{,}000/1.04)$.

The conditional shortfall is $\$14{,}000$, while the expected present cost is $\$538.46$. Neither is automatically “the capital requirement”: that depends on the capital measure and scenario being assessed. The trap is to multiply the conditional shortfall by the death probability when a question asks for a specified death-scenario loss, or to omit that probability when it explicitly asks for expected cost.

Aggregating Shortfalls and Deducting Resources

For a portfolio with several guarantees, calculate the contractual top-up for each benefit and then aggregate according to the specified scenario and valuation basis. Do not add account values to guarantee shortfalls as though both were insurer losses. If the contract pays the account value when it exceeds the guarantee, that account value is part of the policyholder benefit but not an insurer guarantee top-up.

Identify the benefit trigger and guarantee.
Distinguish maturity from death, and write the actual contractual benefit before calculating a loss.

Project the account under the specified scenario.
Use the stated fund return, timing, and any distribution assumptions. Keep the account amount separate from insurer payments.

Calculate each signed gap and positive top-up.
Retain $G-A$ as a signed amount; use $\max(G-A,0)$ for the insurer’s contractual shortfall.

Put amounts on a common basis.
Discount future shortfalls if the capital measure is at an earlier valuation date, and apply probabilities only when the requested measure calls for them.

Compare with eligible resources.
Deduct resources only after defining their date and scope; floor additional capital at zero if the stated measure does so.

Worked Example: Combined Maturity and Death Guarantee Exposure

A hypothetical insurer assesses three one-year linked contracts under a specified adverse scenario at time 0. Contract A survives to maturity with guarantee $\$100{,}000$ and account value $\$106{,}000$. Contract B survives to maturity with guarantee $\$80{,}000$ and account value $\$65{,}000$. Contract C has a death floor of $\$120{,}000$, account value $\$106{,}000$ at death, and dies during the year in the specified scenario. For this conditional scenario, mortality probabilities are not used to weight the death claim; the underlying one-year death probability is $0.03$. All three benefits are payable at the end of the year. The effective annual interest rate is $5\%$, and eligible resources at time 0 are $\$10{,}000$. Under the scenario capital measure defined above, calculate additional capital.

Method. Compute each top-up at the benefit date, add the shortfalls in the specified scenario, discount their sum to time 0, and subtract time-0 eligible resources.

For A, the signed gap is $\$100{,}000-\$106{,}000=-\$6{,}000$, and its top-up is $\max(-\$6{,}000,0)=\$0$. For B, the signed gap and top-up are both $\$80{,}000-\$65{,}000=\$15{,}000$. For C, the death-floor gap and top-up are both $\$120{,}000-\$106{,}000=\$14{,}000$.

The aggregate scenario shortfall at year-end is $\$0+\$15{,}000+\$14{,}000=\$29{,}000$. Its time-0 present value is $\$29{,}000/1.05=\$27{,}619.0476\ldots$, or $\$27{,}619.05$ to the nearest cent. After deducting $\$10{,}000$ of eligible resources, additional capital is $\max(\$27{,}619.05-\$10{,}000,0)=\mathbf{\$17{,}619.05}$, to the nearest cent.

The death probability of $0.03$ is not applied because the question specifies a conditional adverse scenario in which C dies. If the requested quantity were instead expected cost and death were the only uncertain event, the death top-up would be probability-weighted. The check is that A contributes no insurer shortfall despite having a $\$106{,}000$ account value; only B and C create top-ups in this scenario.

Common Mistakes and Interpretation

A signed gap is useful for understanding how far the account lies above or below the guarantee, but it is not itself always the insurer’s claim. In the first worked example, the base account is $\$106{,}000$ and the guarantee is $\$100{,}000$, so the signed gap is $\$100{,}000-\$106{,}000=-\$6{,}000$. Do not replace that signed result by zero when asked for the gap. The top-up, separately, is zero because the contract applies the positive-part operation.

Other frequent errors are applying the guarantee to the wrong benefit date, confusing a death benefit with a maturity benefit, discounting an amount already expressed at time 0, and probability-weighting a conditional stress loss when the scenario is explicitly conditional. A disciplined calculation labels each amount by what it represents: account value, signed gap, insurer top-up, expected cost, present value, or additional capital.

Key takeaway: The policyholder’s account value is not the insurer’s guarantee loss. Calculate the contractual top-up as the positive part of guarantee minus account, preserve negative signed gaps when requested, and apply scenario probabilities, discounting, and resource deductions only as specified by the capital measure.

Check Your Understanding

For each problem, distinguish the policyholder account from the insurer’s top-up and follow the stated valuation basis.

A maturity guarantee is $\$90{,}000$, and the maturity account value is $\$96{,}000$. State the signed gap and the insurer’s top-up. (Answer: signed gap $-\$6{,}000$; top-up $\$0$.)
A one-year death floor is $\$110{,}000$, the account value at death is $\$92{,}000$, the one-year death probability is $0.03$, and $i=4\%$. Assuming year-end payment and no other benefit, find the present value of expected top-up. (Answer: $\$519.23$.)
A maturity guarantee is $\$100{,}000$. The account value is $\$105{,}000$ in the base scenario and $\$78{,}000$ in the adverse scenario. Find the base and adverse top-ups and the increase in maturity-date shortfall. (Answer: base $\$0$; adverse $\$22{,}000$; increase $\$22{,}000$.)
At year-end, a maturity guarantee of $\$70{,}000$ has account value $\$60{,}000$, and a death floor of $\$90{,}000$ has account value $\$83{,}000$ in the specified death scenario. The interest rate is $0\%$, and eligible resources at the same date are $\$6{,}000$. Find additional capital under the stated scenario measure. (Answer: $\$11{,}000$.)
A death benefit is the greater of $1.2$ times the account value or a guaranteed floor of $\$100{,}000$. At death, the account value is $\$80{,}000$. Calculate the total death benefit and insurer top-up relative to the account. (Answer: benefit $\$100{,}000$; top-up $\$20{,}000$.)

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