Equity-Linked Maturity Guarantees Under Falling Markets

Introduction

An equity-linked contract can promise a minimum amount at maturity even when its investment fund has fallen in value. The insurer’s guarantee exposure is not necessarily the whole maturity benefit: the fund provides part or all of the payment, and the insurer supplies a top-up only when the fund is below the guaranteed floor.

The key is to project the fund to the contractual maturity date, apply the guarantee at that date, and keep the resulting insurer shortfall separate from the total amount paid to the policyholder. A fund value can fall below the guarantee before maturity without creating an immediate payment if the contract provides only a maturity guarantee.

Definition: Let $F_T$ be the linked fund value at the maturity date $T$, and let $G$ be the guaranteed maturity amount. The total maturity benefit is $\max(F_T,G)$. The insurer’s incremental maturity guarantee payment, or shortfall, is $\max(G-F_T,0)$. Thus, total benefit equals the fund value plus the top-up: $F_T+\max(G-F_T,0)=\max(F_T,G)$.

The calculations below use hypothetical contracts. Unless a problem states otherwise, assume the contract remains in force and the policyholder survives to maturity; there are no premiums after issue, withdrawals, interim guarantee payments, or charges other than those already reflected in the stated net fund returns. Mortality and lapse probabilities therefore do not enter these conditional maturity calculations. A contract with a death benefit, a guarantee reset, or interim withdrawals requires its own contractual cash-flow rules.

Projecting the Fund and Applying the Floor

If the initial fund is $F_0$ and the net annual fund returns are $r_1,\ldots,r_T$, the fund at the end of year $T$ is found by applying each return in sequence. For a fixed annual return $r$, this reduces to compound accumulation. Negative returns reduce the fund; they do not, by themselves, generate a payment from the insurer.

F_T=F_0\prod_{t=1}^{T}(1+r_t), \qquad B_T=\max(F_T,G), \qquad C_T=\max(G-F_T,0), \qquad B_T=F_T+C_T.

Here $B_T$ is the total maturity benefit and $C_T$ is the insurer’s top-up at maturity. If $F_T\geq G$, the fund covers the benefit and $C_T=0$. If $F_T<G$, the policyholder receives $G$, of which $F_T$ is funded from the linked account and $G-F_T$ is the guarantee shortfall. This distinction is important when reporting insurer outgo: reporting $B_T$ as the guarantee cost would count the fund value as though it were an additional insurer payment.

Worked Example: A Three-Year Fall Below the Floor

A hypothetical contract receives a single premium of $10,000, all allocated to its linked fund. It matures after three years and guarantees 90% of the initial premium. The net fund returns are −10% in year 1, −15% in year 2, and −5% in year 3. Assume the policy remains in force to maturity. Find the fund value, total maturity benefit, and insurer top-up.

Method. Compound the net returns in chronological order, then compare the terminal fund with the guarantee. The guarantee is 90% of the initial premium, not 90% of the fund at maturity.

The guaranteed amount is $G=0.90(10{,}000)=\$9{,}000$. The fund projection is:

\begin{aligned} F_1&=10{,}000(0.90)=\$9{,}000,\\ F_2&=9{,}000(0.85)=\$7{,}650,\\ F_3&=7{,}650(0.95)=\$7{,}267.50. \end{aligned}

Since $\$7{,}267.50<\$9{,}000$, the total maturity benefit is $B_3=\$9{,}000$. The insurer top-up is $C_3=9{,}000-7{,}267.50=\$1{,}732.50$. The cash-flow check is $\$7{,}267.50+\$1{,}732.50=\$9{,}000$, as required.

Common mistake avoided: The fund first equals the guarantee at the end of year 1, but that does not mean a top-up is paid then. Under the stated maturity-only guarantee, the insurer pays only at year 3.

Why the Shortfall Formula Holds

The formula follows directly from the contract’s floor. If the fund is at least $G$, the policyholder receives $F_T$, and the insurer adds nothing. If the fund is below $G$, the policyholder receives $G$, so the insurer must supply exactly the difference between $G$ and the fund. These two cases are exhaustive:

Fund at or above the floor.
If $F_T\geq G$, then $B_T=F_T$ and $C_T=0$. Therefore $F_T+C_T=F_T=B_T$.

Fund below the floor.
If $F_T<G$, then $B_T=G$ and $C_T=G-F_T$. Therefore $F_T+C_T=F_T+(G-F_T)=G=B_T$.

Combine the cases.
The top-up is zero in the first case and $G-F_T$ in the second, which is exactly $C_T=\max(G-F_T,0)$. Adding it to the fund gives $B_T=\max(F_T,G)$.

For several specified market scenarios $j$, calculate the shortfall in each scenario first. If scenario $j$ has probability $p_j$, the expected maturity shortfall is $\sum_j p_j\max(G-F_{T,j},0)$. The maximum must be applied scenario by scenario: averaging the fund values first and then taking a maximum generally gives a different result.

Worked Example: Expected Top-Up Across Maturity Scenarios

A hypothetical three-year contract has an initial premium and guaranteed maturity amount of $15,000. At maturity, three possible net fund outcomes are specified: $12,000 with probability 0.30, $15,000 with probability 0.20, and $18,000 with probability 0.50. Find the expected insurer top-up. Then find its value at issue using an effective annual valuation rate of 4%. Assume each scenario’s top-up, if any, is paid at the end of year 3.

Method. Apply the floor separately to each scenario, weight the resulting top-ups by their probabilities, and discount that expected payment three years. The stated 4% rate is the valuation rate for this calculation; it is not an assumption about the fund’s returns.

The guarantee is $G=\$15{,}000$. The scenario top-ups are:

Fund of $12,000: $\max(15{,}000-12{,}000,0)=\$3{,}000$.
Fund of $15,000: $\max(15{,}000-15{,}000,0)=\$0$.
Fund of $18,000: $\max(15{,}000-18{,}000,0)=\$0$.

Therefore the expected maturity top-up is $0.30(3{,}000)+0.20(0)+0.50(0)=\$900$. At 4% effective annual interest, the three-year accumulation factor is $1.04^3=1.124864$. The value at issue is:

\frac{\$900}{1.124864}=\$800.10 \quad\text{(to the nearest cent).}

The check is that only the scenario below the floor contributes to the expected top-up, and its 0.30 probability gives $30\%$ of $3,000, or $900. The discounted amount is smaller than the maturity expectation because it is valued three years earlier at a positive rate.

Common mistake avoided: The expected fund is $0.30(12{,}000)+0.20(15{,}000)+0.50(18{,}000)=\$15{,}600$. Applying the floor to this average would give a top-up of zero, which is not the expected top-up of $900. The guarantee is nonlinear, so scenario-level shortfalls must be averaged.

Payment Date and Discounting

A maturity shortfall is paid at maturity, not when the fund first crosses below the guarantee. To compare amounts at different dates, discount each shortfall from its own payment date. Under a deterministic effective annual valuation rate $i$, a known payment $C_T$ at time $T$ has value at issue $C_T/(1+i)^T$. This is a time-value calculation using the rate specified for the valuation; it does not, by itself, determine a market-consistent value for a guarantee under uncertain investment returns.

Worked Example: A Falling Fund at Two Possible Maturity Dates

Consider two hypothetical versions of a contract, each with an initial fund of $10,000 and a guarantee of $10,000. One version matures after year 2; the other matures after year 3. Both use the same net fund returns while in force: −10% in year 1, −20% in year 2, and −10% in year 3. Assume no interim payments and use an effective annual valuation rate of 4%. For each version, calculate the maturity top-up and its value at issue.

Method. Project the common fund path, apply the guarantee only at each version’s own maturity, and discount that version’s top-up for its own term.

The fund values are $F_1=10{,}000(0.90)=\$9{,}000$, $F_2=9{,}000(0.80)=\$7{,}200$, and $F_3=7{,}200(0.90)=\$6{,}480$.

For the two-year version, the maturity shortfall is $10{,}000-7{,}200=\$2{,}800$. The two-year accumulation factor is $1.04^2=1.0816$, so its value at issue is $2{,}800/1.0816=\$2{,}588.76$, to the nearest cent.

For the three-year version, the maturity shortfall is $10{,}000-6{,}480=\$3{,}520$. The three-year accumulation factor is $1.04^3=1.124864$, so its value at issue is $3{,}520/1.124864=\$3{,}129.27$, to the nearest cent.

Both checks are consistent with the contractual floor: at each maturity, fund plus top-up equals $10,000. The later maturity has the larger nominal shortfall, and in this case also the larger discounted value. Neither version pays a top-up in year 1 merely because its fund is then below $10,000.

Common mistake avoided: Discounting the three-year shortfall for only two years, or paying a year-1 top-up, uses the wrong payment date. The maturity date determines both when the floor is tested and how long the shortfall is discounted.

Common Exam Traps

A frequent incorrect calculation is to subtract the guarantee from the fund without restricting the result to a nonnegative amount. For example, if $F_T=\$11{,}200$ and $G=\$10{,}000$, the wrong calculation $G-F_T=-\$1{,}200$ treats an excess fund as a negative insurer payment. The correct top-up is $\max(10{,}000-11{,}200,0)=\$0$, and the total maturity benefit is $11,200.

Using the wrong guarantee base. Read whether $G$ is based on the initial premium, accumulated premiums, or another stated amount. Do not assume it is a percentage of the final fund.
Mixing total benefit and insurer exposure. The total benefit is $\max(F_T,G)$; the insurer’s incremental top-up is only $\max(G-F_T,0)$.
Paying at the first crossing. A maturity-only guarantee is tested at maturity, even if the fund is below the floor earlier.
Using the wrong time unit. For annual effective $i$, a payment at the end of year $T$ is discounted by $(1+i)^T$. Match $T$ to the actual payment date.
Averaging before applying the floor. With multiple outcomes, calculate each outcome’s shortfall before taking the probability-weighted average.

Key takeaway: Project the fund to the contractual maturity date, calculate the total benefit as $\max(F_T,G)$, and report the insurer’s maturity shortfall separately as $\max(G-F_T,0)$. Discount that shortfall only from its actual payment date and at the rate specified for the valuation.

Check Your Understanding

For each hypothetical contract, assume the policy remains in force to the stated maturity and apply the guarantee only at maturity. Amounts are in dollars; give discounted values to the nearest cent.

A contract has an initial premium of $12,000 and guarantees 90% of that amount at maturity. Its maturity fund is $10,200. Find the total benefit and insurer top-up. (Answer: total benefit $10,800; top-up $600.)
A $18,000 fund has net returns of −5%, −10%, and +2% over three years. The maturity guarantee is 100% of the initial fund. Project the terminal fund and find the top-up. (Answer: terminal fund $15,697.80; top-up $2,302.20.)
A $20,000 contract guarantees 95% of the initial premium. At maturity, the fund is $16,000 with probability 0.25, $19,000 with probability 0.50, and $22,000 with probability 0.25. Find the expected top-up. (Answer: $750.)
Use the outcomes in question 3, but now value the expected top-up at issue for a four-year term using an effective annual valuation rate of 3%. (Answer: $666.37.)
A two-year contract has an initial fund of $25,000, guarantees 100% of that amount, and earns net returns of −12% and −8%. Using an effective annual valuation rate of 4%, find the maturity top-up and its value at issue. (Answer: maturity top-up $4,760; value at issue $4,400.89.)

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