Guarantee Moneyness Under Joint Market Stresses

Why Guarantee Moneyness Matters

A guarantee on an investment-linked policy becomes costly to the insurer when the policy fund is not enough to meet the promised benefit. Equity returns affect the size of the fund, while interest rates affect how a future shortfall is valued today. The two effects should be tracked together: a scenario can change both the likelihood or size of a top-up and its present value.

The focus here is a hypothetical maturity guarantee with a single payment at time $T$. The policyholder receives the greater of the fund value and the guaranteed benefit. This simple structure isolates the key relationship between the fund and the guarantee; actual products can include additional benefits, charges, cash flows, and guarantees.

Definition: Let $F_T$ be the policy fund at maturity and $G_T$ the guaranteed maturity benefit. The insurer’s maturity top-up is $H_T=\max(G_T-F_T,0)$, and the policyholder’s maturity payment is $\max(F_T,G_T)$. The guarantee is in the money for the insurer when $F_T<G_T$, at the money when $F_T=G_T$, and out of the money when $F_T>G_T$. One useful measure of moneyness is the surplus $F_T-G_T$; a negative surplus indicates a potential top-up.

For a one-period scenario with effective annual interest rate $y$, the present value of the top-up is $H_1/(1+y)$. For a $T$-year scenario with a constant effective annual rate $y$, it is $H_T/(1+y)^T$. If the scenario specifies different rates by year, discount successively using each year’s rate. In every case, first determine whether the guarantee pays; discounting a zero top-up still gives zero.

In a scenario analysis, a present value weighted by scenario probabilities is a useful measure of stressed guarantee exposure. Unless the probabilities and discounting assumptions are appropriate for pricing, it is not automatically a market-consistent option value. A full market-consistent valuation requires a valuation framework and assumptions appropriate to that purpose.

Calculating Moneyness Under a Joint Scenario

A joint scenario specifies the market assumptions that apply together. For example, it may specify an equity return and a discount rate in the same scenario row. Use the equity assumption to project the fund and the interest-rate assumption to discount the resulting top-up. Do not substitute a different rate or equity return from another row.

Worked Example: One-Year Fund Position and Discount Rate

A hypothetical policy has a fund of $100 at the start of the year and a guaranteed benefit of $102 at year-end. Consider two possible one-year scenarios. In Scenario A, the equity fund earns 10% and the effective annual discount rate is 3%. In Scenario B, the equity fund earns -15% and the effective annual discount rate is 1%. Calculate the terminal fund, guarantee shortfall, moneyness, and present value of the top-up in each scenario.

Method. Project the fund as $F_1=100(1+r_E)$, compare it with $G_1=102$, and discount any positive shortfall by $1+y$.

Scenario A. The terminal fund is $100(1.10)=\$110$. The fund surplus is $110-102=\$8$, so the guarantee is out of the money. The top-up is $\max(102-110,0)=\$0$, and its present value is $0/1.03=\$0$.

Scenario B. The terminal fund is $100(0.85)=\$85$. The fund surplus is $85-102=-\$17$, so the guarantee is in the money. The top-up is $\max(102-85,0)=\$17$. Its present value is $17/1.01=\$16.8317$, or $16.83 to the nearest cent.

Check. Scenario A’s fund exceeds the guarantee, so a positive top-up would be inconsistent with the contract. Scenario B’s top-up equals the $17 gap between the guaranteed benefit and the fund. The common mistake is to discount the fund and guarantee separately and then treat their difference as a payment even in the out-of-the-money scenario; the contract pays only the positive shortfall.

Scenario-Weighted Guarantee Exposure

When several mutually exclusive scenarios are given, calculate the top-up in each one before aggregating. Let scenario $j$ have probability $p_j$, terminal fund $F_{T,j}$, guaranteed benefit $G_{T,j}$, and effective annual discount rate $y_j$. If the rate is constant for the $T$-year period, the scenario-weighted present value is:

\sum_j p_j \frac{\max(G_{T,j}-F_{T,j},0)}{(1+y_j)^T}, \qquad \sum_j p_j=1.

This formula assumes the scenario probabilities are given for the analysis and that the specified rate applies to discount the top-up in that scenario. It does not claim that the probabilities are market-implied pricing probabilities. If the guarantee amount varies by scenario, use the scenario-specific $G_{T,j}$, rather than silently treating it as fixed.

Worked Example: Weighted Present Value Across Three Scenarios

A hypothetical one-year contract has a guaranteed maturity benefit of $105 and an initial fund of $100. The prescribed joint scenarios are: (1) probability 0.25, equity return 15%, discount rate 4%; (2) probability 0.50, equity return -5%, discount rate 2%; and (3) probability 0.25, equity return -20%, discount rate 1%. Calculate the scenario-weighted present value of the top-up.

Method. For each scenario, project the fund, apply the positive-part top-up formula, discount at that scenario’s rate, and weight by its probability.

Scenario 1. The fund is $100(1.15)=\$115$. The shortfall is $\max(105-115,0)=\$0$, so the present value is $0/1.04=\$0$. Its weighted contribution is $0.25(0)=\$0$.

Scenario 2. The fund is $100(0.95)=\$95$. The shortfall is $105-95=\$10$, and the present value is $10/1.02=\$9.8039$. Its weighted contribution is $0.50(9.8039)=\$4.9020$.

Scenario 3. The fund is $100(0.80)=\$80$. The shortfall is $105-80=\$25$, and the present value is $25/1.01=\$24.7525$. Its weighted contribution is $0.25(24.7525)=\$6.1881$.

The total is $0+4.9020+6.1881=\$11.0901$, or $11.09 to the nearest cent.

Check. The undiscounted expected top-up is $0.25(0)+0.50(10)+0.25(25)=\$11.25$. The weighted present value is lower, as each positive top-up is discounted at a positive rate. The common mistake is to calculate a top-up from the average fund: the probability-weighted fund is $0.25(115)+0.50(95)+0.25(80)=\$96.25$, which gives a top-up of $8.75, not the expected top-up of $11.25.

Why the Positive-Part Calculation Must Come First

The guarantee payoff is nonlinear because it cannot be negative. This makes the order of operations important. The correct procedure is to apply the guarantee to each scenario’s fund, then aggregate the resulting payments. Averaging the funds first can conceal low-fund scenarios that trigger large top-ups.

\sum_j p_j\max(G-F_j,0) \geq \max\!\left(G-\sum_j p_j F_j,0\right).

To justify the inequality, the function $h(F)=\max(G-F,0)$ is convex: it is the maximum of the two affine functions $G-F$ and $0$. Jensen’s inequality for a convex function gives $\sum_j p_j h(F_j)\geq h(\sum_j p_jF_j)$. The inequality can be strict, as in the preceding example: the expected top-up is $11.25, while the top-up calculated from the average fund is only $8.75.

Discounting does not remove this issue. If rates vary by scenario, first calculate and discount each scenario’s top-up using its own rate, then take the probability-weighted sum. In general, a single average rate applied to an average fund will not reproduce that result.

Separating Moneyness from Discounting

Moneyness is determined by comparing the fund with the guaranteed benefit at the same time. The discount rate does not change that comparison when the terminal fund and guarantee are already specified. It changes the present value of a positive future top-up. If a product specifies that the guarantee itself grows at a credited rate, project that guaranteed benefit as well; do not confuse the credited rate with the scenario discount rate.

Worked Example: A Growing Guarantee and a Two-Year Equity Stress

A hypothetical policy starts with a fund of $100 and a guaranteed benefit of $100. The guaranteed benefit grows at 2% effective annually for two years. Under an adverse equity scenario, the fund earns -5% effective annually in both years. Compare the fund and guarantee at year two, then calculate the top-up’s present value at either a 1% or a 4% effective annual discount rate, constant over the two years.

Method. Project both amounts to the same maturity date. The benefit is $G_2=100(1.02)^2=\$104.04$. The fund is $F_2=100(0.95)^2=100(0.9025)=\$90.25$. The guarantee is in the money by $104.04-90.25=\$13.79$.

At a 1% discount rate, the present value is $13.79/(1.01)^2=13.79/1.0201=\$13.5183$, or $13.52. At a 4% discount rate, it is $13.79/(1.04)^2=13.79/1.0816=\$12.7496$, or $12.75.

As a comparison, if the fund instead earned 5% in both years, it would be $100(1.05)^2=\$110.25$, exceeding the $104.04 guarantee by $6.21. The top-up would then be zero at either discount rate. The common mistake is to grow the fund but leave the guaranteed benefit at its initial $100; both must be projected to year two before moneyness is assessed.

In this example, the fund return and guarantee growth determine whether a top-up occurs. Once the top-up is fixed, the higher discount rate lowers its present value. In other products, market rates may also affect the fund projection or the guarantee design; those effects must be included only when specified by the product and scenario assumptions. Do not change a projected fund return merely because a separate discount rate has changed.

Practical Checks and Common Traps

Compare values at the same date. A current fund should not be compared directly with a future guaranteed benefit. Project both to maturity first.
Apply the floor before averaging. Calculate $\max(G-F,0)$ in each scenario. A guarantee cannot produce a negative top-up in a scenario where the fund exceeds the benefit.
Use each scenario’s rate consistently. Discount a scenario’s top-up with the rate assigned to that scenario and the correct number of years.
Distinguish an exposure measure from a price. A probability-weighted stress calculation is not automatically a market-consistent guarantee value; its interpretation depends on the probability and discounting basis.
Keep scenario assumptions joint. Use the equity return and interest rate specified together in a scenario. Combining values from separate rows creates a different scenario.

Key takeaway: Project the fund and guaranteed benefit under the same joint scenario, calculate the nonnegative top-up, discount it using that scenario’s rate, and only then aggregate across scenarios. This preserves the guarantee’s nonlinear response as fund values cross the benefit floor.

Check Your Understanding

For each problem, calculate the requested quantity using the specified joint assumptions. Round dollar present values to the nearest cent.

A hypothetical one-year policy has a $120 fund and a $126 guarantee at the start of the year. The equity return is -10% and the effective annual discount rate is 2%. Find the terminal fund and present value of the top-up. (Answer: Fund $108; present value $17.65.)
A hypothetical one-year guarantee is $104. The initial fund is $100, the equity return is 8%, and the effective annual discount rate is 3%. Find the present value of the top-up. (Answer: $0.00; the terminal fund is $108 and exceeds the guarantee.)
A one-year hypothetical guarantee is $110. Scenario A has probability 0.40, fund value at maturity $120, and discount rate 5%. Scenario B has probability 0.60, fund value at maturity $90, and discount rate 2%. Calculate the probability-weighted present value of the top-up. (Answer: $0.40(0)+0.60(20/1.02)=\$11.76$.)
A hypothetical policy starts with an $80 fund and an $80 guaranteed benefit. The benefit grows at 3% effective annually and the fund earns -2% effective annually for two years. The discount rate is 4% effective annually for both years. Calculate the present value of the top-up. (Answer: $G_2=\$84.872$, $F_2=\$76.832$, top-up $=\$8.040$, present value $8.040/(1.04)^2=\$7.43$.)
A one-year guarantee is $100. The terminal fund is $130 with probability 0.50 and $70 with probability 0.50; use a 0% discount rate. Compare the expected top-up with the top-up calculated from the expected fund. (Answer: Expected top-up $=0.50(0)+0.50(30)=\$15$; expected fund $=\$100$, so its calculated top-up is $0.)

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