Why Isolate the Guarantee?
An equity-linked benefit may include both a fund-linked payment and a contractual floor. The value of the complete benefit reflects both components. To measure the cost attributable to the floor, compare the value with the guarantee against the value without it, while keeping the investment scenarios, decrements, payment timing, and discount basis identical. The difference is the incremental value of the guarantee on that valuation basis.
This comparison is useful for checking product design and reserves: it separates the guarantee’s contribution from the underlying fund-linked benefit. It does not, by itself, establish a unique market price. The result depends on the scenarios and actuarial assumptions used. The method is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.
Set Up a Paired Comparison
First specify exactly what the unguaranteed and guaranteed contracts pay. For a floor applying to a fund value, the unguaranteed payment is \(F\), while the guaranteed payment is \(\max(F,G)\). Their difference is \(\max(G-F,0)\). A guarantee may apply at maturity, on death, or at both times; do not assume that the same floor applies to every benefit unless the contract says so.
Next state the valuation basis. For example, with annual effective interest rate \(i\), a payment at time \(t\) is discounted by \(v^t\), where \(v=1/(1+i)\). Specify the decrement probabilities, the investment scenarios and their probabilities, and whether investment outcomes are assumed independent of decrements. Use those same inputs in both valuations. If death and survival lead to mutually exclusive end-of-year payments, their probabilities must sum to one.
Write the payment under the contract without the guarantee and the corresponding payment with it. Identify which events trigger the guarantee.
Subtract the unguaranteed payment from the guaranteed payment in each scenario and event. A top-up cannot be negative when the guaranteed benefit is a floor.
Weight each top-up by the probability that its event occurs, then discount it to the valuation date.
Confirm that the difference between the complete guaranteed and unguaranteed values equals the present value of the incremental top-ups.
Why the Difference Is the Guarantee Cost
Suppose all benefits and cash flows other than the guarantee are identical. For each scenario and event, write the guaranteed benefit as the unguaranteed benefit plus the top-up: \(B_t^G=B_t^0+(B_t^G-B_t^0)\). Taking expected present values gives
The equality follows from the linearity of expectation. Common payments cancel in the subtraction; only the guarantee-related increments remain. This is why matching assumptions is essential. If the two calculations use different investment scenarios, mortality assumptions, payment dates, or discount rates, the difference combines the guarantee effect with changes in the valuation basis and is not an isolated guarantee cost.
The same reasoning applies to a prospective reserve at a duration: if the two reserves use the same future assumptions and common cash flows, their difference is the present value of future guarantee top-ups. This statement assumes that premiums, expenses, and other projected cash flows are held constant between the two calculations. If introducing the guarantee also changes those cash flows, their differences must be included rather than assumed to cancel.
Worked Example: A One-Year Floor on Death or Maturity
A hypothetical one-year contract pays the fund value at year-end on death or, if the policyholder survives, at maturity. The guarantee raises either payment to at least $100. At year-end, the fund is $90 in a down scenario and $120 in an up scenario; each scenario has probability 0.50. The scenario is independent of decrement. The probability of death during the year is 0.10, and the probability of survival to maturity is 0.90. Use an annual effective interest rate of 5%. Find the unguaranteed value, the guaranteed value, and the guarantee cost. Assume all payments are made at year-end.
Method and assumptions. Calculate expected payments under each contract using the same scenarios and event probabilities, then discount by \(v=1/1.05\). The floor applies to both death and maturity benefits. Because the death and survival events exhaust the possibilities, their probabilities sum to \(0.10+0.90=1\).
Unguaranteed benefit. In either event, the payment is the fund value. The expected fund value across investment scenarios is
Death and survival lead to the same fund payment here, so the expected year-end benefit is
Thus the unguaranteed present value is
Guaranteed benefit. The down-scenario payment is \(\max(90,100)=100\); the up-scenario payment is \(\max(120,100)=120\). The expected payment conditional on either event is
Weighting by the two events gives \(0.10(110)+0.90(110)=11+99=110\). Therefore
Guarantee cost and check. Only the down scenario produces a top-up, of \(100-90=10\); its probability is 0.50. The event probabilities sum to one, so the expected top-up is \(0.50(10)=5\), and its present value is \(5/1.05=\$4.7619\). This agrees with \(V^G-V^0=104.7619-100.00=\$4.7619\). The common mistake is to weight the top-up by the survival probability alone even though this contract also guarantees the death benefit; doing so would omit the death-event top-up.
Worked Example: A Maturity-Only Guarantee With Two-Year Decrements
A hypothetical two-year contract pays the fund value on death at the end of either year. If the policyholder survives to the end of year 2, the contract pays the fund value at maturity, subject to a guarantee of $100. The guarantee does not apply to death benefits. Use an annual effective interest rate of 10%. The year-1 death probability is 0.10; conditional on surviving year 1, the year-2 death probability is 0.20. There are two investment paths, independent of decrements: path L has probability 0.25 and fund values of $95 at year 1 and $80 at year 2; path H has probability 0.75 and fund values of $110 at year 1 and $130 at year 2. Find the value without the guarantee and the incremental cost of the maturity guarantee.
Method and decrement probabilities. The probability of death during year 1 is 0.10. The unconditional probability of death during year 2 is \(0.90(0.20)=0.18\), and the probability of surviving to maturity is \(0.90(0.80)=0.72\). These probabilities sum to \(0.10+0.18+0.72=1.00\). The scenario-weighted fund values are
Value without the guarantee. The expected death payment in year 1 is \(0.10(106.25)=10.625\). The expected death payment in year 2 is \(0.18(117.50)=21.15\). The expected maturity payment is \(0.72(117.50)=84.60\). Discounting each payment at its own time gives
The total is calculated from unrounded terms and rounded to four decimal places.
Increment due to the guarantee. The guarantee applies only at maturity. On path L, the maturity top-up is \(100-80=20\); on path H, it is \(\max(100-130,0)=0\). The expected top-up conditional on reaching maturity is \(0.25(20)+0.75(0)=5\). Weight by the 0.72 probability of survival, then discount to time 0:
The guaranteed value is \(V^G=V^0+C_G=97.0557851+2.9752066=\$100.0310\), rounded to four decimal places. The check is that the death benefits and the year-2 death probability have not changed; only the maturity payment on path L is increased. A common mistake is to apply the maturity floor to year-2 death benefits as well, even though the stated guarantee is maturity-only.
Worked Example: A Death-Only Floor
A hypothetical one-year contract pays the fund value on death during the year and the fund value at maturity if the policyholder survives. A floor of $110 applies only to the death benefit. The death probability is 0.08 and survival probability is 0.92. The year-end fund is $90 with probability 0.40 and $125 with probability 0.60, independently of decrement. Use an annual effective interest rate of 4%, with all payments at year-end. Calculate the two values and the guarantee cost.
Method. The expected fund value is \(0.40(90)+0.60(125)=36+75=111\). Without the guarantee, both event-contingent benefits equal the fund, so the expected year-end payment is \(0.08(111)+0.92(111)=111\), and
On death, the guaranteed payment is $110 in the low-fund scenario and $125 in the high-fund scenario. Its conditional expected amount is \(0.40(110)+0.60(125)=44+75=119\). The maturity benefit remains unguaranteed, with expected amount $111. Therefore
The incremental death top-up is \(110-90=20\) only in the low-fund scenario. Its expected present value is
The check is \(107.3461538-106.7307692=\$0.6154\), rounded to four decimal places. Applying the floor to the maturity benefit would incorrectly add value to a benefit the contract leaves unchanged.
Interpreting the Result and Avoiding Traps
A guarantee cost is a value under a stated basis, not a guarantee that the insurer will pay exactly that amount in experience. Scenario probabilities, decrements, and discounting determine the result. In particular, a scenario-weighted expected top-up should not be confused with a top-up calculated from the expected fund: in general, \(E[\max(G-F,0)]\) is not equal to \(\max(G-E[F],0)\). The floor is applied in each scenario before the scenario values are averaged.
| Check | Correct treatment | Common error |
|---|---|---|
| Guarantee trigger | Include a top-up only for the event specified in the contract | Apply a maturity floor to death benefits, or a death floor at maturity |
| Investment scenarios | Apply the floor separately to each scenario, then weight | Apply the floor once to the average fund value |
| Decrements | Use unconditional event probabilities at each payment time | Use a conditional second-year death probability as if it were unconditional |
| Comparison basis | Hold assumptions and common cash flows constant | Attribute all differences between two unlike valuations to the guarantee |
| Timing | Discount each top-up from its actual payment time | Discount every benefit as though paid at the same duration |
For an isolated cost calculation, the most efficient check is often to value the incremental payments directly and compare that result with \(V^G-V^0\). A disagreement usually signals a missed event probability, an incorrectly applied guarantee trigger, or inconsistent timing or rounding. Keep full precision through the calculation and round only the final reported value.
Check Your Understanding
For each problem, calculate the incremental guarantee cost. Unless stated otherwise, investment scenarios are independent of decrements, payments occur at the stated duration, and all benefits not changed by the guarantee are identical in the two contracts.
- A one-year maturity-only guarantee sets the benefit to at least $90. The year-end fund is $70 or $120, each with probability 0.50. The probability of surviving to maturity is 0.95; any death benefit is unchanged by the guarantee. Use an annual effective interest rate of 5%. (Answer: $9.05)
- A one-year death-only guarantee sets the death benefit to at least $100. The death probability is 0.06. The year-end fund is $80 with probability 0.30 and $130 with probability 0.70. Use an annual effective interest rate of 4%; the maturity benefit is unchanged. (Answer: $0.35)
- A two-year maturity-only guarantee sets the maturity benefit to at least $95. The probability of surviving to maturity is 0.85. The year-2 fund is $80 with probability 0.40 and $115 with probability 0.60. Use an annual effective interest rate of 5%; death benefits are unchanged. (Answer: $4.63)
- A two-year maturity-only guarantee sets the maturity benefit to at least $100. The year-1 death probability is 0.10, and the conditional year-2 death probability given survival through year 1 is 0.20. The year-2 fund is $75 with probability 0.25 and $125 with probability 0.75, independently of decrement. Use an annual effective interest rate of 8%; death benefits are unchanged. (Answer: $3.86)