Minimum Guarantees in Participating Policy Valuation

Why a Minimum Guarantee Changes the Valuation

A participating policy may provide a benefit that reflects declared or projected bonuses, subject to a minimum amount. At a payment event, the policyholder receives whichever is greater: the guaranteed floor or the projected total benefit. This comparison must be made for the relevant event and outcome before probabilities and discounting are applied.

That order matters. A policy can have several possible payment dates, such as death during the term or survival to maturity. The projected benefit may be above the floor in some cases and below it in others. Valuing the floor and projected benefits separately in aggregate can conceal those differences and produce an incorrect expected present value.

Definition: For a payment event $j$, let $G_j$ be the guaranteed minimum total benefit and let $B_j$ be the projected total benefit under the specified bonus assumptions. The benefit payable for that event and outcome is $$ C_j=\max(G_j,B_j). $$ The floor is applied to the total benefit payable at the event. It is not an additional amount paid on top of $B_j$.

The values $G_j$ and $B_j$ may differ by event, duration, or scenario. A projected benefit should include the bonus components that apply under the stated assumptions. If a guarantee applies only to a particular component, the comparison must instead follow the contract’s definition of that guarantee.

Valuation Method

Assume a discrete model with payment events at times $t_j$. Let $p_j$ be the probability of event $j$ under the specified mortality and other event assumptions. If the benefit at an event also depends on a random bonus outcome $k$, let $q_{jk}$ be the conditional probability of that outcome given event $j$, and let $B_{jk}$ be the resulting projected benefit. At effective annual interest rate $i$, the expected present value is

\operatorname{EPV} = \sum_j p_j v^{t_j} \sum_k q_{jk}\max(G_j,B_{jk}), \qquad v=\frac{1}{1+i}.

For a benefit with no uncertain bonus outcome conditional on event $j$, the inner sum is unnecessary, and that event contributes $p_jv^{t_j}\max(G_j,B_j)$. This calculation assumes that event probabilities and bonus probabilities are specified on a consistent basis, payments occur at the stated times, and the interest rate is effective annually. If payment timing, bonus assumptions, or event probabilities vary by scenario, the corresponding scenario-specific values must be used.

List the payment events.
Identify each possible event, its payment time, and its probability. For bonus scenarios, record their probabilities conditional on the event.

Project the total benefit.
For every event and scenario, calculate the benefit under the stated bonus assumptions before applying the floor.

Apply the guarantee.
Use the larger of the guaranteed minimum and the projected total benefit for each event and scenario.

Weight and discount.
Multiply each payable benefit by its event and scenario probability, discount it to the valuation date, and sum the results.

Why the Comparison Must Come Before Aggregation

The maximum function is nonlinear. In general, the expected value of a maximum is not the maximum of expected values. For any random projected benefit $B$ and fixed floor $G$,

E[\max(G,B)]\ne \max(G,E[B])

in general. To see why, consider outcomes both below and above the floor. In a below-floor outcome, the payment remains $G$; in an above-floor outcome, the payment is $B$. Averaging $B$ first can blur that distinction. More formally, $\max(G,B)$ is at least $G$ and at least $B$ in every outcome, so its expectation is at least both $G$ and $E[B]$. It may be strictly greater than $\max(G,E[B])$ when the outcomes lie on both sides of $G$.

The same issue arises across payment events: each event has its own payment time, probability, and potentially its own guaranteed and projected amounts. A valid valuation applies the floor within each event and scenario, then takes the expected present value.

Worked Example: A Maturity Floor Equal to the Projected Benefit

A hypothetical participating policy pays a maturity benefit in two years. The basic amount is $8,000, and the projected simple bonus is $250 for each completed policy year. There is no terminal bonus. The maturity floor is $8,500. Assume maturity occurs with certainty and the effective annual interest rate is 5.00%. Find the present value of the maturity benefit.

Method. Calculate the projected total benefit, compare it with the floor at the maturity event, and discount the payable amount. This avoids adding the floor to the projected benefit.

There are two completed years, so the projected bonus is $2\times \$250=\$500$. The projected total benefit is

B=\$8{,}000+\$500=\$8{,}500.

The payable benefit is the greater of $8,500 and $8,500, hence $8,500. With $v=1/1.05$, the present value is

\$8{,}500v^2 = \frac{\$8{,}500}{1.05^2} = \frac{\$8{,}500}{1.1025} = \$7{,}709.7506\ldots

Rounded to the nearest cent, the present value is $7,709.75. As a check, the projected benefit equals the floor, so the benefit paid is exactly $8,500; discounting it for two years at 5.00% gives a present value below $8,500.

Worked Example: Applying the Floor at Death and Maturity

A hypothetical policy has three possible payment events. Death in year 1 has probability 0.10; death in year 2 has probability 0.18; and survival to maturity at the end of year 2 has probability 0.72. These events are mutually exclusive and exhaustive. The minimum benefit is $22,000 at every event. Projected total benefits are $20,000 for year-1 death, $22,500 for year-2 death, and $25,000 at maturity. Assume an effective annual interest rate of 4.00%, and that benefits are paid at the end of the year of the event. Calculate the expected present value.

Method. Apply the floor to each event separately, then multiply by that event’s probability and discount to time 0. The common error avoided is comparing only aggregate expected benefits.

At year-1 death, the payable benefit is $\max(\$22{,}000,\$20{,}000)=\$22{,}000$. At year-2 death, it is $\max(\$22{,}000,\$22{,}500)=\$22{,}500$. At maturity, it is $\max(\$22{,}000,\$25{,}000)=\$25{,}000$. Therefore,

\begin{aligned} \operatorname{EPV} &=0.10(\$22{,}000)v +0.18(\$22{,}500)v^2 +0.72(\$25{,}000)v^2\\ &=\frac{\$2{,}200}{1.04} +\frac{\$4{,}050+\$18{,}000}{1.04^2}\\ &=\$2{,}115.3846\ldots +\frac{\$22{,}050}{1.0816}\\ &=\$2{,}115.3846\ldots+\$20{,}386.4643\ldots\\ &=\$22{,}501.8489\ldots \end{aligned}

The expected present value is $22,501.85, rounded to the nearest cent. The probabilities sum to $0.10+0.18+0.72=1.00$, as required.

To check the aggregation trap, first value the $22,000 floor at each event. The year-1 contribution is $0.10(\$22{,}000)/1.04=\$2{,}115.3846\ldots$. The year-2 contribution is

\frac{0.18(\$22{,}000)+0.72(\$25{,}000)}{1.04^2} = \frac{\$3{,}960+\$18{,}000}{1.0816} = \$20{,}303.2544\ldots .

Thus, the present value of the floor-only payments is $22,418.64. The year-2 death benefit increases the event-by-event value by $0.18(\$22{,}500-\$22{,}000)/1.04^2=\$83.21$, giving $22,501.85. In contrast, the present value of the projected benefits without applying the floor is $22,309.54, so taking the larger of that aggregate value and the floor-only value would give $22,418.64. That is not the required valuation: it misses the year-2 death benefit’s excess over the floor.

Worked Example: A Maturity Floor Under Bonus Scenarios

A hypothetical policy is certain to reach maturity in three years. Its basic amount is $10,000. The projected total benefit, including bonuses, is $11,000 with probability 0.20, $12,500 with probability 0.50, and $14,000 with probability 0.30. The maturity floor is $12,000. Assume an effective annual interest rate of 4.00%. Find the expected present value.

Method. Apply the floor separately to each bonus outcome, form the probability-weighted benefit at maturity, and discount for three years. This avoids applying the floor only to the average projected benefit.

The payable benefit is $12,000 in the $11,000 outcome, $12,500 in the $12,500 outcome, and $14,000 in the $14,000 outcome. The expected payable benefit at maturity is

0.20(\$12{,}000)+0.50(\$12{,}500)+0.30(\$14{,}000) = \$2{,}400+\$6{,}250+\$4{,}200 = \$12{,}850.

Discounting to time 0 gives

\operatorname{EPV} = \frac{\$12{,}850}{1.04^3} = \frac{\$12{,}850}{1.124864} = \$11{,}423.603\ldots .

The expected present value is $11,423.60, rounded to the nearest cent. The scenario probabilities sum to 1.00, and the expected payable benefit exceeds the $12,000 floor because the high-benefit outcomes raise the average above the floor.

Common Mistakes and Scope of the Calculation

Adding the guarantee to the projected benefit. A floor on the total benefit means $\max(G,B)$, not $G+B$. Add a guaranteed component to another component only if the contract defines the guarantee as a separate payment.
Applying the floor to the bonus alone. If the guarantee is stated as a minimum total benefit, compare it with the total projected benefit, including the basic amount and applicable bonuses.
Taking the maximum after averaging. Apply the maximum separately to each event and scenario. The expected benefit $E[\max(G,B)]$ cannot generally be replaced by $\max(G,E[B])$.
Using the wrong discount time. A year-2 payment is discounted by $v^2$, even when the policy could instead pay earlier on death.
Mixing unconditional and conditional probabilities. In an event tree, multiply an event probability by a bonus-outcome probability conditional on that event. Do not use a conditional probability as though it were an unconditional probability.

This calculation values benefits on the stated projection and probability basis. If the contract includes other payment choices or features that change the payment according to policyholder decisions, their valuation requires additional assumptions and methods. The key discipline here remains the same: identify the benefit payable at each relevant event before weighting and discounting it.

Key takeaway: For each payment event and bonus outcome, first calculate the actual benefit as the greater of the guarantee and the projected total benefit. Then apply the event probability and discount factor. This sequence preserves the value of the floor in precisely the outcomes where it affects payment.

Check Your Understanding

For each problem, apply the floor at the stated payment event or events before calculating the expected present value.

A hypothetical policy pays a maturity benefit in two years. Its projected total benefit is $9,700 and its floor is $10,000. Assume certain maturity and an effective annual interest rate of 5.00%. Find the present value. (Answer: $9,070.29)
A benefit is paid at the end of year 2. Its floor is $12,000. The projected benefit is $11,000 with probability 0.40 and $15,000 with probability 0.60. At an effective annual interest rate of 4.00%, find the expected present value. (Answer: $12,758.88)
Death in year 1 has probability 0.20 and maturity at the end of year 2 has probability 0.80. The floor is $10,000 at either event; projected benefits are $9,000 on year-1 death and $11,000 at maturity. At an effective annual interest rate of 5.00%, find the expected present value. (Answer: $9,886.62)
A benefit is payable at maturity in four years. Its floor is $15,000. Projected benefits are $12,000 with probability 0.25, $16,000 with probability 0.50, and $20,000 with probability 0.25. At an effective annual interest rate of 3.00%, find the expected present value. (Answer: $14,882.16)
Death in year 1 has probability 0.25, with a $20,000 floor and an $18,000 projected benefit. Otherwise, a year-2 payment occurs with probability 0.75; conditional on that event, the projected benefit is $22,000 with probability 0.40 or $26,000 with probability 0.60. The floor is $20,000 at each event. At an effective annual interest rate of 4.00%, find the expected present value. (Answer: $21,727.07)

What You'll Learn