Worked Valuation of a Guaranteed Maturity Benefit

Introduction

A guaranteed maturity benefit protects a policyholder against a fund value below a specified floor at the end of a contract. Valuing that benefit requires keeping three calculations distinct: projecting the investment fund, determining the payment in each scenario, and discounting the expected payment to the valuation date. The guarantee is a feature of the maturity payment; it does not change the fund projection itself.

The valuation basis must state how investment scenarios are weighted, how any fund deductions are applied, whether payment depends on survival, and when the payment occurs. The examples use hypothetical contracts and specified scenario probabilities. Those probabilities are valuation weights for the calculations shown, not statements about actual investment experience.

Definition: Let $F_T$ be the projected fund immediately before maturity at time $T$, and let $G_T$ be the guaranteed maturity amount. If the benefit is payable only when the policyholder is alive at maturity, the payment at time $T$ is $B_T=\max(F_T,G_T)=F_T+\max(G_T-F_T,0)$. The amount $\max(G_T-F_T,0)$ is the guarantee top-up.

Valuation Method and Assumptions

For a contract with specified investment scenarios $j=1,\ldots,n$, let $p_j$ be the valuation probability of scenario $j$, with $p_j\geq 0$ and $\sum_j p_j=1$. Project a separate maturity fund $F_T^{(j)}$ in each scenario, then calculate $B_T^{(j)}=\max(F_T^{(j)},G_T)$. The expected maturity payment conditional on reaching maturity is the probability-weighted sum of those payments.

E[B_T]=\sum_{j=1}^{n}p_j\max(F_T^{(j)},G_T)

Suppose the maturity benefit is paid only if the policyholder survives to $T$, investment outcomes and survival are independent on the valuation basis, and the effective annual discount rate is $i$. Then ${}_Tp_x$ is the probability of surviving from age $x$ to maturity, and $v=1/(1+i)$. The present value at issue is

\operatorname{PV}_0=v^T\,{}_Tp_x\sum_{j=1}^{n}p_j\max(F_T^{(j)},G_T).

This formula assumes no maturity payment on death before $T$, no lapse, and payment at time $T$. If investment outcomes and survival are not independent, the joint probability of each investment scenario and survival must be used instead of multiplying separate probabilities. If the benefit is payable regardless of survival, the survival factor does not apply.

Project the fund.
Apply each scenario’s returns in order, including any specified charges at their stated times.

Apply the guarantee scenario by scenario.
For each scenario, calculate the top-up and add it to the fund, or equivalently take the larger of the fund and guarantee.

Weight and discount.
Take the probability-weighted payment, adjust for survival if applicable, and discount from the actual payment time.

A fund deduction must be placed at the correct point in the projection. For example, if a charge of $c$ is deducted from the fund at the end of each year after the gross investment return, the annual multiplier is $(1+r)(1-c)$, where $r$ is that year’s gross return. This is not generally the same as subtracting $c$ dollars or subtracting $c$ percentage points from the return.

Worked Example: Scenario-Specific Top-Ups

A hypothetical policy has an initial invested fund of $10,000 and matures after three years. The maturity guarantee is $10,000. The annual effective discount rate is 4%, and assume the policyholder survives to maturity with certainty so that this example isolates investment scenario weighting and discounting. Scenario A has probability 0.40 and annual returns of 5%, −10%, and 8%, respectively. Scenario B has probability 0.60 and annual returns of −8%, 4%, and 2%, respectively. Find the maturity payment in each scenario and its present value.

Method. Project each scenario’s fund by multiplying by one plus the return in each year. Then apply the guarantee, weight the payments, and discount for three years.

Scenario A fund. After year 1, the fund is $10{,}000(1.05)=10{,}500$. After year 2, it is $10{,}500(0.90)=9{,}450$. After year 3, it is $9{,}450(1.08)=10{,}206$. The top-up is $\max(10{,}000-10{,}206,0)=0$, so the payment is $10,206.

Scenario B fund. After year 1, the fund is $10{,}000(0.92)=9{,}200$. After year 2, it is $9{,}200(1.04)=9{,}568$. After year 3, it is $9{,}568(1.02)=9{,}759.36$. The top-up is $10{,}000-9{,}759.36=240.64$, so the payment is $10,000.

The expected maturity payment is $0.40(10{,}206)+0.60(10{,}000)=4{,}082.40+6{,}000=$$10,082.40. Since the payment occurs at the end of year 3, its present value is

\frac{10{,}082.40}{(1.04)^3} =\frac{10{,}082.40}{1.124864} =8{,}963.22.

The result is $8,963.22, rounded to the nearest cent. The method avoids the common mistake of averaging the funds first and then applying the guarantee: the guarantee is applied separately to each scenario’s fund.

Why the Guarantee Is Applied Before Averaging

The order of operations follows directly from the contract. The contract determines a payment for the realized maturity fund; it does not first calculate an average fund across hypothetical scenarios. For each scenario, the top-up is $ (G_T-F_T^{(j)})_+ $, where $z_+=\max(z,0)$. Therefore, averaging the contract payments gives

E[B_T]=\sum_j p_j\left(F_T^{(j)}+(G_T-F_T^{(j)})_+\right).

In the previous example, the probability-weighted fund is $0.40(10{,}206)+0.60(9{,}759.36)=9{,}938.016$. Taking the maximum of this average and the $10,000 guarantee gives $10,000, not the correct expected payment of $10,082.40. Averaging first loses the distinction between scenarios above and below the floor.

The same logic applies when the guarantee increases over time. If the contract specifies an annual guaranteed roll-up rate $g$ on an initial guaranteed amount $G_0$, the maturity floor is $G_T=G_0(1+g)^T$. Use that specified floor in every scenario’s payment calculation; do not add the guarantee to the projected fund.

Worked Example: Fund Deductions and Survival-Contingent Payment

A hypothetical two-year policy has a single premium of $20,000, of which 95% is allocated to the investment fund at issue. At the end of each year, a charge equal to 1% of the fund after that year’s gross return is deducted. The guarantee is $20,000 at maturity. Scenario A has probability 0.25 and gross returns of −20% in year 1 and 10% in year 2. Scenario B has probability 0.75 and gross returns of 5% in each year. The annual effective discount rate is 4%. The annual death probabilities are 0.01 in year 1 and 0.02 in year 2. Assume mortality is independent of investment scenario, no payment is made on death before maturity, and the maturity benefit is paid at the end of year 2. Find its present value at issue.

Initial invested fund. The amount allocated to the fund is $20{,}000(0.95)=$$19,000. In each year, first apply the gross return and then retain 99% of the resulting fund. Thus the annual multiplier is $(1+r)(0.99)$.

Scenario A projection. In year 1, the fund is $19{,}000(0.80)(0.99)=19{,}000(0.792)=15{,}048$. In year 2, it is $15{,}048(1.10)(0.99)=15{,}048(1.089)=16{,}387.272$. The top-up is $20{,}000-16{,}387.272=3{,}612.728$, and the total maturity payment is $20,000.

Scenario B projection. The multiplier in each year is $1.05(0.99)=1.0395$. After year 1, the fund is $19{,}000(1.0395)=19{,}750.50$. After year 2, it is $19{,}750.50(1.0395)=20{,}530.64475$. The fund exceeds the guarantee, so the top-up is zero and the maturity payment is $20,530.64475.

The expected payment conditional on survival is $0.25(20{,}000)+0.75(20{,}530.64475)=5{,}000+15{,}397.9835625=20{,}397.9835625$. The probability of survival is $(1-0.01)(1-0.02)=0.99(0.98)=0.9702$. The present value is therefore

\frac{20{,}397.9835625(0.9702)}{(1.04)^2} =\frac{19{,}790.1236523}{1.0816} =18{,}297.08.

The present value is $18,297.08, rounded to the nearest cent. The calculation uses the net fund multipliers for the projection, but discounts the maturity payment using the separate valuation rate of 4%. Treating the 1% fund charge as a reduction to the discount rate would mix two different parts of the calculation.

Roll-Up Guarantees and Valuation Checks

A guaranteed maturity amount may be fixed at issue or may increase according to a contractually specified formula. The formula should be evaluated for the full term before scenario payments are calculated. The next example includes a guaranteed roll-up and uses survival probabilities in the present-value calculation.

Worked Example: A Rolled-Up Guarantee

A hypothetical policy begins with an invested fund of $15,000 and matures after two years. The initial guarantee of $15,000 rolls up at 2% effective annually, so the maturity guarantee is $15{,}000(1.02)^2$. Scenario A has probability 0.30 and returns of −10% and 5%. Scenario B has probability 0.70 and returns of 12% and 8%. The annual effective discount rate is 3.5%, and annual death probabilities are 0.02 and 0.03. Assume survival and investment scenario are independent, and the benefit is payable only on survival to maturity. Find the present value at issue.

Guarantee. The two-year guarantee is $15{,}000(1.02)(1.02)=15{,}000(1.0404)=$$15,606.

Scenario A fund and payment. The year-1 fund is $15{,}000(0.90)=13{,}500$. The year-2 fund is $13{,}500(1.05)=14{,}175$. The top-up is $15{,}606-14{,}175=1{,}431$, giving a total payment of $15,606.

Scenario B fund and payment. The year-1 fund is $15{,}000(1.12)=16{,}800$. The year-2 fund is $16{,}800(1.08)=18{,}144$. Since this exceeds $15,606, the top-up is zero and the payment is $18,144.

The expected maturity payment conditional on survival is $0.30(15{,}606)+0.70(18{,}144)=4{,}681.80+12{,}700.80=$$17,382.60. The survival probability is $0.98(0.97)=0.9506$, and the two-year discount factor is $1/(1.035)^2=1/1.071225$. Thus

\operatorname{PV}_0 =\frac{17{,}382.60(0.9506)}{1.071225} =\frac{16{,}523.89956}{1.071225} =15{,}425.24.

The present value is $15,425.24, rounded to the nearest cent. The intermediate check is that the scenario A payment equals the guarantee, while the scenario B payment equals the larger fund value.

Common Exam Traps

Applying the floor to an average fund. Calculate $\max(F_T^{(j)},G_T)$ in each scenario before taking the weighted average.
Leaving a scenario’s return unspecified. A fund projection is determined only when the return or other fund movement is specified for every year through maturity.
Adding the guarantee to the fund. The payment is the greater of the fund and the guarantee; it is not the fund plus the full guarantee. Add only the shortfall when the fund is below the floor.
Using the wrong timing. A payment at time $T$ is discounted for $T$ years. Include survival only when the contract’s payment is contingent on survival and use the survival probability to the stated maturity date.
Combining investment charges with the valuation rate. Apply charges in the fund projection at the specified times, then apply the stated discount rate to the contractual payment.

Key takeaway: Project each scenario’s fund to maturity, calculate its contractual payment using the guarantee, and only then weight, survival-adjust, and discount the payments. This order keeps the fund mechanics, guarantee cost, and time value of money distinct.

Check Your Understanding

For each problem, apply the guarantee to scenario-specific fund values before weighting the payments. Unless stated otherwise, ignore expenses and lapse.

A hypothetical two-year contract starts with an $8,000 fund and guarantees $8,500 at maturity. Scenario A has probability 0.40 and returns of −10% and 0%; Scenario B has probability 0.60 and returns of 20% and 0%. Find the payment in each scenario and the expected maturity payment. (Answer: Scenario A $8,500; Scenario B $9,600; expected payment $9,160.)
A hypothetical one-year policy has a $12,000 fund, a maturity guarantee of $11,500, a 5% annual effective discount rate, and a 0.01 probability of death during the year. The benefit is payable only on survival, and the fund return is −10%. Find the present value at issue. (Answer: $10,842.86.)
A two-year policy starts with $10,000 and guarantees $10,000 at maturity. Scenario A has probability 0.25 and returns of −10% and 0%; Scenario B has probability 0.75 and returns of 5% in each year. The annual effective discount rate is 5%, annual death probabilities are 0.02 and 0.01, and survival is independent of investment scenario. The maturity benefit is payable only on survival. Find the present value at issue. (Answer: $9,476.50.)
A three-year contract starts with $5,000 and guarantees $5,500. Scenario A has probability 0.25 and returns of 20%, 0%, and 0%. Scenario B has probability 0.75 and returns of −10%, 10%, and 0%. Find the fund and payment in each scenario and the expected maturity payment. (Answer: Scenario A fund and payment $6,000; Scenario B fund $4,950 and payment $5,500; expected payment $5,625.)
A two-year policy starts with $10,000 and guarantees $10,000 at maturity. Scenario A has probability 0.40 and returns of −20% and 10%; Scenario B has probability 0.60 and returns of 8% in each year. The annual effective discount rate is 4%, annual death probabilities are 0.01 and 0.02, and the benefit is payable only on survival. Find the present value at issue, assuming survival and investment scenario are independent. (Answer: $9,865.61.)

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