Present Values of Guaranteed Participating Benefits

What Is Being Valued?

A participating policy can provide guaranteed benefits as well as benefits that depend on future declarations. This tutorial values the guaranteed amounts: death benefits, maturity benefits, and payments conditional on survival. If an addition has already been declared and is guaranteed under the policy, its effect can be included in the benefit amounts specified for each claim date. Uncertain future additions are not included.

The key task is to match each payment to the event that triggers it and the time at which it is paid. A death benefit is weighted by the probability of death in the relevant period; a survival or maturity benefit is weighted by the probability of surviving to its payment date. Each amount is then discounted from its payment date to the valuation date.

Definition: The actuarial present value (APV) of a set of future benefits is the expected value, at the valuation date, of their discounted payments. For a deterministic effective annual interest rate $i$, a payment at time $k$ is discounted by $v^k$, where $v=1/(1+i)$. In this tutorial, death benefits are paid at the end of the policy year of death, and survival benefits are paid at the stated policy anniversary.

Use an annual discrete model. Let $q_{x+k-1}$ be the probability that a life aged $x+k-1$ dies during policy year $k$, conditional on being alive at its start, and let $p_{x+k-1}=1-q_{x+k-1}$. Then ${}_{k-1}p_x q_{x+k-1}$ is the probability of surviving to the start of year $k$ and dying during that year; ${}_kp_x$ is the probability of surviving to the end of year $k$. Assume the stated mortality model is the only way a policy can terminate. If lapse, disability, or another decrement affects payment eligibility, it must also be represented in the model.

Write $B_k$ for the guaranteed death benefit paid at the end of year $k$, $C_j$ for a guaranteed benefit paid at time $j$ only if the life survives to that time, and $M_n$ for a maturity benefit paid at time $n$ if the life survives to maturity. The amounts can vary by duration. In particular, a declared addition may make $B_k$, $C_j$, or $M_n$ larger, but the benefit schedule must specify the amount payable for each event.

Theorem: Under the annual model, with deterministic annual effective interest and the payment timings just stated, the APV is $$ \operatorname{APV} = \sum_{k=1}^{n} v^k\,{}_{k-1}p_x q_{x+k-1}B_k +\sum_{j\in J} v^j\,{}_jp_x C_j +v^n\,{}_np_x M_n, $$ where $J$ is the set of dates for survival benefits that are distinct from the maturity benefit. Each death, survival, and maturity payment is included once, at its specified date and conditional event.

Why the Present-Value Formula Works

The formula follows by separating outcomes according to the event that produces each payment. For a death benefit in year $k$, the probability of reaching that year alive is ${}_{k-1}p_x$; conditional on that, the probability of dying in the year is $q_{x+k-1}$. Thus the unconditional probability of that payment is their product. Its present value if paid is $v^kB_k$, so its expected present value is the product of that amount and the event probability.

The same reasoning applies to a survival benefit: it is paid only if the life reaches its payment date, an event with probability ${}_jp_x$. The maturity benefit is treated in the same way at time $n$. Finally, expectation is additive, so the expected present values of the separate payments can be summed. This does not require the payments to be independent; it requires each payment and its triggering event to be identified correctly.

E[\text{discounted payment in year }k] = v^kB_k\,P(\text{death in year }k) = v^kB_k\,{}_{k-1}p_xq_{x+k-1}.

Worked Example: Three-Year Guaranteed Endowment

A hypothetical three-year policy pays $10,000 at the end of the year of death if death occurs during the term, or $12,000 at the end of year 3 if the insured survives to maturity. The annual effective interest rate is $i=25\%$. Mortality is independent from year to year, with annual death probability $q=0.20$ and survival probability $p=0.80$. Calculate the APV at issue.

Method. Calculate the probability and discounted value of each mutually exclusive death-year benefit, then add the survival-to-maturity benefit. Here $v=1/1.25=0.80$.

Death in year 1 has probability $0.20$, and its discount factor is $v=0.80$. Its contribution is $10{,}000(0.20)(0.80)=\$1{,}600.00$.

Death in year 2 has probability $0.80(0.20)=0.16$, and its discount factor is $v^2=0.64$. Its contribution is $10{,}000(0.16)(0.64)=\$1{,}024.00$.

Death in year 3 has probability $0.80^2(0.20)=0.128$, and its discount factor is $v^3=0.512$. Its contribution is $10{,}000(0.128)(0.512)=\$655.36$.

Survival to maturity has probability $0.80^3=0.512$. The discounted maturity contribution is $12{,}000(0.512)(0.512)=\$3{,}145.728$.

Adding the contributions gives $\$1{,}600.00+\$1{,}024.00+\$655.36+\$3{,}145.728=\$6{,}425.088$. Rounded to the nearest cent, the APV is $6,425.09.

Check. The probabilities of death in the three years sum to $0.20+0.16+0.128=0.488$, and the probability of surviving to maturity is $0.512$; together they equal 1. The maturity payment is therefore not added to any death outcome.

Common mistake avoided. Adding the $12,000 maturity benefit to the death benefit in year 3 would pay both benefits on the same outcome. The contract specifies death in year 3 or survival to maturity, so those events are separate.

Including Declared Additions in Guaranteed Amounts

A declared addition should be included only as the policy terms specify. At the valuation date, first identify what is already guaranteed and which claim amounts include the declared addition. Then use those total amounts in the APV calculation. Do not add the declared amount again if it is already included in the stated death or maturity benefit.

A useful benefit schedule makes this explicit. It can list the total payable amount for death in each policy year and the total payable at maturity. This is safer than adding a separate “bonus value” to a total benefit whose definition is unclear. The calculation below uses a stated schedule in which the guaranteed death and maturity amounts already reflect declared additions.

Worked Example: Benefit Schedule Including Vested Additions

A hypothetical two-year participating policy has a $10,000 base benefit. A $500 addition is already declared and vested at the valuation date. A further $500 addition becomes guaranteed at the first anniversary if the policy is then in force. The total death benefit is therefore $10,500 for death in year 1 and $11,000 for death in year 2. The total maturity benefit is $12,000, including the vested additions and the policy’s additional guaranteed maturity amount. The annual effective interest rate is $i=10\%$; annual mortality probabilities are $q=0.20$ and $p=0.80$ in each year. Calculate the APV at the valuation date.

Method. Use the total benefit amounts for each event exactly once. The discount factor is $v=1/1.10=10/11$, and $v^2=100/121$.

For death in year 1, the probability is $0.20$. The contribution is $10{,}500(0.20)(10/11)=\$1{,}909.0909\ldots$.

For death in year 2, the probability is $0.80(0.20)=0.16$. The contribution is $11{,}000(0.16)(100/121)=\$1{,}454.5454\ldots$.

For survival to maturity, the probability is $0.80^2=0.64$. The contribution is $12{,}000(0.64)(100/121)=\$6{,}347.1074\ldots$.

The total is $\$1{,}909.0909\ldots+\$1{,}454.5454\ldots+\$6{,}347.1074\ldots=\$9{,}710.7438\ldots$. Rounded to the nearest cent, the APV is $9,710.74.

Check. The death probabilities are $0.20$ and $0.16$, while the maturity survival probability is $0.64$. They sum to 1. The declared $500 and the later guaranteed $500 have not been separately added to the scheduled benefits because the stated totals already include them.

Common mistake avoided. If one were to add the declared additions to the $12,000 maturity amount again, the maturity benefit would be overstated. The key is to establish whether each amount is a base benefit or an already-inclusive total before applying probabilities and discounting.

Survival Benefits at More Than One Date

A policy may pay a guaranteed amount at an interim anniversary if the insured is alive, and a separate maturity amount later. The interim benefit is not conditional on surviving beyond its own payment date. For example, if an amount is due at year 2, its probability weight is ${}_2p_x$, even if the insured could die during year 3. The year 3 death benefit and any later maturity payment are then valued on their own triggering events.

Worked Example: Interim Survival Payment and Maturity Benefit

A hypothetical four-year policy pays a death benefit of $8,000 at the end of the year of death. If the insured survives to the end of year 2, the policy also pays $2,000 then. If the insured survives to the end of year 4, it pays a $10,000 maturity benefit. The annual effective interest rate is $i=1/3$, so $v=0.75$. Annual mortality is independent, with $q=0.25$ and $p=0.75$. Calculate the APV.

Method. Value four possible death benefits, the year 2 survival payment, and the year 4 maturity benefit separately. A year 2 survivor payment can be received even by someone who later dies in year 3 or 4.

The probability of death in year $k$ is $0.25(0.75)^{k-1}$. After discounting by $v^k=0.75^k$, the probability-discount factors for years 1 through 4 are:

Year 1: $0.25(0.75)^0(0.75)^1=0.1875$.
Year 2: $0.25(0.75)^1(0.75)^2=0.10546875$.
Year 3: $0.25(0.75)^2(0.75)^3=0.059326171875$.
Year 4: $0.25(0.75)^3(0.75)^4=0.0333709716796875$.

The death-benefit contribution is $8{,}000(0.1875+0.10546875+0.059326171875+0.0333709716796875)=\$3{,}085.3271484375$.

The probability-discount factor for the year 2 survival payment is ${}_2p_xv^2=(0.75)^2(0.75)^2=0.31640625$. Its contribution is $2{,}000(0.31640625)=\$632.8125$.

The probability-discount factor for the maturity benefit is ${}_4p_xv^4=(0.75)^4(0.75)^4=0.1001129150390625$. Its contribution is $10{,}000(0.1001129150390625)=\$1{,}001.129150390625$.

The total is $\$3{,}085.3271484375+\$632.8125+\$1{,}001.129150390625=\$4{,}719.268798828125$. Rounded to the nearest cent, the APV is $4,719.27.

Check. The year 2 payment is included once for everyone who survives to year 2. The year 4 maturity benefit is included only for those who survive to year 4. Death benefits are paid instead of maturity for deaths during the term.

Common mistake avoided. Applying the year 4 survival probability to the year 2 payment would incorrectly require survival beyond the payment date. The year 2 amount is due at year 2, so its event probability is ${}_2p_x$, not ${}_4p_x$.

Valuation Checks and Limits

A reliable calculation starts with a benefit-event schedule and ends with a probability check. For a policy with only death during years 1 through $n$ or survival to year $n$, the event probabilities must satisfy $\sum_{k=1}^{n}{}_{k-1}p_xq_{x+k-1}+{}_np_x=1$. If they do not, a conditional probability, duration, or survival term has likely been omitted or misapplied.

That identity changes in appearance when the policy has additional exits, such as lapse, or when a survival benefit can be paid before the end of the term. The benefit events still need to be modeled according to the contract, and mutually exclusive termination events must account for all possible outcomes. For an interim survival benefit, do not force the benefit events across the entire policy term to sum to 1: the payment can occur and a later death benefit can also be paid.

Match the time units. Annual mortality probabilities require annual policy periods, and the annual effective rate discounts annual payment dates.
Use the right conditional event. A death benefit in year $k$ uses survival to the start of year $k$, then death during that year.
Separate base and total amounts. A declared addition is either included in a stated total benefit or added to a base amount, not both.
Include only guaranteed amounts. A possible future declaration is not a guaranteed payment under the assumptions used here.

These methods are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus. The same event-by-event structure is useful whenever a participating policy has benefits payable on different contingencies and at different times.

Check Your Understanding

For each problem, use annual effective interest, end-of-year death payments, and the stated mortality assumptions. Round dollar answers to the nearest cent.

A two-year endowment pays $4,000 at the end of the year of death or $5,000 at the end of year 2 if the insured survives. The annual effective interest rate is 25%, and each year $q=0.20$, $p=0.80$. Find the APV. (Answer: $3,097.60)
A three-year policy pays $6,000 at the end of the year of death or $9,000 at the end of year 3 if the insured survives. The annual effective interest rate is 10%, and each year $q=0.10$, $p=0.90$. Find the APV. (Answer: $6,286.25)
A guaranteed $3,000 payment is due at the end of year 2 only if the insured survives to that date. The annual effective interest rate is 5%, and each year $q=0.10$, $p=0.90$. There are no other benefits. Find the APV. (Answer: $2,204.08)
A two-year policy pays $7,000 for death in year 1, $9,000 for death in year 2, and $11,000 at maturity if the insured survives to year 2. It also pays $1,000 at the end of year 1 if the insured survives to that date. The annual effective interest rate is 10%, and each year $q=0.25$, $p=0.75$. Find the APV. (Answer: $8,780.99)

What You'll Learn