Valuing Terminal Bonuses at Claim

Introduction

A terminal bonus is a discretionary amount payable when an eligible policy makes a claim, commonly on death or maturity. Its amount may depend on the policy’s declared additions, duration, or another specified measure. Because payment occurs only if a qualifying claim happens, valuing the bonus requires both a bonus scale and a model for the timing and probability of claims.

The calculation must also keep the scope of the requested value clear. The expected present value of a terminal bonus is not automatically the expected present value of the entire claim benefit: guaranteed benefits and declared reversionary bonuses are separate cash flows. Here, the objective is to value only the terminal bonus specified by the problem.

Definition: A terminal bonus at claim is an additional amount payable on a specified claim event, such as death during the policy term or survival to maturity. The bonus amount is determined by the stated scale and eligibility rules at the time of the claim. Its expected present value is the sum of each possible bonus payment multiplied by the probability and discount factor for that payment event.

Annual Claims Paid at the End of the Year

First consider a discrete annual model for a policy with an $n$-year term. Assume that the policy is in force at issue, death claims occurring during policy year $k$ are paid at the end of that year, and the maturity claim is paid at the end of year $n$ if the policyholder survives the term. Let $q_k$ be the conditional probability of death during year $k$, given survival and in-force status at its start, and let $p_k=1-q_k$. These probabilities may vary by year.

Write $B_k^{D}$ for the terminal bonus payable on a death claim at the end of year $k$, and $B_n^{M}$ for the terminal bonus payable on maturity. Each amount is the bonus scale applied to the eligible policy amount at that claim time. The amounts need not be equal: the death and maturity scales, or the eligible amounts they use, may differ.

Under an effective annual interest rate $i$, define $v=1/(1+i)$. The probability of a death claim in year $k$ is the probability of surviving the first $k-1$ years and then dying in year $k$. The probability of reaching maturity is the probability of surviving all $n$ years. Thus:

P(\text{death in year }k) = \left(\prod_{j=1}^{k-1}p_j\right)q_k, \qquad P(\text{survive to maturity}) = \prod_{j=1}^{n}p_j.

These are mutually exclusive claim events. The expected present value of the terminal bonuses is therefore:

\operatorname{EPV}(\text{terminal bonuses}) = \sum_{k=1}^{n} v^k B_k^{D} \left(\prod_{j=1}^{k-1}p_j\right)q_k + v^n B_n^{M}\prod_{j=1}^{n}p_j.

An empty product, as in the death-in-year-1 term, equals 1. The formula values the bonus cash flows only. If the question instead asks for the full claim benefit, the guaranteed amount and any other claim components must be valued separately and included.

Why the Expected-Present-Value Formula Holds

For each possible claim event, define a present-value random variable equal to its bonus discounted to issue if that event occurs, and equal to zero otherwise. The total present value is the sum of these event-specific random variables. By linearity of expectation, its expected value is the sum of their expected values; independence between the events is not required.

Identify an event.
For a death claim in year $k$, the event requires survival through years $1$ to $k-1$, followed by death in year $k$. Its probability is $\left(\prod_{j=1}^{k-1}p_j\right)q_k$.

Discount its payment.
A payment at the end of year $k$ has present value $v^kB_k^D$. Multiplying this amount by the event probability gives that event’s expected present value.

Add the mutually exclusive events.
Repeat for each death year, then include the maturity event with probability $\prod_{j=1}^np_j$ and discount factor $v^n$. The sum is the expected present value of the terminal bonuses.

This argument depends on matching each probability to the correct event and each discount factor to the actual payment time. It applies to the annual end-of-year claim model just stated. If death benefits are paid immediately at the moment of death, the annual end-of-year discount factors do not represent the payment timing; a continuous-time calculation is needed instead.

Worked Example: A Three-Year Death-and-Maturity Scale

A hypothetical three-year participating policy is eligible for a terminal bonus based on $5,000 of declared additions. The death-bonus scale is 8% at the end of year 1, 10% at the end of year 2, and 11% at the end of year 3. The maturity-bonus scale is 12%. Death claims are paid at the end of the year of death; a surviving policy pays the maturity bonus at the end of year 3. The conditional annual death probabilities are $q_1=0.01$, $q_2=0.02$, and $q_3=0.03$. Use $i=5\%$. Find the expected present value of the terminal bonuses only.

Method. Apply each scale to the eligible additions, calculate the probability of each claim event, and discount that event’s bonus. The maturity payment is a bonus-only cash flow; this calculation does not include a guaranteed maturity benefit.

The death bonuses are $0.08(5{,}000)=\$400$, $0.10(5{,}000)=\$500$, and $0.11(5{,}000)=\$550$. The maturity bonus is $0.12(5{,}000)=\$600$. The annual survival probabilities are $p_1=0.99$, $p_2=0.98$, and $p_3=0.97$.

The death-in-year probabilities are $0.01$ in year 1, $0.99(0.02)=0.0198$ in year 2, and $0.99(0.98)(0.03)=0.029106$ in year 3. The probability of surviving to maturity is $0.99(0.98)(0.97)=0.941094$. The discount factors are $v=1/1.05$, $v^2=1/1.1025$, and $v^3=1/1.157625$.

\begin{aligned} \operatorname{EPV} &= \frac{400(0.01)}{1.05} +\frac{500(0.0198)}{1.1025} +\frac{550(0.029106)}{1.157625} +\frac{600(0.941094)}{1.157625}\\ &= 3.80952381+8.97959184+13.82857143+487.77142857\\ &=514.38911565. \end{aligned}

Rounded to the nearest cent, the expected present value is $514.39. As a check, the four claim-event probabilities sum to $0.01+0.0198+0.029106+0.941094=1$, and each bonus is discounted to its stated end-of-year payment time.

Common mistake avoided: The probability of death in year 2 is not $q_2=0.02$ by itself. That is a conditional probability given survival to the start of year 2. The unconditional probability of a year-2 death claim is $p_1q_2=0.99(0.02)=0.0198$.

Eligibility and Portfolio Valuation

For a group of policies, apply the eligibility rule before totaling the bonus amounts. If a scale is a percentage of eligible declared additions, an ineligible policy contributes zero terminal bonus even if it has declared additions. For eligible policies, calculate the scale-based amount for each relevant claim event and sum those amounts. If every policy has the same mortality and interest assumptions, the common event probabilities and discount factors can be applied to the aggregate eligible amounts.

That shortcut does not justify treating ineligible policies as eligible or averaging different claim-time scales without checking the amounts. If policies have different mortality assumptions, terms, or payment timing, calculate the expected present value separately for the applicable groups and then add the results.

Worked Example: Aggregating Only Eligible Policies

A hypothetical two-year portfolio has four eligible policies with declared additions of $1,000, $1,500, $2,000, and $900, plus one ineligible policy with $1,600 of additions. The death-bonus scale is 5% of eligible additions for a year-1 death claim and 8% for a year-2 death claim. The maturity scale is 10%. Death claims and maturity claims are paid at the ends of their respective years. Assume $q_1=q_2=0.04$ and $i=3\%$. Find the portfolio expected present value of terminal bonuses only.

Method. First total the eligible additions: $1{,}000+1{,}500+2{,}000+900=\$5{,}400$. The ineligible policy’s $1,600 is excluded. The bonus amounts are therefore $270 for a year-1 death, $432 for a year-2 death, and $540 at maturity.

The event probabilities are $0.04$ for death in year 1, $0.96(0.04)=0.0384$ for death in year 2, and $0.96^2=0.9216$ for survival to maturity. Using $v=1/1.03$, the expected present value is:

\begin{aligned} \operatorname{EPV} &= \frac{270(0.04)}{1.03} +\frac{432(0.0384)}{1.03^2} +\frac{540(0.9216)}{1.03^2}\\ &= \frac{10.8}{1.03} +\frac{16.5888}{1.0609} +\frac{497.664}{1.0609}\\ &= 10.48543689+15.63584221+469.09793484\\ &=495.21921394. \end{aligned}

Rounded to the nearest cent, the portfolio expected present value is $495.22. The event probabilities sum to $0.04+0.0384+0.9216=1$. The calculation excludes the ineligible policy and includes no guaranteed benefits.

Common mistake avoided: Applying the 10% maturity scale to all five policies would incorrectly include the ineligible policy’s additions. Eligibility is part of the definition of the amount payable, not an adjustment made after valuing the portfolio.

Death Benefits Paid Immediately

Some models treat death as occurring continuously and pay the death bonus immediately. Suppose the force of mortality is constant at $\mu$ per year, the force of interest is constant at $\delta$, and the term is $T$ years. Under this model, the probability of surviving to time $t$ is $e^{-\mu t}$, and the density of the time of death is $\mu e^{-\mu t}$. If a fixed bonus $B^D$ is paid immediately on death before time $T$, its expected present value is:

B^D\int_0^T e^{-\delta t}\mu e^{-\mu t}\,dt = B^D\frac{\mu}{\mu+\delta} \left(1-e^{-(\mu+\delta)T}\right).

If a maturity bonus $B^M$ is paid at time $T$ only upon survival, its expected present value is $B^Me^{-\delta T}e^{-\mu T}$. These expressions assume constant forces, a fixed bonus for each type of claim, and payment at the stated times. They do not apply unchanged to a duration-dependent scale or varying forces; in that case, the time-varying bonus and claim density must remain inside the integral.

Worked Example: Immediate Death Payment Under Constant Forces

A hypothetical three-year policy pays a $400 terminal bonus immediately on death and a $600 terminal bonus at maturity if the policyholder survives. Assume a constant force of mortality $\mu=0.03$ per year and a constant force of interest $\delta=0.02$ per year. Find the expected present value of the terminal bonuses.

Method. Value immediate death payments by integrating the discounted death-time density, then add the discounted survival-contingent maturity bonus. Since $\mu+\delta=0.05$, the death-bonus value is:

400\frac{0.03}{0.05}\left(1-e^{-0.05(3)}\right) = 240\left(1-e^{-0.15}\right) = \$33.43008566.

The maturity-bonus value is:

600e^{-0.02(3)}e^{-0.03(3)} = 600e^{-0.15} = \$516.42478563.

Adding the unrounded components gives $33.43008566+516.42478563=\$549.85487129$, or $549.85 to the nearest cent. As a check, the death-event probability before maturity is $1-e^{-0.09}$, while the maturity survival probability is $e^{-0.09}$; together they sum to 1. The different present values reflect the different bonus amounts and payment times.

Common mistake avoided: Discounting every death bonus to the end of its policy year would contradict the stated immediate-payment assumption. The integral discounts each possible death payment from its actual time $t$.

Common Valuation Traps

The expected present value is sensitive to the claim event, the eligible amount, and the payment time. Before calculating, identify each of these explicitly. The following distinctions prevent frequent errors:

Issue	Incorrect approach	Correct approach
Conditional mortality	Use $q_k$ as the unconditional probability of death in year $k$.	Multiply $q_k$ by the probability of surviving to the start of year $k$.
Maturity claim	Pay a maturity bonus to every policy, including policies that died earlier.	Multiply the maturity bonus by the probability of survival to maturity.
Eligibility	Apply the scale to all policies and adjust the result afterward.	Determine eligible policies and eligible amounts before applying the scale.
Scope of value	Call the terminal-bonus EPV the value of the whole claim benefit.	Include guaranteed and other claim components only if the question asks for the full benefit and supplies the needed amounts.
Timing	Use an end-of-year discount factor for an immediate death payment.	Discount at the specified payment time, using a continuous-time model when appropriate.

These calculations are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus. In an exam-style valuation, a compact event table can help organize the work: list the bonus amount, event probability, and discount factor for each possible claim, then sum the event-specific expected present values.

Key takeaway: A terminal-bonus valuation is a claim-contingent cash-flow calculation. Apply the stated scale only to eligible amounts, use the probability of the actual death or maturity event, and discount to the specified payment time. State clearly whether the result covers terminal bonuses alone or all claim benefits.

Check Your Understanding

For each problem, calculate only the terminal-bonus expected present value unless the question states otherwise. Round monetary answers to the nearest cent.

A two-year policy has $i=10\%$ and conditional annual death probability $q=0.10$ in each year. Its death bonuses are $110 at the end of year 1 and $121 at the end of year 2; its maturity bonus is $121. Find the expected present value. (Answer: $100.00)
Five policies have identical assumptions, but only four are eligible. Each eligible policy has $1,000 of additions and pays a 5% bonus on those additions only at maturity after two years. Assume $i=5\%$ and $q=0.02$ in each year. Find the portfolio expected present value of the terminal bonuses. (Answer: $174.22)
A three-year policy pays a $400 terminal bonus immediately on death and $600 at maturity if the policyholder survives. Under constant forces $\mu=0.03$ and $\delta=0.02$ per year, find the expected present value. (Answer: $549.85)
A three-year policy has $i=10\%$ and conditional annual death probability $q=0.10$ in each year. Death bonuses paid at each year-end are $100, $200, and $300 for claims in years 1, 2, and 3, respectively; the maturity bonus is $400. Find the expected present value. (Answer: $261.31)

Issue	Incorrect approach	Correct approach
Conditional mortality	Use \(q_k\) as the unconditional probability of death in year \(k\).	Multiply \(q_k\) by the probability of surviving to the start of year \(k\).
Maturity claim	Pay a maturity bonus to every policy, including policies that died earlier.	Multiply the maturity bonus by the probability of survival to maturity.
Eligibility	Apply the scale to all policies and adjust the result afterward.	Determine eligible policies and eligible amounts before applying the scale.
Scope of value	Call the terminal-bonus EPV the value of the whole claim benefit.	Include guaranteed and other claim components only if the question asks for the full benefit and supplies the needed amounts.
Timing	Use an end-of-year discount factor for an immediate death payment.	Discount at the specified payment time, using a continuous-time model when appropriate.

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