Reserve Changes After a Bonus Declaration

Why a Bonus Declaration Changes the Reserve

A reversionary bonus is a declared addition to policy benefits. Once declared, it is guaranteed under the policy terms: it is not merely a possible future allocation. The prospective reserve therefore needs to reflect the present value of the additional guaranteed payments, assessed using the same interest, mortality and other valuation assumptions as before the declaration.

The reserve increase is not automatically equal to the bonus amount. A bonus may be payable only on death, only at maturity, or on either event, and each payment may occur at a different time. The reserve change depends on both the amount added to the benefit and the probability and timing of the payment. This tutorial considers the effect of a declaration at a specified valuation date, with the valuation basis and other contractual terms held fixed. It does not include possible future, undeclared bonuses.

Definition: A prospective reserve at a valuation time is the expected present value of future benefits less the expected present value of future premiums, calculated under a stated valuation basis. A declared reversionary bonus changes the reserve by changing the guaranteed future benefits. If the future premiums and all other terms are unchanged, the reserve increase is the expected present value of the additional guaranteed benefits.

Valuing the Increase, Not the Whole Contract Again

Let $V^{-}$ be the prospective reserve immediately before declaration and $V^{+}$ the reserve immediately after declaration. Let $B^{-}$ and $B^{+}$ denote the future benefit cash flows before and after declaration, and let $P$ denote the unchanged future premium cash flows. On the same valuation basis,

V^{-}=\operatorname{EPV}(B^{-})-\operatorname{EPV}(P), \qquad V^{+}=\operatorname{EPV}(B^{+})-\operatorname{EPV}(P).

Subtracting the first reserve from the second cancels the common premium value. If the declaration adds benefits $D=B^{+}-B^{-}$, the change is

\Delta V=V^{+}-V^{-} =\operatorname{EPV}(B^{+}-B^{-}) =\operatorname{EPV}(D).

This is the central calculation: isolate the incremental benefit and value it under the existing basis. The formula relies on the stated conditions. In particular, premiums, expenses, valuation assumptions and other contractual cash flows must be unchanged. If any of these change at declaration, their reserve effects must also be included.

Result: When a declared bonus adds a fixed amount $b$ to a benefit paid at time $t$ only if a specified event occurs, its reserve contribution is the probability of that event multiplied by $b$ and discounted to the valuation time. Sum the contributions over mutually exclusive event times. For annual effective interest $i$, a payment at the end of year $k$ is discounted by $v^k$, where $v=1/(1+i)$.

Worked Examples

Worked Example: A Death-Only Addition to a Three-Year Term Benefit

A hypothetical three-year term policy has just received a reversionary bonus that adds $1,200 to the death benefit. The addition is paid at the end of the year of death and is not payable if the insured survives the term. The conditional probability of death in each year, given survival to its start, is $q=0.20$. The effective annual interest rate is $i=10\%$. Calculate the increase in the prospective reserve at the declaration date.

Method. Value the added $1,200 payment separately for each possible year of death. The payment amount is the same in every year, but the probability of reaching each year alive changes.

The probability of death in year 1 is $0.20$. The probability of surviving year 1 and dying in year 2 is $0.80(0.20)=0.16$. The probability of surviving two years and dying in year 3 is $0.80^2(0.20)=0.128$. The discount factor is $v=1/1.10$.

\begin{aligned} \Delta V &=1200\left(0.20v+0.16v^2+0.128v^3\right)\\ &=1200\left(\frac{0.20}{1.10} +\frac{0.16}{1.10^2} +\frac{0.128}{1.10^3}\right)\\ &=1200\left(0.1818182+0.1322314+0.0961683\right)\\ &=1200(0.4102179)\\ &=\$492.26. \end{aligned}

The reserve increases by $492.26, rounded to the nearest cent. As a check, the three death probabilities sum to $0.20+0.16+0.128=0.488$, the probability of dying during the term. The remaining probability, $0.512$, is survival to the end and produces no bonus payment under the stated contract.

Common mistake avoided: Applying $q=0.20$ directly to each of the three years treats the annual deaths as if each were unconditional. That incorrect calculation gives $1200(0.20v+0.20v^2+0.20v^3)=\$596.84$. Later-year death probabilities must also include survival to the start of those years.

Worked Example: A Compound Reversionary Bonus on an Endowment

A hypothetical two-year endowment policy has a sum insured of $40,000 and $4,000 of previously declared reversionary bonuses. A new compound bonus is declared at 3% of the sum insured plus previously declared bonuses. The new addition is payable on death during the two-year term or on survival to maturity. The conditional probability of death in each year is $0.10$, and the effective annual interest rate is $5\%$. Find the reserve increase at declaration.

Step 1: Calculate the newly declared benefit. The bonus base is $40,000+$4,000=$44,000. Thus the new guaranteed addition is $0.03(44,000)=\$1,320$. The $1,320 is the incremental benefit to value; the existing sum insured and bonuses are not part of the reserve increase.

Step 2: Determine the event probabilities and payment times. Death in year 1 has probability $0.10$. Death in year 2 has probability $0.90(0.10)=0.09$. Survival to maturity has probability $0.90^2=0.81$. The year-2 death benefit and maturity benefit are both paid at the end of year 2, so their probabilities can be combined for discounting.

With $v=1/1.05$, the reserve increase is

\begin{aligned} \Delta V &=1320\left(0.10v+(0.09+0.81)v^2\right)\\ &=1320\left(\frac{0.10}{1.05}+\frac{0.90}{1.05^2}\right)\\ &=1320(0.0952381+0.8163265)\\ &=1320(0.9115646)\\ &=\$1,203.27. \end{aligned}

The reserve increases by $1,203.27, rounded to the nearest cent. The event probabilities $0.10$, $0.09$ and $0.81$ sum to 1.00, as required: exactly one of death in year 1, death in year 2, or survival to maturity occurs.

Common mistake avoided: A compound bonus rate applies to the specified bonus base, including the previously declared bonuses in this example. Applying 3% only to the $40,000 sum insured would give an addition of $1,200 instead of $1,320 and understate the reserve increase.

Worked Example: Before-and-After Reserves With an Unchanged Premium

At a valuation date, a hypothetical policy has one year remaining. At the end of that year it pays $60,000 on death or survival to maturity. If the policyholder survives to the end of the year, a $3,000 premium is also payable then; no premium is payable on death. A declared bonus adds $2,000 to the benefit payable on either event. The probability of death during the year is $0.10$, and the effective annual interest rate is $4\%$. Calculate the prospective reserve before and after declaration and verify the reserve increase.

Before declaration. A claim of $60,000 is paid at year-end with probability 1, because either death or survival produces a benefit. The premium is paid only on survival, with probability $0.90$. With $v=1/1.04$,

\begin{aligned} V^{-} &=60{,}000v-0.90(3{,}000)v\\ &=(60{,}000-2{,}700)\frac{1}{1.04}\\ &=\frac{57{,}300}{1.04}\\ &=\$55{,}096.15. \end{aligned}

After declaration. The benefit payable at year-end is now $62,000 on either event. The premium amount and its survival condition are unchanged.

\begin{aligned} V^{+} &=62{,}000v-0.90(3{,}000)v\\ &=(62{,}000-2{,}700)\frac{1}{1.04}\\ &=\frac{59{,}300}{1.04}\\ &=\$57{,}019.23. \end{aligned}

Therefore, the reserve change is $V^{+}-V^{-}=\$57{,}019.23-\$55{,}096.15=\$1,923.08$. Directly valuing the added benefit gives $2,000v=2,000/1.04=\$1,923.08$, which checks the result. Both reserves and the difference are rounded to the nearest cent.

Common mistake avoided: The bonus addition is payable whether the insured dies or survives, so its probability of payment is 1. Using the death probability $0.10$ alone would value only a death-contingent addition, not the stated death-or-maturity benefit.

A General Calculation Procedure

The reserve change is easiest to calculate reliably when the bonus amount, payment condition and timing are kept distinct. The following sequence applies whether the bonus is simple or compound: first determine the declared addition from the policy’s bonus rule, then value that addition according to the contract’s payment conditions.

Fix the valuation date and basis.
Record the interest and mortality assumptions and identify which cash flows are already past. Use the same basis before and after declaration.

Calculate the declared increment.
Apply the stated bonus rule to its stated base. Do not include previously declared amounts twice, and do not include possible future undeclared bonuses in this increment.

List each payment condition and time.
Separate death payments by year where needed, and distinguish death benefits from maturity benefits when their amounts or timing differ.

Calculate event probabilities and discount.
Use unconditional probabilities from the valuation date. Discount each incremental payment from its actual payment time.

Check the reserve difference.
Where practical, calculate both prospective reserves and subtract them. The result should match the present value of the incremental benefit if all other cash flows are unchanged.

For a death-only addition $b$ paid at the end of the year of death, with annual conditional death probabilities $q_k$, the probability of death in year $k$ is the probability of surviving the preceding $k-1$ years multiplied by $q_k$. If $i$ is an effective annual rate, the reserve contribution is therefore

b\sum_{k=1}^{n} {}_{k-1}p_x q_{x+k-1}v^k.

This expression assumes annual periods, death benefits paid at the end of the year of death, and no payment of this addition on survival to the end of the term. If a bonus is also payable at maturity, include a separate survival-to-maturity payment, discounted to maturity. If payment timing is different, discount using the actual timing rather than this year-end formula.

Interpretation and Exam Traps

A declaration can increase the guaranteed benefit by a fixed dollar amount while increasing the reserve by a smaller amount, because the reserve is a present value and a death-only benefit may not be paid if the insured survives. Conversely, if an addition is payable at maturity on survival as well as on death, the event probabilities across all possible claims sum to 1, but payments at different times still have different present values.

Do not use a conditional probability as an unconditional probability. A year-2 death probability must include survival through year 1.
Do not confuse the bonus amount with its reserve value. The reserve contribution reflects the probability and timing of payment as well as discounting.
Do not omit a maturity payment. Check whether the declared bonus attaches to death benefits, maturity benefits, or both.
Do not change the valuation basis midway. The before-and-after comparison isolates the declaration only when the basis and other terms are held fixed.
Do not deduct a premium twice. If future premiums are unchanged, they cancel in the difference between reserves; they still belong in each reserve considered separately.

The reserve increase after a declaration is a useful control on a participating-policy valuation: the before-and-after reserves should differ by the value of the newly guaranteed cash flows, not by an unrelated change in assumptions or projected future distributions. This treatment concerns the bonus already declared; the valuation of possible future discretionary benefits is a separate question.

Check Your Understanding

For each problem, calculate the reserve increase using the stated valuation basis and payment conditions.

A two-year term policy receives a $500 addition payable at the end of the year of death only. Conditional death probability is $0.10$ in each year, and $i=5\%$. Find the reserve increase. (Answer: $88.44)
A three-year endowment receives an $800 addition payable on death during the term or on survival to maturity. Conditional annual death probability is $0.20$, and $i=0\%$. Find the reserve increase. (Answer: $800.00)
A one-year term policy receives a $1,500 addition payable at year-end only if death occurs during the year. The death probability is $0.04$, and $i=2\%$. Find the reserve increase. (Answer: $58.82)
A two-year endowment has a $25,000 sum insured and $5,000 of previously declared bonuses. A new simple bonus is 2% of the sum insured only. The addition is payable on death or maturity. Conditional annual death probability is $0.10$, and $i=4\%$. Find the reserve increase. (Answer: $463.76)
A policy with one year remaining pays its benefit at year-end on death or survival. A declared bonus adds $1,000 to either payment, and no premium or other cash flow changes. The death probability is $0.20$, and $i=5\%$. Find the reserve increase. (Answer: $952.38)

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