Reversionary Bonuses and Guaranteed Liability Growth

Why a Reversionary Bonus Changes the Guarantee

A reversionary bonus is a declared addition to a participating policy benefit. Once declared and vested under the product terms, it is generally guaranteed: it increases the benefit payable on specified future events, such as death or maturity. The declaration therefore does more than allocate surplus in the current year. It can increase future expected claims, the reserve held for those claims, and the funds needed to meet a chosen solvency measure.

The key calculation is to keep three quantities distinct: the original sum assured, the benefit already guaranteed after declarations, and the reserve needed at a particular valuation date for future guaranteed payments. The reserve change is not necessarily equal to the bonus amount. A bonus may be payable years later, and the reserve reflects the probabilities and timing of the covered payments, together with the valuation interest rate and any future premiums.

Definition: Let $S$ be the original sum assured and $B_t$ the guaranteed benefit in force immediately after declarations at time $t$. Under a simple reversionary bonus rate $b_t$ applied to the original sum assured, the addition declared at time $t$ is $b_t S$, so $B_t=B_{t-1}+b_tS$. Under a compound bonus rate $c_t$ applied to the current guaranteed benefit, $B_t=B_{t-1}(1+c_t)$. These are different bonus bases. A declared addition is included in $B_t$ only from the time it vests under the stated product terms.

In a real contract, the declaration date, benefit basis, and covered events must be read from the policy terms. The calculations below use explicitly stated hypothetical terms. They are not intended to describe a particular insurer or regulatory regime.

Worked Example: Projecting Simple and Compound Bonuses

A hypothetical participating policy has an original sum assured of $120,000. Three annual bonuses are declared, each at 2% of the original sum assured. Project the guaranteed benefit after each declaration. Then compare it with three annual compound bonuses of 2% of the current guaranteed benefit.

Method. For simple bonuses, each addition is $0.02(120{,}000)=\$2{,}400$. Starting from $B_0=\$120{,}000$, the sequence is:

After declaration 1: $B_1=120{,}000+2{,}400=\$122{,}400$.
After declaration 2: $B_2=122{,}400+2{,}400=\$124{,}800$.
After declaration 3: $B_3=124{,}800+2{,}400=\$127{,}200$.

For compound bonuses, each declaration multiplies the current benefit by $1.02$:

$B_1=120{,}000(1.02)=\$122{,}400$.
$B_2=122{,}400(1.02)=\$124{,}848$.
$B_3=124{,}848(1.02)=\$127{,}344.96$.

The simple-bonus guarantee is $127,200; the compound-bonus guarantee is $127,344.96. The compound result is larger because later bonuses apply to earlier vested additions as well as to the original sum assured. The common mistake avoided here is treating a rate on the original sum assured as though it were a rate on the current guaranteed benefit.

Reserving for a Vested Addition

For a prospective reserve, specify the valuation date, the benefits payable after that date, the mortality assumption, the interest rate, and any future premiums. In the next calculation, the policy is paid-up, so there are no future premiums. A benefit is paid at the end of the year of death, or at the end of year 2 if the policyholder survives to maturity. There are no further bonuses. At the valuation date, the same guaranteed amount is payable on death during either year and on survival to maturity.

Model: At time 0, let the probability of death in each of the next two years, conditional on being alive at the start of that year, be $q=0.10$; thus $p=1-q=0.90$. Let the effective annual valuation interest rate be $i=0.04$, with $v=1/(1+i)$. For a fixed benefit $B$, the reserve is the expected present value of the benefit: a payment at the end of year 1 if death occurs in year 1, or a payment at the end of year 2 if death occurs in year 2 or the policyholder survives to maturity.

The present-value factor for this model is:

$$ qv+pqv^2+p^2v^2=qv+p(q+p)v^2=qv+pv^2. $$

The first term covers death in year 1. In the second and third terms, the policyholder survives year 1 and receives a payment at the end of year 2, whether by death in year 2 or by survival to maturity. Since $q+p=1$, those two year-2 outcomes combine to probability $p$.

Worked Example: The Reserve Increase from a Declared Addition

Using the model above, suppose the guaranteed benefit is $100,000 before a declaration. A $3,000 reversionary addition vests immediately, increasing the guaranteed benefit to $103,000. Calculate the reserves before and after declaration and the reserve increase.

Method. The benefit is payable at the end of year 1 on a first-year death, and at the end of year 2 on any outcome following survival through year 1. Therefore, the present-value factor is $qv+pv^2$, not simply $v^2$.

First, $v=1/1.04=0.96153846$ and $v^2=0.92455621$. The factor is:

$0.10(0.96153846)+0.90(0.92455621)=0.09615385+0.83210059=0.92825444$.

Before the declaration, the reserve is $100{,}000(0.92825444)=\$92{,}825.44$. After the declaration, it is $103{,}000(0.92825444)=\$95{,}610.21$. The increase is $95{,}610.21-92{,}825.44=\$2{,}784.77$, with amounts rounded to the nearest cent.

Check. Applying the same factor directly to the additional $3,000 gives $3{,}000(0.92825444)=\$2{,}784.76$. The one-cent difference from subtracting the rounded reserves is due to rounding each reserve separately; using unrounded values gives a reserve increase of $2,784.76. The common mistake avoided is discounting the whole guarantee for two years, which would omit the possibility of a year-1 death claim.

Why the Reserve Change Has This Form

In the two-year model, the reserve is linear in the guaranteed benefit: every possible claim amount is $B$, multiplied by the present-value weight for its payment time and probability. This explains why the reserve effect of a vested addition can be calculated by valuing the addition itself, provided the addition has the same payment conditions as the existing benefit.

List mutually exclusive outcomes.
Death in year 1 has probability $q$ and payment time 1. Survival through year 1 followed by death in year 2 has probability $pq$ and payment time 2. Survival to maturity has probability $p^2$ and payment time 2.

Discount each outcome.
For a fixed benefit $B$, its expected present value is $B(qv+pqv^2+p^2v^2)$.

Combine the year-2 outcomes.
Since $pq+p^2=p(q+p)=p$, the reserve is $B(qv+pv^2)$.

Value the vested addition.
If a bonus $C$ has the same payment conditions, the change in reserve is $C(qv+pv^2)$, because replacing $B$ by $B+C$ adds exactly $C$ to each possible payment.

This justification depends on the stated model: the addition is guaranteed for the same events and paid at the same times as the existing benefit, and there are no future premiums or other benefit differences. If a bonus applies only to maturity, has a different death-benefit treatment, or changes a premium, the cash flows must be valued separately. Likewise, mortality rates that vary by year require the year-specific probabilities rather than a constant $q$.

From Liability Growth to Profit and Capital

A bonus declaration can reduce current distributable surplus because the insurer has made a larger future guarantee. In a simplified balance-sheet view, if assets are unchanged at the declaration date, the change in surplus is the negative of the change in the measured liability. This is a timing statement, not a claim that every declared bonus creates an equal ultimate economic loss: future investment income, premiums, expenses, experience, and valuation bases can also affect profit.

Capital is not a single universal formula. To make the effect measurable, a hypothetical case can define a transparent capital proxy, such as the present value of guaranteed benefits under a stated stress. The stressed liability is then compared before and after the bonus. This illustrates how a larger guarantee can increase required backing; it should not be confused with a particular jurisdiction's formal capital rules.

Worked Example: Bonus Declaration, Surplus, and Stressed Liability

A hypothetical paid-up policy has the same two-year benefit structure as before, a valuation interest rate of 4%, and base annual mortality probabilities of $q=0.10$ and $p=0.90$. Assets allocated to the policy are $98,000. A $3,000 vested addition increases the guarantee from $100,000 to $103,000. For a simple capital proxy, value the guaranteed benefit under stressed mortality $q_s=0.15$, $p_s=0.85$, retaining the 4% interest rate. Define the proxy as this stressed present value; do not interpret it as a prescribed regulatory capital formula.

Base reserve and surplus before declaration. The base factor is $0.92825444$, as calculated above. The reserve is $100{,}000(0.92825444)=\$92{,}825.44$, so assets less this reserve are $98{,}000-92{,}825.44=\$5{,}174.56$.

Base reserve and surplus after declaration. The reserve is $103{,}000(0.92825444)=\$95{,}610.21$. With assets unchanged, assets less reserve are $98{,}000-95{,}610.21=\$2{,}389.79$. The immediate surplus reduction is approximately $2,784.76, equal to the reserve increase calculated using unrounded values.

Stressed liability before declaration. Under stress, the factor is $q_sv+p_sv^2$. Using $v=0.96153846$ and $v^2=0.92455621$, it is $0.15(0.96153846)+0.85(0.92455621)=0.93010355$. The stressed liability is $100{,}000(0.93010355)=\$93{,}010.36$.

Stressed liability after declaration. It is $103{,}000(0.93010355)=\$95{,}800.67$. The defined capital proxy therefore increases by $95{,}800.67-93{,}010.36=\$2{,}790.31$, using unrounded values before rounding to cents. If one instead reports assets less stressed liability as a solvency-surplus measure, that amount falls from $98{,}000-93{,}010.36=\$4{,}989.64$ to $98{,}000-95{,}800.67=\$2{,}199.33$. The common mistake avoided is to use the base reserve increase as if it were automatically the stressed capital increase: the stress calculation has its own probability weights.

Exam Traps and Practical Checks

When projecting a reversionary bonus, a reliable calculation starts with the declaration mechanics and ends with a cash-flow check. In particular:

Use the stated bonus base. A percentage of original sum assured gives a fixed addition each year under a simple-bonus model. A percentage of the current guaranteed benefit compounds.
Apply the correct vesting date. A bonus declared at an anniversary may affect benefits only from that point onward. Do not include it in an earlier death claim unless the contract says it applies.
Value the benefit events actually covered. Death benefits and maturity benefits can differ. A single reserve factor is valid only when the payment amount and conditions support it.
Keep valuation bases separate. Base reserves and stressed liabilities can use different mortality assumptions. State the interest and mortality assumptions for each calculation.
Distinguish liability from capital. A reserve increase is a liability change. A capital result requires a stated capital definition or stress measure.

A useful final check is to ask whether the vested addition appears in every future payment that the policy terms require, and nowhere before it vests. Then verify that probabilities across the possible outcomes sum to 1 and that the liability change has the same sign as the added guarantee.

Key takeaway: A declared reversionary bonus converts allocated surplus into a larger guaranteed benefit. Project the benefit using the correct bonus base, then value the resulting future payments under an explicitly stated reserve or stress model. The reserve and capital effects need not equal the face amount of the bonus.

Check Your Understanding

For each problem, state the relevant model assumptions and check whether the bonus changes the payment amount, payment time, or both.

A policy has original sum assured $75,000 and four simple annual bonuses of 1.6% of original sum assured. What is the guaranteed benefit after the fourth declaration? (Answer: $79,800.)
A paid-up policy has one year remaining, with a guaranteed $64,000 payable at year-end on death or survival. The effective annual interest rate is 5%. What is the reserve immediately after a declaration that does not change the benefit amount? (Answer: $60,952.38.)
A paid-up policy has two years remaining. The benefit is paid at the end of the year of death or at maturity at the end of year 2. Each year's conditional death probability is 0.20, the effective annual interest rate is 5%, and a vested addition is $2,500. What is the reserve increase? (Answer: $2,290.25.)
An original sum assured of $90,000 receives two annual bonuses of 3%. Calculate the final guarantee under (a) simple bonuses on original sum assured and (b) compound bonuses on the current guarantee. (Answer: (a) $95,400; (b) $95,481.)
In the two-year model with $q=0.10$, $p=0.90$, and $i=0.04$, a vested addition of $5,000 has the same death and maturity payment conditions as the existing benefit. What is its reserve increase? (Answer: $4,641.27.)

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