Introduction
A declared reversionary bonus changes the guaranteed benefit of a participating policy. Once attached under the contract, the addition is generally payable on a future insured event, such as death or maturity. The valuation task is therefore to identify exactly which additions are already guaranteed, determine which claims are entitled to them, and discount the resulting benefits according to their possible payment dates.
This tutorial values the guaranteed benefits created by declared reversionary bonuses. It does not assume that future discretionary bonuses will be declared. A future illustration or an expected bonus rate is not a guaranteed addition unless the contract has already made it guaranteed.
Establish the Guaranteed Amount Before Valuing It
Let \(S\) be the basic sum assured and let \(b_j\) be the bonus addition declared at the \(j\)th eligible declaration date. If all those additions are vested at the valuation date, the guaranteed amount is
The bonus ledger must follow the contract. For a simple bonus rate \(r_j\) per unit of the original sum assured, the addition may be \(b_j=S r_j\), with the rate expressed as a proportion. For a compound bonus rate \(c_j\) applied to the then-current guaranteed amount, the calculation is \(G_j=G_{j-1}(1+c_j)\), where \(G_0=S\). The declarations are historical facts at the valuation date: do not project a discretionary future declaration as though it were already vested.
After establishing \(G\), identify the benefit payable on each possible claim outcome. A contract may pay the same vested amount on death and maturity, but the valuation should not assume that without checking the terms. The death benefit may depend on the date the policy became entitled to an addition, and a maturity benefit may be subject to a separate eligibility condition. Record the payable amounts as \(B_k^D\) for death in policy year \(k\) and \(B_n^M\) for survival to maturity at the end of year \(n\).
Present Value of the Guaranteed Claims
For death during year \(k\), the policyholder must first survive \(k-1\) years and then die during the next year. The probability of this outcome is \({}_{k-1}p_x q_{x+k-1}\). If the death benefit is paid at the year-end, its discount factor is \(v^k\). If the policyholder instead survives to maturity, the probability is \({}_np_x\), and the maturity payment is discounted by \(v^n\). Thus the expected present value is
This is an outcome-by-outcome valuation. It is valid when the stated annual mortality probabilities describe the policyholder’s future lifetime under the valuation model and the benefit amounts correctly reflect the contract. If every claim outcome pays the same vested amount \(G\), substitute \(B_k^D=B_n^M=G\). If the contract produces different death and maturity benefits, retain the separate amounts rather than forcing them into one \(G\).
Start with the basic sum assured and add only declarations to which the policy is entitled and that are guaranteed at the valuation date.
For each death or maturity outcome, determine which declared additions are payable under the contract and record that benefit amount.
Multiply each outcome’s benefit by its probability and the discount factor for its payment time, then sum the present values.
Why the Valuation Formula Holds
The formula follows by partitioning the possible outcomes over the remaining term. There can be death in exactly one of years \(1,\ldots,n\), or survival to the maturity date. These outcomes are mutually exclusive and exhaustive under the annual model. For death in year \(k\), survival through year \(k-1\) has probability \({}_{k-1}p_x\), and conditional death in year \(k\) has probability \(q_{x+k-1}\). Their product is the probability of that claim outcome. Its present value is \(v^kB_k^D\), so its contribution to expected present value is \(v^k{}_{k-1}p_xq_{x+k-1}B_k^D\). The maturity outcome contributes \(v^n{}_np_xB_n^M\). Adding the contributions of all disjoint outcomes gives the stated expression.
As a check on the probability model, the death probabilities and maturity survival probability must sum to one:
This probability check does not say the present-value factors sum to one: discounting makes earlier and later payments have different values.
Worked Example: Simple Declared Additions on a Three-Year Endowment
A hypothetical policy has a basic sum assured of $80,000. Four simple reversionary bonuses have already been declared, each at 1.5% of the original sum assured. The policy has three years remaining. Assume a 10% effective annual interest rate and a conditional probability of death of 0.10 in each remaining year, independent of prior survival. The declared additions are payable on death or maturity, and claims are paid at the end of the relevant year. Find the expected present value of the guaranteed benefit.
Method: establish the vested amount, then value each mutually exclusive claim outcome. Each declared addition is \(0.015(80{,}000)=\$1{,}200\). The total declared additions are \(4(1{,}200)=\$4{,}800\), so the guaranteed benefit is \(G=80{,}000+4{,}800=\$84{,}800\).
The probabilities of death in years 1, 2, and 3 are respectively \(0.10\), \(0.90(0.10)=0.09\), and \(0.90^2(0.10)=0.081\). The probability of surviving to maturity is \(0.90^3=0.729\). These sum to \(0.10+0.09+0.081+0.729=1.000\).
With \(v=1/1.10\), the four present-value contributions are:
- Death in year 1: \(\$84{,}800(0.10)/1.10=\$7{,}709.09\).
- Death in year 2: \(\$84{,}800(0.09)/1.10^2=\$6{,}307.44\).
- Death in year 3: \(\$84{,}800(0.081)/1.10^3=\$5{,}160.63\).
- Survival to maturity: \(\$84{,}800(0.729)/1.10^3=\$46{,}445.68\).
Adding the contributions gives an expected present value of \(\$7{,}709.09+\$6{,}307.44+\$5{,}160.63+\$46{,}445.68=\$65{,}622.84\). The calculation is checked by the outcome probabilities summing to one and by each payment being discounted for its actual year-end payment time. The common mistake avoided here is adding the $4,800 of declarations once to \(G\) and then adding it again as a separate benefit.
Compound Additions and Valuation at a Later Duration
The same valuation method applies when bonuses were declared using compound rates. Compound rates change the amount in the bonus ledger, not the probability or discounting method. At a later valuation date, use the policy’s accumulated guaranteed amount as it stands on that date; do not restart the bonus calculation from the original sum assured.
Worked Example: Compound Bonuses Already Vested at Valuation
A hypothetical policy began with a $50,000 sum assured. At each of three past anniversaries, a compound reversionary bonus of 4% of the current guaranteed amount was declared and vested. At the valuation date, two years remain to maturity. Assume an effective annual interest rate of 10%, a conditional annual death probability of 0.10 in each remaining year, and payment of the same guaranteed amount on death or maturity. Find the expected present value at the valuation date.
Method: roll forward the declared additions, then apply the remaining-term outcome probabilities. The amounts after each declaration are \(G_1=50{,}000(1.04)=\$52{,}000\), \(G_2=52{,}000(1.04)=\$54{,}080\), and \(G_3=54{,}080(1.04)=\$56{,}243.20\). Thus the guaranteed claim amount at the valuation date is $56,243.20.
For the two remaining years, the death probabilities are \(0.10\) and \(0.90(0.10)=0.09\), while the maturity probability is \(0.90^2=0.81\). At \(i=0.10\), the present-value factor for these outcomes is
The expected present value is therefore \(\$56{,}243.20(101/121)=\$46{,}946.80\), rounded to the nearest cent. As a check, the outcome probabilities \(0.10+0.09+0.81\) equal 1.00. The common mistake avoided is applying the three historical bonus rates to the original $50,000 as simple additions; the declared rates were compound rates, so each was applied to the then-current guaranteed amount.
Entitlement Rules and Undeclared Bonuses
A declaration date and an entitlement date are not automatically the same thing. The contract might require the policy to be in force on a specified anniversary for a bonus to be allocated. It might also say that an already declared addition remains payable if the policyholder later dies or surrenders, or set different conditions for death and maturity claims. The valuation must follow those provisions rather than infer entitlement from a bonus illustration.
A useful distinction is:
| Item at the valuation date | Treatment in a guaranteed-benefit valuation |
|---|---|
| Declared and vested addition, with the policy meeting the contract conditions | Include it in the applicable guaranteed benefit. |
| Illustrated or expected future bonus not declared | Exclude it from the guaranteed benefit. |
| Declared addition subject to an unmet eligibility condition | Apply the contract condition; do not assume it is vested. |
| Addition already included in the stated guaranteed sum assured | Count it once, not again as a separate payment. |
Worked Example: Excluding a Bonus That Has Not Been Declared
A hypothetical policy has a basic sum assured of $72,000 and two declared, vested additions of $1,500 each. The policy has two years remaining. An illustration shows a possible further $1,500 addition at the next anniversary, but that bonus has not been declared and is not contractually guaranteed. Assume the two vested additions are payable on death or maturity, the annual effective interest rate is 5%, and the conditional death probabilities are 0.04 in year 1 and 0.06 in year 2. Claims are paid at year-end. Find the expected present value of the guaranteed benefits.
Method: apply the entitlement rule before valuing the outcomes. The guaranteed amount is \(72{,}000+2(1{,}500)=\$75{,}000\). The illustrated $1,500 is excluded because it is undeclared and not guaranteed. The death probabilities are \(0.04\) in year 1 and \(0.96(0.06)=0.0576\) in year 2; the maturity probability is \(0.96(0.94)=0.9024\). Their sum is \(0.04+0.0576+0.9024=1.0000\).
The year-1 death contribution is \(\$75{,}000(0.04)/1.05=\$2{,}857.14\). The year-2 death contribution is \(\$75{,}000(0.0576)/1.05^2=\$3{,}918.37\). The maturity contribution is \(\$75{,}000(0.9024)/1.05^2=\$61{,}387.76\). Their sum is \(\$68{,}163.27\), rounded to the nearest cent. The common mistake avoided is treating an illustrated future bonus as a guaranteed addition simply because it appears in a projection.
Common Exam Traps
- Counting a bonus twice. If a guaranteed sum assured already includes declared additions, do not add the bonus ledger to that amount again.
- Valuing an illustration as a declaration. A possible future bonus is not guaranteed merely because a projection displays it. Include only amounts guaranteed under the valuation assumptions and contract.
- Using the wrong entitlement date. A bonus declared at an anniversary may require the policy to be in force on that date. Check whether the claim occurs before or after the contract’s qualifying event.
- Discounting every claim for the full term. A death benefit paid at the end of year \(k\) is discounted by \(v^k\), not \(v^n\). Maturity is discounted by \(v^n\).
- Ignoring the death-versus-maturity distinction. Use separate benefit amounts if the contract provides different entitlements for death and maturity.
Check Your Understanding
For each problem, use the stated annual model and treat death claims as payable at the end of the year of death.
- A policy has a $40,000 sum assured and three declared simple additions of $500 each. What is the vested guaranteed amount? (Answer: $41,500)
- A vested guaranteed benefit of $60,000 is payable on death in either of the next two years or on maturity at the end of year 2. Assume \(i=10\%\) and annual conditional death probabilities of 0.10. Find the expected present value. (Answer: $50,082.64)
- A policy has a vested guaranteed benefit of $90,000, two years remaining, \(i=5\%\), and conditional death probabilities 0.02 in year 1 and 0.04 in year 2. The same amount is payable on death or maturity. Find the expected present value. (Answer: $81,714.29)
- A $50,000 basic sum assured has had two compound bonuses of 3% declared and vested, each applied to the current guaranteed amount. No other bonuses have been declared. What is the guaranteed amount now? (Answer: $53,045)
- A policy has an $80,000 basic sum assured and two declared, vested additions of $1,000 each. A further $1,000 appears in an illustration but has not been declared or guaranteed. What amount should be used as the guaranteed benefit, assuming all declared additions are payable on death and maturity? (Answer: $82,000; exclude the illustrated addition.)