Consistency Between Bonus Projections and Valuation

Why Consistency Matters

A participating policy valuation is only meaningful if its projected bonus cash flows describe the same benefits as the contract being valued. A projection might use the correct bonus scale but include policyholders who are not eligible, place a payment on the wrong date, or discount it at a rate different from the valuation basis. Each mismatch changes the value, even if the arithmetic is otherwise correct.

The key is to check four items together: the amount of each bonus, who qualifies to receive it, when it is paid, and the assumptions used to discount it. This is a consistency check on the projected cash flows, not a new bonus scale or a new allocation rule.

Definition: A bonus projection and valuation are consistent when they use the same contractual bonus amount and benefit base, the same eligibility conditions, the same payment event and date, and the same valuation assumptions for discounting and any probabilities of payment.

A Cash-Flow Test for Consistency

For each possible bonus payment, identify its amount $B_k$, its payment time $t_k$, and an indicator $I_k$ that equals 1 if the policy satisfies the eligibility rule and 0 otherwise. If the annual effective valuation interest rate is $i$, then $v=1/(1+i)$. The expected present value of the payment per policy at the valuation date is

\operatorname{EPV}_k=E[I_k B_k v^{t_k}].

If the amount and payment time are fixed whenever the policy is eligible, this simplifies to

\operatorname{EPV}_k=P(I_k=1)B_kv^{t_k}.

This simplified expression requires the eligibility probability to refer to the exact event specified by the contract. For example, being in force at an anniversary is not always the same as surviving to a later claim date. If a projected amount is already an expected cash flow per policy issued, its eligibility probability has already been included and must not be applied a second time.

The formula follows directly from conditioning on eligibility. When the policy is ineligible, the payment is zero; when eligible, the discounted payment is $B_kv^{t_k}$. Therefore,

E[I_k B_k v^{t_k}] =P(I_k=1)B_kv^{t_k}+P(I_k=0)\cdot 0 =P(I_k=1)B_kv^{t_k}.

For a sequence of payments, add the expected present values of the individual payments. This addition does not require the eligibility events to be independent. The events and amounts must, however, be specified consistently for each payment.

Write the contractual payment.
State the bonus amount, the base on which it is calculated, and the event that makes it payable.

Identify eligible policies.
Determine whether eligibility depends on being in force at declaration, at payment, at claim, or at maturity.

Place the payment on the timeline.
Use the actual payment date, not automatically the declaration date or the end of the projection interval.

Apply probabilities and discounting once.
Check whether a projected amount is conditional or already expected per policy issued, then discount from the payment date using the stated valuation rate.

Worked Applications

The following hypothetical contracts use annual periods and a discrete mortality model: a death in year $k$ is treated as occurring at the end of that year. For each year, $q_k$ is the probability of death during that year conditional on being in force at its start, and $p_k=1-q_k$. These conventions are part of each example; a different payment timing or mortality model would require corresponding changes.

Worked Example: A Reversionary Addition Payable on Later Claims

A hypothetical two-year endowment has a guaranteed benefit of $10,000, payable at the end of the year of death or at maturity. At the end of year 1, a $200 reversionary addition is declared for policies then in force. It is added to the benefit payable on a subsequent claim or maturity, but does not apply to a death in year 1. Assume $q_1=0.10$, $q_2=0.20$, and an annual effective valuation interest rate of 5%. Calculate the present value at issue of the guaranteed benefit and of the addition.

Method. Separate the guaranteed benefit from the addition, and value each using its own eligibility rule and payment time. The addition only applies to policies that are in force at the end of year 1. A death in year 2 and maturity are both paid at time 2 under the stated convention.

The discount factor is $v=1/1.05=0.95238095$, so $v^2=0.90702948$. The probabilities of the three contract outcomes are:

P(\text{death in year 1})=0.10,\quad P(\text{death in year 2})=0.90(0.20)=0.18,\quad P(\text{maturity})=0.90(0.80)=0.72.

These probabilities sum to $0.10+0.18+0.72=1.00$. The guaranteed benefit is paid at time 1 for a year-1 death and at time 2 for the other outcomes. Its present value is

10{,}000[0.10v+0.90v^2] =10{,}000[0.10(0.95238095)+0.90(0.90702948)] =10{,}000(0.91156463) =\$9{,}115.65.

The addition is not payable for the year-1 death. The probability it is payable at time 2 is $0.18+0.72=0.90$, the probability the policy is in force when it is declared. Thus its present value is

200(0.90)v^2 =180(0.90702948) =\$163.27.

The total present value of these benefits is $\$9{,}115.65+\$163.27=\$9{,}278.92$. As a check on eligibility, the addition's expected undiscounted payment is $200(0.90)=\$180$, and discounting that payment for two years gives $\$163.27$.

Common mistake avoided: Adding the $200 to the year-1 death benefit would incorrectly make it payable to a policy that ended before the declaration. That error adds $200(0.10)v=\$19.05$ to the present value.

Worked Example: Cash Dividends Paid Only to Policies in Force

A hypothetical participating contract pays a cash dividend of $100 at the end of year 1 if the policy is then in force, and $150 at the end of year 2 if it is then in force. Assume $q_1=0.05$, $q_2=0.10$, and an annual effective valuation interest rate of 4%. Find the expected present value per policy issued.

Method. Value each payment separately. The year-1 payment requires survival through year 1; the year-2 payment requires survival through both years. The conditional mortality assumptions give

P(\text{in force at }1)=0.95,\qquad P(\text{in force at }2)=0.95(0.90)=0.855.

With $v=1/1.04=0.96153846$, the year-1 dividend has present value

100(0.95)v=\frac{95}{1.04}=\$91.35.

The year-2 dividend has present value

150(0.855)v^2 =\frac{128.25}{1.0816} =\$118.57.

Therefore the expected present value per policy issued is $\$91.35+\$118.57=\$209.92$. The expected undiscounted payments are $95 at time 1 and $128.25 at time 2, which also confirms that the in-force probabilities have been applied to the correct amounts.

Common mistake avoided: Discounting the full $100 and $150 without applying the in-force probabilities gives $100v+150v^2=\$234.84$. That values payments to policies that have already ended.

Worked Example: Expected Bonus Projection and Valuation Rate

A hypothetical two-year contract pays a terminal bonus of $400 at time 2 to each policy that is in force at the start of year 2 and whose claim or maturity payment occurs at the end of year 2. A policy that dies in year 1 receives no terminal bonus. Assume $q_1=0.10$, $q_2=0.20$, and an annual effective valuation interest rate of 4%. A projection reports the expected bonus payment per policy issued at time 2. Calculate that projected amount and its present value. Then compare it with the value using a 3% rate.

Method. First convert the conditional bonus amount into the expected payment per policy issued. Then discount that expected payment from time 2. The policy must be in force at the start of year 2; under the stated end-of-year claim timing, a death during year 2 still receives the bonus.

The probability of reaching the start of year 2 is $1-q_1=0.90$. Conditional on reaching that date, the policy exits in year 2 by death with probability $0.20$ or by maturity with probability $0.80$. Thus the total probability of a year-2 claim or maturity is $0.90(0.20)+0.90(0.80)=0.18+0.72=0.90$. The expected bonus payment at time 2 per policy issued is

400(0.90)=\$360.

Using the stated 4% valuation rate, the present value is

360v^2=\frac{360}{1.04^2} =\frac{360}{1.0816} =\$332.84.

Using 3% instead would give $360/1.03^2=360/1.0609=\$339.33$, which is $6.49 higher. The 3% calculation does not match the specified valuation basis, even though it discounts the same projected payment to the correct date.

Common mistake avoided: The $360 is already the expected payment per policy issued, so multiplying it again by 0.90 would count year-1 survival twice. The incorrect result would be $360(0.90)/1.0816=\$299.56$. Alternatively, if the projection records the conditional amount of $400 per eligible policy, the eligibility probability must be included once: $400(0.90)/1.0816=\$332.84$.

Common Consistency Checks

A useful valuation review compares the projection and valuation line by line. The amount in the projection must reflect the stated bonus base: for example, a percentage of the original sum assured is not interchangeable with the same percentage of a benefit that has already been increased by bonuses. Eligibility must also be tied to the correct event. A bonus declared only for policies in force at an anniversary should not automatically be assigned to policies that ended before that anniversary.

Payment timing is equally important. A bonus declared at one date may be payable immediately, at a later claim, or only at maturity. Discounting from the declaration date when the contract pays at a later claim overstates present value. Conversely, moving an immediate payment to a later date understates it. On a discrete annual model, a payment at the end of year $k$ is discounted by $v^k$; if the contractual payment occurs at another time, the discount period must reflect that time.

Finally, distinguish an amount conditional on eligibility from an expected amount per policy issued. The first still needs an eligibility probability; the second already incorporates it. A clear projection should label which kind of amount it presents. When projected cash flows are provided as expected amounts at future dates, use those amounts directly in the present-value calculation rather than reapplying the probability assumptions.

Key takeaway: For every projected bonus, reconcile the contractual amount and benefit base, the eligibility event, the payment date, and the valuation assumptions. A mismatch in any one of these can produce a precise calculation of the wrong cash flow.

Check Your Understanding

For each hypothetical contract, identify eligibility and timing before calculating. Amounts are in dollars, deaths occur at the end of each year, and the stated annual effective rate is the valuation rate.

A cash dividend of $80 is paid at the end of year 1 only if the policy is in force then. If $q_1=0.15$ and $i=5\%$, find its expected present value per policy issued. (Answer: $64.76)
A $300 reversionary addition is declared at the end of year 1 for policies then in force. It is payable on a death in year 2 or at maturity at time 2, but not on a year-1 death. If $q_1=0.10$, $q_2=0.25$, and $i=4\%$, find its present value at issue. (Answer: $249.63)
A contract pays $120 at time 1 and $180 at time 2, each only if the policy is in force at that payment date. If $q_1=0.20$, $q_2=0.10$, and $i=3\%$, find the combined expected present value per policy issued. (Answer: $215.36)
A $500 terminal bonus is paid at time 2 only to policies that survive to time 2; a death during year 2 is not eligible. If $q_1=0.10$, $q_2=0.20$, and $i=4\%$, find its present value at issue. (Answer: $332.84)

What You'll Learn