Comparing Bonus Allocation Bases

Why the Allocation Base Matters

A participating fund may have a specified amount of surplus available for policyholder bonuses, but the allocation rule determines how that amount is divided among policies. A premium-based rule gives greater weight to policies with more eligible premiums; a reserve-based rule gives greater weight to policies with larger reserves; and a guaranteed-benefit-based rule gives greater weight to policies with larger guaranteed benefits. These rules can assign different bonus amounts to one policy even when the total surplus being distributed is identical.

To compare allocation bases fairly, hold the projected surplus, valuation assumptions, and bonus payment timing fixed. Then calculate each policy’s allocated bonus under each rule and, if needed, the expected present value of those payments. The comparison is about distribution: it does not imply that one allocation rule creates additional surplus.

Definition: For a year in which total distributable surplus is $S$, let $b_j$ be the eligible exposure for policy $j$ under a chosen allocation basis. The allocation rate is $r=S/\sum_j b_j$, and policy $j$'s bonus is $B_j=rb_j=S b_j/\sum_j b_j$. The basis $b_j$ may be eligible premium, a specified reserve, or a specified guaranteed benefit. The basis and its valuation date must be defined consistently for all policies.

The formula assumes that the entire specified surplus is allocated in proportion to nonnegative exposures and that there is no separate cap, minimum, or other adjustment. If a contract or projection specifies such a feature, the simple proportional formula does not apply without modification. A basis of zero for every policy also cannot support a proportional allocation; the allocation rule would need a separately specified fallback.

Calculating and Comparing the Allocation Rates

The denominator is the total exposure for the chosen basis across the eligible policies. It is not the exposure of the policy being valued. Once the allocation rate is calculated, multiplying it by a policy’s exposure gives that policy’s share. As a basic check, the policy shares must sum to the available surplus:

\sum_j B_j = \sum_j \frac{S b_j}{\sum_k b_k} = \frac{S}{\sum_k b_k}\sum_j b_j = S.

The rate has units that depend on the basis: dollars of bonus per dollar of eligible premium, reserve, or guaranteed benefit. Rates from different bases should not be compared as though they had the same denominator. The policy bonus amounts, and then their present values under common assumptions, provide the useful comparison.

Worked Example: Comparing Three Bases for One Year

A hypothetical participating portfolio has $12,000 of surplus to allocate at the end of year 1. Its aggregate eligible premiums are $90,000, its aggregate reserves are $150,000, and its aggregate guaranteed benefits are $250,000. A target policy has an eligible premium of $900, a reserve of $1,500, and a guaranteed benefit of $5,000. Each bonus is paid at the end of year 1 only if the policy is in force. The probability it remains in force to that date is 0.98, and the effective annual interest rate is 5%. Find the target policy’s bonus and expected present value under each basis.

Method. For each basis, divide the same $12,000 surplus by the portfolio’s total exposure, then multiply that rate by the target policy’s exposure. The expected present value is the bonus multiplied by the probability of being in force and discounted one year.

Premium basis. The allocation rate is $12{,}000/90{,}000=0.133333\ldots$ dollars of bonus per dollar of eligible premium. The target policy’s bonus is $12{,}000(900/90{,}000)=\$120$. Its expected present value is

\frac{120(0.98)}{1.05} = \frac{117.60}{1.05} = \$112.00.

Reserve basis. The rate is $12{,}000/150{,}000=0.08$. The target policy’s bonus is $0.08(1{,}500)=\$120$, and its expected present value is $120(0.98)/1.05=\$112.00$.

Guaranteed-benefit basis. The rate is $12{,}000/250{,}000=0.048$. The target policy’s bonus is $0.048(5{,}000)=\$240$, and its expected present value is $240(0.98)/1.05=\$224.00$.

The check is that each allocation rule distributes the same $12,000 in total: its rate multiplied by the corresponding aggregate exposure equals $12,000. The target’s present values differ because its share of the portfolio exposure differs by basis, not because the total surplus changed. Amounts are reported to the nearest cent.

Expected Present Value Over Multiple Years

When bonuses are projected in multiple years, apply the chosen allocation rule separately to each year’s surplus and exposure totals. If $B_t$ is the bonus due at the end of year $t$, ${}_tp$ is the probability the policy is in force at that time, and the effective annual interest rate is $i$, the expected present value at the start of year 1 is:

\operatorname{EPV} = \sum_{t=1}^{n} \frac{B_t\,{}_tp}{(1+i)^t}.

This expression assumes the stated bonus is paid at the end of year $t$ only if the policy is in force then, and that the annual effective interest rate is constant. The in-force probability is cumulative from the valuation date; do not multiply by it again if the bonus amount has already been defined as an unconditional expected payment. If a benefit is guaranteed to be paid even after the policy leaves the in-force state, the relevant payment condition and probability must be changed.

Justification: A payment $B_t$ made only when the policy is in force has expected amount $B_t\,{}_tp$, because the payment is $B_t$ with probability ${}_tp$ and zero otherwise. Discounting that expected payment $t$ years at effective annual rate $i$ gives $B_t\,{}_tp/(1+i)^t$. Adding the discounted expected payments across years gives the displayed expected present value. This uses linearity of expectation and does not require payments in different years to be independent.

Worked Example: Multi-Year Values Under Each Basis

A hypothetical insurer projects surplus of $12,000 at the end of year 1 and $15,000 at the end of year 2. For year 1, portfolio exposures are $90,000 of eligible premiums, $150,000 of reserves, and $250,000 of guaranteed benefits. The target policy exposures are respectively $900, $1,800, and $5,000. For year 2, the corresponding portfolio exposures are $150,000, $200,000, and $300,000, while the target exposures are $1,200, $2,400, and $6,000. Bonuses are payable only if the policy is in force at the end of the relevant year. The cumulative in-force probabilities are 0.96 at year 1 and 0.91 at year 2. The effective annual interest rate is 4%. Compare the target’s expected present value under the three bases.

Method. First calculate the target’s allocated bonus in each year using that year’s surplus and exposures. Then weight each payment by its in-force probability, discount it to the valuation date, and add the two amounts.

Premium basis. In year 1, the target receives $12{,}000(900/90{,}000)=\$120$. In year 2, it receives $15{,}000(1{,}200/150{,}000)=\$120$. Therefore:

\operatorname{EPV}_{\text{premium}} = \frac{120(0.96)}{1.04} + \frac{120(0.91)}{1.04^2} = 110.7692\ldots+100.9615\ldots = \$211.73.

Reserve basis. The year 1 bonus is $12{,}000(1{,}800/150{,}000)=\$144$, and the year 2 bonus is $15{,}000(2{,}400/200{,}000)=\$180$. Thus:

\operatorname{EPV}_{\text{reserve}} = \frac{144(0.96)}{1.04} + \frac{180(0.91)}{1.04^2} = 132.9231\ldots+151.4423\ldots = \$284.37.

Guaranteed-benefit basis. The year 1 bonus is $12{,}000(5{,}000/250{,}000)=\$240$, and the year 2 bonus is $15{,}000(6{,}000/300{,}000)=\$300$. Therefore:

\operatorname{EPV}_{\text{benefit}} = \frac{240(0.96)}{1.04} + \frac{300(0.91)}{1.04^2} = 221.5385\ldots+252.4038\ldots = \$473.94.

As a check, the year 1 allocation rates multiplied by their aggregate exposures each give $12,000; the year 2 rates multiplied by their aggregate exposures each give $15,000. The target policy has the largest value under the guaranteed-benefit basis because its share of that basis is larger in both years. Results are rounded only at the final step to cents.

Portfolio Shares and the Distributional Effect

A basis comparison is also useful for seeing which policies gain or lose relative weight when an allocation rule changes. The following example tracks one policy class against the remainder of a hypothetical portfolio. It holds the available surplus fixed and checks both the target’s share and the total allocated amount.

Worked Example: Checking a Policy’s Share of the Allocation

A hypothetical portfolio has $18,000 of surplus to allocate. Aggregate eligible premiums are $120,000, aggregate reserves are $300,000, and aggregate guaranteed benefits are $600,000. A target policy has eligible premiums of $2,400, a reserve of $9,000, and a guaranteed benefit of $30,000. Its probability of remaining in force until the end-of-year payment is 0.95, and the effective annual interest rate is 5%. Calculate the target’s allocated bonus and expected present value under each basis, and verify the portfolio allocation.

Premium basis. The target’s share is $2{,}400/120{,}000=0.02$, or 2%. Its bonus is $18{,}000(0.02)=\$360$. The other policies receive $\$18{,}000-\$360=\$17{,}640$, so the allocations sum to $18,000. The target’s expected present value is $360(0.95)/1.05=\$325.71$.

Reserve basis. The target’s share is $9{,}000/300{,}000=0.03$, or 3%. Its bonus is $18{,}000(0.03)=\$540$, leaving $\$18{,}000-\$540=\$17{,}460$ for the other policies. The total again equals $18,000. The target’s expected present value is $540(0.95)/1.05=\$488.57$.

Guaranteed-benefit basis. The target’s share is $30{,}000/600{,}000=0.05$, or 5%. Its bonus is $18{,}000(0.05)=\$900$, and the remaining allocation is $\$18{,}000-\$900=\$17{,}100$. The target’s expected present value is $900(0.95)/1.05=\$814.29$.

The arithmetic confirms that each basis allocates the full surplus, while the target’s share rises from 2% to 3% to 5%. This illustrates a distributional effect, not a change in total portfolio surplus. All present values are rounded to cents.

Common Mistakes and Interpretation

Using the target’s exposure as the rate denominator. The allocation rate uses total eligible exposure across the portfolio. Dividing the surplus by one policy’s premium, reserve, or benefit does not produce the portfolio allocation rate.
Changing the surplus while comparing bases. If the objective is to isolate the allocation rule, use the same projected surplus for each basis in each year.
Discounting from the wrong date. An end-of-year 2 payment is discounted by $(1+i)^2$, not by $1+i$.
Omitting or duplicating the in-force probability. Apply ${}_tp$ once when the payment is conditional on being in force. Do not apply it again to an amount already expressed as an unconditional expected payment.
Making an arithmetic error before discounting. For example, if a $120 bonus is payable with probability 0.98 at the end of year 1 and $i=5\%$, then $120(0.98)=117.60$, and $117.60/1.05=\$112.00$. Replacing $117.60$ with $112$ before discounting would give the wrong result.
Comparing rates without considering their units. A rate per dollar of premium is not directly comparable with a rate per dollar of reserve. Compare the resulting allocated amounts or present values.

Allocation-basis analysis is useful when a valuation must explain how the same projected surplus produces different policy-level discretionary benefits. The comparison is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus. State the exposure definition, valuation date, surplus, payment condition, and timing before calculating; these details determine whether the comparison is meaningful.

Check Your Understanding

For each question, use the stated basis and timing. Round monetary present values to the nearest cent.

A hypothetical portfolio has $9,000 of surplus and $75,000 of total eligible premiums. A policy has $1,000 of eligible premium. Find its premium-based bonus. (Answer: $120)
A hypothetical portfolio allocates $16,000 in proportion to reserves. Total reserves are $400,000, and a policy’s reserve is $6,000. Find the policy’s bonus. (Answer: $240)
A guaranteed-benefit-based allocation distributes $20,000 across a portfolio with $500,000 of total guaranteed benefits. A policy has a guaranteed benefit of $8,000. Find its bonus and its share of the surplus. (Answer: $320; 1.6%)
A policy receives a $100 bonus at the end of year 1 and a $150 bonus at the end of year 2. Its cumulative in-force probabilities are 0.96 and 0.91, respectively. The effective annual interest rate is 4%. Find the expected present value. (Answer: $100(0.96)/1.04+150(0.91)/1.04^2=\$218.51$)
A hypothetical portfolio allocates $10,000 using, in turn, total eligible premiums of $100,000, total reserves of $250,000, and total guaranteed benefits of $500,000. A target policy has eligible premiums of $800, reserves of $3,000, and guaranteed benefits of $6,000. Calculate the target bonus under each basis and identify the largest. (Answer: Premium basis $80; reserve basis $120; guaranteed-benefit basis $120; the reserve and guaranteed-benefit bases tie.)

What You'll Learn