Why Compare Terminal-Bonus Scales?
A terminal bonus is a discretionary amount payable when an eligible policy claim occurs, such as on death or maturity. A specified scale determines how that amount is calculated from the claim, the claim type, or both. Two scales can therefore produce different bonus amounts on the same claim, and their relative values can also depend on which claims are more likely and when the payments occur.
The comparison is an expected-present-value calculation for the terminal bonuses alone. It is not, by itself, a full policy reserve or a measure of whether a scale is affordable. The calculation requires a clear statement of eligibility, claim outcomes, probabilities, payment times, the interest assumption, and the rules of each scale. These ingredients prevent a common error: comparing scale amounts without allowing for the chance and timing of each eligible claim.
The probabilities \(p_j\) are unconditional probabilities measured from the valuation date: they include the chance that the policy remains in force until the relevant claim. If the outcomes describe mutually exclusive ways the policy can end, their probabilities sum to one when the list includes every possible outcome. An outcome with no eligible terminal bonus still belongs in the probability model; its bonus amount is zero.
A scale may be a percentage of the eligible claim, a fixed amount, a tiered percentage, or a rule that differs by claim type. The function \(g_s\) must represent the scale exactly as specified. For example, “4% of the first $10,000 and 6% of the excess” does not mean 6% of the entire claim when the claim exceeds $10,000.
Valuation Method
For each scale, calculate the bonus separately for every possible eligible claim. Then multiply each bonus by the probability and discount factor for its outcome. The sum is the scale’s expected present value. A reliable comparison uses the same outcomes, probability assumptions, interest rate, and valuation date for both scales.
List each claim type, eligible claim amount, probability, and payment time. Include outcomes with no bonus if they are part of the model.
Calculate the terminal bonus payable in each outcome, following any eligibility, threshold, or claim-type conditions exactly.
For outcome \(j\), multiply its bonus by \(p_jv^{t_j}\). Use \(v^{t_j}\) for a payment at time \(t_j\), not \(v\) automatically.
Add the weighted present values for each scale, then subtract the smaller EPV from the larger to state the difference and its direction.
The formula follows directly from the definition of expected value. Let the random present value under scale \(s\) be \(Z_s\). In outcome \(j\), the present value is \(g_s(C_j,j)v^{t_j}\). Since the outcomes are mutually exclusive and exhaustive, the expectation of \(Z_s\) is the sum of each possible present value multiplied by its probability:
This argument also explains why a bonus amount cannot simply be discounted and added across possible claim types: only one mutually exclusive outcome occurs, so each amount must be weighted by the probability of that outcome. If outcomes are instead represented by a complete sequence of possible policy paths, the same expectation principle applies to those paths.
Worked Example: Different Percentages for Death and Maturity
A hypothetical participating policy has a $15,000 eligible claim. From the valuation date, death at the end of year 1 has probability 0.30, and maturity at the end of year 2 has probability 0.70. Exactly one of these outcomes occurs. The effective annual interest rate is 10%. Scale A pays 2% of the claim on death and 6% on maturity. Scale B pays 4% of the claim for either outcome. Compare the expected present values.
Method. Calculate each scale’s bonus separately by claim type, then weight and discount each payment. The interest discount factors are \(v=1/1.10\) for year 1 and \(v^2=1/(1.10)^2=1/1.21\) for year 2.
Scale A. The death bonus is \(0.02(15{,}000)=\$300\), and the maturity bonus is \(0.06(15{,}000)=\$900\). The weighted present value of the death bonus is \(0.30(300)/1.10=\$81.8182\). For maturity it is \(0.70(900)/1.21=\$520.6612\). Therefore:
Scale B. The bonus is \(0.04(15{,}000)=\$600\) for either claim type. The weighted present value at death is \(0.30(600)/1.10=\$163.6364\), and at maturity it is \(0.70(600)/1.21=\$347.1074\). Thus:
Comparison and check. Scale A exceeds Scale B by \(\$602.48-\$510.74=\$91.74\), to the nearest cent. The result is plausible: Scale A’s bonus is lower than Scale B’s on death but higher on maturity, and maturity has a substantial probability and a later payment time. Both EPVs are below the corresponding undiscounted expected bonus amounts, as positive interest requires.
Common mistake avoided: treating each scale’s bonus as if it were certain, or swapping the death and maturity probabilities. The probability attached to each amount must match the outcome in which that amount is payable.
Applying Scales to Claim Amounts
When the scale depends on claim size, apply its rule to each claim before taking the expectation. This order matters for tiered scales: the bonus on a large claim may not be the same percentage of the whole claim as the bonus on a smaller one. A probability-weighted average claim amount can be useful for a linear percentage scale, but it must not be substituted for applying a nonlinear or tiered rule to each possible claim.
An outcome with no eligible claim contributes zero to the terminal-bonus EPV. It should not be removed and the remaining probabilities renormalized unless the quantity being calculated is explicitly conditional on a claim occurring. The EPV from the valuation date is unconditional.
Worked Example: A Flat Percentage Compared With a Tiered Scale
A hypothetical policy pays an eligible terminal claim at the end of year 2. The claim is $10,000 with probability 0.40, $25,000 with probability 0.20, and no eligible claim with probability 0.40. The effective annual interest rate is 5%. Scale A pays 5% of the eligible claim. Scale B pays 4% of the first $10,000 and 6% of any amount above $10,000. Compare the EPVs.
Method. Apply each rule to each positive claim, retain the zero-bonus outcome in the probability model, and discount each possible payment by \(1/(1.05)^2=1/1.1025\).
Scale A. The bonuses on claims of $10,000 and $25,000 are \(0.05(10{,}000)=\$500\) and \(0.05(25{,}000)=\$1{,}250\). The no-claim outcome has a bonus of zero. The expected bonus before discounting is \(0.40(500)+0.20(1{,}250)+0.40(0)=\$450\). Hence:
Scale B. On a $10,000 claim the bonus is \(0.04(10{,}000)=\$400\). On a $25,000 claim it is \(0.04(10{,}000)+0.06(25{,}000-10{,}000)=\$400+\$900=\$1{,}300\). Its expected bonus before discounting is \(0.40(400)+0.20(1{,}300)+0.40(0)=\$420\). Therefore:
Comparison and check. Scale A exceeds Scale B by \(408.16-380.95=\$27.21\), to the nearest cent. The no-claim probability contributes zero, not an additional payment. Also, each EPV is below its scale’s expected undiscounted bonus, as both are discounted for two years at a positive rate.
Common mistake avoided: applying the tiered scale’s 6% rate to the entire $25,000 claim. The stated rate applies only to the $15,000 above the first $10,000, so the correct bonus is $1,300, not $1,500.
Why a Higher Rate Need Not Mean a Higher EPV
A scale with a higher percentage for one type of claim need not have the larger EPV overall. The comparison depends on the claim amounts, the probability of each outcome, and the time until payment. A large bonus payable only on a rare or distant claim can contribute less than a smaller bonus payable more often or sooner.
The difference between two scales can be calculated outcome by outcome. If the same interest and outcome assumptions apply, subtracting the EPVs gives:
This form makes the source of the difference visible. Some outcomes may favor Scale A and others Scale B; the discounted probability-weighted differences determine the overall result. It is not enough to compare the scale percentages in isolation.
Worked Example: The Scale With the Higher EPV Changes With Claim Mix
A hypothetical policy has two mutually exclusive claim outcomes. A $10,000 death claim occurs at the end of year 1 with probability 0.50; otherwise a $40,000 maturity claim occurs at the end of year 2 with probability 0.50. The effective annual interest rate is 10%. Scale A pays 8% of the claim on death and 2% on maturity. Scale B pays 5% of the claim in either case. Compare their EPVs.
Method. First calculate each bonus by outcome. Then use the outcome-specific probability and discount factor. The death and maturity discount factors are \(1/1.10\) and \(1/1.21\), respectively.
Scale A. The death bonus is \(0.08(10{,}000)=\$800\); the maturity bonus is \(0.02(40{,}000)=\$800\). Thus:
Scale B. The death bonus is \(0.05(10{,}000)=\$500\), and the maturity bonus is \(0.05(40{,}000)=\$2{,}000\). Therefore:
Comparison and check. Scale B exceeds Scale A by \(1{,}053.72-694.21=\$359.50\), to the nearest cent. Scale A has the higher rate on death, but both scales pay $800 on the death outcome. On maturity, Scale B pays $1,200 more, and that difference outweighs the effect of the later payment time. The EPVs are below their undiscounted expected bonuses: $800 for Scale A and $1,250 for Scale B.
Common mistake avoided: deciding that Scale A must be more valuable because its death percentage is higher. The scale must be evaluated across all eligible outcomes, not ranked by one rate alone.
Checks Before Reporting a Comparison
Before reporting an EPV, verify the scale rule, outcome weights, and payment times independently. These checks are especially useful when a scale has eligibility conditions or more than one claim tier.
- Check eligibility. Confirm which claim events receive a terminal bonus and whether any amount is excluded from the scale base.
- Check the probability model. Ensure probabilities are from the valuation date and assigned to the correct outcomes. Include zero-bonus outcomes when appropriate.
- Check the timing. Use the discount factor for the actual payment time. A claim paid at the end of year 2 is discounted by \(v^2\), not \(v\).
- Check the scale arithmetic. For a threshold scale, separate the amount below the threshold from the excess and apply each rate to the specified portion.
- Check the comparison. State which scale has the larger EPV and calculate the difference from unrounded values. Round monetary results to the cent only at the end.
The EPV comparison is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus. Its central discipline is to keep three operations distinct: determine the bonus under the scale, weight it by the probability of its eligible outcome, and discount it for the time of payment.
Check Your Understanding
For each problem, compare the specified terminal-bonus values from the valuation date. Round monetary answers to the nearest cent.
- A hypothetical policy has an eligible claim of $12,000 at the end of year 2 with probability 0.25, a claim of $20,000 then with probability 0.75, and no other outcomes. At an effective annual interest rate of 4%, Scale A pays 3% of the claim. Find its EPV. (Answer: $499.26.)
- A terminal claim is $10,000 at the end of year 1 with probability 0.20, or $25,000 at the end of year 3 with probability 0.80. At an effective annual rate of 5%, a scale pays 2% of each eligible claim. Find the EPV. (Answer: $383.63.)
- At the end of year 2, an eligible claim is $8,000 with probability 0.30 or $30,000 with probability 0.70. The effective annual interest rate is 8%. A scale pays 4% of the claim. Find the EPV. (Answer: $802.47.)
- A claim of $15,000 occurs at the end of year 1 with probability 0.50; otherwise a $35,000 claim occurs at the end of year 2. The effective annual rate is 10%. A tiered scale pays 3% of the first $15,000 and 5% of any excess. Find the EPV. (Answer: $803.72.)
- A hypothetical policy has a $10,000 claim at the end of year 1 with probability 0.40, or a $20,000 claim at the end of year 2 with probability 0.60. At an effective annual rate of 5%, Scale A pays 6% of the claim in either outcome, while Scale B pays a flat $800 in either outcome. Which scale has the higher EPV, and by how much? (Answer: Scale A, by $141.50.)