Valuation Under Alternative Surplus Allocations

Introduction

A participating policy may generate distributable surplus in several policy years. The insurer can allocate that surplus among policyholders, shareholders, and, where specified, a retained or unallocated account. Even when the projected surplus, interest, mortality, and contractual guarantees are unchanged, a different allocation rule can change the value assigned to each recipient.

The key is to compare like with like. First value the same projected surplus at the same payment times and under the same survival assumptions. Then apply each allocation rule to those amounts. A change in the split does not, by itself, change the value of the total surplus. It changes who receives that value. The calculations below are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Definition: An alternative surplus allocation is a different stated division of the same projected distributable surplus among recipients, such as policyholders, shareholders, and a retained account. A comparison holds the projected surplus, its timing, the interest and decrement assumptions, and contractual guarantees constant; only the allocation proportions change.

Valuation Framework

Suppose $D_t$ is the distributable surplus at the end of policy year $t$, stated per policy that remains in force to that date. Let $p_t$ be the probability that the policy is still in force at the end of year $t$, conditional on being in force at valuation. Let $i$ be the effective annual interest rate and $v=1/(1+i)$. If the policyholder, shareholder, and retained-account proportions at duration $t$ are respectively $\alpha_t$, $\beta_t$, and $\gamma_t$, then these proportions must be nonnegative and satisfy $\alpha_t+\beta_t+\gamma_t=1$.

V_{\mathrm{PH}}=\sum_{t=1}^{n}p_t v^t\alpha_tD_t,\qquad V_{\mathrm{SH}}=\sum_{t=1}^{n}p_t v^t\beta_tD_t,\qquad V_{\mathrm{R}}=\sum_{t=1}^{n}p_t v^t\gamma_tD_t.

These expressions assume that the allocated amounts are delivered at the end of year $t$ if the policy remains in force to that date. The $p_t$ factor therefore accounts for the possibility that the policy has ended before the payment. If surplus is payable on death, on surrender, or at another time, the relevant payment probability and discount time must instead match that contract. Likewise, if the amounts $D_t$ are already unconditional expected values rather than amounts conditional on the policy being in force, applying $p_t$ again would count survival twice.

A contractual guarantee is valued separately using its own payment conditions. For example, a guaranteed maturity amount payable only if the policy survives to year $n$ is discounted and multiplied by the probability of surviving to year $n$. It is not reduced merely because a smaller proportion of distributable surplus is allocated to policyholders.

Why the Total Value Is Unchanged

The reconciliation across recipients follows directly from the allocation rule. It is useful both as a short justification and as a check on numerical work.

Value the total surplus at each duration.
At duration $t$, its expected present value is $p_t v^tD_t$, because the amount is payable at the end of year $t$ only if the policy remains in force then.

Apply the stated allocation proportions.
The recipient values at that duration sum to $p_t v^tD_t(\alpha_t+\beta_t+\gamma_t)$.

Use the proportions’ sum.
Since $\alpha_t+\beta_t+\gamma_t=1$, the three recipient values sum to $p_t v^tD_t$. Summing over all durations gives the present value of total distributable surplus.

V_{\mathrm{PH}}+V_{\mathrm{SH}}+V_{\mathrm{R}} =\sum_{t=1}^{n}p_t v^tD_t.

Thus, if the projected surplus and its timing are fixed, changing only the allocation proportions redistributes value but does not change this total. The result does not say that all policy measures are unchanged: the policyholder value and shareholder value can each change, and the value of a guaranteed benefit remains a separate component.

Worked Example: Comparing Two Constant Allocation Rules

A hypothetical participating policy is valued at issue for three years. The effective annual interest rate is 5%. The annual probability of remaining in force for each year is 0.90, independently of earlier years, so the probabilities of being in force at the ends of years 1, 2, and 3 are 0.90, 0.81, and 0.729. Distributable surplus per policy in force at each year-end is $100, $120, and $140. Compare these two allocations: Rule A assigns 70% to policyholders, 20% to shareholders, and 10% to retention at every duration; Rule B assigns 80%, 15%, and 5%, respectively.

Method. Calculate the present value of the same surplus stream first, then multiply that value by each constant allocation proportion. This avoids recalculating or changing the underlying surplus projection.

The discount factor is $v=1/1.05$. The survival-weighted discount factors are:

p_1v=0.90/1.05=0.8571429,\quad p_2v^2=0.81/1.1025=0.7346939,\quad p_3v^3=0.729/1.157625=0.6297376.

The value of total distributable surplus is therefore:

100(0.8571429)+120(0.7346939)+140(0.6297376) =85.7143+88.1633+88.1633 =\$262.0409.

Under Rule A, the policyholder value is $0.70(\$262.0409)=\$183.4286$; the shareholder value is $0.20(\$262.0409)=\$52.4082$; and the retained value is $0.10(\$262.0409)=\$26.2041$.

Under Rule B, the policyholder value is $0.80(\$262.0409)=\$209.6327$; the shareholder value is $0.15(\$262.0409)=\$39.3061$; and the retained value is $0.05(\$262.0409)=\$13.1020$.

Check. Rule A gives $183.4286+52.4082+26.2041=\$262.0409$, subject to rounding. Rule B gives $209.6327+39.3061+13.1020=\$262.0408$, with the one ten-thousandth difference due to rounding. Rounded to cents, the recipient values are $183.43, $52.41, and $26.20 under Rule A, and $209.63, $39.31, and $13.10 under Rule B. The common mistake this method avoids is treating the allocation change as a change in projected surplus; the total value remains $262.04.

Duration-Specific Allocations and Payment Timing

Allocation proportions need not be constant over the policy term. When they vary by duration, each surplus amount must be multiplied by the proportion stated for that duration before summing. This matters because a dollar allocated in one year need not have the same present value as a dollar allocated in another year: the discount period and probability of payment can differ.

Worked Example: A Later Policyholder Allocation

Consider a hypothetical two-year policy. The effective annual interest rate is 4%, and the probability of remaining in force for each year is 0.95, independently, giving end-of-year survival probabilities 0.95 and 0.9025. Distributable surplus is $80 at the end of year 1 and $120 at the end of year 2, conditional on being in force at the relevant year-end. A $500 contractual maturity guarantee is payable at the end of year 2 if the policy survives. Compare Rule A, which allocates 50% of each year’s surplus to policyholders, with Rule B, which allocates 75% in year 1 and 50% in year 2. Shareholders receive the remainder; there is no retention.

The survival-weighted discount factors are $0.95/1.04=0.9134615$ for year 1 and $0.9025/1.0816=0.8344120$ for year 2. Hence total distributable-surplus value is:

80(0.9134615)+120(0.8344120) =73.0769+100.1294 =\$173.2063.

Under Rule A, the policyholder allocation value is:

(0.50)(80)(0.9134615)+(0.50)(120)(0.8344120) =36.5385+50.0647 =\$86.6032.

The shareholder allocation value under Rule A is also $86.6032. Under Rule B, the policyholder value is:

(0.75)(80)(0.9134615)+(0.50)(120)(0.8344120) =54.8077+50.0647 =\$104.8724.

The Rule B shareholder value is $0.25(80)(0.9134615)+0.50(120)(0.8344120)=18.2692+50.0647=\$68.3339$. In both cases, policyholder and shareholder values sum to $173.2063, up to rounding.

The guaranteed maturity value is common to both rules. Its expected present value is:

500(0.9025)(1.04)^{-2}=451.25/1.0816=\$417.2060.

Including that guarantee, total policyholder value is $503.81 under Rule A and $522.08 under Rule B, rounded to cents. These figures add the same guaranteed value to each rule’s policyholder allocation value. The example illustrates why the allocation must be applied year by year and why the guarantee must not be scaled by a discretionary allocation percentage.

Timing Can Reverse an Intuitive Comparison

A rule with a larger policyholder percentage in later years is not automatically more valuable to policyholders than a rule with the same average percentage. Later amounts are discounted for longer and, when payment depends on remaining in force, are also subject to a smaller survival probability. A useful comparison therefore evaluates the survival-weighted present value rather than averaging allocation percentages without regard to duration.

Worked Example: Equal Average Shares, Different Present Values

A hypothetical three-year policy has an effective annual interest rate of 0% and a constant annual probability of remaining in force of 0.80. The end-of-year survival probabilities are 0.80, 0.64, and 0.512. Distributable surplus is $50 at each year-end, conditional on being in force. Compare Rule A, with policyholder shares of 40%, 60%, and 80% by year, to Rule B, with a 60% policyholder share in every year. Shareholders receive the balance and no amount is retained.

The total value of surplus is $50(0.80+0.64+0.512)=50(1.952)=\$97.60$. Under Rule A, the policyholder value is $20(0.80)+30(0.64)+40(0.512)=16+19.20+20.48=\$55.68$. Shareholder value is $30(0.80)+20(0.64)+10(0.512)=24+12.80+5.12=\$41.92$.

Under Rule B, policyholder value is $30(1.952)=\$58.56$, and shareholder value is $20(1.952)=\$39.04$. Both rules allocate 60% of the undiscounted $150 total surplus to policyholders when the annual amounts are simply added. Their policyholder present values differ because Rule A places more of the policyholder share in the later years, when fewer policies are expected to remain in force. Here Rule A’s 80% final-year share does not compensate for its lower early-year shares.

Check. Under either rule, recipient values sum to $97.60. The common mistake this example avoids is comparing unweighted average percentages as though every year-end payment had the same probability and present value.

Practical Checks and Common Traps

For a reliable comparison, write down the assumptions and cash-flow timing before doing arithmetic. The survival probability should correspond to the event required for the payment, and each discount factor should match the payment date. If a value is already an expected present value, do not apply survival or discounting again.

Changing more than the allocation. If one comparison also changes projected surplus, mortality, interest, or timing, it is not isolating the effect of the allocation rule.
Applying one percentage to every duration. Use $\alpha_t$, $\beta_t$, and $\gamma_t$ for the proportions actually stated at each duration; do not assume they are constant unless the problem says so.
Ignoring survival. A surplus amount conditional on being in force at year-end must be weighted by the probability of reaching that year-end.
Scaling a guarantee with discretionary surplus. A contractual guarantee is valued from its own terms. An allocation change does not change it when the guarantee and valuation assumptions are held fixed.
Failing the reconciliation check. At each duration, the allocation proportions must sum to 1. Consequently, the present values allocated among recipients must sum to the present value of total distributable surplus.

Key takeaway: Under fixed experience, fixed surplus, fixed timing, and fixed guarantees, alternative surplus allocations redistribute the present value among recipients. Value each duration using its stated allocation, survival probability, and discount factor; then reconcile the recipient totals to the value of the common surplus stream.

Check Your Understanding

For each problem, assume payments occur at the stated year-end only if the policy remains in force then. Give monetary answers to the nearest cent.

A hypothetical two-year policy has an effective annual interest rate of 5%, annual survival probability 0.90, and year-end surpluses of $100 in each year. The policyholder receives 60% of each amount. Find the policyholder value. (Answer: $95.51)
At a 0% effective annual interest rate, a hypothetical policy has end-of-year survival probabilities 0.90 and 0.81, with surplus of $100 at each year-end. The policyholder shares are 40% in year 1 and 80% in year 2. Find the policyholder and shareholder values and verify their total. (Answer: Policyholder $100.80; shareholder $70.20; total $171.00)
A hypothetical policy has a $1,000 guaranteed benefit payable at the end of year 2 if it survives to that date. The effective annual interest rate is 10%, and annual survival probability is 0.90. Find the guarantee’s expected present value. (Answer: $669.42)
A hypothetical two-year policy has an effective annual interest rate of 5%, annual survival probability 0.90, and end-of-year surpluses of $50 and $80. The policyholder shares are 50% in year 1 and 75% in year 2; shareholders receive the remainder. Find both recipient values and check their sum against the total surplus value. (Answer: Policyholder $65.51; shareholder $36.12; total $101.63)
A hypothetical three-year policy has an effective annual interest rate of 0%, annual survival probability 0.80, and surplus of $50 at each year-end. Rule A assigns policyholder shares of 40%, 60%, and 80%, respectively. Rule B assigns a 60% share at every duration. Find both policyholder values and identify which rule gives the larger value. (Answer: Rule A $55.68; Rule B $58.56; Rule B is larger)

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