Sensitivity of Surplus Allocation to Experience

Why Measure Sensitivity?

A surplus result is not determined by the allocation rule alone. It also depends on how actual experience compares with the assumptions used to project premiums, claims, investment income, and expenses. A change in mortality may increase or reduce total surplus; a change in investment return or expenses may do the same. The allocation rule then determines how much of that changed surplus is attributed to policyholders and shareholders.

This tutorial uses a one-year block model to isolate those effects. It is a deliberately specified hypothetical model, not a description of any particular insurer. The model assumes a fixed number of policies, a fixed benefit per death, a fixed opening asset amount, and a fixed expense exposure. It ignores tax, lapses, changes in policy count, timing differences within the year, and interactions between experience sources. Those effects require a more detailed projection.

Definition: Let $S_0$ be surplus under the valuation or pricing basis. The experience change in surplus is the difference between surplus under specified actual experience and $S_0$. Under a fixed linear allocation rule with policyholder fraction $\alpha$, where $0\leq\alpha\leq1$, positive allocable surplus $S$ is divided as $H=\alpha S$ for policyholders and $Q=(1-\alpha)S$ for shareholders. Here $H$ and $Q$ denote the allocated amounts. Unless a different rule is stated, the examples assume $S>0$, no allocation cap, and no retained portion.

A One-Year Sensitivity Model

Suppose a block has $N$ policies. Let $q_0$ be the assumed one-year death probability, $q$ the actual death proportion used in the model, and $B$ the benefit paid for each death. The expected number of deaths on the basis is $Nq_0$; the actual number is $Nq$. A lower-than-assumed death count creates a mortality gain in this model because fewer fixed death benefits are paid.

Let $A$ be the assets exposed to investment return during the year, $r_0$ the assumed annual effective return, and $r$ the actual annual effective return. Let $e_0$ and $e$ be assumed and actual expense per policy. All monetary amounts below use the same currency and the same one-year period.

\Delta S =NB(q_0-q)+A(r-r_0)+N(e_0-e), \qquad S=S_0+\Delta S.

The three terms are, respectively, the mortality result, investment result, and expense result. The signs follow from the cash flows: fewer deaths than assumed save benefit payments; higher investment return produces more investment income; and lower expenses save outgo. Each result is measured against the stated basis, not against an unstated benchmark.

Model condition: The formula treats $N$, $B$, $A$, and the one-year exposure as fixed when experience assumptions change. It is appropriate for a controlled sensitivity calculation under those conditions. It should not be used unchanged if mortality changes the asset exposure, expenses vary with claims, policy counts change, or the timing of cash flows materially affects investment income.

Why the Formula and Allocation Sensitivities Hold

For mortality, the basis claim amount is $NBq_0$ and the actual claim amount is $NBq$. Their difference, basis claims minus actual claims, is $NBq_0-NBq=NB(q_0-q)$. Similarly, basis investment income is $Ar_0$ and actual investment income is $Ar$, giving $A(r-r_0)$. Basis expenses are $Ne_0$ and actual expenses are $Ne$, giving $N(e_0-e)$. Adding these changes to the basis surplus gives the stated expression for $S$.

Under the fixed allocation rule, $H=\alpha S$ and $Q=(1-\alpha)S$. Therefore, as long as the rule remains linear and applies to the surplus being considered, a change $\Delta S$ produces the following changes in allocated amounts:

\Delta H=\alpha\Delta S, \qquad \Delta Q=(1-\alpha)\Delta S, \qquad \Delta H+\Delta Q=\Delta S.

The last equality is an important check: the change allocated to policyholders plus the change allocated to shareholders must equal the change in total surplus. For small changes in the model inputs, the corresponding sensitivities are

\frac{\partial S}{\partial q}=-NB,\qquad \frac{\partial S}{\partial r}=A,\qquad \frac{\partial S}{\partial e}=-N.

For example, increasing the actual death proportion by one percentage point changes surplus by $-0.01NB$, not by $-NB$. The derivative is per unit change in the proportion; the percentage-point change must still be converted to a decimal.

Worked Example: Separating the Three Experience Effects

A hypothetical participating block has 1,000 policies. The basis assumes a one-year death probability of 0.020, a death benefit of $50,000, investment return of 4.0% on $10,000,000 of exposed assets, and annual expense of $300 per policy. The basis surplus is $100,000. Actual deaths are 16, actual investment return is 4.5%, and actual expense is $310 per policy. The stated allocation rule gives 60% of positive surplus to policyholders and 40% to shareholders. Calculate the actual surplus and both allocations, and compare them with the basis allocations.

Method. Calculate each experience result against its own basis, add the results to basis surplus, and then apply the allocation rule. This avoids treating the actual surplus as if it were all experience gain.

Mortality. Expected deaths are $1{,}000(0.020)=20$. There are 16 actual deaths, so the mortality gain is $(20-16)(\$50{,}000)=\$200{,}000$.

Investment. The return difference is $0.045-0.040=0.005$. The investment result is $\$10{,}000{,}000(0.005)=\$50{,}000$.

Expenses. Actual expenses exceed basis expenses by $\$310-\$300=\$10$ per policy. The expense result is $1{,}000(\$300-\$310)=-\$10{,}000$.

The total experience change is $\$200{,}000+\$50{,}000-\$10{,}000=\$240{,}000$. Actual surplus is therefore $\$100{,}000+\$240{,}000=\$340{,}000$, which is positive and eligible for the stated allocation.

The basis allocations are $0.60(\$100{,}000)=\$60{,}000$ to policyholders and $0.40(\$100{,}000)=\$40{,}000$ to shareholders. The actual allocations are $0.60(\$340{,}000)=\$204{,}000$ and $0.40(\$340{,}000)=\$136{,}000$, respectively. The increases are $\$144{,}000$ and $\$96{,}000$; they sum to the $\$240{,}000$ experience change.

Check. Actual allocations sum to $\$204{,}000+\$136{,}000=\$340{,}000$. The common mistake is to call $\$340{,}000$ the experience gain; the gain is only the difference from basis surplus, $\$240{,}000$.

Combined Changes and Attribution

When several assumptions change at once, the same calculation can be performed term by term. Under the fixed-exposure model, the terms add, so the combined change equals the sum of the separate changes. This makes an attribution table useful: it shows which sources increase surplus and which reduce it. It does not, by itself, prove that the sources are independent in a more realistic model. If an assumption change alters another exposure, interactions must be modelled rather than silently ignored.

Worked Example: A Lower Return Partly Offsets Other Gains

A hypothetical block has 2,000 policies, basis death probability 0.015, and death benefit $40,000. The basis surplus is $180,000. Exposed assets are $8,000,000, with basis return 3.5%. Basis expense is $250 per policy. Actual deaths are 24, the actual return is 3.0%, and actual expenses are $240 per policy. Allocate positive surplus 60% to policyholders and 40% to shareholders. Find the total change and actual allocations.

Mortality result. Basis deaths are $2{,}000(0.015)=30$. The six fewer actual deaths produce a gain of $(30-24)(\$40{,}000)=\$240{,}000$.

Investment result. The return difference is $0.030-0.035=-0.005$. Thus the investment result is $\$8{,}000{,}000(-0.005)=-\$40{,}000$.

Expense result. Expenses are $\$10$ lower per policy, producing a gain of $2{,}000(\$250-\$240)=\$20{,}000$.

The combined change is $\$240{,}000-\$40{,}000+\$20{,}000=\$220{,}000$. Actual surplus is $\$180{,}000+\$220{,}000=\$400{,}000$. Policyholders receive $0.60(\$400{,}000)=\$240{,}000$, and shareholders receive $0.40(\$400{,}000)=\$160{,}000$.

The allocation increases are found from the change, not from the full actual surplus: policyholders’ increase is $0.60(\$220{,}000)=\$132{,}000$, and shareholders’ increase is $0.40(\$220{,}000)=\$88{,}000$. These increases sum to $\$220{,}000$, as required. The investment loss offsets part, but not all, of the mortality and expense gains.

Check. The basis allocations are $108,000 and $72,000; adding the respective increases gives $240,000 and $160,000. A common error is to use $0.035-0.030$ for the investment result, reversing the sign and incorrectly turning a loss into a gain.

When the Allocation Rule Is Not Linear

The sensitivity of allocated shares need not remain a fixed percentage of the change in total surplus if the allocation rule includes a cap, threshold, or other condition. In that case, calculate allocations at each surplus level under the rule, then take their difference. Applying a fixed sharing percentage to the entire surplus change can be wrong when the change crosses a boundary.

Worked Example: Expense Sensitivity With a Policyholder Cap

A hypothetical block has current positive surplus of $210,000. Its allocation rule gives policyholders 70% of surplus, subject to a maximum policyholder allocation of $150,000; shareholders receive the remaining surplus. There are 1,000 policies. Consider a $10 reduction in expense per policy, with all other experience unchanged. Calculate the change in total surplus and in each allocation.

Current allocation. Seventy percent of $210,000 is $0.70(\$210{,}000)=\$147{,}000$, below the $150,000 cap. Policyholders receive $147,000 and shareholders receive $\$210{,}000-\$147{,}000=\$63{,}000$.

Surplus change. The expense reduction is a saving of $1{,}000(\$10)=\$10{,}000$. New surplus is $\$210{,}000+\$10{,}000=\$220{,}000$.

New allocation. Seventy percent of $220,000 is $154,000, which exceeds the cap. Policyholders therefore receive $150,000, and shareholders receive $\$220{,}000-\$150{,}000=\$70{,}000$.

Policyholders’ allocation increases by only $\$150{,}000-\$147{,}000=\$3{,}000$, while shareholders’ allocation increases by $\$70{,}000-\$63{,}000=\$7{,}000$. Together these changes equal the $10,000 surplus increase. The common mistake is to allocate $70\%$ of the increase, or $7,000, to policyholders without checking the cap.

Common Traps in Sensitivity Calculations

Reversing the mortality sign. In this model, fewer deaths than the basis expectation increase surplus because fewer death benefits are paid. Use $NB(q_0-q)$, or basis claims minus actual claims.
Using percentage points as whole numbers. A return increase from 4.0% to 4.5% is $0.005$, not $0.5$, when rates are entered as decimals.
Changing the exposure without stating it. Investment sensitivity is based on the exposed asset amount $A$; expense sensitivity is based on the policy count $N$. If either exposure changes, recalculate it explicitly.
Allocating the experience change as though it were total surplus. First add the change to $S_0$ to obtain $S$. Allocate $S$ for actual shares, or allocate $\Delta S$ only when calculating changes in shares under a linear rule.
Ignoring rule boundaries. Caps and thresholds can change the marginal share of a surplus change. Reapply the complete allocation rule at each scenario.

Sensitivity analysis matters because a surplus figure without its drivers can conceal opposing effects. A mortality gain may be partly offset by lower investment return; expense savings may reinforce or weaken the result. Reporting both the source-by-source changes and their effect on each allocated share makes the calculation auditable and clarifies who bears the effect under the stated rule. These calculations are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Check Your Understanding

For each problem, use the allocation rule stated in that problem and check that the allocated changes reconcile to the total surplus change.

A block has 800 policies, basis death probability 0.025, actual deaths 18, and death benefit $30,000. Assets of $5,000,000 earn 4.4% rather than the 4.0% basis return. Expense increases from $180 to $185 per policy. Find the total experience change and its allocation under a 65% policyholder share. (Answer: total change $76,000; policyholder change $49,400; shareholder change $26,600.)
A block has 1,200 policies and a basis death probability of 0.010, so 12 deaths are expected; 15 deaths occur, with a benefit of $25,000 each. On $6,000,000 of assets, return is 4.2% rather than 4.0%. Expense falls from $90 to $85 per policy. Find the total change and changes in allocated shares under a 70% policyholder share. (Answer: total change $-\$57,000$; policyholder change $-\$39,900$; shareholder change $-\$17,100$.)
Basis surplus is $250,000. A mortality gain of $30,000, an investment loss of $50,000, and an expense gain of $12,000 occur. Find actual surplus and both actual allocations under a 55% policyholder share. (Answer: actual surplus $242,000; policyholders $133,100; shareholders $108,900.)
Current surplus is $190,000, and the allocation rule gives policyholders 80% subject to a $140,000 cap, with shareholders receiving the remainder. For 1,000 policies, expenses fall by $15 per policy. Find the changes in each allocation. (Answer: total surplus increases by $15,000; policyholder allocation changes by $0; shareholder allocation increases by $15,000.)
For 1,500 policies, the actual death proportion increases from 0.012 to 0.014, with a fixed benefit of $20,000. All other experience is unchanged. Under a linear rule allocating 60% of surplus to policyholders, find the mortality-driven change in total surplus and each allocated share. (Answer: total surplus decreases by $60,000; policyholder allocation decreases by $36,000; shareholder allocation decreases by $24,000.)

What You'll Learn