Combining Experience Variances into Surplus

Why Combining Experience Variances Requires Care

A participating policy’s period result can differ from its expected result for several reasons at once. Claims may be fewer than assumed, investment returns may exceed the pricing basis, expenses may be higher than expected, or lapse and morbidity experience may differ from the assumptions. The individual calculations are manageable; the harder task is combining them without mixing signs, exposure bases, or payment dates.

Here, an experience variance means the monetary difference between actual experience and an explicitly stated expected basis. It is not the same as the statistical variance $\operatorname{Var}(X)$ of a random variable. We first use “variance” in the actuarial experience sense, then distinguish the statistical concept when discussing uncertainty in the combined result.

Definition: Use a favorable-surplus sign convention: a positive experience variance increases surplus relative to the expected basis, and a negative variance reduces it. For a period with expected surplus $S_0$ and components $G_j$, the experience bridge is $S_{\mathrm{actual}}=S_0+\sum_j G_j$. Each $G_j$ must be measured in the same currency and at the same valuation time. The bridge is additive only when the component definitions do not double count the same financial effect.

Put Every Component on a Common Basis

A claim count alone is not a monetary variance. It must be compared with an expected count for a clearly defined exposure, multiplied by the appropriate claim amount, and valued at a consistent time. The same principle applies to expenses and investment income. A practical sign convention for common sources is:

Mortality: for a fixed death benefit, fewer deaths than expected are favorable. If expected and actual death counts are $D_e$ and $D_a$, and each death produces a benefit $B$, then $G_m=(D_e-D_a)B$ when both counts and payments are valued at the same time.
Investment: when the same opening assets $A_0$ earn expected rate $r_e$ or actual rate $r_a$, the investment variance is $G_i=A_0(r_a-r_e)$, assuming the return difference is recognized at period-end.
Expenses: actual expenses above the expected amount reduce surplus. If $E_e$ and $E_a$ are comparable period-end expense amounts, then $G_e=E_e-E_a$.
Lapse: compare actual with expected lapse counts and multiply their difference by the defined net surplus effect per lapse. The effect’s sign depends on the contract and must be stated; it can include surrender payments and reserve release, but not an amount already included in another component.
Morbidity: for a fixed claim amount, fewer morbidity claims than expected are favorable, using the same logic as for mortality claims.

The exposure basis matters. For example, expected claims may be calculated using policy-years at risk, while an investment variance may use opening assets. Do not compare a count expected over one exposure with an actual count observed over another without stating the intended decomposition. One useful approach is to compare actual claims with the expected claim count for the same observed exposure, then measure lapse effects separately using a definition that excludes claim and expense effects already captured elsewhere.

Timing matters as well. If one component is measured at mid-period and another at period-end, the amounts are not yet on a common value basis. At effective annual interest rate $i$, a payment at time $t$ years can be accumulated to period-end by multiplying by $(1+i)^{1-t}$, provided the chosen interest basis is appropriate for that cash flow.

Worked Example: Combining Five Experience Sources

A hypothetical participating block has 1,000 policies exposed for a full year. All amounts below are measured at year-end; claim payments are assumed to occur at year-end. The expected surplus before experience variances is $120,000. Expected mortality is 12 deaths, compared with 9 actual deaths, with a $15,000 death benefit per claim. Expected investment return is 4.8% on opening assets of $2,500,000; the actual return is 5.2%. Expected expenses are $84 per policy, while actual expenses total $87,000. Expected lapses are 40 and actual lapses are 46; each additional lapse has a separately defined favorable net surplus effect of $250. Expected morbidity claims are 20 and actual claims are 24, each costing $2,500. Calculate actual surplus.

Method. Calculate each source as a period-end monetary gain or loss, using the favorable-positive sign convention.

Mortality. Expected claim cost is $12\times \$15{,}000=\$180{,}000$. Actual claim cost is $9\times \$15{,}000=\$135{,}000$. Thus $G_m=\$180{,}000-\$135{,}000=+\$45{,}000$.

Investment. The return difference is $0.052-0.048=0.004$. Therefore $G_i=\$2{,}500{,}000\times0.004=+\$10{,}000$.

Expenses. Expected expenses are $1{,}000\times \$84=\$84{,}000$. Hence $G_e=\$84{,}000-\$87{,}000=-\$3{,}000$.

Lapses. There are $46-40=6$ more lapses than expected. By the stated definition of the net effect, $G_l=6\times \$250=+\$1{,}500$. This amount is separate from the expense and claim components.

Morbidity. Expected claim cost is $20\times \$2{,}500=\$50{,}000$, while actual claim cost is $24\times \$2{,}500=\$60{,}000$. Thus $G_h=\$50{,}000-\$60{,}000=-\$10{,}000$.

The combined experience variance is $+\$45{,}000+\$10{,}000-\$3{,}000+\$1{,}500-\$10{,}000=+\$43{,}500$. Actual surplus is $\$120{,}000+\$43{,}500=\$163{,}500$. The calculation checks: the favorable sources total $56,500 and the unfavorable sources total $13,000, leaving a net gain of $43,500. A common mistake is to treat every excess count as unfavorable; here, the defined net lapse effect makes additional lapses favorable.

Why the Components Add

The bridge is an accounting identity when each component is defined as its own non-overlapping difference from the expected basis. For example, expected total surplus can be adjusted by the difference in claim costs, the difference in investment income, and the difference in expenses. Adding these differences reconstructs the corresponding actual total. A lapse component can also be added if it measures a distinct effect rather than repeating a claim, expense, or investment difference already counted.

The sign of a claim variance follows directly from subtracting actual cost from expected cost. With fixed benefit $B$, expected cost is $D_eB$ and actual cost is $D_aB$. Their difference is:

$$ G_m=D_eB-D_aB=(D_e-D_a)B. $$

If actual deaths are fewer, $D_a<D_e$, so $G_m>0$. If actual deaths exceed expectation, $G_m<0$. This derivation assumes a fixed benefit and a common payment time. If claim amounts vary or payments occur at different times, compare the expected and actual monetary claim amounts after valuing each on the same basis; the simple count-times-benefit expression may not apply.

Fix the period and valuation time.
Choose the measurement period and a common date, such as year-end, for every monetary component.

State the exposure basis.
Identify the relevant exposure for each expected count or amount, such as policy-years, opening assets, or policies exposed for the full period.

Calculate signed components.
Use a consistent favorable-positive convention and define any contract-specific lapse effect explicitly.

Check for overlap and add.
Confirm that no financial effect appears in two components, then add the variances to expected surplus.

Worked Example: Matching Exposure and Payment Timing

A hypothetical block has 360 actual policy-years at risk during a year. Use an effective annual interest rate of 4%. Expected mortality is 0.01 per policy-year and actual deaths are 3; each death benefit is $10,000, paid on average halfway through the year. Expected morbidity is 0.05 claims per policy-year and actual claims are 16; each morbidity benefit is $4,000, also paid halfway through the year. Expected expense is $100 per policy-year on this observed exposure, while actual expense is $37,800. Expected investment return is 4.0% on opening assets of $1,000,000; actual return is 4.3%. Expected lapses are 20 and actual lapses are 30. Each additional lapse has a net period-end cost of $500, excluding claim and expense effects already measured. Find the total experience variance at year-end.

Method. First use the observed 360 policy-years for expected mortality, morbidity, and expenses. Then accumulate mid-year claim differences to year-end. The separate lapse amount is defined to avoid including the claim and expense effects again.

The accumulation factor for a payment at $t=0.5$ is $(1.04)^{1-0.5}=\sqrt{1.04}=1.0198039$ (shown to seven decimal places for the intermediate calculation).

Mortality. Expected deaths on the observed exposure are $360\times0.01=3.6$. The favorable claim difference is $3.6-3=0.6$ deaths. Its year-end value is $0.6\times \$10{,}000\times1.0198039=\$6{,}118.82$, rounded to the nearest cent.

Morbidity. Expected claims are $360\times0.05=18$. There are $18-16=2$ fewer claims than expected, giving a year-end gain of $2\times \$4{,}000\times1.0198039=\$8{,}158.43$, rounded to the nearest cent.

Expenses. Expected expenses are $360\times \$100=\$36{,}000$. Therefore $G_e=\$36{,}000-\$37{,}800=-\$1{,}800$.

Investment. The return difference is $0.043-0.040=0.003$, so $G_i=\$1{,}000{,}000\times0.003=+\$3{,}000$.

Lapses. There are $30-20=10$ additional lapses. At a defined net cost of $500 each, $G_l=-10\times \$500=-\$5{,}000$.

Adding the period-end amounts gives $\$6{,}118.82+\$8{,}158.43-\$1{,}800+\$3{,}000-\$5{,}000=+\$10{,}477.25$. The result is a favorable net variance of $10,477.25. The exposure check is that both expected claim counts use 360 policy-years, not the original number of policies. A tempting but incorrect morbidity calculation is $2\times \$4{,}000=\$8{,}000$; it leaves the mid-year payments at the wrong valuation time. Accumulating them gives $8,158.43.

Combining Statistical Variances

Sometimes the aim is not to explain a realized period result, but to measure the uncertainty of the combined result. Let $Z_m,Z_i,Z_e,Z_l,Z_h$ be random monetary deviations from expected surplus, each already expressed at the same date. Then:

\operatorname{Var}\left(\sum_j Z_j\right) =\sum_j\operatorname{Var}(Z_j)+2\sum_{j<k}\operatorname{Cov}(Z_j,Z_k).

To justify this identity, center each component by subtracting its mean and expand the square of the sum. The squared terms contribute the individual variances; each cross-product occurs twice and contributes twice its covariance. Consequently, simply adding component variances is correct only when all pairwise covariances are zero. The relevant covariances are between the signed monetary deviations, not necessarily between raw event counts.

Worked Example: Allowing for Dependence Between Sources

A hypothetical insurer models five period-end surplus deviations, all measured in thousands of dollars: mortality, investment, expense, lapse, and morbidity. Each has mean zero. Their standard deviations are respectively 12, 8, 3, 5, and 10. Assume all pairs are uncorrelated except mortality and morbidity, whose correlation is 0.25, and investment and lapse, whose correlation is $-0.20$. The expected surplus is $25,000. Calculate the statistical variance and standard deviation of total surplus, and state its expected value.

Method. Add the five individual variances, then include twice each nonzero covariance. All calculations are in squared thousands of dollars until the standard deviation is found.

The sum of individual variances is $12^2+8^2+3^2+5^2+10^2=144+64+9+25+100=342$.

The mortality-morbidity covariance is $0.25\times12\times10=30$. The investment-lapse covariance is $-0.20\times8\times5=-8$. Thus the total variance is $342+2(30)+2(-8)=386$ in squared thousands of dollars.

The standard deviation is $\sqrt{386}=19.6469$ thousand dollars, or $19,646.88 to the nearest cent. Since each deviation has mean zero, expected total surplus remains $25,000. The check is that positive dependence adds $60$ squared-thousand units, while negative dependence subtracts $16$, giving a net increase of $44$ over the uncorrelated variance of $342$. A common error is to add the standard deviations or to omit the factor of two on covariance terms.

Common Checks Before Reporting a Result

Before accepting a combined calculation, verify that every term has the same favorable-positive sign convention, currency, and valuation date. Confirm that expected claim counts use the stated exposure and that actual counts refer to the same risk period or to an explicitly defined comparison basis. If a cash flow is valued at a different time, convert it before adding it. For a statistical variance calculation, include covariance terms unless independence or zero covariance has been justified.

These checks matter because a plausible-looking total can be wrong even when each individual multiplication is correct. In particular, a lapse effect may already be reflected in the exposure used for claims or expenses, and a surrender cost may already be included in a separately calculated expense or benefit amount. Define component boundaries before summing. This variance bridge explains how experience departs from expectation; it is not a substitute for defining the contract cash flows or the valuation basis.

Check Your Understanding

For each problem, use a favorable-positive convention and respect the stated exposure and timing basis.

A block has 600 policy-years exposed. Expected mortality is 0.015 per policy-year, actual deaths are 7, and each death benefit is $12,000 paid at year-end. Find the mortality variance. (Answer: +$24,000)
Opening assets are $3,200,000. The expected annual investment return is 3.5% and the actual return is 3.2%, with both recognized at year-end. Find the investment variance. (Answer: -$9,600)
For 2,500 policies, expected expenses are $110 per policy and actual expenses are $285,000. Find the expense variance. (Answer: -$10,000)
There are 240 observed policy-years. Expected morbidity is 0.04 claims per policy-year, actual claims are 8, and each claim pays $5,000 at mid-year. At an effective annual interest rate of 4%, find the morbidity variance at year-end, rounded to cents. (Answer: +$8,158.43)
Expected surplus is $30,000. Five non-overlapping period-end experience variances are mortality +$12,000, investment -$4,500, expense +$1,200, lapse -$3,000, and morbidity -$2,400. Find actual surplus. (Answer: $33,300)

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