Why Compare Expected and Actual Experience?
A participating insurer needs to understand how observed experience differs from the assumptions underlying its expected results. That comparison helps explain whether the insurer’s financial result improved or worsened during a period. The comparison is useful only when the expected and actual amounts are measured on a consistent basis: the same exposure, time period, cash-flow definitions, and valuation basis must be used.
In this tutorial, a variance is measured from the insurer’s perspective. A positive variance means actual experience produced a better financial result for the insurer than expected; a negative variance means it produced a worse result. The sign depends on whether the item is income or outgo. More income than expected is favorable, while more outgo than expected is unfavorable.
The expected result is calculated using the relevant expected assumptions and the exposure being analyzed. The actual result uses the observed financial outcomes for that same exposure. An expectation for an original plan may not be the right comparator if the actual number of policies or time exposed differs from the plan. In that case, distinguish changes in volume from changes in experience per unit of exposure.
A Consistent Sign Convention
For an income item, calculate actual income minus expected income. For an outgo item, calculate expected outgo minus actual outgo. This makes a positive value favorable in both cases. Equivalently, combine all items by writing income with a positive sign and outgo with a negative sign, then subtract the expected result from the actual result.
For example, an increase in claims is an increase in outgo, so it reduces the insurer’s result. An increase in premium income, by contrast, increases the insurer’s result. The word “variance” by itself does not tell you whether a deviation is favorable; identify the item’s effect on income or outgo first.
| Item being compared | Favorable to insurer when... | Insurer variance |
|---|---|---|
| Income | Actual income exceeds expected income | Actual income minus expected income |
| Outgo | Actual outgo is below expected outgo | Expected outgo minus actual outgo |
| Combined result | Actual net result exceeds expected net result | Actual net result minus expected net result |
This convention describes the insurer’s financial result before any subsequent decision about how surplus is used. A favorable variance is not, by itself, a calculation of a dividend or an amount available for distribution.
Deriving the Combined Variance Formula
The combined formula follows directly from the definition of net result. Let \(I_A\) and \(O_A\) be actual income and outgo, and let \(I_E\) and \(O_E\) be expected income and outgo. Actual net result is \(I_A-O_A\); expected net result is \(I_E-O_E\). Subtracting the latter from the former gives:
The first term is the income variance, favorable when actual income is higher. The second is the outgo variance, favorable when actual outgo is lower. Adding these signed components must equal the total variance. This equality is a useful arithmetic check and a way to explain which components caused the overall result.
Worked Example: Reconciling Income and Outgo Variances
A hypothetical participating policy block has expected receipts of $52,000, expected claim outgo of $38,000, and expected expenses of $9,000 for a period. Actual receipts are $52,000, actual claims are $41,000, and actual expenses are $8,000. Calculate the expected and actual net results, each component variance, and the total insurer variance.
Method. Calculate income minus outgo for each basis, then calculate signed component variances. Treat claims and expenses as outgo.
Expected net result:
Actual net result:
The receipts variance is \( \$52{,}000-\$52{,}000=\$0\). The claim variance, stated as expected outgo minus actual outgo, is \( \$38{,}000-\$41{,}000=-\$3{,}000\), unfavorable. The expense variance is \( \$9{,}000-\$8{,}000=\$1{,}000\), favorable.
Adding the component variances gives \( \$0-\$3{,}000+\$1{,}000=-\$2{,}000\). Directly, actual net result minus expected net result is \( \$3{,}000-\$5{,}000=-\$2{,}000\). The two calculations agree, so the total insurer variance is an unfavorable $2,000.
Common mistake avoided: Subtracting actual claims from expected claims gives the correct favorable-sign convention for outgo. Reversing that subtraction would incorrectly label the higher actual claims as favorable.
Match the Exposure Before Comparing Amounts
Expected amounts often depend on how much exposure occurred. Exposure might be measured in policy-years, covered lives, or another unit appropriate to the item. If actual exposure differs from a planned amount, comparing actual total outgo with an expected total calculated for the original plan mixes two effects: the difference in exposure volume and the difference in experience per exposure unit.
To assess experience per unit, apply the expected assumption to the actual exposure. Then compare actual outgo with that matched expectation. If the purpose is instead to explain the difference from an original business plan, show the volume effect separately rather than folding it into the experience variance.
Worked Example: Matching Expected Cost to Actual Exposure
A hypothetical block was planned to have 100 policy-years of exposure. The expected expense assumption is $300 per policy-year. Actual exposure is 84 policy-years, and actual total expenses are $24,360. Calculate the expense experience variance at actual exposure and explain why the original plan amount is not the appropriate direct comparator.
Method. Calculate expected expenses for the 84 actual policy-years, then subtract actual expenses from that matched expected amount because expenses are outgo.
Expected expenses for actual exposure are \(84\times \$300=\$25{,}200\). The expense variance is:
Thus, the expense experience is favorable to the insurer by $840. As a check, actual expense per policy-year is \( \$24{,}360/84=\$290\). The expected amount is $300 per policy-year, so the difference is $10 per policy-year; \(84\times \$10=\$840\), the same result.
The original plan amount is \(100\times \$300=\$30{,}000\). Comparing that directly with actual expenses would give \( \$30{,}000-\$24{,}360=\$5{,}640\), which is not the matched experience variance. It includes the effect of having 16 fewer policy-years than planned. That difference may be relevant to a plan-versus-actual analysis, but it should not be confused with the expense variance at the actual exposure.
Common mistake avoided: Using 100 planned policy-years as the expected exposure when the question asks about experience on the actual block. The correct matched basis is 84 policy-years.
Compare Cash Flows at a Common Valuation Date
When actual and expected cash flows occur at different times, nominal totals alone may not measure their financial impact consistently. If the comparison is explicitly on a present-value basis, value both sets of cash flows at the same date using the same interest assumption. This is a comparison of discounted outgo, not a claim that the timing difference is irrelevant: the discounting is precisely how timing enters the calculation.
For a cash flow \(C_t\) paid at the end of year \(t\), with effective annual interest rate \(i\), its present value at time 0 is \(C_t v^t\), where \(v=1/(1+i)\). For outgo, expected present value minus actual present value is the insurer’s favorable-sign variance.
Worked Example: Comparing Expected and Actual Payment Timing
A hypothetical insurer expects to pay two claims: $5,000 at the end of year 1 and $5,000 at the end of year 2. Actual claims instead total $9,000, paid at the end of year 1, with no other payment. Compare expected and actual claim outgo at time 0 using an effective annual interest rate of 5%.
Method. Discount every payment to time 0 at the same 5% rate, then calculate expected present value of outgo minus actual present value of outgo.
The discount factors are \(v=1/1.05=0.95238095\) and \(v^2=1/(1.05)^2=0.90702948\). Expected present value is:
Actual present value is:
Therefore, the present-value variance in claim outgo is \( \$9{,}297.05-\$8{,}571.43=\$725.62\), favorable to the insurer. The calculations retain unrounded discount factors until the final monetary amounts, which are rounded to the nearest cent.
Check. The expected nominal total is $10,000 and the actual nominal total is $9,000, but the present-value comparison also reflects that the actual amount is paid earlier. Both cash-flow patterns were discounted to the same date at the same rate.
Common mistake avoided: Comparing the expected present value with the actual nominal total, or discounting the two patterns using different rates. Neither comparison is on a consistent valuation basis.
A Practical Calculation Sequence
A reliable analysis can be organized as a short sequence. The first decisions are about definition and basis, not arithmetic: specify what result is being measured, the exposure it relates to, and when the financial amounts are valued. Then calculate and reconcile the signed differences.
List the relevant income and outgo items, and state whether the comparison is nominal or on a discounted basis.
Use the same period, exposure units, cash-flow definitions, and valuation date for expected and actual amounts.
For income use actual minus expected; for outgo use expected minus actual.
Add the component variances, compare with actual net result minus expected net result, and explain what a positive or negative total means for the insurer.
This framework is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus. In exam calculations, explicitly stating the basis and sign convention makes the result easier to verify and reduces the risk of a correct magnitude being given the wrong interpretation.
Common Traps
- Reversing the outgo sign. For an insurer, actual outgo above expected outgo is unfavorable. Use expected minus actual for the outgo variance.
- Comparing different exposures. Expected cost for a planned number of policies is not automatically comparable with actual cost for a different number. Recalculate expected cost at actual exposure when measuring experience per unit.
- Mixing periods or valuation dates. Align the period, and discount cash flows to the same date if the comparison is on a present-value basis.
- Adding favorable-looking component magnitudes without signs. Keep each component signed before summing; an unfavorable claim variance can outweigh a favorable expense variance.
- Calling every deviation a gain or loss without context. The sign describes the effect on the insurer’s result under the stated convention; it does not by itself determine any later allocation.
Check Your Understanding
For each question, use the insurer-perspective convention stated above and keep the comparison basis consistent.
- Expected income is $24,000 and expected outgo is $18,000. Actual income is $23,500 and actual outgo is $17,200. What is the total insurer variance, and is it favorable or unfavorable? (Answer: +$300, favorable.)
- A block has 60 actual policy-years. Expected cost is $450 per policy-year, and actual total cost is $26,100. Find the matched outgo variance. (Answer: $27,000 − $26,100 = +$900, favorable.)
- Expected outgo of $12,000 and actual outgo of $12,600 are both paid at the end of year 1. Using a nominal, undiscounted comparison, find the insurer variance. (Answer: $12,000 − $12,600 = −$600, unfavorable.)
- At an effective annual interest rate of 4%, expected outgo is $8,000 at the end of year 2 and actual outgo is $7,500 at the end of year 2. Compare present values at time 0. (Answer: \((\$8{,}000-\$7{,}500)/(1.04)^2=\$462.28\), favorable.)
- Expected income is $100,000, expected claims are $65,000, and expected expenses are $20,000. Actual amounts are $98,000, $70,000, and $18,000, respectively. Calculate the total variance and identify the component variances. (Answer: income −$2,000, claims −$5,000, expenses +$2,000; total −$5,000, unfavorable.)