Reperformance and Calculation Checks

Why Reperformance Matters

A calculation can be documented, reviewed, and still be implemented incorrectly. An exposure may be converted using the wrong number of days, a model coefficient may be applied to the wrong category, or rounded rates may fail to reproduce the intended aggregate. Reperformance is the independent reconstruction of selected calculations from their stated inputs and methods. Its purpose is to establish whether the reported result follows from those inputs—not to assume that an output is correct because it came from a model or spreadsheet.

A useful check starts with a specific expected result: for example, an earned-exposure total, a fitted claim count, or an exposure-weighted average rate. The reviewer independently applies the documented rule, compares the result with the recorded output, and investigates any difference. The comparison should use adequate precision and a tolerance appropriate to the calculation. Rounding a displayed number may explain a small difference; it does not excuse a material difference in the underlying calculation.

Definition: A reperformance check independently recalculates a selected transformation or result from its inputs, using the stated method and assumptions. A calculation check is a test of a particular computation, such as verifying a total, an offset, or a relativity. Neither term, by itself, means that the model is appropriate for its intended use.

A disciplined check records the input values, units, applicable assumptions, calculation steps, expected output, observed output, and disposition of any difference. When possible, use more than one kind of check: recalculate selected individual records, then reconcile segment or portfolio totals. A total can agree by coincidence even when individual records are wrong in offsetting directions.

Reperforming a Model Calculation

Consider a frequency model with a Poisson response, log link, and exposure offset. For policy segment $i$, let $N_i$ be the claim count and $e_i$ the exposure in policy-years. Suppose counts are conditionally independent and $\mathbb{E}[N_i\mid x_i]=\mu_i$, with $\log(\mu_i)=\log(e_i)+\beta_0+\beta_1x_i$. Here $x_i$ is an indicator equal to 1 for a specified class and 0 otherwise; the model coefficients and exposure are treated as given. This calculation checks the model’s stated output, not whether the distribution, predictors, or fitted coefficients are suitable.

\mu_i=e_i\exp(\beta_0+\beta_1x_i)

The equation follows by exponentiating both sides of the log-link equation. The offset contributes $\exp(\log(e_i))=e_i$, so the fitted claim count is exposure multiplied by the fitted frequency per policy-year. Omitting the offset gives a rate per policy-year where the requested quantity is a count; applying exposure twice overstates the count.

Worked Example: Reproducing Fitted Claim Counts from Day-Based Exposure

A hypothetical insurer converts covered days to policy-years using 365 days per year. Segment A has 7,300 covered days and segment B has 10,950 covered days. A Poisson frequency model with log link has $\beta_0=\log(0.05)$ and $\beta_1=\log(1.4)$, where $x=0$ for A and $x=1$ for B. Reproduce the fitted frequency and expected count for each segment, then check the total. Assume the day counts are complete, the stated 365-day convention applies, and each segment’s exposure is the sum of its covered days divided by 365.

Method and unit check. Convert days to policy-years before applying the offset: $e_A=7{,}300/365=20$ and $e_B=10{,}950/365=30$. The model’s baseline rate is $\exp(\beta_0)=0.05$ claims per policy-year. For B, the rate multiplier is $\exp(\beta_1)=1.4$.

Segment A. Since $x_A=0$, its fitted frequency is $\exp(\log(0.05)+\log(1.4)\cdot0)=0.05$. Its expected claim count is $20(0.05)=1.00$.

Segment B. Since $x_B=1$, its fitted frequency is $\exp(\log(0.05)+\log(1.4))=0.05(1.4)=0.07$. Its expected claim count is $30(0.07)=2.10$.

Aggregate check. The independently reproduced total is $1.00+2.10=3.10$ expected claims. Equivalently, total exposure is $20+30=50$ policy-years, and the exposure-weighted fitted frequency is $3.10/50=0.062$ claims per policy-year. Multiplying back gives $50(0.062)=3.10$, so the two calculations reconcile.

Common mistake avoided. Treating the rates 0.05 and 0.07 as segment claim counts would give $0.12$, which omits the offset’s exposure contribution. Using 360 rather than 365 days would also change both exposures and the reproduced total. The unit conversion must follow the stated convention.

Checking Indicated Relativities and Rounding

An indicated class rate is often the product of a base rate and a class relativity. A basic aggregate check is to multiply each class rate by its exposure and sum the resulting premium or expected cost. The exposure-weighted average class rate is the aggregate divided by total exposure. These checks are especially useful when results are normalized or displayed at reduced precision.

Worked Example: Reconciling Class Rates Before and After Rounding

A hypothetical rating analysis has a base rate of $500 per exposure unit. Class A has relativity $2/3$ and 100 exposure units; class B has relativity $7/6$ and 200 exposure units. Reproduce the unrounded class rates and aggregate indication, then test the effect of displaying rates to the nearest cent. Assume the stated base rate and relativities are exact for this calculation.

Method. Multiply the base rate by each relativity. For A, the unrounded rate is $\$500(2/3)=\$333.333333\ldots$ per exposure unit. For B, it is $\$500(7/6)=\$583.333333\ldots$ per exposure unit.

Unrounded aggregate. Class A contributes $100(\$333.333333\ldots)=\$33{,}333.333333\ldots$. Class B contributes $200(\$583.333333\ldots)=\$116{,}666.666667\ldots$. The sum is exactly $\$150{,}000$ because the unrounded rates are $ \$1{,}000/3$ and $\$1{,}750/3$, respectively. Total exposure is $100+200=300$, so the exposure-weighted average rate is $\$150{,}000/300=\$500$.

Cent-rounded aggregate. The displayed rates are $333.33 and $583.33. Their exposure-weighted amounts are $100(\$333.33)=\$33{,}333$ and $200(\$583.33)=\$116{,}666$. Thus the cent-rounded rates produce $\$149{,}999$ in aggregate and a weighted average of $\$149{,}999/300=\$499.9967$ per exposure unit, approximately. The unrounded rates, not the cent-rounded rates, produce the stated $150,000 aggregate and $500 weighted average.

Common mistake avoided. Rounding each class rate to cents and then asserting that the rates still produce $150,000 is incorrect. Recompute the aggregate from the rounded rates: the one-dollar difference is real at that displayed precision. A production check should establish whether calculations use unrounded rates and round only for display, or whether rounded rates are actually used downstream.

If class relativities are normalized, an additional check is that their exposure-weighted average equals the intended normalization target, often 1. This is a calculation property of the chosen weights and normalization method; it does not establish that the classes or relativities are actuarially appropriate.

Reproducing a Premium Indication

A calculation check should preserve the stated basis of each provision. For example, a fixed expense stated per exposure is added to the loss cost per exposure. A variable expense and a profit provision stated as percentages of premium are different: together they are portions of the indicated premium, so the remaining premium share funds losses and fixed expenses. The following formula applies under those assumptions and when the combined variable expense and profit proportions are less than 1.

\text{Indicated rate} = \frac{\text{loss cost per exposure}+\text{fixed expense per exposure}} {1-\text{variable expense proportion}-\text{profit proportion}}

Worked Example: Checking the Denominator in a Rate Indication

For a hypothetical product, expected losses are $240,000 for 600 exposure units. Fixed expenses are $40 per exposure unit. Variable expenses are 20% of premium and the target profit provision is 5% of premium. Reproduce the indicated rate per exposure unit and verify the premium allocation. Assume all amounts use the same period and exposure basis, and both percentages are applied to indicated premium.

Method. The loss cost per exposure is $\$240{,}000/600=\$400$. Add fixed expenses of $40 to obtain $440 per exposure that must be funded by the share of premium remaining after variable expenses and profit. That share is $1-0.20-0.05=0.75$. Therefore the indicated rate is $\$440/0.75=\$586.666666\ldots$ per exposure unit, or $586.67 when rounded to cents.

Check the unrounded indication. Total premium is $600(\$586.666666\ldots)=\$352{,}000$. Variable expenses are $0.20(\$352{,}000)=\$70{,}400$, and the profit provision is $0.05(\$352{,}000)=\$17{,}600$. Losses plus fixed expenses total $\$240{,}000+600(\$40)=\$264{,}000$. The allocation reconciles: $\$264{,}000+\$70{,}400+\$17{,}600=\$352{,}000$.

Common mistake avoided. Dividing $440 by $1+0.20+0.05=1.25$ gives $352 per exposure, but this incorrectly treats premium-based provisions as additions to the numerator. The proper equation is $R=440+0.25R$, so $0.75R=440$ and $R=440/0.75$. The formula would not apply as written if the expense or profit bases were different from premium.

Designing Useful Reperformance Tests

A strong test case is selected to expose a particular failure mode. Testing only a typical record can miss errors involving category boundaries, zero values, unusual exposures, or rounding. Reperformance should be selective but purposeful: test cases should be traceable to a documented input and should have an independently computed expected result.

Fix the scope and inputs.
Identify the transformation or calculation being checked, the input version, units, applicable model parameters, and any exclusions or assumptions.

Choose discriminating test cases.
Include ordinary cases and cases that test boundaries or likely errors, such as a zero indicator, a nonzero indicator, a short exposure period, or a value that rounds at the stated precision.

Calculate independently.
Reproduce the stated calculation from the documented method. Keep intermediate values and retain sufficient precision so that a display-rounding difference can be distinguished from a calculation error.

Compare and explain differences.
Compare record-level and aggregate results. Document the size and source of each discrepancy; do not label a difference immaterial solely because it appears small.

Record the conclusion and evidence.
State what was reproduced, the expected and observed outputs, the precision or tolerance used, and whether any difference was resolved. Preserve enough detail for another reviewer to repeat the check.

Reperformance is most informative when paired with reconciliation. For example, if a total exposure is wrong, independently recomputing the total may identify the discrepancy, while checking several underlying records can locate its cause. Conversely, matching totals alone do not prove that each transformation was correctly applied. A good check ties the numerical result back to the data and the documented calculation rule.

Common Calculation Traps

Mixing units. Confirm whether exposure is in policy-years, days, or another unit before applying an offset or rate.
Applying an adjustment twice or not at all. Trace each factor from input through output and verify where it enters the calculation.
Confusing a rate with an aggregate amount. Multiply by exposure when the requested result is an expected count, premium, or loss total.
Rounding too early. Keep unrounded values for intermediate calculations unless the specified process requires rounding at that stage; explicitly check the impact of rounded production values.
Using a percentage with the wrong base. A percentage of premium belongs in the denominator under the assumptions shown, while an amount per exposure belongs in the per-exposure numerator.

Key takeaway: A defensible calculation check is reproducible from stated inputs, follows the stated units and method, retains enough precision to assess rounding, and reconciles its result to an appropriate total. An unexplained difference is a finding to investigate, not a result to smooth away.

Check Your Understanding

For each problem, state the calculation basis and check the result using the relevant aggregate or unit reconciliation.

A hypothetical frequency model has a fitted rate of 0.08 claims per policy-year for a class with 25 policy-years of exposure. Reproduce its expected claim count and explain what happens if the rate is mistakenly reported as the count. (Answer: 2.00 expected claims; reporting 0.08 omits the exposure offset.)
A base rate is $450 per exposure unit. Two classes have relativities 0.8 and 1.2, with exposures 80 and 120. Calculate the class rates and aggregate indication, then check the weighted average. (Answer: Rates are $360 and $540; aggregate is $93,600 and weighted average is $468 per exposure unit.)
A portfolio has 10,950 covered days, converted using 365 days per policy-year. Its indicated frequency is 0.06 claims per policy-year. Reproduce the exposure and expected count. (Answer: 30 policy-years and 1.80 expected claims.)
Expected losses are $310 per exposure unit, fixed expenses are $25 per exposure unit, variable expenses are 18% of premium, and profit is 7% of premium. Calculate the indicated rate and, for 500 exposure units, the total indicated premium. (Answer: Rate is $\$335/0.75=\$446.6667$, or $446.67 to cents; unrounded total premium is $223,333.33 to cents.)
Two classes have raw relativities 0.75 and 1.25, with exposures 40 and 60. Normalize them so their exposure-weighted average is 1, then apply a $420 base rate. Check the indicated aggregate using unrounded values. (Answer: Weighted raw average is 1.05; normalized relativities are $5/7$ and $25/21$; rates are $300 and $500; aggregate is $42,000 and weighted average is $420.)

What You'll Learn