Why Analyze Experience Plan Sensitivity?
An experience modification depends on choices such as how much credibility to assign to an insured's history, how much of a large claim to recognize, and which years to include. A small parameter change can have a large premium effect for one risk and almost none for another. Sensitivity analysis makes those effects visible before a plan is adopted or revised.
The purpose is not to declare that a more responsive plan is automatically better. A parameter change can make modifications more predictive while also making them less stable from one renewal to the next. It can also redistribute premium among risks without changing the total premium collected. These are distinct consequences and should be measured separately.
A Simple Modification Framework
For the examples, let \(L\) be the insured's experience loss measure on the plan's selected basis, and let \(E\) be the expected loss measure on that same basis. Assume \(E>0\), let \(Z\) be a fixed credibility weight with \(0\leq Z\leq1\), and suppose the plan blends observed and expected losses. The modification is the blended loss divided by expected loss:
This is a simplified experience-modification model, not a universal rating-plan formula. It assumes the observed and expected amounts have already been put on a consistent basis, including any applicable exposure, trend, loss-limit, and classification adjustments. It does not describe the effect of a separate expense provision or of plan-specific caps and floors unless those are added explicitly.
The formula gives a direct interpretation of credibility sensitivity. Write \(R=L/E\). Holding the loss ratio fixed, the change in modification for a change from \(Z_1\) to \(Z_2\) is:
For example, if \(R<1\), increasing credibility lowers the indicated modification; if \(R>1\), it raises it. If \(R=1\), changing credibility alone has no effect. The result follows directly by subtracting \(1+Z_1(R-1)\) from \(1+Z_2(R-1)\). It is a local sensitivity result for a fixed \(R\), not a claim that the underlying experience will remain unchanged as a plan is redesigned.
Record the current parameter values, loss basis, expected losses, and resulting modifications or premiums.
For example, change the credibility weight while holding the experience period and loss limits fixed.
Recompute the affected loss measure and its matching expected-loss denominator where the parameter requires it.
Review modification changes, premium changes, dispersion across risks, and relevant predictive or stability measures.
Worked Example: Changing the Credibility Weight
Worked Example: Changing the Credibility Weight
A hypothetical insured has expected losses of $120,000 and experience losses of $84,000. The base premium before modification is $200,000. Compare credibility weights of 0.25, 0.50, and 0.75. Assume the loss basis and base premium stay fixed, and the final premium is the base premium times the modification.
Method. Use \(R=L/E\), then \(M=1+Z(R-1)\). The observed-to-expected ratio is:
At \(Z=0.25\):
At \(Z=0.50\):
At \(Z=0.75\):
Increasing \(Z\) from 0.25 to 0.50 reduces the modification by \(0.075\) and the indicated premium by $15,000.00. This agrees with the sensitivity formula: \((0.50-0.25)(-0.30)=-0.075\). The three premiums are checked by multiplying each modification by the stated $200,000 base premium.
Common mistake avoided: Treating increased credibility as an automatic premium increase. Here the loss ratio is below one, so giving the favorable experience more weight decreases the modification.
Loss Limits and Primary-Excess Split Points
A per-claim loss limit caps the amount of each claim recognized in the experience measure. For a claim amount \(x\) and limit \(c\), the capped amount is \(\min(x,c)\). A primary-excess split-point method instead separates each loss at a threshold \(s\): the primary amount is \(\min(x,s)\), and the excess amount is \(\max(x-s,0)\). A plan may assign the excess component a different weight.
Changing either parameter can change the experience numerator, but a fair comparison also needs a corresponding expected-loss basis. Comparing capped observed losses with uncapped expected losses, for instance, confounds the effect of the cap with a denominator mismatch. In practice, the expected losses must be calculated using the plan's applicable limit and split-point rules.
Worked Example: Comparing a Loss Limit and a Split Point
A hypothetical account has three observed claims, in thousands of dollars: 20, 70, and 180. Its matching benchmark claim amounts, on the same exposure basis, are 40, 80, and 120. Use fixed credibility \(Z=0.40\) and a base premium of $500,000. First compare per-claim caps of $100,000 and $150,000. Then compare a primary-excess method with excess weight \(q=0.25\) at split points of $50,000 and $100,000. All calculations below use thousands of dollars for losses.
Part A: Per-claim caps. At the $100,000 cap, the observed capped amounts are 20, 70, and 100, totaling \(190\). The benchmark capped amounts are 40, 80, and 100, totaling \(220\). Thus:
The indicated premium is \( \$500{,}000(0.945454\ldots)=\$472{,}727.27\), rounded to the nearest cent.
At the $150,000 cap, observed capped amounts are 20, 70, and 150, totaling \(240\). Benchmark capped amounts are 40, 80, and 120, totaling \(240\), not 270: each benchmark claim is already below $150,000. Therefore:
Raising the cap increases the indicated premium by $27,272.73. The calculation is checked against the benchmark amounts: at the higher cap, the observed and benchmark totals are both 240 thousand dollars, so the modification must be 1.
Part B: Primary-excess split points. For each claim \(x\), define its weighted amount as \(\min(x,s)+q\max(x-s,0)\). At \(s=50\), the observed primary amounts total \(20+50+50=120\), and the observed excess amounts total \(0+20+130=150\). The weighted observed total is \(120+0.25(150)=157.5\). The benchmark primary total is \(40+50+50=140\), and its excess total is \(0+30+70=100\), giving a weighted benchmark total of \(140+0.25(100)=165\). Hence:
At \(s=100\), the observed primary total is \(20+70+100=190\), and its excess total is \(0+0+80=80\). The weighted observed total is \(190+0.25(80)=210\). The benchmark primary total is \(40+80+100=220\), and its excess total is \(0+0+20=20\), giving a weighted benchmark total of \(220+0.25(20)=225\). Therefore:
For this split-point method, increasing the threshold from $50,000 to $100,000 lowers the indicated premium by $4,242.42. A higher threshold does not have a universal directional effect: the result depends on how the observed and benchmark losses are distributed across primary and excess layers.
Common mistake avoided: Applying a limit to observed claims but not to the matching expected claims, or treating a split point as if all dollars above it receive full weight. Here the excess weight is explicitly 0.25, so each excess dollar contributes one quarter of a dollar to the weighted measure.
Experience-Period Choices and Stability
Changing the experience period changes which observations enter the loss measure. A longer period can reduce the influence of a single unusual year, but it can also make the indication less responsive to a real change in risk. Before comparing periods, amounts from different years must be put on a consistent basis, and expected losses must reflect the exposures included in each period. The example below assumes those adjustments have already been made.
Worked Example: Adding Older Years to the Experience Period
A hypothetical insured is rated using the same credibility weight, \(Z=0.50\), and the same modification formula under either period. Adjusted observed and expected losses, in thousands of dollars, are:
| Year group | Observed losses | Expected losses |
|---|---|---|
| Most recent year | 90 | 100 |
| Second most recent year | 120 | 100 |
| Third most recent year | 150 | 100 |
| Fourth most recent year | 70 | 100 |
| Fifth most recent year | 80 | 100 |
Compare a three-year period with a five-year period. For the three-year period, observed losses total \(90+120+150=360\), and expected losses total \(100+100+100=300\). Thus \(R_3=360/300=1.20\), so:
For the five-year period, the two additional years contribute \(70+80=150\) of observed losses and \(100+100=200\) of expected losses. The totals are \(360+150=510\) observed and \(300+200=500\) expected. Therefore \(R_5=510/500=1.02\), and:
On a hypothetical $300,000 base premium, the indicated premiums are \( \$300{,}000(1.10)=\$330{,}000.00\) for three years and \( \$300{,}000(1.01)=\$303{,}000.00\) for five years, a decrease of $27,000.00. The five-year modification is closer to 1 because the additional years have a combined observed-to-expected ratio of \(150/200=0.75\), below the three-year ratio of 1.20. The totals and both premium multiplications verify the comparison.
Common mistake avoided: Averaging the annual loss ratios without regard to expected-loss volume. Here the correct combined ratio is total observed losses divided by total expected losses. An unweighted average of annual ratios can give small- and large-exposure years equal influence when their expected losses differ.
Interpreting Accuracy, Stability, and Premium Distribution
A sensitivity calculation describes how an output changes under a specified parameter change. It does not establish that the changed plan is more accurate. Accuracy must be judged against an appropriate outcome, such as later losses or another stated prediction target; a lower modification or a smoother premium pattern is not, by itself, evidence of better prediction.
Stability concerns variation across renewals. In the simplified formula, if \(R\) is a random experience ratio with fixed expected loss and fixed credibility, then:
This follows because \(M=1+Z(R-1)\), and adding a constant does not change variance while multiplying a random variable by \(Z\) multiplies its variance by \(Z^2\). Thus higher credibility amplifies the variation in the experience ratio, all else equal. A loss cap may reduce variation by limiting the influence of severe claims, but it may also reduce responsiveness to genuine differences in severe-loss risk. The formula does not account for changing \(E\), changing \(Z\), or interactions among parameters.
Premium distribution asks who pays more or less under the change. A portfolio-wide average can conceal substantial increases for one group and decreases for another. Useful summaries include the count and share of risks whose premiums rise, the distribution of percentage changes, and total indicated premium by risk segment. If premiums are subject to modification floors or caps, calculate both the uncapped indication and the actual capped premium: a floor or cap can make observed premium sensitivity zero even when the underlying indication changes.
A disciplined comparison therefore states the baseline, the parameter change, the fixed assumptions, and the measured outputs. Changing credibility, limits, split points, and period length all at once may produce a result, but it does not show which choice caused it. One-at-a-time analysis isolates effects; it should not be mistaken for a complete assessment of interactions or for a counterfactual evaluation of an implemented plan.
Check Your Understanding
For each problem, assume the stated simplified modification formula applies and that all losses and expected losses are on a consistent basis.
- An account has \(L=\$72{,}000\), \(E=\$90{,}000\), \(Z=0.40\), and base premium $150,000. Calculate the modification and indicated premium. (Answer: \(M=0.92\); premium = $138,000.00)
- For a fixed experience ratio with \(\operatorname{Var}(R)=0.09\), calculate the standard deviation of \(M\) at \(Z=0.20\) and \(Z=0.60\). (Answer: 0.06 at \(Z=0.20\) and 0.18 at \(Z=0.60\))
- Observed claims are 30, 90, and 140 thousand dollars; matching expected claims are 50, 100, and 100 thousand dollars. With \(Z=0.50\), calculate the modification using a $100,000 per-claim cap. (Answer: capped totals are 220 and 250 thousand dollars; \(M=0.94\))
- Observed primary and excess amounts at split point \(s=40\) thousand dollars are 100 and 80 thousand dollars. Matching benchmark primary and excess amounts are 120 and 40 thousand dollars. If excess weight is \(q=0.25\) and \(Z=0.50\), calculate the modification. (Answer: weighted amounts are 120 and 130 thousand dollars; \(M\approx0.9615\))
- A two-year period has observed losses of 240 and expected losses of 200 thousand dollars. Adding an older year with observed losses of 60 and expected losses of 100 thousand dollars gives a three-year period. With \(Z=0.50\), calculate both modifications and explain the direction of change. (Answer: \(M_2=1.10\), \(M_3=1.00\); the added year lowers the combined ratio to 1.00)