Why Smooth Adjacent Class Factors?
Factors for ordered classes—such as bands of annual equipment use or ranges of insured value—may be expected to change gradually. Yet indicated factors can jump up and down because experience is limited, large claims are unevenly distributed, or estimates are otherwise imprecise. Smoothing can reduce unsupported fluctuations and make a rating structure more stable. It is not a reason to assume that neighboring classes have the same risk: a real difference may be large, and a smooth-looking pattern may still be wrong.
The decision therefore has two parts. First, assess whether irregularities are plausibly explained by estimation noise or whether the data support the differences. Second, if smoothing is justified, specify a reproducible rule and check its effect on the overall factor average. The method should be documented, not chosen after trying alternatives until the factors look appealing.
Smoothing is applied to relativities on the scale used by the rating plan. A simple, transparent choice is a local moving average. For each class, average its indicated factor with those of nearby classes; at an endpoint, use the available adjacent classes. The local averaging weights may be equal or may reflect the relative precision of the estimates. The choice, including how endpoints are handled, must be stated.
A moving average does not generally preserve the exposure-weighted average. A separate multiplicative rebase can restore it. Define \(h_j\) as the locally averaged factor, and let \(n_j\) be the exposures used to represent the portfolio mix. Then set
The weights used to average neighboring estimates and the exposures used to preserve the portfolio average need not be identical. For example, inverse-variance weights may be more appropriate for the local estimates, while the actual projected exposure mix is appropriate for the portfolio-average constraint. State both sets of weights when they differ.
Assessing Whether a Difference Is Noise
The size of an adjacent-factor difference alone does not show whether it is credible. If standard errors are available, a useful diagnostic is the difference divided by its standard error. For estimates \(r_j\) and \(r_{j+1}\), the variance of their difference is
If the estimates are independent, the covariance term is zero. If they share data, a base, or a normalization constraint, independence may not be reasonable. Ignoring a material covariance can misstate the uncertainty in the contrast. A standardized contrast is a diagnostic, not an automatic decision rule: the appropriate interpretation depends on the estimation method, the number of comparisons, and the intended use of the factors.
Also examine the pattern across the ordered classes and, where available, across experience periods. A single reversal between neighbors may be weak evidence of a real reversal if its uncertainty is large. A consistent progression, or a sharp difference supported by substantial experience, deserves more protection from smoothing. The experience volume by itself is not a complete measure of precision: claim variability, exposure homogeneity, and the method used to estimate the factors matter too.
Worked Example: Moving-Average Smoothing With a Portfolio Rebase
A hypothetical rating plan has five ordered equipment-use classes. The indicated factors are \(0.80, 1.10, 0.90, 1.30, 1.20\), with projected exposures \(100, 200, 300, 200, 100\). Apply an equal-weight, three-class moving average, using two classes at each endpoint. Then rebase to preserve the exposure-weighted average. Report factors to four decimal places.
Method and local averages. Let \(h_j\) be the mean of the indicated factor for class \(j\) and its available immediate neighbors. Thus,
Original weighted average. The total exposure is \(100+200+300+200+100=900\). The exposure-weighted factor total is
The original weighted average is \(950/900=1.055555\ldots\). For the preliminary smoothed factors, the weighted total is
So their weighted average is \((2890/3)/900=1.070370\ldots\), which is not the original average. The required multiplier is
Rebased factors. Multiplying each preliminary factor by \(c\) gives \(s_1=0.95(285/289)=0.9369\), \(s_2=(14/15)(285/289)=0.9204\), \(s_3=1.10(285/289)=1.0848\), \(s_4=(17/15)(285/289)=1.1176\), and \(s_5=1.25(285/289)=1.2327\).
Check. The rebased weighted total is \(c(2890/3)=950\), so the weighted average is \(950/900=1.055555\ldots\), the same as before smoothing. The result is less locally erratic, but that alone does not establish that this particular degree of smoothing is appropriate.
Common mistake avoided: Reporting the moving averages without rebasing would raise the portfolio-weighted factor average from \(1.055555\ldots\) to \(1.070370\ldots\). Local smoothing and average preservation are separate steps.
Selective Pooling When Only One Irregularity Is Weak
A full moving average can change factors in classes whose estimates are already well supported. An alternative is selective adjacent pooling: combine only neighboring estimates for which the observed contrast is weak relative to its uncertainty, and leave better-supported contrasts intact. For equal-precision estimates, their pooled value is their arithmetic mean. With unequal precision, a precision-weighted mean is more appropriate, provided the precision weights are justified and documented.
Worked Example: Smoothing a Weak Middle Contrast
A hypothetical four-class plan has indicated factors \(0.80, 1.05, 1.00, 1.35\). Their standard errors are \(0.08, 0.07, 0.07, 0.09\), respectively. Assume the estimates are independent and each class has the same projected exposure. Use adjacent standardized contrasts to decide whether to pool the middle pair, then check average preservation.
Method. For independent estimates, divide the absolute difference by the square root of the sum of their variances. For classes 1 and 2,
For classes 2 and 3, the contrast is
For classes 3 and 4, it is
The middle contrast is small relative to its estimated uncertainty, while the two outer contrasts are materially larger. These diagnostics do not impose a universal cutoff, but they support investigating the middle pair for selective pooling rather than smoothing all four factors.
Pool and check. The middle estimates have equal standard errors, so their equal-precision pooled value is \((1.05+1.00)/2=1.025\). The revised factors are \(0.80, 1.025, 1.025, 1.35\). Before pooling, their equally weighted average was \((0.80+1.05+1.00+1.35)/4=4.20/4=1.05\). After pooling, it is \((0.80+1.025+1.025+1.35)/4=4.20/4=1.05\). No rebase is needed in this example.
Common mistake avoided: Smoothing every class merely because the factors are ordered would also pull down the high last factor and pull up the low first factor, despite their larger adjacent contrasts. The documented selective rule changes only the weak middle contrast.
When Smoothing Would Hide a Supported Pattern
Ordered classes do not have to have equal or nearly equal factors. A steadily rising pattern may represent a real increase in expected cost across the class definitions. Smoothing such a pattern can reduce the indicated differences even when the estimates are precise. In that case, retaining the unsmoothed factors may be the more defensible choice.
Worked Example: Do Not Smooth a Supported Progression
A hypothetical three-band rating plan has factors \(0.72, 1.00, 1.28\), with standard error \(0.05\) for each factor. Assume the estimates are independent and the three classes have equal projected exposures. Compare each adjacent contrast with its uncertainty, and calculate what an endpoint-aware moving average would do.
Assess the contrasts. Each adjacent difference is \(0.28\), and the standard error of either difference is \(\sqrt{0.05^2+0.05^2}=\sqrt{0.005}=0.0707107\ldots\). Therefore, each standardized contrast is
Both adjacent differences are large relative to their estimated standard errors. This is evidence against treating the visible progression as a small random fluctuation, subject to the limits of the uncertainty assumptions and the broader evidence.
Quantify the possible distortion. The endpoint-aware three-class moving average produces \(h_1=(0.72+1.00)/2=0.86\), \(h_2=(0.72+1.00+1.28)/3=1.00\), and \(h_3=(1.00+1.28)/2=1.14\). The original equal-exposure average is \(1.00\), and the smoothed average is also \(1.00\), so the method preserves the average without rebasing. But the factors change from \(0.72, 1.00, 1.28\) to \(0.86, 1.00, 1.14\), substantially compressing the supported class differences.
Decision. Do not apply this smoothing rule solely to make the sequence look less steep. The diagnostic evidence favors retaining the progression unless additional analysis identifies a reason to distrust it. Preserving the average is necessary for this chosen approach when required; it is not proof that the revised factors are appropriate.
Common mistake avoided: A smooth sequence is not automatically a better estimate. Smoothing can remove real risk differences as well as random variation.
Documentation and Practical Checks
A defensible smoothing decision records what was considered and why. At a minimum, document the original indicated factors, class order, exposure or precision weights, uncertainty diagnostics, smoothing rule, endpoint treatment, and final average check. If several reasonable rules are considered, show their effects rather than selecting one solely because it produces a preferred pattern.
Also state whether factors were estimated on a common basis. Estimates normalized to an overall average or rebased to a chosen base class can share a constraint, and their uncertainties may therefore be dependent. When this matters, use the covariance of the factor estimates in the adjacent-contrast calculation or use an estimation method that directly accounts for the dependence. A standard error comparison based on an unsupported independence assumption can give misleading evidence.
The exposure weights in the rebase should represent the portfolio mix for which the average is to be preserved. If projected business mix changes, the weighted average under the new mix can differ even though each class factor is unchanged. Conversely, changing the weights simply to make the rebase convenient does not preserve the intended portfolio average.
Check Your Understanding
For each problem, use the stated weights and smoothing rule; answers are provided for checking.
- A hypothetical three-class plan has factors \(0.90, 1.20, 1.10\) and equal exposures. Apply a three-class moving average to the middle class and two-class averages at the endpoints. What are the preliminary factors and their average? (Answer: Preliminary factors \(1.05, 1.0667, 1.15\); preliminary average \(1.0889\).)
- Two independent adjacent factors are \(0.96\) and \(1.08\), with standard errors \(0.04\) and \(0.05\). Calculate the standardized contrast to two decimals. (Answer: \(0.12/\sqrt{0.04^2+0.05^2}=1.87\).)
- Four equally exposed classes have factors \(0.80, 1.00, 1.10, 1.50\). Selectively pool only classes 2 and 3 using their arithmetic mean. What are the revised factors, and is the equally weighted average preserved? (Answer: \(0.80, 1.05, 1.05, 1.50\); yes, both averages are \(1.10\).)
- Three classes have factors \(0.85, 1.00, 1.25\), exposures \(100, 200, 100\), and preliminary smoothed factors \(0.925, 1.0333, 1.125\). Calculate the original weighted average and the rebase multiplier using the displayed preliminary factors. (Answer: Original weighted average \(1.025\); preliminary weighted average \(1.0292\); multiplier \(1.025/1.0292\approx0.9959\).)
- Adjacent independent factor estimates are \(1.10\) and \(1.22\), each with standard error \(0.03\). A separate pair is \(1.10\) and \(1.22\), each with standard error \(0.10\). Calculate both standardized contrasts and state which pair has less support for a real difference based on this diagnostic. (Answer: The contrasts are \(2.83\) and \(0.85\), respectively; the second pair has less support for a real difference based on this diagnostic.)