Smoothing Relativities Across Neighboring Territories

Why Smooth Territory Relativities?

Territory indications can vary sharply from one area to the next, especially when local experience is limited. Some variation may reflect genuine differences in expected cost; some may be random fluctuation. Smoothing is a way to moderate variation between areas judged to be related. It does not establish that neighboring territories have the same risk, and adjacency alone is not evidence that they do.

The method must therefore make its assumptions visible. The analyst specifies which territories count as neighbors, how strongly each neighbor contributes, and how much smoothing to apply. The resulting factors should then be checked against the rating objective, the underlying experience, and plausible geographic or underwriting differences. This topic is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

The preliminary relativities may already reflect the experience measure, credibility approach, and complement selected for the analysis. Smoothing is a separate geographic adjustment to those inputs; it should not silently replace those choices or count the same information twice. In particular, territory factors are not smoothed merely because adjacent areas can be drawn on a map.

A One-Pass Neighbor-Averaging Rule

One transparent rule blends each preliminary relativity with a weighted average of its neighbors. For territory \(j\), let \(a_{jk}\) be the weight assigned to neighbor \(k\), where the weights are nonnegative and sum to one across \(N(j)\). Let \(0\leq\alpha\leq1\) be the smoothing strength. Then

When \(\alpha=0\), there is no smoothing; when \(\alpha=1\), the preliminary factor is replaced by the weighted neighbor average. Values between zero and one produce a convex blend, so the result lies between the territory’s own indication and its neighbor average. This rule is a one-pass calculation: all neighbor averages use the preliminary \(r_k\), not factors updated earlier in the calculation. Repeatedly updating factors would be a different procedure and must be stated as such.

The rebalance in this example multiplies all smoothed factors by one constant. It preserves their ordering and their ratios to one another, but changes their absolute levels. The appropriate target depends on the rating plan’s balancing requirement; it should not be assumed to be 1.00 without checking that requirement.

Choosing the Neighborhood Is an Actuarial Judgment

A geographic neighborhood is defensible when location plausibly conveys information about expected cost after considering the other features already reflected in the indications. Potential reasons for similarity could include a continuous area with comparable exposure conditions. Potential reasons against it include a major geographic barrier, a sharp change in urban density, different weather or traffic patterns, or a boundary that separates substantially different risks. These are hypotheses to evaluate, not facts established by a shared boundary.

When local experience is sparse, nearby territories can be considered as a complement, but proximity should not be used as an automatic substitute for assessing the selected complement or credibility. A defensible analysis asks whether the relationship is plausible for the insured population and coverage, whether the chosen neighbors are consistent with that relationship, and whether the degree of smoothing is proportionate to the evidence. If the geographic relationship is weak or contradicted by relevant risk differences, the analyst should reduce the smoothing, change the neighbor set, or leave the factors unsmoothed rather than force similarity.

One practical way to represent unequal neighbor influence is to assign weights \(a_{jk}\), such as weights that give more influence to a genuinely similar adjacent territory than to a less similar one. The weights must be specified and normalized to sum to one for each territory with neighbors. A territory with no selected neighbors has no neighbor average under this rule; the analyst must not invent one by treating every territory as a neighbor.

Penalized Smoothing: Balancing Fit and Neighbor Similarity

A second formulation makes the tradeoff explicit. Let \(w_j>0\) measure the relative precision assigned to preliminary indication \(r_j\), and let \(c_{jk}\geq0\) be the strength of an included neighbor link. A penalized method chooses factors \(q_j\) to minimize a weighted measure of departure from the indications plus a penalty for differences across linked territories:

Each pair \(\{j,k\}\) is included once in the second sum. The first term rewards fidelity to the preliminary indications; the second discourages differences across selected links. Larger \(\lambda\) imposes stronger similarity. This is a different rule from one-pass averaging: \(\lambda\) is a penalty strength relative to the \(w_j\) and \(c_{jk}\), not a direct percentage blend. The weights need an actuarial rationale and should reflect the reliability assigned to each indication, not be chosen just to obtain a desired result.

When Smoothing Helps—and When It Does Not

Smoothing can reduce unstable jumps when adjacent areas are plausibly related and local indications contain substantial random variation. It can also obscure a real difference if the neighbor set or penalty is poorly chosen. Neither a visually smooth map nor a single numerical fit demonstrates that the factors are actuarially appropriate.

Before applying a rule, make the following checks:

• Define the input. State whether the starting factors are raw indications or already adjusted for credibility, and what they represent.
• State the graph and rule. Identify included neighbors, weights, smoothing strength, and whether the calculation is one-pass or iterative.
• Assess plausibility. Ask whether proximity is meaningful for this coverage and population, and whether barriers or sharp risk changes undermine the link.
• Check the rating constraint. Recalculate the required weighted average after smoothing and rebalance if the rating objective requires it.
• Retain judgment. A mathematical smoothing rule cannot resolve an unsupported geographic assumption. If the relationship is not defensible, do not impose it.

The central distinction is between using geography as evidence and using geography as a convenient label. Proximity may be a useful complement to sparse local experience when the underlying risk relationship is credible. It is not, by itself, a reason to move one territory’s factor toward another’s.