Rating Variable

Study guide: Exam 8

Rating Variable is a characteristic of the insured, the insured object or the coverage — driver age, territory, construction, payroll class, deductible — whose level decides which rating factor a risk receives. Together, the rating variables define the rating classes.

\text{Rate}_i = B \times \prod_{k=1}^{K} R_k(v_{ik})

B_{\text{new}} = B_{\text{old}} \times \frac{\bar{R}_{\text{old}}}{\bar{R}_{\text{new}}}

• v_{ik} is risk i's level of variable k, R_k that level's factor and B the base rate. \bar R is the exposure-weighted average factor across the book, and the second formula is the base-rate off-balance that keeps the overall rate level unchanged when a variable's factors are revised.
• What makes a good variable. It must be statistically related to expected losses; a causal story strengthens the case, but ASOP No. 12 does not require one. It should be objective and hard to manipulate, practical to collect and verify, and legal (Risk Classification Restrictions, Unfair Discrimination) and socially acceptable. See the Werner criteria under Classification Ratemaking.
• Form in a model. A continuous variable may be transformed, binned or fitted with polynomials or splines. A categorical one uses the levels of a categorical predictor, possibly with interactions. Variables with hundreds of levels need grouping, credibility (random effects) or a separate model — see High Dimensional Variables and Workers Compensation Classification.
• Evaluating a model's variables.
  • Benchmark: compare against the current plan or a competing model, for example with a double lift chart (Model Benchmarking).
  • Recalibrate: refresh the coefficients on newer data while keeping the structure. Models lose accuracy as the world changes, so they are also rebuilt periodically.
  • Implement: the variable must be obtainable at the point of sale, and the resulting factors are capped, rounded, off-balanced and checked for dislocation (Model Implementation).
• Coverage options (deductible, limit) are variables the insured chooses. Their factors belong outside the fitted model, set by loss elimination and entered as offsets, so that selection effects do not leak into them.

Should Self-Reported Mileage Be a Rating Variable? A personal auto insurer finds that self-reported annual mileage is strongly predictive in its GLM. Evaluate it as a rating variable.

Answer

• Actuarial: it passes. Exposure to driving plausibly drives loss frequency, the signal is statistically significant, and it is not already captured by other variables.
• Operational: this is the weak point. The insured reports the figure, has every reason to under-report it, and the insurer cannot cheaply verify it. A variable the insured controls by statement is manipulable. That erodes the factor over time, as low-mileage declarations rise without losses following.
• Social: it passes. The variable is controllable by the insured, causally intuitive and not a proxy for a protected class.
• Legal: it is generally permitted, subject to local rules.

Adopt it only in a verifiable form — odometer readings at renewal or telematics — or with a moderated factor and audit rights. A strong model coefficient does not by itself make a usable rating variable.

Recalibrating One Variable Without Changing the Rate Level Vehicle-age factors are refitted on newer data. New vehicles (base) stay at 1.00 with 30\% of exposure. Ages 3–7 move from 0.88 to 0.92 (45\%), and ages 8+ move from 0.80 to 0.85 (25\%). Hold the overall rate level constant. Find the base-rate adjustment and each level's premium change.

Answer

\begin{align*}
\bar{R}_{\text{old}} &= 0.30 + 0.45(0.88) + 0.25(0.80) \\
&= 0.8960 \\[4pt]
\bar{R}_{\text{new}} &= 0.30 + 0.45(0.92) + 0.25(0.85) \\
&= 0.9265 \\[4pt]
\frac{B_{\text{new}}}{B_{\text{old}}} &= \frac{0.8960}{0.9265} \\
&= 0.9671
\end{align*}

The premium changes by level are:

• New vehicles: 0.9671 - 1 = -3.3\%
• Ages 3–7: \tfrac{0.92}{0.88} \times 0.9671 - 1 = +1.1\%
• Ages 8+: \tfrac{0.85}{0.80} \times 0.9671 - 1 = +2.8\%

Weighted by current premium, these net to zero. A "revenue-neutral" recalibration still moves individual premiums: new-vehicle owners get a decrease they did not earn through their own factor. That dislocation is what gets reviewed before implementation.