Actuarial Notes

Classification Ratemaking

Study guides: Exam ASTAM, Exam 5, Exam 8, Exam GI-301

Classification Ratemaking is the segmentation of a book into rating classes and the derivation of relativities — the factors by which each class's rate differs from the base class — so that each risk pays according to its own expected cost.

\text{Relativity}_i = \frac{\text{Pure Premium}_i}{\text{Pure Premium}_{\text{base}}}

\text{Rate}_i = \text{Base Rate} \times \prod_k R_{i,k}

Why classify at all. In a competitive market an insurer that charges one average rate to a heterogeneous group loses the better risks to competitors that price them properly, and retains the worse ones — the adverse selection spiral. Classification is what keeps the pool viable, and Principle 3 requires the rate to reflect the individual risk transfer.

Criteria for a rating variable (Werner, after Finger's Risk Classification):

  • Statistical — statistical significance, homogeneity within levels, and credibility.
  • Operational — objective, inexpensive to obtain and verify, not manipulable by the insured.
  • Social — privacy, causality, controllability and affordability; these are where public and regulatory objections arise.
  • Legal — some variables are prohibited outright in some jurisdictions.

Univariate vs. multivariate.

  • A univariate (one-way) pure premium or loss ratio analysis attributes to a variable the effect of everything correlated with it. If young drivers also drive older cars, a one-way age relativity contains part of the vehicle-age effect, and rating both double-counts.
  • Minimum bias procedures were the historical bridge; modern practice uses a GLM, which estimates all variables simultaneously and reports each net of the others.
  • The loss ratio method for relativities has an advantage over the pure premium method: because its denominator is premium, it is already adjusted for the other rating variables in force, so it measures the residual effect.

Off-balance. Relativity changes almost always change the average rate. The base rate must be adjusted so that the combined effect equals the intended overall indication — see Rate Change.

Class Relativity from Pure Premiums Base class: 10{,}000 exposures, \$2{,}000{,}000 losses. Class A: 5{,}000 exposures, \$1{,}500{,}000 losses. The target loss ratio is 60\%.

Compute the relativity and the class rates.

Answer

\begin{align*}
\text{PP}_{\text{base}} &= \frac{\$2{,}000{,}000}{10{,}000} = \$200 \\[4pt]
\text{PP}_{A} &= \frac{\$1{,}500{,}000}{5{,}000} = \$300 \\[6pt]
\text{Relativity}_A &= \frac{\$300}{\$200} = 1.50
\end{align*}
\begin{align*}
\text{Base rate} &= \frac{\$200}{0.60} = \$333.33 \\[4pt]
\text{Class A rate} &= \$333.33 \times 1.50 = \$500.00
\end{align*}

Two checks before this becomes a filing: is Class A's 5{,}000 exposures enough to be credible at 1.50, and is the difference caused by the class variable or by something correlated with it that is also being rated?

Univariate Analysis Double-Counts An insurer rates on two variables. One-way pure premiums:

Pure premium Relativity
Youthful driver \$900 1.80
Adult driver \$500 1.00
Old vehicle \$780 1.56
New vehicle \$500 1.00

A GLM run on the same data gives a youthful relativity of 1.55 and an old-vehicle relativity of 1.20. Explain the gap and its consequence.

Answer The two variables are correlated: youthful drivers disproportionately drive old vehicles. The one-way youthful figure therefore contains part of the old-vehicle effect, and the one-way old-vehicle figure contains part of the youthful effect. Each is right about its own cell and wrong about the variable's own contribution.

Rating a youthful driver in an old vehicle:

\begin{align*}
\text{Univariate} &= 1.80 \times 1.56 = 2.81 \\
\text{GLM} &= 1.55 \times 1.20 = 1.86
\end{align*}

The univariate structure over-charges that cell by 51\% — and, because the relativities are balanced back to the overall level, correspondingly under-charges adult drivers in new vehicles.

The market consequence is predictable: a competitor using the multivariate result quotes the youthful/old-vehicle segment far cheaper and takes it, while sending its own adult/new-vehicle business here. The insurer keeps exactly the risks it has under-priced. Multivariate methods are not a refinement; on a correlated book they are a competitive necessity.