Calculate risk classification differential changes, including balancing back.

You change the youthful-driver relativity from 2.50 to 2.20 and senior from 1.40 to 1.50. Total written premium just moved, even though you only meant to redistribute. Balance back fixes that.

The rate equation. Most short-term lines price each policy as base rate times the product of class relativities.

$$\text{Rate}_i = B \times R_i, \qquad P_i = E_i \times B \times R_i$$

Total written premium is $P = B \sum_i E_i R_i$. The base rate $B$ anchors absolute level. The relativities $R_i$ carve policyholders into rating classes (age, territory, vehicle use, prior claims).

Why differentials drift. Indicated relativities come from a univariate or GLM analysis on recent loss experience. The new indicated $R_i^{new}$ seldom equals the in-force $R_i^{old}$. Class A may indicate 1.20 vs in-force 1.00. Class B may indicate 0.90 vs in-force 1.00. Even when one moves up and one moves down, the exposure-weighted average rarely stays put.

The off-balance factor. Hold the base rate fixed at $B^{old}$ and recompute premium with the new relativities.

Common mistakes

• Forgetting the $1+t$ factor. Revenue-neutral is one target. Non-zero $t$ needs $B^{old}(1+t)/OB$.
• Using historical exposures when projected exposures are materially different. Use the projection-period mix.
• Absorbing the off-balance by rescaling every $R_i^{new}$. This destroys indicated structure. Move $B$, not $R$.

Bottom line

• Class differentials (relativities) multiply a base rate to produce class rates; changing any differential shifts total premium.
• Off-balance factor $OB = \sum E_i R_i^{new} / \sum E_i R_i^{old}$ quantifies how much total premium moved at the old base rate.
• Balance back divides the proposed base by $OB$: revenue-neutral new base = old base / $OB$; targeted new base = old base $\times (1+t)$ / $OB$.
• Balance back is a base-rate adjustment applied after differentials are set; never absorb the off-balance by rescaling relativities.

Exam shortcut

Build two columns: $E_i R_i^{old}$ and $E_i R_i^{new}$. Sum each. Ratio is $OB$. Then $B^{new} = B^{old}(1+t)/OB$. Three steps, done. If the question gives a "+$t$%" overall target, multiply $B^{old}$ by $1+t$ first, then divide by $OB$. Forgetting the multiplier loses the target every time. Sanity check direction: if $OB > 1$, then $B^{new} < B^{old}$ at $t = 0$; if $OB < 1$, then $B^{new} > B^{old}$. If the number moves the wrong way, you inverted the ratio.

The full lesson (about 1,777 words, 12 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• 6e

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