Mix of Business

Study guide: Exam 5

Mix of Business is the distribution of a book across segments — territory, class, limit, deductible, coverage, distribution channel. A shift in the mix changes aggregate results even when nothing about any individual segment has changed, and it is one of the standard explanations an actuary must rule out before concluding that costs or adequacy have moved.

\overline{\text{LR}} = \sum_i w_i \times \text{LR}_i

• The aggregate is a weighted average, so it moves with the weights w_i as well as with the segment results \text{LR}_i. Diagnosing which has moved requires segment-level data; the aggregate alone cannot distinguish them.
• In ratemaking, a mix shift shows up as premium trend (average premium moving without a rate change), as apparent loss trend, and as a change in the overall loss ratio. The response is at the relativity level, not the overall rate level — raising everyone's rate because the mix moved over-corrects the segments that did not change.
• In reserving, a mix shift breaks the homogeneity assumption behind the triangle. If the segments have different emergence patterns, the blended development factors drift as the weights drift — and the triangle gives no signal that this is what is happening.
• Mix shifts are frequently endogenous to pricing: a large rate increase drives away the risks with the most alternatives, which are usually the better ones. The indication that caused the shift does not anticipate it, so the following year's experience deteriorates for reasons the analysis attributed to the market.
• Other common causes: a new distribution channel, entering or exiting a territory, a change in the deductible or limit profile, a growth push in one segment, and underwriting tightening.

A Mix Shift That Looks Like Rate Inadequacy An insurer writes two territories, each with a stable loss ratio:

Territory	Loss ratio	Year 1 weight	Year 2 weight
Urban	80\%	40\%	70\%
Rural	55\%	60\%	30\%

What happens to the overall loss ratio, and what is the correct response?

Answer

\begin{align*}
\text{Year 1} &= 0.40(80\%) + 0.60(55\%) = 65.0\% \\
\text{Year 2} &= 0.70(80\%) + 0.30(55\%) = 72.5\%
\end{align*}

The overall loss ratio rose 7.5 points and neither territory's rates became inadequate. The whole movement is the shift toward urban business.

An across-the-board increase would be the wrong response twice over: it over-charges rural policyholders whose rates were already adequate, and it under-diagnoses the urban territory, whose 80\% loss ratio may itself be a problem the relativity structure is failing to price. The correct action is at the territory relativity level.

Note also that the mix shift is probably not random — a book that grew 75\% in its urban territory in one year was pushing there, and someone should ask why.

A Mix Shift Hiding Inside a Triangle An insurer's commercial package book is reserved on one combined triangle. Its composition by premium:

Coverage	12–24 factor	2020 weight	2024 weight
Property	1.05	70\%	45\%
Liability	1.60	30\%	55\%

What does the combined triangle show, and what does it miss?

Answer The true blended factor in each year:

\begin{align*}
2020: \; 0.70(1.05) + 0.30(1.60) &= 1.215 \\
2024: \; 0.45(1.05) + 0.55(1.60) &= 1.353
\end{align*}

A combined triangle selecting an all-year average would land near 1.28 — too high for the older years and too low for the current one, which is the year the factor is actually applied to.

What makes this dangerous is that the triangle looks fine. There is no diagonal effect, no jump in average case outstanding, no change in disposal rates. The factors simply drift upward year by year, and an actuary watching for the usual distortions sees nothing wrong — the drift reads as a slow change in development pattern rather than as a change in what is being averaged.

The only fix is structural: split the triangle by coverage (Reserving Data Organization). The blended answer then falls out of the sum, computed at the current mix, instead of being baked into factors estimated at an old one.