Rate Level Change

Study guide: Exam 5

Rate Level Change is the effect that past rate changes have on the comparability of data across years — the reason historical premium must be brought to a common level before any loss ratio, trend or a priori expectation built on it can be believed.

$\text{On-Level Factor} = \frac{\text{Current cumulative index}}{\text{Index in force during the period}}$

$\text{Adjusted ELR}_i = \text{ELR}_{\text{base}} \times \frac{\text{Rate level}_{\text{base}}}{\text{Rate level}_i} \times \frac{\text{Loss trend}_i}{\text{Loss trend}_{\text{base}}}$

In ratemaking the correction is on-levelling: restate historical earned premium to current rates so that a rising loss ratio means rising costs rather than yesterday's cheaper prices.
In reserving the same distortion attacks the a priori loss ratio. The BF and expected claims techniques need an expected loss ratio per accident year, and a year written after a $+15\%$ rate change should carry a materially lower ELR than the year before it. Friedland's standard adjustment moves a base-year ELR across years by the ratio of rate level changes and loss trend, as above.
The Cape Cod technique is exposed in the same way, because its used-up premium denominator is actual premium: without on-levelling, years written at low rates get too much weight in the derived ELR.
Rate changes also shift the mix. A large increase drives away the risks with the most alternatives — typically the better ones — so the remaining book's experience deteriorates in a way no on-level factor corrects. See Mix of Business.
Reconstructing rate level history is genuine work: changes may apply to new business only, to renewals at their anniversaries, or to policies in force; tier and underwriting-rule changes move effective price without appearing as a filed rate change at all.

Adjusting a Priori Loss Ratios for Rate Level An actuary sets a BF expected loss ratio for accident year $2024$ . The base year is AY $2021$ , whose ultimate loss ratio was $68\%$ and is considered a fair a priori. Since then rates rose $+6\%$ ( $1/1/2022$ ), $+10\%$ ( $1/1/2023$ ) and $+4\%$ ( $1/1/2024$ ); annual loss trend is $5\%$ .

Derive the AY 2024 ELR.
Answer Cumulative rate level change from $2021$ to $2024$ :

$1.06 \times 1.10 \times 1.04 = 1.2126$

Cumulative loss trend over three years:

$1.05^{3} = 1.1576$
 $\begin{align*} \text{ELR}_{2024} &= 0.68 \times \frac{1.1576}{1.2126} \\ &= 0.68 \times 0.9546 \\ &= 64.9\% \end{align*}$ 
Rates outran loss trend over the period, so the a priori loss ratio falls from $68\%$ to $64.9\%$ . Carrying $68\%$ into the BF calculation would overstate AY 2024 IBNR — and would keep overstating it every year the insurer stayed ahead of trend, producing a reserve that runs redundant while the analysis appears stable.

A Rate Increase Hiding a Deteriorating Book A commercial auto book reports:

AY Earned premium Ultimate losses Reported LR

$2022$ $\$20{,}000{,}000$ $\$14{,}400{,}000$ $72.0\%$

$2023$ $\$23{,}000{,}000$ $\$15{,}870{,}000$ $69.0\%$

$2024$ $\$27{,}600{,}000$ $\$18{,}216{,}000$ $66.0\%$

Rates rose $+12\%$ effective $1/1/2023$ and $+12\%$ effective $1/1/2024$ ; exposures are flat. Management concludes the book is improving.

Test the conclusion.
Answer With $1/1$ changes and uniform annual writings, each calendar year earns half at each level. Taking $2022$ 's level as $1.000$ , the average indices are $1.000$ , $1.060$ and $1.187$ ; the current level is $1.2544$ .

On-level factors: $1.2544$ , $1.1834$ , $1.0567$ .
 $\begin{align*} \text{OL EP}_{2022} &= \$25{,}088{,}000 \Rightarrow \text{LR} = 57.4\% \\ \text{OL EP}_{2023} &= \$27{,}218{,}000 \Rightarrow \text{LR} = 58.3\% \\ \text{OL EP}_{2024} &= \$29{,}165{,}000 \Rightarrow \text{LR} = 62.5\% \end{align*}$ 
On a constant rate level the loss ratio has deteriorated by five points, not improved by six. The apparent improvement was the two rate increases masking loss trend running faster than the rate action.

This is exactly why an indication is built on on-level premium and why reserving ELRs are adjusted for rate level: the raw loss-ratio series says the opposite of the truth.

AY	Earned premium	Ultimate losses	Reported LR
$2022$	$\$20{,}000{,}000$	$\$14{,}400{,}000$	$72.0\%$
$2023$	$\$23{,}000{,}000$	$\$15{,}870{,}000$	$69.0\%$
$2024$	$\$27{,}600{,}000$	$\$18{,}216{,}000$	$66.0\%$