Actuarial Notes

Overall Rate Level Indication

Study guides: Exam FAM, Exam ASTAM, Exam 5

Overall Rate Level Indication is the answer to the ratemaking question: by what percentage must the rate level change so that expected premium covers expected losses, expenses and profit for the future policy period? It is the aggregate result, before any classification work distributes it.

\text{Indicated Change} = \frac{\text{Projected Loss \& LAE Ratio} + F\%}{1 - V - Q_T} - 1

\text{Indicated Change} = \frac{\text{Indicated Rate}}{\text{Current Rate}} - 1

The build-up, following Werner & Modlin's chapters:

  1. Premium — historical earned premium, brought to current rate level, developed to ultimate where it is still changing, and trended.
  2. Losses — reported losses with extraordinary large losses and catastrophes replaced by a long-term provision, developed to ultimate, trended to the future cost level, restated for any benefit or coverage change, and including ALAE and ULAE. Werner prescribes no single order for these adjustments.
  3. Expenses — split into fixed (numerator, per exposure or as a ratio) and variable (denominator); the net cost of non-proportional reinsurance may be included as an expense item.
  4. Profit — the target underwriting profit provision Q_T.
  5. Credibility — weight the indication against a complement where the data is thin.
  6. Selection — the filed change, which may differ for the reasons on Rate Change.

Further points:

  • Steps 1 and 2 are the two halves of "put everything on a common basis". Premium is adjusted for rate level; losses for maturity and cost level. Skipping either produces a systematically wrong answer, and the two errors do not offset.
  • The Loss Ratio Method and Pure Premium Method are alternative routes to the same indication; the first gives a change, the second an absolute rate.
  • Weighting across years in the experience period is a judgment: many companies sum projected losses and premium across all years, which weights each year's loss ratio by its premium; others select weights, often giving more to the most recent years. More years buys credibility, fewer years buys responsiveness.
  • The indication is an estimate with a distribution, not a fact. A sensitivity test shows how the answer moves under alternative trend, development and profit selections.

Full Overall Indication Two accident years, valued 12/31/2024. Loss trend 6\%; trend periods 3.0 and 2.0 years. LAE 10\% of loss. Variable expenses 23\%, fixed expense ratio 6\%, target profit 5\%. Credibility Z = 0.80 against a complement of +2.0\%.

AY Earned premium On-level factor Reported loss CDF
2023 \$12{,}000{,}000 1.085 \$7{,}100{,}000 1.16
2024 \$13{,}500{,}000 1.030 \$6{,}200{,}000 1.38

Compute the selected rate change.

Answer Losses:

\begin{align*}
2023: \; 7{,}100 \times 1.16 \times 1.06^{3.0} \times 1.10 &= \$10{,}790 \text{K} \\
2024: \; 6{,}200 \times 1.38 \times 1.06^{2.0} \times 1.10 &= \$10{,}575 \text{K}
\end{align*}

Premium:

\begin{align*}
2023: \; 12{,}000 \times 1.085 &= \$13{,}020 \text{K} \\
2024: \; 13{,}500 \times 1.030 &= \$13{,}905 \text{K}
\end{align*}

Loss ratios: 82.9\% and 76.1\%; combined \frac{\$21{,}365}{\$26{,}925} = 79.35\%.

\begin{align*}
\text{Indicated factor} &= \frac{0.7935 + 0.06}{1 - 0.23 - 0.05} \\
&= \frac{0.8535}{0.72} \\
&= 1.1854
\end{align*}

an indication of +18.5\%. Credibility-weighting:

0.80 \times 18.5\% + 0.20 \times 2.0\% = +15.2\%

Before filing +15.2\%, two things demand attention: the two years disagree by nearly seven points while the less mature year rests on a 1.38 CDF, and an increase of this size will move the mix through retention. Both belong in the documentation.

Where the Indication Is Sensitive Using the book above, test the +18.5\% indication against alternative selections: loss trend 4\% instead of 6\%; AY 2024 CDF 1.30 instead of 1.38; profit provision 3\% instead of 5\%.

Answer Changing one assumption at a time (before credibility weighting):

Change New indication
Base +18.5\%
Loss trend 4\% +13.4\%
AY 2024 CDF 1.30 +15.4\%
Profit provision 3\% +15.3\%

A two-point change in the trend selection moves the indication five points — more than either of the other two assumptions — because trend compounds over a 2–3 year period and applies to the entire loss provision.

ASOP 13 sets the standard for the trending procedure behind that selection, including documentation of its methods, assumptions and data in a form another actuary could assess for reasonableness.