Loss Ratio Method

Study guide: Exam 5

Loss Ratio Method derives an indicated rate change by comparing the projected loss and LAE ratio against the permissible loss ratio. It works entirely in ratios, so it needs premium on level rather than exposure counts.

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

\text{Indicated Change} = \frac{\text{Projected Loss Ratio}}{\text{PLR}} - 1 \quad \text{(all-variable form)}

• The first form is Werner's: fixed expenses as a ratio in the numerator, variable expenses V and profit Q_T in the denominator. The second is the all-variable simplification, where \text{PLR} = 1 - V - F\% - Q_T. Using F\% in both places double-counts it.
• The numerator's loss ratio must be projected: ultimate losses (developed), trended to the future policy period, over earned premium brought to current rate level (On-Leveling) and trended (Premium Trend).
• The method's advantage is that it needs no exposure data, which makes it preferable where exposure data is unavailable or the product has no clearly defined exposure base (Werner & Modlin's example: commercial general liability, whose sub-lines use different exposure bases). Its requirement is premium at current rate level, which is hard to calculate accurately after many changes to a complex rating algorithm.
• It produces a change, not a rate — so it cannot price a new line of business with no current rates to adjust. It is the more widely used of the two methods, and preferred where the indicated change itself is a critically important statistic for the pricing decision.
• The method is equivalent to the Pure Premium Method given consistent data and assumptions, because a loss ratio is the pure premium divided by the average premium; an approximation such as the parallelogram method in the on-level premium can make the two disagree.

Straightforward Loss Ratio Indication Projected ultimate trended losses and LAE are \$720{,}000; trended on-level earned premium is \$1{,}000{,}000. Variable expenses are 22\%, fixed expenses 6\% of premium, target profit 5\%.

Compute the indicated change.

Answer

\begin{align*}
\text{Projected LR} &= \frac{\$720{,}000}{\$1{,}000{,}000} = 72.0\% \\[6pt]
\text{Indicated factor} &= \frac{0.72 + 0.06}{1 - 0.22 - 0.05} \\
&= \frac{0.78}{0.73} \\
&= 1.0685
\end{align*}

\text{Indicated change} = +6.8\%

Under the all-variable shortcut, \text{PLR} = 1 - 0.22 - 0.06 - 0.05 = 0.67 and the indication is 0.72/0.67 - 1 = +7.5\%. Both are defensible; using 0.78/0.67 = +16.4\% is not.

A Full Indication from Raw Data Three accident years, each with reported losses at 12/31/2024, earned premium, and the on-level factors and CDFs shown. Loss trend is 5\%; trend periods are 3.5, 2.5 and 1.5 years. LAE is 11\% of loss. Variable expenses 21\%, fixed expense ratio 7\%, target profit 4\%.

AY	EP	Reported loss	OLF	CDF
2022	\$8{,}000{,}000	\$4{,}600{,}000	1.120	1.08
2023	\$8{,}600{,}000	\$4{,}500{,}000	1.075	1.19
2024	\$9{,}300{,}000	\$3{,}900{,}000	1.030	1.42

Compute the indicated rate change.

Answer Losses — develop, trend, load LAE:

\begin{align*}
2022: \; 4{,}600 \times 1.08 \times 1.05^{3.5} \times 1.11 &= \$6{,}541 \text{K} \\
2023: \; 4{,}500 \times 1.19 \times 1.05^{2.5} \times 1.11 &= \$6{,}715 \text{K} \\
2024: \; 3{,}900 \times 1.42 \times 1.05^{1.5} \times 1.11 &= \$6{,}614 \text{K}
\end{align*}

Premium — on level:

\begin{align*}
2022: \; 8{,}000 \times 1.120 &= \$8{,}960 \text{K} \\
2023: \; 8{,}600 \times 1.075 &= \$9{,}245 \text{K} \\
2024: \; 9{,}300 \times 1.030 &= \$9{,}579 \text{K}
\end{align*}

Loss ratios by year: 73.0\%, 72.6\%, 69.0\% — a stable series, which is itself a reasonableness check.

\begin{align*}
\text{Weighted LR} &= \frac{\$19{,}870}{\$27{,}784} \\
&= 71.5\% \\[6pt]
\text{Indicated factor} &= \frac{0.715 + 0.07}{1 - 0.21 - 0.04} \\
&= \frac{0.785}{0.75} \\
&= 1.047
\end{align*}

\text{Indicated change} = +4.7\%

Two judgment calls sit inside this: whether to weight the three years by premium volume (as here — summing losses and premium across years, which weights each year's loss ratio by its premium) or to select weights that give more to the most recent years, and whether the 69.0\% in AY 2024 is a genuine improvement or an artefact of the 1.42 CDF on the least mature year. Weighting the loss ratios 20/30/50 toward recency would give 70.9\% here, and a slightly lower indication of +3.9\%.