Permissible Loss Ratio

Study guide: Exam 5

Permissible Loss Ratio (PLR, or target loss ratio) is the share of premium available to pay losses once every other cost has been provided for. It is the benchmark against which a projected loss ratio is judged in the Loss Ratio Method.

Werner & Modlin define two permissible loss ratios, both with LAE included in losses:

\text{PLR} = 1 - V - F\% - Q_T \quad \text{(total permissible loss ratio)}

\text{VPLR} = 1 - V - Q_T \quad \text{(variable permissible loss ratio)}

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

• V is the variable expense ratio, Q_T the target underwriting profit provision, and F\% the fixed expense ratio. The PLR is the share of each premium dollar intended for loss and LAE; the VPLR is the share intended for loss, LAE and fixed expenses. The loss ratio indication compares the projected loss & LAE ratio plus the fixed expense ratio to the VPLR — fixed expenses sit in the numerator with the losses, because they are assumed to be the same for each risk rather than a percentage of premium, and so do not scale with the rate being solved for.
• If all expenses are treated as variable, the VPLR and the PLR are the same, and the indication is simply:

\text{Indicated Change} = \frac{\text{Projected Loss \& LAE Ratio}}{\text{PLR}} - 1

This is the all variable expense method. It reproduces the fixed-expense result only for the average risk — Werner & Modlin's \$20 fixed expense is exactly 8\% of the \$250 average premium — and undercharges risks with below-average premium while overcharging those above it. Mixing the two conventions — subtracting F\% in the denominator and adding it in the numerator — double-counts fixed expenses.

• The PLR is the complement of the target combined ratio: a 70\% PLR with a 25\% total expense ratio implies a target combined ratio of 95\% and a 5\% underwriting profit.
• A higher variable expense ratio lowers the PLR, leaving less room for losses. Insurers with cheap distribution can run higher loss ratios profitably — which is why a loss ratio comparison across companies with different channels says little about relative pricing discipline.
• Where the losses and premium are not already net of non-proportional reinsurance, its net cost (cost of the reinsurance minus expected recoveries) may be included as an expense item in the indication.

Computing the PLR and the Indication An insurer's provisions: variable expenses 22\% of premium, fixed expenses 8\% of premium, target underwriting profit 5\%. The projected loss and LAE ratio is 60\%.

Compute the indicated rate change with fixed expenses in the numerator, and check it against the all variable expense method.

Answer Fixed expenses in the numerator — against the VPLR:

\begin{align*}
\text{VPLR} &= 1 - 0.22 - 0.05 = 0.73 \\[6pt]
\text{Indicated factor} &= \frac{0.60 + 0.08}{0.73} \\
&= \frac{0.68}{0.73} \\
&= 0.9315
\end{align*}

an indicated change of -6.8\%.

All variable expense method — against the total PLR:

\begin{align*}
\text{PLR} &= 1 - 0.22 - 0.08 - 0.05 = 0.65 \\[4pt]
\text{Indicated factor} &= \frac{0.60}{0.65} = 0.9231
\end{align*}

an indicated change of -7.7\%.

The two differ by about a point because the fixed expense ratio was measured at the current premium level; when rates change, the fixed dollars per exposure stay put while the all variable method lets them move with premium. The fixed-expense form is the one to use whenever the rate change is material.

The error to avoid entirely is \dfrac{0.60 + 0.08}{0.65} = 1.046 — a +4.6\% indication from charging fixed expenses twice, which flips the sign of the answer.

Reading Adequacy from the PLR Three books, each with a projected loss and LAE ratio of 68\% and a target underwriting profit of 5\%:

Book	Variable expenses	Fixed expense ratio
Independent agency	27\%	9\%
Captive agency	20\%	11\%
Direct	9\%	16\%

Which are adequately rated?

Answer

\begin{align*}
\text{Agency: } \frac{0.68 + 0.09}{1 - 0.27 - 0.05} &= \frac{0.77}{0.68} = 1.132 \Rightarrow +13.2\% \\[6pt]
\text{Captive: } \frac{0.68 + 0.11}{1 - 0.20 - 0.05} &= \frac{0.79}{0.75} = 1.053 \Rightarrow +5.3\% \\[6pt]
\text{Direct: } \frac{0.68 + 0.16}{1 - 0.09 - 0.05} &= \frac{0.84}{0.86} = 0.977 \Rightarrow -2.3\%
\end{align*}

Identical loss ratios, three different answers. The direct book can sustain a 68\% loss ratio and still return its target profit; the agency book needs a 13\% rate increase to reach the same place.

The wider lesson: a loss ratio is meaningless as a measure of pricing adequacy without the expense structure behind it. This is also why the direct writer can quote below the agency writer on identical risks without under-pricing — and why the agency book will lose its best risks first.