Basic Insurance Ratios

Study guides: Exam DISC-RM, Exam 5

Basic Insurance Ratios are the standard ratios — frequency, severity, pure premium, average premium, loss ratio, LAE ratio, expense ratios, combined ratio, retention and close ratios — that insurers, regulators, rating agencies and investors use to monitor whether a book's rates are adequate. Werner & Modlin introduce them together in Chapter 1 of Basic Ratemaking; the rest of ratemaking is largely about projecting them.

$\begin{aligned} \text{Pure Premium} &= \frac{\text{Losses}}{\text{Exposures}} \\ &= \text{Frequency} \times \text{Severity} \end{aligned}$

$\begin{aligned} \text{Loss Ratio} &= \frac{\text{Losses}}{\text{Premium}} \\ &= \frac{\text{Pure Premium}}{\text{Average Premium}} \end{aligned}$

$\begin{aligned} \text{Combined Ratio} &= \frac{\text{Losses} + \text{LAE}}{\text{EP}} \\ &\quad + \frac{\text{UW Expenses}}{\text{WP}} \end{aligned}$

Loss side. Frequency is claims ÷ exposures (normally reported claims over earned exposures); Severity is losses ÷ claims (a paid severity uses paid losses on closed claims over closed counts, a reported severity uses reported losses over reported counts, with ALAE in or out); Pure Premium is losses ÷ exposures.
Premium side. Average premium is premium ÷ exposures, with both on the same basis — written, earned or in-force. Once adjusted for rate changes, a moving average premium signals a shift in the mix of business (Premium Trend).
The LAE ratio is to losses, not premium: $\text{LAE ratio} = \text{LAE}/\text{Losses}$ (allocated and unallocated). So the loss and LAE ratio is $\text{LR} \times (1 + \text{LAE ratio})$ — not the sum of the two.
Expense ratios. The underwriting expense ratio measures expenses incurred at inception (commissions, other acquisition, taxes, licenses, fees) against written premium and expenses incurred through the term (general) against earned premium, then adds them. The operating expense ratio is $\text{OER} = \text{UW expense ratio} + \text{LAE}/\text{EP}$ , and $\text{Combined Ratio} = \text{LR} + \text{OER}$ with the loss ratio excluding LAE — otherwise LAE is counted twice. See Combined Ratio.
Marketing ratios. $\text{Retention ratio} = \text{policies renewed} / \text{potential renewal policies}$ and $\text{Close ratio} = \text{accepted quotes} / \text{quotes}$ measure competitiveness rather than cost; they are watched after every rate change and feed lifetime value models.
Measuring the insurer as a whole. For the company rather than one book, the combined ratio measures underwriting; the investment income ratio is investment income ÷ earned premium, and the operating ratio subtracts it from the combined ratio, so a combined ratio above 100% can still leave an operating ratio below it (Insurance Operations). Return on equity, net income ÷ net worth, measures the return to owners. The combined ratio a line needs to break even depends on how much investment income it earns: long-tail liability lines hold funds longer and can break even at a higher combined ratio than property lines.
Measuring the other goals. An insurer's goals include meeting customer needs and legal requirements (Insurer Goals and Constraints), which are measured differently. A state complaint index divides the insurer's share of the state's complaints in a line by its share of the line's premium: 1.00 means complaints in proportion to its business, 2.00 twice as many as its share. Retention ratios show whether customers stay, and regulatory examinations and fines show compliance.
Every ratio has variants (reported vs. paid, accident vs. calendar year, with or without ALAE, how a claim or a quote is counted), so its inputs must be documented — the same caution as on Loss Ratio.

Computing the Full Set for One Year A personal lines book for one year: $40{,}000$ earned exposures; $2{,}400$ reported claims; reported losses $\$18{,}000{,}000$ ; LAE (ALAE and ULAE) $\$2{,}160{,}000$ ; earned premium $\$28{,}000{,}000$ ; written premium $\$30{,}000{,}000$ ; commissions, other acquisition, taxes, licenses and fees $\$5{,}400{,}000$ ; general expenses $\$1{,}680{,}000$ .

Compute the basic ratios.
Answer
 $\begin{align*} \text{Frequency} &= 2{,}400 / 40{,}000 = 6.0\% \\ \text{Severity} &= \$18{,}000{,}000 / 2{,}400 = \$7{,}500 \\ \text{Pure premium} &= 0.060 \times \$7{,}500 = \$450 \\ \text{Average premium} &= \$28{,}000{,}000 / 40{,}000 = \$700 \\ \text{Loss ratio} &= \$450 / \$700 = 64.29\% \end{align*}$ 
 $\begin{align*} \text{LAE ratio} &= \$2{,}160{,}000 / \$18{,}000{,}000 = 12.0\% \\ \text{Loss \& LAE ratio} &= 64.29\% \times 1.12 = 72.0\% \\ \text{UW expense ratio} &= \tfrac{5.40}{30.0} + \tfrac{1.68}{28.0} \\ &= 18.0\% + 6.0\% = 24.0\% \\ \text{OER} &= 24.0\% + \tfrac{2.16}{28.0} = 31.71\% \\ \text{Combined ratio} &= 64.29\% + 31.71\% = 96.0\% \end{align*}$ 
The book keeps four cents of each premium dollar before investment income. Note that adding the $12\%$ LAE ratio straight onto the loss ratio would give $76.3\%$ , overstating the loss and LAE ratio by more than four points.

Decomposing a Loss Ratio Movement From one year to the next, with no rate change: frequency fell from $6.00\%$ to $5.82\%$ , severity rose from $\$7{,}000$ to $\$7{,}560$ , and average earned premium rose from $\$660$ to $\$673.20$ .

How much did the loss ratio move, and why?
Answer
 $\begin{align*} \text{LR}_1 &= \frac{0.0600 \times \$7{,}000}{\$660} = \frac{\$420}{\$660} = 63.6\% \\ \text{LR}_2 &= \frac{0.0582 \times \$7{,}560}{\$673.20} = \frac{\$440}{\$673.20} = 65.4\% \\ \frac{\text{LR}_2}{\text{LR}_1} &= \frac{0.97 \times 1.08}{1.02} \\ &= 1.0271 \end{align*}$ 
The loss ratio rose $2.7\%$ . Severity ( $+8\%$ ) is the driver; lower frequency ( $-3\%$ ) offsets part of it, so pure premium rose $4.8\%$ . Average premium rose $2\%$ without any rate change — a shift toward higher-rated risks — which absorbed part of the pure premium increase. Each ratio points at a different question: severity at claim costs and trend, frequency at underwriting and the claim environment, average premium at the mix of business.

Measuring an Insurer's Overall Performance With illustrative figures in $ millions, an insurer has earned premium of 500, incurred losses and LAE of 340, underwriting expenses of 150 and investment income of 45; its net income is 60 and its policyholders' surplus 400. In one state it received 30 of the 1,200 complaints filed in personal auto, where it writes 2% of the premium. Find its combined ratio, investment income ratio, operating ratio, return on equity and complaint index.
Answer For simplicity every ratio here uses earned premium.
 $\begin{align*} \text{CR} &= \frac{340 + 150}{500} \\ &= 98\% \\ \text{Investment income ratio} &= \frac{45}{500} \\ &= 9\% \\ \text{Operating ratio} &= 98\% - 9\% \\ &= 89\% \\ \text{ROE} &= \frac{60}{400} \\ &= 15\% \end{align*}$ 
 $\begin{align*} \text{Complaint index} &= \frac{30 / 1{,}200}{0.02} \\ &= \frac{0.025}{0.02} \\ &= 1.25 \end{align*}$ 
The insurer is profitable on underwriting and more so after investment income, but it draws 25% more complaints than its share of the market would suggest, a sign it may not be meeting customers' needs as well as its profit goal.

How it appears on the exam

Exam 5: The ratios open Basic Ratemaking, and questions compute them from a year of data or decompose a loss ratio change into frequency, severity and average premium, as in the examples above.

DISC-RM: The overview of insurance operations covers measuring insurer performance. Questions compute a combined, investment income or operating ratio, explain why a combined ratio above 100% need not mean a loss, or ask how performance against non-financial goals, such as meeting customer needs, is measured.

Exam traps

The LAE ratio is to losses, not premium: Adding it to the loss ratio double-counts; multiply instead, or divide LAE by premium for the operating expense ratio.

Combined ratio is not total profit: Investment income is outside it. The operating ratio brings it in.

Same basis: Average premium and the expense ratios must use premium on the basis stated, written or earned, or the ratio drifts with growth.

A complaint index is relative: It compares the insurer's share of complaints with its share of business. A large insurer with many complaints can still have an index below 1.00.