Relativities from Class Loss Ratios

Why Compare Class Loss Ratios?

A class loss ratio puts losses in relation to the earned premium collected for that class. Comparing it with the portfolio loss ratio gives an initial indication of whether the class has produced more or less loss per premium dollar than the portfolio overall. The comparison is useful for classification ratemaking, but it is a loss-based indication, not by itself a complete rate recommendation.

The calculation depends on a sound match between the data: losses must be allocated to the same classes whose earned premiums are being used, and both should represent compatible periods and bases. For example, if losses have been adjusted to a common rate level or trend basis, the premium basis must be appropriate for that comparison. This tutorial treats the supplied losses and earned premiums as already matched and suitable for the intended comparison.

Definition: For class $j$, let $L_j$ be its losses and $P_j$ its earned premium. When $P_j>0$, its class loss ratio is $LR_j=L_j/P_j$. For a portfolio of classes with positive total earned premium, the portfolio loss ratio is $LR_{\mathrm{port}}=\sum_j L_j/\sum_j P_j$. The initial loss-ratio relativity indication for class $j$ is $r_j=LR_j/LR_{\mathrm{port}}$, provided $LR_{\mathrm{port}}>0$.

A class indication above 1.00 means the class loss ratio exceeds the portfolio benchmark; below 1.00 means it is lower. Interpreted as a preliminary multiplier on the class’s current premium, an indication above 1.00 points upward and one below 1.00 points downward. It does not establish that the indicated factor should be adopted unchanged: the calculation does not account for expenses, other rating considerations, or uncertainty in the observed experience.

Why the Portfolio Benchmark Is Premium-Weighted

The portfolio loss ratio is the ratio of total losses to total earned premium. It is not generally the simple average of the class loss ratios. For classes with positive earned premiums, its identity as a weighted average follows by writing each class’s losses as its loss ratio times its premium:

LR_{\mathrm{port}} =\frac{\sum_j L_j}{\sum_j P_j} =\frac{\sum_j P_j LR_j}{\sum_j P_j} =\sum_j \left(\frac{P_j}{\sum_k P_k}\right) LR_j.

Thus, each class’s weight is its share of portfolio earned premium. A class with more premium contributes more to the portfolio benchmark. This identity is also a useful calculation check: the premium-weighted average of the class indications should equal 1, as long as the portfolio loss ratio is positive and the ratios are defined.

Deriving the Initial Indication

The indication can be justified by asking what premium multiplier would bring a class to the portfolio loss ratio if its observed losses were held fixed. Let $k_j$ multiply the class’s current earned premium. Its loss ratio after that premium adjustment would be $L_j/(k_jP_j)$. Setting this equal to the portfolio benchmark and solving for $k_j$ gives:

\frac{L_j}{k_jP_j}=LR_{\mathrm{port}} \quad\Longrightarrow\quad k_j=\frac{L_j/P_j}{LR_{\mathrm{port}}} =\frac{LR_j}{LR_{\mathrm{port}}} =r_j.

This derivation requires $P_j>0$, $LR_{\mathrm{port}}>0$, and $L_j>0$ for the premium-adjustment interpretation: then $k_j$ is positive, and the adjusted class loss ratio is defined and equals the benchmark. If $L_j=0$, the formula gives $r_j=0$, but applying that multiplier makes adjusted premium zero, so the resulting loss ratio is $0/0$, which is undefined. No positive premium multiplier can turn a zero observed loss amount into a positive benchmark loss ratio while holding losses fixed. A zero-loss indication therefore needs separate judgment rather than this algebraic interpretation. If all portfolio losses are zero, the portfolio loss ratio is zero and the relativity division is undefined. A class with zero earned premium also has no defined class loss ratio.

Key result: When class premiums and losses are positive and the portfolio loss ratio is positive, $r_j=LR_j/LR_{\mathrm{port}}$ is the premium multiplier that would make the class’s fixed observed losses produce the portfolio loss ratio. For zero-loss classes, the ratio may be calculated as an indication, but it cannot be interpreted as a positive premium adjustment that achieves the benchmark.

Worked Applications

Worked Example: Comparing Three Classes With the Portfolio

A hypothetical insurer has the following earned premium and allocated losses, in thousands of dollars. Calculate each class loss ratio, the portfolio loss ratio, and each initial relativity indication. This is a deterministic ratio calculation: the interest rate is 0% because no discounting is used; there is no mortality assumption; and no probability distribution or distribution parameters are used.

Class	Earned premium ($000)	Losses ($000)
A	400	220
B	300	210
C	300	150

Step 1: Calculate class loss ratios. For A, $LR_A=220/400=0.55$, or 55%. For B, $LR_B=210/300=0.70$, or 70%. For C, $LR_C=150/300=0.50$, or 50%.

Step 2: Calculate the portfolio benchmark. Total losses are $220+210+150=580$ thousand dollars, and total earned premium is $400+300+300=1{,}000$ thousand dollars. Therefore $LR_{\mathrm{port}}=580/1{,}000=0.58$, or 58%.

Step 3: Divide each class ratio by the benchmark. The indications are $r_A=0.55/0.58=0.948275\ldots$, $r_B=0.70/0.58=1.206896\ldots$, and $r_C=0.50/0.58=0.862068\ldots$. Rounded only at the final step to four decimal places, they are 0.9483, 1.2069, and 0.8621.

Check. Using the exact ratios, the premium-weighted average indication is $(400/1{,}000)(0.55/0.58)+(300/1{,}000)(0.70/0.58)+(300/1{,}000)(0.50/0.58)=1$. Rounding the displayed indications gives a value extremely close to 1. Class B is above the portfolio benchmark; A and C are below it. The method avoids treating the class ratios as equally weighted when calculating the portfolio result.

Worked Example: Interpreting an Indication as a Premium Adjustment

A hypothetical portfolio has three classes, with amounts in thousands of dollars: Class X has premium 240 and losses 144; Class Y has premium 360 and losses 252; Class Z has premium 400 and losses 204. Find the portfolio benchmark and each indication. Then, for Class Y, calculate the premium that would produce the benchmark loss ratio if its observed losses were held fixed. This is a deterministic calculation with a 0% interest rate and no discounting, no mortality assumption, and no probability distribution or distribution parameters.

Step 1: Find the class ratios. Class X has $144/240=0.60$. Class Y has $252/360=0.70$. Class Z has $204/400=0.51$.

Step 2: Find the portfolio ratio. Total losses are $144+252+204=600$ thousand dollars, and total premium is $240+360+400=1{,}000$ thousand dollars. Thus $LR_{\mathrm{port}}=600/1{,}000=0.60$.

Step 3: Calculate the indications. Class X’s indication is $0.60/0.60=1.00$. Class Y’s is $0.70/0.60=7/6=1.166666\ldots$, or 1.1667 to four decimal places. Class Z’s is $0.51/0.60=0.85$, or 0.8500 to four decimal places.

Step 4: Apply the indication to Class Y’s premium. Its indicated multiplier is $7/6$, so the corresponding premium is $360(7/6)=420$ thousand dollars. Checking directly, $252/420=0.60$, the portfolio benchmark. The calculation illustrates why the indication is a multiplier on premium, not on losses. The common mistake is to multiply the losses by $7/6$; that would change the numerator instead of solving for the premium needed to match the benchmark.

Worked Example: Pooling Experience Across Different Premium Volumes

A hypothetical class’s first period has earned premium of $100,000 and losses of $70,000; its second period has earned premium of $200,000 and losses of $130,000. A comparison class has premium of $300,000 and losses of $180,000. Pool the first class’s amounts, then calculate both class ratios, the portfolio benchmark, and the indications. This deterministic calculation uses a 0% interest rate with no discounting, no mortality assumption, and no probability distribution or distribution parameters.

Step 1: Pool the first class’s amounts. Its combined losses are $70{,}000+130{,}000=200{,}000$, and its combined premium is $100{,}000+200{,}000=300{,}000$. Its pooled loss ratio is $200{,}000/300{,}000=2/3=0.666666\ldots$.

Step 2: Calculate the comparison class ratio and portfolio ratio. The comparison class has loss ratio $180{,}000/300{,}000=0.60$. Total losses are $200{,}000+180{,}000=380{,}000$, and total premium is $300{,}000+300{,}000=600{,}000$. Therefore the portfolio loss ratio is $380{,}000/600{,}000=19/30=0.633333\ldots$.

Step 3: Calculate indications. The first class’s indication is $(2/3)/(19/30)=20/19=1.052631\ldots$, or 1.0526. The comparison class’s indication is $0.60/(19/30)=18/19=0.947368\ldots$, or 0.9474. The premium-weighted check is $0.5(20/19)+0.5(18/19)=1$.

Method and trap avoided. Pool losses and premium separately before dividing. Averaging the first class’s two period ratios equally would give $(70\%+65\%)/2=67.5\%$, which is not its pooled loss ratio: the periods had different premium volumes. The correct pooled result is 66.6667%.

Worked Example: A Zero-Loss Class and the Limit of the Formula

A hypothetical portfolio has Class M with earned premium of $100,000 and losses of $0, and Class N with earned premium of $400,000 and losses of $240,000. Calculate the benchmark and indications, and assess whether a positive premium multiplier can bring Class M to the benchmark while holding its losses fixed. This is a deterministic calculation with a 0% interest rate and no discounting, no mortality assumption, and no probability distribution or distribution parameters.

Step 1: Calculate the ratios. Class M’s ratio is $0/100{,}000=0$. Class N’s ratio is $240{,}000/400{,}000=0.60$. The portfolio ratio is $240{,}000/(100{,}000+400{,}000)=0.48$.

Step 2: Calculate the indications. Since the benchmark is positive, the formula gives $r_M=0/0.48=0$ and $r_N=0.60/0.48=1.25$.

Step 3: Interpret the zero-loss result correctly. For any positive multiplier $k_M$, the adjusted Class M ratio is $0/(k_M \times 100{,}000)=0$, not 0.48. A multiplier of zero would make the adjusted premium zero and the ratio undefined. Thus the zero indication is not a feasible positive premium adjustment that attains the benchmark. This example avoids applying the fixed-loss adjustment derivation outside its conditions.

Common Calculation Traps

Taking a simple average of class ratios. The portfolio ratio uses total losses divided by total premium. Equivalently, it is a premium-weighted average of class ratios, not an unweighted one.
Reversing the indication. The initial indication is class ratio divided by portfolio ratio. If a class ratio is 70% and the benchmark is 60%, its indication is $70\%/60\%=1.1667$, not $60\%/70\%$.
Mixing data bases. Losses allocated to one set of classes cannot be compared meaningfully with premiums from a different grouping or incompatible period basis.
Treating an indication as a final selected factor. This comparison isolates relative loss experience against the portfolio benchmark. It does not, by itself, address expenses, uncertainty, or all considerations in setting a rate.
Ignoring zero or undefined ratios. A class with no earned premium has no defined loss ratio. A zero-loss class has a zero ratio when its premium is positive, but it cannot reach a positive benchmark through a positive premium multiplier with losses fixed.

Key takeaway: Calculate each class ratio from its losses and earned premium, calculate the portfolio ratio from total losses and total earned premium, and divide the class ratio by the portfolio ratio. Check the result against the premium-weighted portfolio identity, and interpret it as an initial indication subject to the data conditions and limitations stated above.

Check Your Understanding

For each hypothetical problem, calculate from the supplied loss and premium amounts. Unless stated otherwise, assume losses and earned premiums are matched. Monetary amounts are in thousands of dollars where shown.

Class A has earned premium of $200,000 and losses of $110,000; Class B has earned premium of $300,000 and losses of $210,000. Calculate the portfolio ratio and each class indication. (Answer: portfolio loss ratio 64%; A indication 0.8594; B indication 1.0938.)
Three classes have: M, premium 450 and losses 270; N, premium 250 and losses 200; O, premium 300 and losses 180. Calculate the portfolio ratio and the three indications. (Answer: portfolio loss ratio 65%; M indication 0.9231; N indication 1.2308; O indication 0.9231.)
Class A has premium 100 and losses 80; Class B has premium 900 and losses 450. Calculate the portfolio loss ratio and both indications. Also state why the simple average of the class loss ratios is not the portfolio ratio. (Answer: portfolio loss ratio 53%; A indication 1.5094; B indication 0.9434; the simple average is 65% and ignores the unequal premium weights.)
Class R has premium 100 and zero losses; Class S has premium 400 and losses 240. Find the benchmark and indications, then determine whether a positive premium multiplier can bring R to the benchmark with losses held fixed. (Answer: portfolio loss ratio 48%; R indication 0; S indication 1.2500; no positive multiplier can do so because R’s adjusted loss ratio remains zero.)
A class has current earned premium of $360,000 and losses of $252,000. The portfolio loss ratio is 60%. Calculate its indication and the premium amount implied by applying that indication to the current premium; verify the resulting ratio using fixed losses. (Answer: indication 1.1667; indicated premium $420,000; resulting loss ratio 60%.)

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