Why Matching Bases Comes First
A rate indication compares projected losses with the premium available to fund them. That comparison is meaningful only when both quantities describe the same coverage, exposures, and time period, and when the premium is stated at the appropriate rate level. A calculation can be arithmetically correct and still be misleading if, for example, it divides losses for one policy year by written premium from another period.
The loss ratio rate indication method uses a target loss ratio to translate projected losses into a premium need. The method presumes that the losses and premium being compared have already been put on compatible bases. Here, the main task is to check that presumption and make the necessary adjustments before calculating the indication.
The loss amount used in an indication may already have been developed to ultimate or trended to a prospective period. Those adjustments do not, by themselves, ensure that its basis matches the premium. For example, developing losses to ultimate addresses when the losses are expected to be known; it does not make calendar-year losses correspond to a particular set of earned exposures. Likewise, trending losses does not convert losses under one deductible to losses under another.
A Basis Checklist
Before calculating, write down what each amount represents. A compact basis ledger helps expose mismatches before they become hidden in a ratio.
Identify the period represented by the losses and by the premium. Check whether each is earned, written, incurred, or otherwise defined, and confirm that the two amounts refer to compatible periods or exposures.
Specify the unit, such as one earned vehicle-year or one policy-month. Do not compare a loss cost per annual exposure with a premium for a fraction of a year.
Check that the losses and premium reflect the same limits, deductibles, and other coverage terms. If they do not, use appropriate data or an explicit adjustment to put them on a common coverage basis.
Adjust historical premium to the rate level required for the comparison. An earned premium at an earlier rate level is not automatically comparable with losses projected for a later rate level.
Only after the checks above, compare projected losses with the adjusted premium and the target loss ratio.
A common form of the loss ratio indication is
A factor above 1 indicates an increase; a factor below 1 indicates a decrease. The indicated percentage change is the factor minus 1. This calculation is only interpretable if the numerator and denominator use consistent bases.
Why the Comparison Formula Works
Suppose the projected losses are \(L\), the adjusted premium at the rate level being evaluated is \(P\), and the target loss ratio is \(r\). The target loss ratio means that, at the indicated premium, the projected losses should equal the fraction \(r\) of premium. If \(F\) is the factor by which premium must change, the target condition is
Solving for \(F\) gives \(F=L/(Pr)\). The derivation also shows why the basis checks are essential: the equality requires \(L\) and \(PF\) to describe the same risk and coverage over comparable exposures. If the losses refer to a broader coverage than the premium, the equation does not express a valid target loss ratio, even though the division can still be performed.
Worked Example: On-Leveling Earned Premium
A hypothetical block has 1,000 earned annual exposures. Its historical earned premium is $1,200,000 at an average rate level equal to 90% of the rate level being evaluated. Losses have already been developed and projected to the prospective period on the same coverage and exposure basis, and total $720,000. The target loss ratio is 60%. Find the indicated rate-level factor and percentage change. Also show the result of using the unadjusted earned premium.
Method: put premium at the comparison rate level, then apply the loss ratio formula. Because the historical rate level is 90% of the comparison level, the premium on the comparison level is
The premium amount is now on the same rate level as the one being evaluated. The projected losses, coverage, and 1,000 earned annual exposures are already on the stated comparison basis. The premium required at a 60% target loss ratio is
Therefore, the indicated factor is
The indicated percentage change is \(0.90-1=-0.10\), or a 10.00% decrease. As a check, applying the factor to the comparison-level premium gives \(0.90(\$1{,}333{,}333.33)=\$1{,}200{,}000\); 60% of that premium is $720,000, the projected loss amount.
Common mistake. Using the unadjusted $1,200,000 premium gives \( \$720{,}000/[\$1{,}200{,}000(0.60)]=1.00 \), suggesting no rate change. That result treats premium written at 90% of the comparison rate level as though it were already at the comparison level. The arithmetic is valid, but the basis is not.
Coverage Terms Must Match Too
Losses under different deductibles or limits are different measurements of the same underlying claims. For a ground-up loss \(X\) subject to an ordinary deductible \(d\), with no payment limit, the insurer payment per loss is \((X-d)_+=\max(X-d,0)\). Thus, changing the deductible changes the loss amount used in the indication. The premium must also be expressed for the coverage being evaluated; otherwise the loss ratio compares unlike products.
In practice, coverage adjustments require suitable information, such as claim-level losses or an appropriate coverage factor. An aggregate loss total alone generally does not reveal how losses would change under another deductible: the number and size distribution of claims matter.
Worked Example: Changing the Deductible Basis
A hypothetical portfolio has 100 claims, each with a ground-up loss of $2,000. There is no payment limit. The historical premium is $500,000 for the portfolio at a $500 deductible. The rate indication is needed for a $100 deductible. Assume the premium for that target coverage at the current rate level is 1.20 times the premium at the $500 deductible. The target loss ratio is 50%. All amounts refer to the same period and exposures. Find the rate-level factor for the $100-deductible coverage.
Method: calculate losses at the target deductible, then adjust premium to the same coverage. At the $500 deductible, the insurer payment per claim is
At the target $100 deductible, the payment per claim is
The premium on the target coverage at the current rate level is
The premium required at the 50% target loss ratio is \(\$190{,}000/0.50=\$380{,}000\). The indicated factor is
To four decimal places, the factor is 0.6333, corresponding to a 36.67% decrease. As a check, \(0.6333(\$600{,}000)\) is approximately $380,000, and 50% of $380,000 is $190,000, apart from rounding.
Common mistake. Using the $150,000 loss at the $500 deductible with the $600,000 target-coverage premium gives \(150{,}000/[600{,}000(0.50)]=0.50\). This mixes the old deductible’s losses with the target deductible’s premium. The loss amount must first be recalculated on the target coverage basis.
Exposure Units and Time Periods
A policy is not automatically one annual exposure. A policy written for six months contributes one-half of an annual exposure if the chosen unit is a policy-year and the policy is earned for the full six months. A premium and its associated losses can be compared as aggregate amounts over the same earned period without annualizing either; alternatively, both can be expressed per annual exposure. The mistake is to annualize one quantity but not the other.
The same discipline applies to earned and written amounts. Written premium reflects policies written during a period, whereas earned premium reflects the portion attributable to coverage provided during that period. If the losses represent earned experience, written premium should not be substituted without a reasoned adjustment that aligns the periods and exposures.
Worked Example: Six-Month Policies and Annual Exposure Units
A hypothetical portfolio has 80 policies, each earned for six months, with identical coverage. The 80 policies produce $24,000 of projected losses and $32,000 of earned premium for those six-month terms. The target loss ratio is 65%. Find the indicated rate-level factor. Show the calculation both per annual exposure and per six-month policy.
Method: express both losses and premium in the same time unit. Each six-month policy is one-half of an annual exposure, so the total annual exposure is
The projected loss per annual exposure is \(\$24{,}000/40=\$600\), and the premium per annual exposure is \(\$32{,}000/40=\$800\). The target loss amount per annual exposure is
Therefore,
The factor is 1.1538 to four decimal places, or a 15.38% increase. The same answer follows from the six-month amounts: each policy has an average loss of \(\$24{,}000/80=\$300\) and premium of \(\$32{,}000/80=\$400\). Its target loss amount is \(\$400(0.65)=\$260\), and \(\$300/\$260=1.153846\ldots\). The aggregate check also agrees: \(\$24{,}000/[\$32{,}000(0.65)]=1.153846\ldots\).
Common mistake. Comparing the six-month loss of $300 per policy with the annual target loss amount of $520 gives a factor below 1. That compares a half-year numerator with a full-year denominator. Either use $300 and $260 per six-month policy or $600 and $520 per annual exposure.
What a Ratio Can and Cannot Fix
A ratio removes common units only when its numerator and denominator were defined consistently in the first place. Dividing total losses by total premium does not automatically correct for differences in coverage or time period. Likewise, counting policies cannot replace counting earned exposure units when policy terms differ.
| Mismatch | Why it distorts the indication | Basis correction |
|---|---|---|
| Historical premium at a different rate level | The premium does not represent the rate level being evaluated. | On-level premium to the comparison rate level. |
| Losses and premium with different deductibles or limits | The amounts describe different insurance protection. | Recalculate or adjust one side to the chosen coverage basis using appropriate information. |
| Annual losses with partial-year premium, or vice versa | The amounts cover different durations of exposure. | Use matching periods, or express both amounts per the same exposure unit. |
| Earned losses compared with unadjusted written premium | The amounts may represent different coverage periods and exposure sets. | Use premium and losses defined for compatible earned exposures, or make an explicit adjustment. |
A useful exam habit is to annotate every amount before substituting it into a formula: “earned,” “at current rate level,” “$100 deductible,” or “per annual exposure” can prevent a substantial error. If a required adjustment cannot be determined from the information given, state what information is missing rather than silently assuming that the bases match.
Check Your Understanding
For each problem, identify the relevant basis adjustment and calculate the requested indication.
- A hypothetical portfolio has $324,000 of projected losses and $720,000 of earned premium. The premium is at 80% of the comparison rate level; losses and premium otherwise match in coverage, period, and exposures. The target loss ratio is 60%. Find the indicated factor and percentage change. (Answer: factor 0.6000; 40.00% decrease.)
- There are 50 identical claims, each with a ground-up loss of $1,500 and no payment limit. Historical premium is $200,000 at a $300 deductible. For a $100 deductible, the premium at the current rate level is 1.05 times the historical premium. With a 50% target loss ratio and all other bases matched, find the rate-level factor for the $100 deductible. (Answer: factor 0.6667; 33.33% decrease.)
- A hypothetical book has 120 policies, each earned for three months. Projected losses are $18,000 and earned premium is $24,000 for those terms. The target loss ratio is 60%. Find the indicated factor, using annual exposures as an intermediate check. (Answer: 30 annual exposures; factor 1.2500; 25.00% increase.)
- At a $250 deductible, 60 claims each have a ground-up loss of $1,750, and the matching earned premium is $300,000. At the target $50 deductible, premium at the current rate level is 1.10 times that amount. If the target loss ratio is 60%, find the indicated factor for the target coverage. (Answer: target losses $102,000; target-coverage premium $330,000; factor 0.5152; 48.48% decrease.)
- A hypothetical portfolio has 75 policies earned for six months each, with $45,000 of projected losses and $67,500 of earned premium for those terms. The target loss ratio is 65%. Calculate the factor per annual exposure and verify it using the aggregate amounts. (Answer: 37.5 annual exposures; factor 1.0256; 2.56% increase.)