Why Compare These Two Ratios?
A loss ratio provides a compact way to compare losses with the premium available to pay them. In a loss ratio indication, the actual loss ratio is compared with the permissible loss ratio: the target loss ratio consistent with the expense and profit provisions at the current rate level. Their relationship indicates whether rates need to increase or decrease, and by how much under the method’s assumptions.
The comparison is easy to misread. If the actual loss ratio is 8 percentage points above the permissible loss ratio, the indicated rate change is generally not an 8% increase. The indication is based on the ratio of the two loss ratios, not their difference. This distinction matters whenever an exam problem asks for the percentage rate change rather than just the gap between ratios.
Here, “actual” names the loss ratio being assessed; it does not mean that the ratio must consist only of already-paid claims. In an indication, the selected loss amount and premium base must be appropriate to the calculation and expressed on a consistent basis. The earlier material on actual loss ratios and the indication framework addresses those inputs. In the comparisons below, the ratios or their underlying amounts are supplied, so the task is to compare them rather than construct or adjust the underlying experience.
Meaning of the Indicated Factor
The actual loss ratio measures losses relative to the premium currently represented by the comparison. The permissible loss ratio describes the share of premium that may be allocated to losses while still allowing for the applicable expense and profit provisions. If \(A>P\), losses are high relative to the permissible share of premium, so the indication is an increase. If \(A<P\), the indication is a decrease. If \(A=P\), the indication is zero.
The percentage-point gap is \(A-P\). The indicated percentage change is \(A/P-1\). These answer different questions: the first describes the distance between two ratios; the second describes the proportional premium change indicated by the loss ratio method.
Worked Example: Turning a Ratio Gap Into a Rate Change
A hypothetical coverage has selected losses of $924,000 and corresponding premium of $1,200,000. Its permissible loss ratio is 70%. Calculate the actual loss ratio, the indicated factor, and the indicated percentage change. State the ratio gap and explain why it is not the percentage change.
Method. Calculate actual losses divided by premium, then divide that actual loss ratio by the permissible loss ratio.
Actual loss ratio. \(A=924{,}000/1{,}200{,}000=0.77=77\%\).
Indicated factor. \(A/P=0.77/0.70=1.10\). The factor is 1.10.
Indicated percentage change. \(1.10-1=0.10\), so the indicated change is a 10% increase.
Ratio gap. \(77\%-70\%=7\) percentage points. That gap is not the indicated 10% increase: the rate change is the ratio \(77\%/70\%\), less 1.
Check and common mistake avoided. If the premium increases by 10% while the selected losses remain $924,000, the premium becomes \(1{,}200{,}000(1.10)=$1{,}320{,}000\). The resulting loss ratio is \(924{,}000/1{,}320{,}000=70\%\), exactly the permissible ratio. Subtracting the two ratios and calling the 7-point gap a 7% increase would not produce the target ratio.
Why the Formula Holds
The loss ratio indication can be derived by comparing the current premium with the premium needed for the selected losses to equal the permissible share of premium. Let \(L\) be the selected losses, \(C\) the current premium on the comparison basis, and \(T\) the target premium. By definition, \(A=L/C\). At the target premium, the loss ratio is \(P=L/T\), so \(T=L/P\). Therefore:
Thus the indicated factor is \(A/P\), and the percentage change from current premium to target premium is \(T/C-1=A/P-1\). The derivation assumes the selected loss amount is held fixed when calculating the target premium, that premium scales proportionally with the rate change, and that the actual and permissible ratios refer to compatible bases. It does not model a change in exposure volume, policyholder behavior, or loss costs caused by the rate change. If the loss and premium measures do not match in scope or basis, the quotient may not provide a meaningful indication.
For positive \(P\), the direction follows directly from the quotient. If \(A>P\), then \(A/P>1\), so the indicated percentage change is positive. If \(A<P\), then \(A/P<1\), so the change is negative. Equality gives a factor of 1 and no indicated change.
Interpreting a Decrease Correctly
A factor below 1 is not itself the percentage decrease. The percentage change is the factor minus 1, expressed as a percentage. For example, a factor of 0.90 means the target premium is 90% of the current premium, which is a 10% decrease. This is particularly important when the actual ratio is below the permissible ratio: the gap in percentage points can look similar in size to the decrease while representing a different calculation.
Worked Example: A Below-Target Actual Loss Ratio
For a hypothetical product, selected losses are $459,000 and comparable premium is $850,000. The permissible loss ratio is 60%. Find the actual loss ratio, indicated factor, and percentage change. Verify the result using the target premium.
Method. Calculate \(A\), then use \(A/P\) for the factor and subtract 1 for the change.
Actual loss ratio. \(A=459{,}000/850{,}000=0.54=54\%\).
Indicated factor and change. \(A/P=0.54/0.60=0.90\). Hence the indicated percentage change is \(0.90-1=-0.10\), or a 10% decrease.
Target-premium check. The premium needed for a 60% loss ratio is \(T=459{,}000/0.60=$765{,}000\). Relative to the current $850,000, the change is \((765{,}000-850{,}000)/850{,}000=-0.10\), confirming a 10% decrease.
Common mistake avoided. The gap is \(60\%-54\%=6\) percentage points. It is not a 6% decrease. Using the difference as the rate change would give a target premium of \(850{,}000(0.94)=$799{,}000\), for which the loss ratio would be \(459{,}000/799{,}000\), approximately 57.45%, not the 60% target.
Combining Experience Segments
When several segments share one permissible loss ratio and one overall indication is requested, calculate the combined actual loss ratio from total losses divided by total premium. This is equivalent to weighting each segment’s actual loss ratio by that segment’s share of total premium. A simple arithmetic average of segment ratios generally gives segments equal weight even when their premium volumes differ.
Worked Example: Premium-Weighted Actual Loss Ratio
A hypothetical book has two segments. Segment 1 has premium of $250,000 and selected losses of $190,000. Segment 2 has premium of $750,000 and selected losses of $525,000. Both use a permissible loss ratio of 65%. Calculate the combined actual loss ratio and overall indicated change.
Method. Add losses and premiums before calculating the combined actual ratio, then compare that ratio with the common permissible ratio.
Segment ratios. Segment 1 has \(190{,}000/250{,}000=0.76=76\%\). Segment 2 has \(525{,}000/750{,}000=0.70=70\%\).
Combined amounts. Total losses are \(190{,}000+525{,}000=$715{,}000\). Total premium is \(250{,}000+750{,}000=$1{,}000{,}000\).
Combined actual loss ratio. \(A=715{,}000/1{,}000{,}000=0.715=71.5\%\). Equivalently, the premium weights are 25% and 75%, so \(0.25(76\%)+0.75(70\%)=19\%+52.5\%=71.5\%\).
Indication. The factor is \(0.715/0.65=1.10\), so the indicated change is \(1.10-1=0.10\), or a 10% increase.
Check and common mistake avoided. Applying a 10% increase to the $1,000,000 premium gives $1,100,000. The combined loss ratio is then \(715{,}000/1{,}100{,}000=0.65\), the permissible ratio. An unweighted average of 76% and 70% would be 73%, not the correct 71.5%, because the segments have different premium volumes.
Conditions and Common Exam Traps
The quotient is simple, but it is meaningful only when its inputs are comparable. For example, the numerator and denominator should reflect compatible coverage and experience bases, and the permissible ratio must correspond to the rate level and provisions being evaluated. This tutorial takes those inputs as given; it does not prescribe how to construct the actual ratio from detailed experience data.
The loss ratio method also answers a specific question: what proportional premium change would bring the selected loss ratio to the permissible ratio under the stated assumptions? It does not, by itself, explain which input caused a gap or whether an indicated change should be adopted without judgment. The indication is a calculation, not an automatic decision.
Identify the actual and permissible loss ratios and check that they refer to compatible bases.
Divide the actual loss ratio by the permissible loss ratio: \(A/P\).
Subtract 1 from the factor. A positive result is an increase; a negative result is a decrease.
If \(A>P\), the factor must exceed 1. If \(A<P\), it must be below 1. Use \(T/C=A/P\) as a premium-level check when appropriate.
A frequent exam trap is to subtract the permissible ratio from the actual ratio and report the result as the indicated percentage change. Another is to calculate the factor correctly but forget to subtract 1: a factor of 1.08 is an 8% increase, not a 108% increase. For combined data, adding losses and premiums before dividing avoids giving a small segment the same influence as a much larger one.
Check Your Understanding
For each problem, calculate the requested indication from the stated ratios or amounts. Treat all scenarios as hypothetical.
- A hypothetical line has selected losses of $544,000 and comparable premium of $800,000. Its permissible loss ratio is 64%. Find the actual loss ratio, indicated factor, and percentage change. (Answer: Actual ratio 68%; factor 1.0625; 6.25% increase.)
- A product has actual and permissible loss ratios of 70% and 75%, respectively. Calculate the indicated percentage change, then state the percentage-point gap. (Answer: Factor \(70/75=0.9333\ldots\); 6.67% decrease; the actual ratio is 5 percentage points below the permissible ratio.)
- Two segments have premiums of $300,000 and $700,000 and selected losses of $210,000 and $525,000. If they share a permissible loss ratio of 70%, find the combined actual loss ratio and indication. (Answer: Total losses $735,000 and premium $1,000,000 give an actual ratio of 73.5%; factor 1.05; 5% increase.)
- A smaller segment has premium of $300,000 and losses of $180,000; a larger segment has premium of $700,000 and losses of $560,000. The common permissible loss ratio is 68%. Calculate the correct combined indication and explain why the unweighted average of the segment ratios is unsuitable. (Answer: Segment ratios are 60% and 80%; combined actual ratio is \((180{,}000+560{,}000)/(300{,}000+700{,}000)=74\%\); factor \(0.74/0.68=1.088235\ldots\), or an 8.82% increase. The unweighted average, 70%, ignores the different premium volumes.)
- An actual loss ratio is 84%, and the permissible loss ratio is 70%. State both the percentage-point gap and the indicated percentage change. (Answer: 14 percentage points; factor \(0.84/0.70=1.20\), so a 20% increase.)