What an Actual-to-Expected Loss Ratio Measures
Experience rating compares observed losses with a stated expectation. The actual-to-expected loss ratio, often written as an A/E ratio, expresses the actual losses as a multiple of expected losses. It provides a direct comparison with the plan’s expected-loss baseline: a ratio of 1.00 means actual and expected losses are equal, while a ratio above or below 1.00 indicates how actual losses compare with that baseline.
This tutorial focuses on the uncapped ratio calculated from supplied plan inputs. It does not apply a credibility adjustment, a stabilizing amount, or a minimum or maximum modification. Those are separate operations; they should not be silently folded into this basic calculation.
The numerator is actual losses, not expected losses or premium. The denominator is expected losses, not the number of claims or exposure units. Expected losses may be supplied as a total or calculated from supplied exposures and expected-loss rates. If they are calculated, the units and rate must correspond: for example, exposure units multiplied by dollars of expected loss per exposure unit.
A ratio does not have to be close to 1.00. Its meaning is relative to the baseline: 0.90 means actual losses are 90% of expected losses, while 1.10 means they are 110% of expected losses. The difference from 1.00 is often expressed as a percentage deviation:
For instance, an A/E ratio of 1.10 corresponds to a 10% excess over expected, not a 110% excess. The ratio itself is 110% of expected; its deviation from the baseline is 10%.
Calculation and Interpretation
Use totals on a consistent basis. First identify the actual-loss total and the expected-loss total specified by the plan inputs. If the expected total must be calculated from exposure units and rates, calculate each component and add them. Then divide actual losses by expected losses, and check the result by translating it back into a comparison with 1.00.
Record the actual losses and the expected losses that the supplied plan inputs define for the experience being evaluated.
For each exposure group, multiply its exposure by its expected-loss rate, then add the group totals.
Compute \(A/E\), keeping the order of the numerator and denominator explicit.
Compare it with 1.00, and, if useful, subtract 1 and express the difference as a percentage.
| Ratio | Comparison | Interpretation |
|---|---|---|
| A/E < 1.00 | A < E | Actual losses are below expected losses. |
| A/E = 1.00 | A = E | Actual losses equal expected losses. |
| A/E > 1.00 | A > E | Actual losses are above expected losses. |
The direction of the ratio follows directly from division by a positive expected-loss total. If \(E>0\), then \(A/E<1\) exactly when \(A<E\); equality holds exactly when \(A=E\); and \(A/E>1\) exactly when \(A>E\). This is why the baseline for interpretation is 1.00 rather than 0.00.
Worked Calculations
Worked Example: Comparing Actual Losses With a Supplied Expected-Loss Rate
A hypothetical service-contract portfolio has 80 earned exposure units. The supplied expected-loss rate is $500 per exposure unit. The actual losses for the evaluation period are $12,600, $18,900, and $7,500. Calculate the uncapped A/E ratio and interpret it.
Assumptions and inputs. Use the 80 exposure units and $500 rate as supplied, with no further adjustment to either total. No interest discounting is applied, no mortality assumption or model is relevant, and no probability distribution is assumed for the observed losses; this is a deterministic calculation from the stated inputs.
Step 1: Calculate expected losses.
Step 2: Total the actual losses.
Step 3: Divide actual by expected.
The ratio is below 1.00. Actual losses are \(0.975\times100\%=97.5\%\) of expected losses, so they are \(100\%-97.5\%=2.5\%\) below expected. Check: expected minus actual is \(\$40{,}000-\$39{,}000=\$1{,}000\), and \(\$1{,}000/\$40{,}000=0.025=2.5\%\). The difference agrees with the ratio interpretation.
Method and mistake avoided. This is a total-to-total comparison. Dividing the average actual loss by the expected loss for the whole portfolio would mix a per-claim quantity with a portfolio total.
Worked Example: Summing Claim Amounts Before Comparing
A hypothetical equipment coverage program has 320 exposure units and an expected-loss rate of $125 per exposure unit. Four actual claims are $9,500, $12,000, $8,500, and $14,500. Find the A/E ratio and state the percentage deviation from expected.
Assumptions and inputs. The plan’s supplied rate applies to all 320 exposure units, and the four listed amounts are the actual losses to be compared. No interest discounting, mortality assumption, or probability distribution is used.
Step 1: Calculate the expected-loss total.
Step 2: Add the actual claims.
Step 3: Calculate and interpret the ratio.
Actual losses are \(1.1125\times100\%=111.25\%\) of expected. The deviation is \(1.1125-1=0.1125\), or \(11.25\%\) above expected. Check: \(\$44{,}500-\$40{,}000=\$4{,}500\), and \(\$4{,}500/\$40{,}000=0.1125\), confirming the percentage deviation.
Method and mistake avoided. Sum the actual claim amounts, then compare that total with the expected losses for all exposure units. The number of claims does not replace the expected-loss denominator.
Worked Example: Combining Expected Losses Across Exposure Groups
A hypothetical property program has two exposure groups. Group A has 150 exposure units with an expected-loss rate of $280 per unit; Group B has 60 units with a rate of $450 per unit. Actual losses are $33,000 for Group A and $39,000 for Group B. Calculate the overall A/E ratio and interpret the result.
Assumptions and inputs. Each group’s supplied rate applies to that group’s exposure, and the two actual-loss amounts are the corresponding group totals. No interest discounting, mortality assumption, or probability distribution is used.
Step 1: Calculate expected losses by group.
Step 2: Add each side across both groups.
Step 3: Calculate the overall ratio. The unrounded quotient is \(72{,}000/69{,}000=24/23=1.043478\ldots\). Rounded to four decimal places:
The ratio exceeds 1.00, so actual losses are above expected. The unrounded percentage deviation is \((3{,}000/69{,}000)\times100\%=4.347826\ldots\%\), or 4.35% to two decimal places. Check: the actual total exceeds the expected total by \(\$3{,}000\), and that difference is about 4.35% of \(\$69{,}000\).
Method and mistake avoided. Add actual losses across groups and expected losses across groups before dividing. Dividing each group’s actual losses by its expected losses and taking an unweighted average would not, in general, equal the ratio of the combined totals.
Deriving the Percentage Deviation
The ratio and the percentage deviation are two ways to report the same comparison. Starting from the definition and adding and subtracting 1 gives:
Thus, the amount by which the ratio differs from 1.00 is \((A-E)/E\). Multiplying that relative difference by 100 expresses it as a percentage of expected losses. The denominator is \(E\), because the question is how large the actual-versus-expected difference is relative to expected. Using \(A\) instead would answer a different question: the difference relative to actual losses.
For example, if actual losses are $44,500 and expected losses are $40,000, the difference is $4,500. Relative to expected, that is \(4,500/40,000=11.25\%\), matching the ratio \(1.1125\). This identity is also a useful audit: the sign of \(A-E\) must agree with whether A/E is above or below 1.00.
Common Exam Trap: Reversing the Ratio
A frequent error is to divide expected losses by actual losses. For the first worked example, the incorrect calculation would be:
That result is not the A/E ratio. It reverses the specified comparison and incorrectly suggests a value above 1.00 even though actual losses are less than expected. The correction is to put actual losses in the numerator:
Another check is to compare the dollar totals before interpreting the quotient. If \(A<E\), the correctly ordered ratio must be below 1.00; if it is not, recheck the division and the totals.
Check Your Understanding
For each problem, calculate the uncapped A/E ratio and interpret it using the stated inputs.
- A hypothetical portfolio has actual losses of $27,600 and expected losses of $30,000. What is its A/E ratio, and are actual losses above or below expected? (Answer: 0.9200; actual losses are 8.00% below expected.)
- A hypothetical liability group has 90 exposure units, an expected-loss rate of $360 per unit, and actual claims of $8,100, $12,150, and $14,310. Calculate expected losses, actual losses, and A/E. (Answer: expected losses $32,400; actual losses $34,560; A/E = 1.0667, or 6.67% above expected.)
- A hypothetical account has two groups: 50 units at an expected-loss rate of $520 and 40 units at a rate of $350. Actual losses are $24,800 and $16,400, respectively. Calculate the combined ratio. (Answer: expected losses $40,000; actual losses $41,200; A/E = 1.0300, or 3.00% above expected.)
- A hypothetical portfolio has expected losses of $75,000 and an A/E ratio of 0.88. Find actual losses and their dollar difference from expected. (Answer: actual losses $66,000; they are $9,000 below expected, or 12.00% below.)
- A hypothetical program has actual losses of $48,600 and expected losses of $48,000. Calculate the ratio and percentage deviation, and state the dollar excess. (Answer: A/E = 1.0125; 1.25% above expected; dollar excess $600.)