Why Reconciliation Comes Before the Ratio
An actual-to-expected ratio is meaningful only when its numerator and denominator describe the same risk. If actual losses include a coverage that expected losses exclude, or if the losses and expected amount refer to different periods or cost levels, dividing one by the other produces a number but not a fair comparison. The essential work is to reconcile the bases first.
The reconciliation applies to the inputs, not just to the final arithmetic. Confirm the covered lines and loss terms, the valuation date, the experience period, and the exposure basis. A consistent comparison can then be summarized by the ratio of aligned actual losses to aligned expected losses. This tutorial concerns that comparison; it does not apply a credibility weighting or a stabilizing adjustment.
The Four Reconciliation Checks
Use these checks before calculating a ratio. The precise adjustment depends on the data and plan rules; do not apply a factor merely because it is available.
Include the same coverage components on both sides. Check whether losses and expected losses are on a consistent gross or net basis and use compatible limits, deductibles, and other plan definitions.
Determine whether actual losses are paid, incurred, or otherwise valued, and what development is needed to reach the chosen valuation basis. If losses are brought to a different cost level, expected losses must be at that same level.
Match the dates represented by actual losses with the dates represented by exposures and expected losses. Include partial periods only to the extent that their exposures and losses are included consistently.
Use the exposure measure specified by the expected loss rate. Convert the exposure to the rate’s unit before multiplying; for example, a rate per $100 of payroll is multiplied by payroll divided by $100.
An adjustment must be applied on the correct side of the comparison. For example, if actual losses are developed and trended to a selected cost level, expected losses must already be expressed at that level or be adjusted to it as well. Applying a trend factor to actual losses while leaving expected losses at an incompatible level does not complete the reconciliation.
Worked Example: Excluding an Unmatched Coverage
A hypothetical employer has $600,000 of payroll for a one-year experience period. The expected loss rate for general liability is $0.80 per $100 of payroll. At the selected valuation date, the recorded actual losses are $5,200 for general liability and $1,500 for property coverage. The expected loss calculation covers general liability only. Calculate the aligned ratio.
Method. First match the coverage. The property loss is not part of the general-liability expected loss basis, so it must not be added to the general-liability actual loss total.
Step 1: Convert payroll to units of $100.
Step 2: Calculate expected general-liability losses.
Step 3: Use only the actual losses for the matching coverage. Aligned actual general-liability losses are $5,200. Therefore,
Rounded to four decimal places, the aligned ratio is 1.0833. As a check, actual general-liability losses are $400 greater than expected, and $400 divided by $4,800 is 0.0833, giving a ratio of 1.0833.
Common mistake avoided: Adding both coverages gives $6,700 divided by $4,800, or 1.3958. That calculation compares actual losses for two coverages with expected losses for one, so the result is not an aligned ratio.
Reconciling Valuation Date and Cost Level
Actual losses may be valued at different stages of maturity. Paid losses omit amounts not yet paid; incurred losses generally include paid losses and case reserves, but may still change as claims develop. If a calculation requires ultimate losses, a development adjustment may be needed. A separate trend adjustment may then bring losses from their historical cost level to the cost level selected for the comparison. These adjustments have different purposes: development addresses the emergence of loss amounts over time, while trend addresses changes in costs between periods.
The factors must be appropriate to the losses and dates in the problem. If a development factor brings losses to ultimate and a trend factor brings ultimate losses to the comparison level, the combined adjustment is their product. Do not apply a factor twice, and do not trend expected losses a second time when their rate is already stated at the target level.
Worked Example: Bringing Losses to a Common Cost Level
A hypothetical fleet’s incurred losses for a one-year experience period are $18,000 at the stated valuation date. The supplied development factor to ultimate is 1.20. The supplied trend factor from the experience period’s ultimate loss level to the comparison cost level is 1.10. Expected losses are $15,000 and are already stated at that comparison cost level, for the same coverage and experience period. Calculate the aligned ratio.
Method. Develop the actual incurred losses to ultimate first, then trend those ultimate losses to the stated comparison level. Since expected losses are already at that level, leave the expected amount unchanged.
Step 1: Develop actual losses to ultimate.
Step 2: Trend ultimate losses to the comparison level.
Step 3: Divide aligned actual by aligned expected losses.
The aligned ratio is 1.5840. The calculation uses the supplied factors in sequence, and the expected amount remains $15,000 because it is already at the comparison cost level.
Common mistake avoided: Applying only the trend factor gives $19,800 of actual losses and a ratio of 1.3200, but that leaves the actual amount at the wrong development basis. Applying the trend factor again to expected losses would also be wrong: its stated rate is already at the target level.
Matching Experience Period and Exposure Basis
The expected loss total is typically built from exposure and an expected loss rate. If \(E_j\) denotes the exposure in segment \(j\), and \(r_j\) is its expected loss rate per exposure unit, then the expected loss for that segment is \(E_j r_j\), after expressing \(E_j\) in the rate’s stated units. Expected losses for multiple segments are summed. This calculation is valid only if the exposures cover the same period and risks as the actual losses being compared.
The relationship also explains why a multi-year ratio is generally calculated from summed losses rather than by taking an unweighted average of annual ratios. If year \(j\) has actual losses \(A_j\) and expected losses \(L_j\), the combined ratio is
To justify the equality, multiply each annual ratio \(A_j/L_j\) by its weight \(L_j/\sum_k L_k\). The product is \(A_j/\sum_k L_k\); summing these products gives \(\sum_j A_j/\sum_k L_k\). Thus the combined ratio is an expected-loss-weighted average of the annual ratios, not necessarily their simple average.
Worked Example: Reconciling Two Years of Payroll Exposure
A hypothetical account has two complete experience years. Actual losses are $2,400 in year 1 and $3,200 in year 2. Payroll is $200,000 in each year, and the expected loss rate is $0.50 per $100 of payroll in both years. Assume the actual losses and expected loss rates use the same coverage, loss terms, and valuation basis. Calculate the two-year ratio.
Method. Calculate expected losses for each year from that year’s payroll, then aggregate actual and expected losses over the same two years.
Step 1: Convert the annual payroll to $100 units.
Step 2: Calculate each year’s expected losses.
Two-year expected losses are \(\$1{,}000+\$1{,}000=\$2{,}000\). Two-year actual losses are \(\$2{,}400+\$3{,}200=\$5{,}600\). Therefore,
The aligned two-year ratio is 2.8000. As a check, the annual ratios are 2.4 and 3.2; because expected losses are equal in the two years, their expected-loss-weighted average is \((2.4+3.2)/2=2.8\), agreeing with the aggregate calculation.
Common mistake avoided: Dividing the $5,600 of actual losses by only one year’s $1,000 of expected losses gives 5.6. The denominator must cover both years, just as the numerator does. Conversely, doubling the actual losses would be equally incorrect because the stated actual total already includes both years.
Common Reconciliation Traps
- Combining unmatched coverages. Do not include a loss component unless the expected loss basis includes that component under compatible terms.
- Mixing valuation bases. Incurred, paid, developed, and ultimate amounts are not interchangeable without a stated basis and appropriate adjustment.
- Using inconsistent periods. A two-year actual total cannot be compared with one year of expected losses. Partial-period exposures should be counted consistently with the period represented by the losses.
- Misreading the exposure unit. A rate per $100 of payroll is multiplied by payroll divided by $100, not by payroll itself.
- Averaging annual ratios without weights. A simple average is only equal to the aggregate ratio in special cases, such as equal expected losses in every year.
- Treating the raw ratio as the final modification. Reconciliation establishes a comparable actual-to-expected measure; it does not by itself supply any additional plan adjustment.
Check Your Understanding
For each problem, identify the compatible basis and calculate the requested ratio. Amounts and factors are hypothetical.
- A one-year general-liability experience period has $7,200 of actual general-liability losses and $900 of actual property losses. The expected loss rate is $1.20 per $100 of receipts, and receipts are $500,000. Expected losses cover general liability only. What is the aligned ratio? (Answer: 1.2000)
- A hypothetical account has $350,000 of payroll and $1,750 of actual losses for the same one-year period and coverage. The expected loss rate is $0.40 per $100 of payroll. Assume the valuation bases match. What is the ratio? (Answer: 1.2500)
- Actual incurred losses are $12,000 at the stated valuation date. The development factor to ultimate is 1.25, and the trend factor to the comparison cost level is 1.08. Expected losses for the same coverage and period are $15,000 at that comparison level. What is the aligned ratio? (Answer: 1.0800)
- For year 1, actual losses are $900 and payroll is $100,000; for year 2, actual losses are $2,100 and payroll is $300,000. In both years the expected loss rate is $0.60 per $100 of payroll. The coverage and valuation bases match. What is the two-year aggregate ratio, and why should it not be replaced by an unweighted average of annual ratios? (Answer: 1.2500; aggregate the two-year totals, since expected losses differ by year.)
- A hypothetical two-coverage account has actual losses of $4,000 for coverage A and $2,000 for coverage B. Payroll is $400,000 for A and $200,000 for B. Expected rates are $0.75 per $100 for A and $0.50 per $100 for B. The requested ratio covers both coverages, with matching periods and valuation bases. What is the ratio? (Answer: 1.5000)