Why Factor-Selection Errors Matter
A development factor is meaningful only when its data, age definition, and intended use are clear. A ratio calculated across mismatched development ages can be numerically correct but actuarially irrelevant. A selection that combines different averaging windows may give inconsistent weight to experience. A single unusual observation can pull an average away from the pattern in the other years. And a well-supported tail estimate can still be used incorrectly if it is omitted or applied twice.
The purpose of this tutorial is to make those errors visible before a selected factor is used. The examples use hypothetical cumulative loss data and selected development factors. They focus on auditing selections, not on projecting an entire triangle. The methods and checks are aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.
A Reliable Audit Before Using a Factor
A factor review should start with the definition of the quantity being selected, not with a spreadsheet average. For each factor, record its starting age, ending age, eligible accident years, averaging method, and any adjustments. Then test whether the data actually match that description.
For a \(j\)-to-\(j+1\) ratio, use the cumulative amount at age \(j+1\) over the cumulative amount at age \(j\), for the same accident year. Do not divide values from different rows or different age pairs.
List which accident years contribute to each selected factor. If the windows differ, establish whether the difference is intentional and explain why the available experience supports it.
Look at individual ratios and their source data. A large or small ratio is a prompt to investigate, not automatic evidence that the observation should be excluded.
Determine whether the selected tail is from the last observed age to ultimate or is an additional link ratio between two specific ages. Apply it only to the age interval it represents.
Age Alignment: Ratios Must Compare Like With Like
A development triangle is arranged so each row represents an accident year and each column represents a development age. A link ratio is formed horizontally within one row. This simple rule prevents two common errors: dividing adjacent columns from different accident years, and labeling a ratio with the wrong development ages.
Worked Example: Finding the Correct Age-to-Age Ratio
A hypothetical paid-loss triangle contains the following cumulative amounts, in thousands of dollars. Select the simple average of the observed 12-to-24-month ratios and identify an apparently plausible but invalid calculation.
| Accident year | 12 months | 24 months | 36 months |
|---|---|---|---|
| 2021 | 100 | 130 | 143 |
| 2022 | 120 | 150 | — |
| 2023 | 90 | — | — |
Method: match the age pair within each eligible row. The 2021 ratio is \(130/100=1.300\). The 2022 ratio is \(150/120=1.250\). The 2023 row has no 24-month value, so it supplies no 12-to-24-month ratio.
The selected simple average is
A tempting but incorrect calculation is \(143/150=0.9533\). The numerator is the 2021 amount at 36 months, but the denominator is the 2022 amount at 24 months. It mixes accident years and does not describe development for any row. For comparison, the valid 2021 ratio from 24 to 36 months is \(143/130=1.100\), a different age pair. The selected 12-to-24-month factor is therefore 1.275, based on two eligible observations.
Check: both ratios used in the average compare 24-month cumulative loss with 12-month cumulative loss in the same accident year. The 36-month value is not used in this age-pair selection.
The reason for this alignment requirement can also be seen algebraically. For one accident year, the ratio \(C_{i,j+1}/C_{i,j}\) measures the change between two successive cumulative amounts. Its numerator is the later amount and its denominator is the earlier amount for that same experience. If the age pairs are consecutive, multiplying the ratios for that row cancels the intermediate cumulative amounts:
This telescoping relationship explains why adjacent-age factors can be combined to describe development across multiple ages. It also shows why a mismatched ratio is not a harmless labeling error: the cancellation, and thus the interpretation of the combined factor, no longer holds.
Averaging Windows and Selection Consistency
A factor can be calculated using several methods, including a simple average of individual link ratios or a volume-weighted ratio formed by dividing the sum of later-age amounts by the sum of earlier-age amounts. These methods need not give the same result. In addition, different development ages may reasonably have different eligible years or selection methods. The error is not automatically choosing different windows; it is failing to recognize, justify, or communicate the difference.
Worked Example: Comparing Unequal Windows
A hypothetical triangle has 12-to-24-month ratios of 1.20, 1.30, and 1.50 for three accident years. Only the first two years have observed 24-to-36-month ratios, which are 1.10 and 1.20. Compare a selection using all available observations with a comparison restricted to the two years observed at both age pairs.
Method: state the window and averaging method for each result. The volume-weighted 12-to-24-month factor using all three years is calculated from the underlying amounts represented by the ratios: \(120/100\), \(156/120\), and \(180/120\). Thus the pooled numerator is \(120+156+180=456\), and the pooled denominator is \(100+120+120=340\). The result is
The 24-to-36-month simple average uses its two available ratios:
These selections use different accident-year windows and different averaging methods. To make a like-window comparison, use the first two years at both ages. Their simple 12-to-24-month average is \((1.20+1.30)/2=1.25\), while the 24-to-36-month average remains 1.15.
The paired-window comparison does not prove that the all-year volume-weighted selection is wrong. It reveals what changes when the window and method change. For example, multiplying the all-year selection by 1.15 gives \(1.3412\times1.15=1.5424\) (rounded to four decimals); using the paired-window selections gives \(1.25\times1.15=1.4375\). The difference is material enough to require explanation. A defensible selection should state why the third year is included for one age pair but unavailable for the next, and why the volume-weighted method is appropriate there.
Common mistake avoided: describing both results as if they represented the same years and the same selection method. Always label the window and method before comparing or combining factors.
Outliers: Investigate Before Adjusting
An unusually high or low link ratio can reflect a data error, a genuine large claim, a change in claim handling, a coverage or settlement pattern, or random variation. The ratio alone cannot distinguish these explanations. Automatically retaining an unusual value can distort a simple average; automatically removing it can discard valid information. Review the underlying amounts and circumstances, quantify the observation’s influence, and document any adjustment or alternative selection.
Worked Example: Measuring the Influence of One High Ratio
Four hypothetical accident-year ratios for the same development age are 1.10, 1.12, 1.13, and 1.65. A review confirms that the high ratio is supported by valid loss emergence, but its cause may not be representative of future development. Calculate the simple average and median, and explain how to use the comparison.
Method: calculate both summaries and treat their difference as a sensitivity check. The simple average is
After sorting, the two middle ratios are 1.12 and 1.13, so the median is
The difference, \(1.25-1.125=0.125\), shows that the high observation strongly affects the simple average. Because the review confirms that the observation is valid, it should not be deleted merely to obtain a lower factor. A median selection of 1.125 could be considered as a robust alternative, but it is not automatically the correct selection either. The actuary should assess whether the event is relevant to future development, review other experience and operational changes, and state the basis for the choice.
Common mistake avoided: treating “outlier” as a sufficient reason for exclusion. The numerical sensitivity is evidence to investigate and disclose; it is not, by itself, a selection rule.
Tail Factors: Check the Starting Age and Endpoint
A tail factor accounts for development beyond an age at which ordinary age-to-age factors have been selected. Its label must make clear what interval it covers. A tail-to-ultimate factor from 36 months to ultimate already includes all development after 36 months. It is not another factor from 36 months to a later fixed age unless explicitly defined that way. Confusing these meanings can omit development or count it twice.
Worked Example: Applying a Tail Factor Once
A hypothetical portfolio has selected factors of 1.20 from 12 to 24 months and 1.10 from 24 to 36 months. A separately selected tail-to-ultimate factor from 36 months is 1.04. Starting from a 12-month cumulative amount of \$5,000, calculate the combined multiplier through ultimate and show the effects of omitting or duplicating the tail.
Method: multiply each factor once, in the order of the age intervals it represents. The two ordinary age-to-age factors give
The tail-to-ultimate factor starts at 36 months, so the combined multiplier from 12 months through ultimate is
Applied to \$5,000, the amount implied by this multiplier is \(5{,}000\times1.3728=\$6{,}864.00\). If the tail is omitted, the multiplier is only 1.32 and the amount is \$6,600.00, a difference of \$264.00. If the tail is mistakenly applied twice, the multiplier is \(1.3728\times1.04=1.427712\), and the amount is \(5{,}000\times1.427712=\$7{,}138.56\).
Check: the tail appears exactly once because it covers all development beyond 36 months. The three results differ only in the treatment of that stated tail. The calculation is rounded to the nearest cent only at the final dollar amounts.
Practical Review and Common Traps
A concise factor audit should make the selection reproducible. For each development age, retain the individual ratios, the years included, the selected method, and the selected value. For the tail, record its starting age, endpoint, and whether it is a cumulative tail-to-ultimate factor or a particular additional link ratio. If an adjustment is made, record its reason and show the effect of a reasonable alternative when the decision is influential.
- Wrong age or row: verify each numerator and denominator against the triangle, accident year, and age labels.
- Unstated averaging window: report the years used for every age pair; different windows may be appropriate, but should not be hidden.
- Automatic outlier treatment: investigate the source and test the influence before deciding to retain, adjust, or use a robust method.
- Tail omitted: check whether the final selected factor reaches ultimate or stops at a later observed development age.
- Tail duplicated: distinguish an additional link ratio from a tail-to-ultimate factor, then apply only the factor covering the remaining interval.
Check Your Understanding
For each problem, identify the relevant check and calculate the requested result.
- For one accident year, cumulative losses are \$80 at 12 months, \$104 at 24 months, and \$117 at 36 months. What is the valid 12-to-24-month ratio? (Answer: 1.3000)
- Three eligible ratios are \(1.18\), \(1.22\), and \(1.30\). Calculate their simple average. (Answer: 1.2333)
- Four ratios are \(1.08\), \(1.10\), \(1.12\), and \(1.50\). Calculate the median and the simple average, and identify which is more affected by the high observation. (Answer: Median \(=1.1100\); simple average \(=1.2000\); the simple average is more affected.)
- A selected tail-to-ultimate factor from 48 months is 1.04. A 48-month cumulative amount is \$3,200. What amount results from applying the tail once? (Answer: \$3,328.00)
- Selected factors are 1.15 from 12 to 24 months, 1.08 from 24 to 36 months, and a tail-to-ultimate factor of 1.03 from 36 months. What is the combined multiplier from 12 months through ultimate? (Answer: \(1.15\times1.08\times1.03=1.27926\))