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CAS Exam 5 · Development Factors and Tail · Tutorial 646 of 989

Select Coherent Factors Across Development Ages

Review factor selections across development ages as one coherent projection, while preserving justified exceptions and avoiding mechanical smoothing.

Intermediate 10 min read

What You'll Learn

  • Define selected age-to-age factors and their relationship to cumulative development factors
  • Check that factors are assigned to the correct consecutive development ages
  • Assess whether a sequence agrees with observed emergence and known claim-handling conditions
  • Calculate cumulative factors and projected ultimate losses from a selected sequence
  • Identify why summing factors or forcing a smooth decline can produce poor selections

Why Review Factors as a Sequence?

An age-to-age factor is selected for a particular transition, such as 12 to 24 months or 24 to 36 months. Those selections are often reviewed one at a time, but their purpose is to work together: applied in order, they describe how losses at an earlier age are expected to develop to later ages. A factor that looks reasonable on its own may still be inconsistent with the observed pattern or with the other selected factors.

A sequence review does not mean that factors must decline smoothly or follow a fixed shape. Development often slows at later ages, so factors commonly move closer to 1.00. But an increase at a later age can be credible when supported by the data and a plausible explanation. The question is whether the full sequence is defensible, not whether it looks cosmetically smooth.

Definition: Let \(C_{i,j}\) denote the cumulative paid or incurred loss for origin period \(i\), evaluated at development age \(j\). The observed link ratio from age \(j\) to the next age \(j+1\) is \(L_{i,j}=C_{i,j+1}/C_{i,j}\), when the denominator is suitable for forming a ratio. A selected age-to-age factor \(f_j\) is the factor chosen to represent development from age \(j\) to age \(j+1\). It is applied to losses at age \(j\), not to losses at some other age.

The cumulative development factor from age \(j\) through age \(k\), using selected factors for each intervening age, is the product of those factors. If the latest observed age is \(m\), the selected cumulative factor to age \(n\) is \(f_m f_{m+1}\cdots f_{n-1}\). The age labels matter: the factor for 12-to-24 months is applied before the factor for 24-to-36 months.

$$ \operatorname{CDF}_{m\to n}=\prod_{j=m}^{n-1} f_j $$

For a triangle built from cumulative paid losses, the cumulative amounts usually do not decrease when ordinary claim payments are added; therefore, positive link ratios are generally at least 1.00. Recoveries, reclassifications, or other adjustments can make paid amounts decrease. Incurred amounts can also decrease when case estimates are reduced. Thus, a factor below 1.00 is not automatically impossible, but it calls for attention to the loss measure and the circumstances that produced it.

What Makes a Sequence Coherent?

A coherent selection is one whose factors fit together in a way that can reasonably represent the observed development on a consistent basis. Review both the individual transitions and their combined implication.

  • Correct age alignment: Each selected factor must connect the stated consecutive ages. A factor for 24-to-36 months cannot be used as though it described 36-to-48 months.
  • Compatibility with observed emergence: Compare the selected factors with the observed link ratios and the shape of development at each age. A selected factor need not equal any one observation, but a material difference should have a reason.
  • Plausible progression: Consider whether the sequence is consistent with how the underlying paid or incurred losses are expected to emerge. Later factors often approach 1.00, but this is a diagnostic expectation, not a rule that overrides evidence.
  • Consistent basis: Confirm that the factors refer to the same loss definition, valuation basis, and relevant claim population. A change in case reserving, claim handling, or the underlying mix may make adjacent-age experience less comparable.
  • Reasonable cumulative implication: Multiply the selected factors and consider the total development they imply. A set of individually plausible values can still imply an unexpectedly large or small cumulative factor when combined.

The product calculation is not a separate selection method. It is an essential consistency check: it shows what the entire sequence does to a loss observed at an earlier age. The earlier material on link ratio interpretation and cumulative development factors provides the underlying calculations; the focus here is on reviewing the selections together.

Why the Factors Multiply

The product follows directly from the meaning of each factor. Suppose a loss amount at age \(j\) is \(C_j\), and the selected factor for the next transition is \(f_j\). The projected amount at age \(j+1\) is \(C_j f_j\). Applying the next selected factor gives \(C_j f_j f_{j+1}\). Repeating this step through age \(n\) gives the product of all intervening factors.

$$ C_{j+1}=C_j f_j,\qquad C_{j+2}=C_{j+1}f_{j+1}=C_j f_j f_{j+1} $$

More generally, since the observed link ratio is \(C_{i,j+1}/C_{i,j}\), multiplying consecutive observed ratios cancels the intermediate cumulative amounts:

$$ \frac{C_{i,j+1}}{C_{i,j}} \cdot \frac{C_{i,j+2}}{C_{i,j+1}} \cdots \frac{C_{i,k}}{C_{i,k-1}} = \frac{C_{i,k}}{C_{i,j}} $$

Selected factors approximate the corresponding transitions rather than necessarily matching one origin period’s ratios exactly. Their product therefore gives the selected cumulative development implied by those choices. This interpretation assumes the factors apply sequentially to the same cumulative loss measure and that the underlying experience is sufficiently relevant to the projection being made. A material change in the data basis weakens that interpretation and should be investigated rather than concealed by arithmetic.

Worked Example: A Sequence That Follows Observed Deceleration

An actuary reviews paid loss link ratios for a hypothetical book. Across the usable origin periods, the observed ratios are 1.58, 1.62, and 1.60 for 12-to-24 months; 1.28, 1.31, and 1.30 for 24-to-36 months; 1.16, 1.14, and 1.15 for 36-to-48 months; and 1.05, 1.07, and 1.06 for 48-to-60 months. The proposed selected factors are 1.60, 1.30, 1.15, and 1.06. A latest 12-month cumulative paid loss of $4,000,000 is to be projected to 60 months. Review the sequence and calculate the projection.

Method: inspect the transition selections, then calculate the cumulative factor. For 12-to-24 months, the observations range from 1.58 to 1.62 and the selection is 1.60. For 24-to-36 months, the observations range from 1.28 to 1.31 and the selection is 1.30. For 36-to-48 months, the observations range from 1.14 to 1.16 and the selection is 1.15. For 48-to-60 months, they range from 1.05 to 1.07 and the selection is 1.06. Each selected value is within its corresponding observed range.

The selected sequence also has a plausible pattern: the amount of additional development declines with age, and each successive factor is closer to 1.00. This is consistent with the supplied observations; it is not a rule that every portfolio must follow this shape.

Calculate the cumulative factor in age order:

$$ 1.60\times1.30=2.08,\qquad 2.08\times1.15=2.392,\qquad 2.392\times1.06=2.53552 $$

Apply the 12-to-60-month cumulative factor to the 12-month cumulative paid loss:

$$ \$4{,}000{,}000\times2.53552=\$10{,}142{,}080 $$

The projected 60-month cumulative paid loss is $10,142,080. As a check, the cumulative factor exceeds 1.00, as expected for this paid-loss scenario with positive observed development, and the projection is greater than the starting amount. This review avoids treating the four factors as alternatives: they represent consecutive transitions and must be multiplied.

Investigate Anomalies; Do Not Smooth Them Away

An unusual factor should prompt questions about the data and the process that generated it. For example, a large late-age incurred ratio might reflect a change in case reserve adequacy rather than a sudden change in ultimate claim cost. A late-age paid ratio might rise because a block of claims was settled or paid earlier than usual. Other possible explanations include data corrections, recoveries, changes in claim handling, or a small and unrepresentative amount of experience.

An explanation is not proof that a factor should be retained. The actuary should check that the explanation is consistent with the timing and type of observed development, and then decide whether the selected factor should reflect that experience. Conversely, changing a factor merely to make the sequence decline smoothly can discard real information. The decision should be tied to the evidence and the purpose of the projection.

Worked Example: Reviewing a Late-Age Outlier

For a hypothetical incurred-loss triangle, the selected sequence is proposed as 1.55 for 12-to-24 months, 1.28 for 24-to-36 months, 1.13 for 36-to-48 months, and 1.05 for 48-to-60 months. The observed 36-to-48-month ratios are 1.12, 1.11, and 1.45. The 1.45 ratio belongs to an origin period affected by a documented, one-time case reserve strengthening. A 24-month incurred amount of $6,000,000 is being projected to 60 months. Explain the concern and calculate the projection using the proposed sequence.

Method: compare the anomalous observation with the operational information, then evaluate the full selected sequence. The first two observed ratios, 1.12 and 1.11, are close to the selected 36-to-48-month factor of 1.13. The third ratio, 1.45, is substantially higher. Because the high ratio coincides with a documented reserve strengthening, it may describe an unusual change in case estimates rather than repeatable development for the projected book.

The selected factor of 1.13 is not simply justified by being smoother than 1.45. It is defensible here only if the actuary concludes that the one-time strengthening is not representative of future development and that the other observations and relevant conditions support the selection. If the change reflects an ongoing reserving practice, the selection would need further analysis.

The loss is at 24 months, so apply only the factors for 24-to-36, 36-to-48, and 48-to-60 months:

$$ 1.28\times1.13=1.4464,\qquad 1.4464\times1.05=1.51872 $$

Then project the 24-month incurred amount:

$$ \$6{,}000{,}000\times1.51872=\$9{,}112{,}320 $$

The projected 60-month incurred amount is $9,112,320. The check is the age alignment: the 12-to-24-month factor of 1.55 is excluded because the observed loss is already at 24 months. A common mistake is to multiply by every factor in the selected sequence and project from the wrong starting age.

Worked Example: Keeping a Supported Increase in the Sequence

A hypothetical paid-loss portfolio has proposed factors of 1.45 for 12-to-24 months, 1.20 for 24-to-36 months, 1.24 for 36-to-48 months, and 1.04 for 48-to-60 months. Observed 36-to-48-month ratios for the relevant origin periods cluster between 1.22 and 1.26. The actuary has confirmed that the increase reflects a recurring payment pattern, not a data error. Project a 12-month cumulative paid amount of $2,500,000 to 60 months and assess whether the increase alone makes the sequence incoherent.

Method: test each selected transition against the evidence and calculate the implied cumulative factor. The 36-to-48-month selection of 1.24 lies within the observed range of 1.22 to 1.26. The stated investigation supports the interpretation that this is a recurring payment pattern. Although the factor rises from 1.20 to 1.24, the increase is not by itself a reason to replace it with a smaller factor.

Multiply the four consecutive factors:

$$ 1.45\times1.20=1.74,\qquad 1.74\times1.24=2.1576,\qquad 2.1576\times1.04=2.243904 $$

Apply the resulting 12-to-60-month cumulative factor:

$$ \$2{,}500{,}000\times2.243904=\$5{,}609{,}760 $$

The projected 60-month cumulative paid amount is $5,609,760. The sequence is coherent on the facts provided: the non-monotone transition is supported by observations and a recurring operational explanation. Forcing the 36-to-48-month factor below 1.20 solely to make every factor decline would ignore the evidence.

Common Errors in Sequence Review

  • Adding rather than multiplying. In the first worked example, adding the factors gives \(1.60+1.30+1.15+1.06=5.11\). Using 5.11 as a multiplier would produce $20,440,000 from $4,000,000, not the correct $10,142,080. Consecutive development factors compound, so the cumulative factor is their product.
  • Forcing a smooth decline. Later factors often move closer to 1.00, but a rising factor can be credible when the data and process support it. A visual pattern is a diagnostic, not a selection rule.
  • Ignoring the loss basis. Paid and incurred development can differ. A pattern caused by changing case reserve practices should not be interpreted as though it were simply a paid emergence pattern.
  • Using a factor at the wrong starting age. Projection begins with the factor for the next age transition after the observed age. Including earlier factors counts development that has already occurred.
  • Explaining away every outlier. A plausible narrative must be checked against the timing, data, and ongoing conditions. An unsupported explanation does not make an anomalous factor representative.
Key takeaway: Review selected factors as a chronological sequence: compare each factor with the relevant experience, assess the pattern without imposing mechanical smoothness, and multiply the correctly aligned factors to test the total implied development.

Check Your Understanding

For each problem, review both the individual transitions and the cumulative implication.

  1. A hypothetical paid-loss selection is 1.40 from 12 to 24 months, 1.20 from 24 to 36 months, and 1.10 from 36 to 48 months. Calculate the 12-to-48-month cumulative factor and the projected 48-month amount from a 12-month amount of $3,200,000. (Answer: cumulative factor \(1.848\); projected amount $5,913,600.)
  2. A 36-month incurred amount is $8,000,000. Selected factors are 1.50 from 12 to 24 months, 1.25 from 24 to 36 months, 1.12 from 36 to 48 months, and 1.05 from 48 to 60 months. Which factors apply, and what is the projected 60-month amount? (Answer: use 1.12 and 1.05; cumulative factor \(1.176\); projected amount $9,408,000.)
  3. For a paid triangle, the selected factors are 1.32, 1.18, 1.21, and 1.03 in consecutive age order. The 1.21 selection is supported by observations between 1.19 and 1.23 and a recurring payment pattern. Is the increase from 1.18 to 1.21 alone grounds to reject the sequence? (Answer: No; the increase is supported by the observations and the stated recurring pattern, though the complete sequence and basis still require review.)
  4. A proposed 12-to-48-month sequence is 1.50, 1.25, and 1.10. A reviewer adds them and reports a cumulative factor of 3.85. Calculate the correct factor and explain the error. (Answer: \(1.50\times1.25\times1.10=2.0625\); factors compound and must be multiplied, not added.)
  5. A proposed 24-to-60-month sequence uses 1.30 for 24-to-36 months, 1.15 for 36-to-48 months, and 1.08 for 48-to-60 months. The 24-month amount is $5,000,000. Calculate the projected 60-month amount and the cumulative factor to three decimals. (Answer: cumulative factor \(1.30\times1.15\times1.08=1.6146\), or 1.615 to three decimals; projected amount $8,073,000.)

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