Balance Recent and Historical Link Ratio Experience

Why Balance Recent and Historical Experience?

A link ratio selection uses observed development experience to estimate how losses at one development age will grow by the next age. Choosing the observation window matters: a short, recent window may reflect current claims handling or settlement conditions, while a longer window usually contains more experience and may be less sensitive to individual unusual observations. Neither window is automatically preferable.

The task is to judge which observations are relevant to the development being estimated. This tutorial focuses on comparing recent and historical experience at the same development age. It does not prescribe a universal number of years for either window. The appropriate length depends on the data, the stability of the underlying process, and the purpose of the estimate.

Definition: For accident year $i$, let $C_{i,j}$ be the cumulative loss at development age $j$. The observed age-to-age link ratio from age $j$ to age $j+1$ is $f_{i,j}=C_{i,j+1}/C_{i,j}$, provided $C_{i,j}$ is a suitable positive denominator. A recent window uses the latest selected accident years with observations at that age; a historical window uses a longer span of experience. Always compare ratios for the same age-to-age interval and on a consistent loss and valuation basis.

The arithmetic can produce different answers for different windows, but it cannot establish which answer is appropriate. Before selecting a window, consider whether the underlying conditions have changed, whether a recent result is representative or unusually volatile, and whether all observations are comparable. For example, if a change in claims handling began at a known date, older experience may describe a process that is no longer in force. Conversely, one unusually high or low recent ratio does not, by itself, prove that the underlying pattern changed.

When an average is used, its meaning depends on the averaging method. A simple average gives each observed link ratio equal weight. A volume-weighted average gives greater weight to ratios with more loss volume in their denominators. The examples below state the method used; the central judgment remains which observations belong in the selection.

A Practical Comparison

A disciplined comparison makes the selection easier to explain and review. Begin by calculating the recent and longer-term estimates separately. Then examine how far apart they are, what observations drive the difference, and whether the evidence supports a persistent change or only short-term variation. If neither window is clearly superior, a stated blend can make the judgment explicit.

Hold the development age and basis fixed.
Compare like with like: the same age-to-age interval, type of loss, valuation basis, and treatment of unusual observations.

Calculate the candidate windows.
Show the recent estimate and the longer-term estimate separately, using the chosen averaging method consistently.

Assess the evidence behind the difference.
Ask whether a documented and continuing change makes recent experience more relevant, or whether the apparent difference is driven by limited or volatile observations.

Select and record the rationale.
Use the better-supported window or state a transparent blend. Record why the selected observations represent the expected development process.

This is a judgment framework, not a formula that mechanically identifies the correct window. A selected factor should be explainable in terms of relevance and reliability, not merely described as “the most recent” or “the longest available.”

Worked Examples

Worked Example: A Recent Window With One Volatile Observation

A hypothetical insurer is estimating the link ratio from 12 to 24 months for a stable line of business. The simple-average link ratios by accident year, from oldest to newest, are 1.16, 1.18, 1.17, 1.19, 1.18, 1.31, and 1.15. The latest two accident years form the recent window. There is no documented change in the underlying process, and the 1.31 observation is known to be unusually volatile. Compare the recent and full-period averages and recommend a selection.

Method. Calculate simple arithmetic averages at the same development age. The recent-window average is

\frac{1.31+1.15}{2}=\frac{2.46}{2}=1.23.

For the full-period average, sum all seven observations:

$$ 1.16+1.18+1.17+1.19+1.18+1.31+1.15=8.34. $$

Therefore, the full-period average is $8.34/7=1.191428\ldots$, or 1.1914 to four decimal places. The five older observations average $5.88/5=1.176$, which also shows that the recent two-year result is elevated mainly because it includes the 1.31 observation.

Given the stated lack of a process change and the identified volatility, the longer window is more representative than the recent two-year window. A reasonable selection from the two candidates is 1.1914, subject to the actuary’s review of the underlying data. This is not a claim that every volatile observation should be retained; the point is that selecting the recent average solely because it is recent would not be supported here.

Check. The recent average, 1.23, exceeds the full-period average, 1.1914, as expected because the recent window contains the unusually high 1.31 observation. The common mistake avoided is treating a short window as automatically more current and therefore more credible.

Worked Example: A Sustained Change Supports Recent Experience

A hypothetical insurer estimates the 24-to-36-month link ratio. The five older accident-year ratios are 1.10, 1.11, 1.12, 1.11, and 1.10. After a documented and continuing change in case-reserving practice, the next three ratios are 1.20, 1.19, and 1.21. Use simple averages to compare the older and recent periods. If the change is expected to continue, select a factor and calculate the projected next-age cumulative loss from an observed $C_{i,24}$ of $8,000,000.

The older-period total is $1.10+1.11+1.12+1.11+1.10=5.54$. Its average is $5.54/5=1.108$. The recent-period total is $1.20+1.19+1.21=3.60$, so its average is $3.60/3=1.20$.

The combined eight-year average is $(5.54+3.60)/8=9.14/8=1.1425$. That average mixes observations from before and after the change. Since the problem states that the change is continuing and relevant to future development, select the recent average, 1.20, rather than the combined average.

By the definition of a link ratio, the projected cumulative loss at 36 months is the 24-month cumulative loss multiplied by the selected factor:

\widehat{C}_{i,36}=C_{i,24}\times 1.20 =

\$8{,}000{,}000\times 1.20=\$9{,}600{,}000.

As a comparison, using the combined average would give $\$8{,}000{,}000\times1.1425=\$9{,}140{,}000$. The difference is $\$9{,}600{,}000-\$9{,}140{,}000=\$460{,}000$. Thus, the window choice has a material effect on the projection.

Check. The selected factor exceeds 1, so projected cumulative losses increase from $8,000,000 to $9,600,000. The common mistake avoided is allowing a longer window to dominate automatically even though it includes experience from a process stated to have changed.

Worked Example: A Transparent Blend When Evidence Is Mixed

For a hypothetical coverage, the four older link ratios at a particular development age are 1.14, 1.12, 1.13, and 1.15. The two most recent ratios are 1.18 and 1.16. Recent experience is somewhat higher, but the available information does not establish whether the difference is a lasting change or short-term variation. An actuary decides to give the recent average 40% weight and the older-period average 60% weight. Calculate the selected factor and the projection from a cumulative loss of $12,000,000.

The older-period sum is $1.14+1.12+1.13+1.15=4.54$, so its average is $4.54/4=1.135$. The recent-period sum is $1.18+1.16=2.34$, so its average is $2.34/2=1.17$.

For the stated blend, multiply each window average by its assigned weight and add:

$$ 0.60(1.135)+0.40(1.17) =0.681+0.468 =1.149. $$

The selected factor is 1.149. Applying it to the observed cumulative loss gives

\$12{,}000{,}000\times1.149=\$13{,}788{,}000.

For comparison, the older average alone would project $\$12{,}000{,}000\times1.135=\$13{,}620{,}000$, while the recent average alone would project $\$12{,}000{,}000\times1.17=\$14{,}040{,}000$. The blended projection lies between these results, as the selected factor must because 1.149 lies between 1.135 and 1.17.

The 60% and 40% weights are an explicit judgment for this hypothetical case, not a universal standard. A blend is useful only when its weights have a defensible rationale and are documented. The common mistake avoided is calculating an equal-weight average of the two window averages, $(1.135+1.17)/2=1.1525$, despite the stated 60%/40% weights.

Why the Window Choice Changes the Projection

The link ratio is a multiplicative factor. If $C_{i,j}$ is the observed cumulative loss at age $j$ and $f$ is the selected factor to age $j+1$, then the projected next-age cumulative loss is $C_{i,j}f$. For two candidate factors, $f_R$ from a recent window and $f_H$ from a historical window, the difference between projected values is

C_{i,j}f_R-C_{i,j}f_H=C_{i,j}(f_R-f_H).

This identity follows by factoring out the same observed $C_{i,j}$ from both projections. It shows why a modest difference between candidate factors can matter when the cumulative loss base is large. It also gives a direct way to explain the financial impact of a window selection without implying that the larger projection is necessarily more appropriate.

A sound selection therefore needs both a numerical comparison and a reasoned interpretation of the data. The calculation tells you what each window implies; the conditions of the hypothetical or actual portfolio determine which experience is more relevant.

Common Traps and a Selection Checklist

Several errors can make a comparison look more decisive than it is. Keep the following checks in view:

Comparing different development ages. A recent average for one age-to-age interval is not directly comparable with a historical average for another interval.
Assuming recency alone proves relevance. A small or volatile recent sample may not represent the expected pattern.
Assuming a long window is always safer. It can include experience from a process that has changed and may no longer be relevant.
Blending without stating weights. A blended estimate should show the component estimates, the weights, and why those weights are reasonable.
Ignoring the projection base. A factor difference should be interpreted in relation to the cumulative loss to which it is applied.

In practice, state the age-to-age interval, the observations included in each window, the averaging method, and the reason for preferring one window or blending them. If the evidence does not establish a continuing change, describe that uncertainty rather than presenting a judgment as a mechanical result.

Key takeaway: Recent experience can better represent a continuing current process; longer-term experience can temper short-window volatility. Compare both at the same development age, explain the relevance of the observations, and make any blend explicit. There is no universally correct window length or automatic weighting rule.

Check Your Understanding

For each problem, calculate the requested comparison before deciding which evidence is relevant.

At one development age, the three most recent ratios are 1.08, 1.10, and 1.12. The five earlier ratios are 1.04, 1.06, 1.08, 1.10, and 1.12. Calculate both window averages and the simple average across all eight observations. If no process change is known and recent volume is limited, which value is the more cautious candidate for stabilizing a selection? (Answer: Recent average = 1.1000; earlier average = 1.0800; all-eight average = 1.0875; the all-eight average is a reasonable stabilizing candidate given the stated conditions.)
Four older ratios average 1.05. Three recent ratios are 1.16, 1.17, and 1.18, following a documented continuing change in claim handling. Find the recent average and state which window better represents future development if that change is expected to persist. (Answer: Recent average = 1.1700; the recent window is more relevant under the stated assumption.)
Three older ratios are 1.12, 1.14, and 1.16; two recent ratios are 1.20 and 1.18. Give the recent average 25% weight and the older-period average 75% weight. What is the blended factor? (Answer: Older average = 1.1400; recent average = 1.1900; blended factor = $0.75(1.14)+0.25(1.19)=1.1525$.)
A recent-window factor is 1.14, while a historical-window factor is 1.10. For an observed cumulative loss of $25,000,000, calculate each next-age projection and their difference. (Answer: Recent projection = $28,500,000; historical projection = $27,500,000; difference = $1,000,000.)
At one age-to-age interval, the older-period average is 1.08 and the recent-period average is 1.15. A reviewer gives the older estimate 30% weight and the recent estimate 70% weight. Calculate the selected factor and explain whether it should be treated as an automatic standard. (Answer: $0.30(1.08)+0.70(1.15)=1.129$; it is a transparent judgment for the stated case, not an automatic standard.)

What You'll Learn