Age to Age Factor

Study guide: Exam 5

Age-to-Age Factor (link ratio) is the ratio of a cohort's cumulative amount at one maturity to its amount at the previous maturity — the observed development between two ages.

f_{n \to n+1}^{\text{AY}} = \frac{C_{\text{AY},\,n+1}}{C_{\text{AY},\,n}}

f_{n \to n+1}^{\text{vol-wtd}} = \frac{\sum_{\text{AY}} C_{\text{AY},\,n+1}}{\sum_{\text{AY}} C_{\text{AY},\,n}}

Selecting a factor from the column of observations is the central judgment of the chain ladder. The candidates:

• Volume-weighted (all-year) — the ratio of column sums. Weights each year by its size, so it is stable and dominated by the large years. The default choice.
• Simple average — equal weight to each year, so a small, volatile year counts as much as a large one. Useful when volume differs for reasons unrelated to the pattern.
• Latest k-year average (volume-weighted or simple) — responsive to a genuine change in the pattern, at the cost of credibility.
• Medial average (drop high and low) — robust to a single distorted year.
• Judgmental selection, documented — the honest answer when the column contains a known one-off.

Further points:

• Factors need not exceed 1.000. Salvage and subrogation recoveries, conservative initial case reserving, and claims closing below their reserves all produce downward development. Flooring factors at 1.000 because "losses only go up" is a standard error.
• Look at the column before averaging it. A steady column supports a volume-weighted all-year selection; a column trending over time is a signal, not noise, and calls for the latest years or an investigation.
• A factor that moves along the diagonal rather than down the column is a calendar-year effect (Development Triangle) and should be addressed by restating the triangle, not by averaging it away.
• The same selection process runs on paid, reported, count and ALAE triangles, and the selections should be mutually consistent — a paid factor implying faster emergence than the reported factor at the same age needs an explanation.

Selecting Factors from a Column Cumulative reported losses ($000s):

AY	12 mo	24 mo	36 mo
2021	500	750	875
2022	550	825	950
2023	600	900

Compute the individual and volume-weighted factors and select.

Answer 12–24:

\frac{750}{500} = 1.500 \qquad \frac{825}{550} = 1.500 \qquad \frac{900}{600} = 1.500

f^{\text{vol-wtd}} = \frac{750 + 825 + 900}{500 + 550 + 600} = \frac{2{,}475}{1{,}650} = 1.500

24–36:

\frac{875}{750} = 1.167 \qquad \frac{950}{825} = 1.152

f^{\text{vol-wtd}} = \frac{875 + 950}{750 + 825} = \frac{1{,}825}{1{,}575} = 1.159

Selections: 1.500 and 1.159. With only two observations at 24–36 and a 1.5\% spread between them, the volume-weighted average is a reasonable pick — but the column is thin, and a benchmark from a wider data source is worth comparing against before it is trusted.

When the Averages Disagree The 12–24 column of a liability triangle:

AY	Factor	Losses at 12 mo (000s)
2018	1.42	4{,}100
2019	1.45	4{,}300
2020	1.38	900
2021	1.61	4{,}800
2022	1.63	5{,}200
2023	1.66	5{,}600

(2020 was a partial year of writings after a mid-year book transfer.) What should be selected?

Answer

\begin{align*}
\text{Simple all-year} &= 1.525 \\
\text{Volume-weighted all-year} &= 1.545 \\
\text{Latest 3-year simple} &= 1.633 \\
\text{Medial (drop high/low)} &= 1.528
\end{align*}

The averages span eleven points, and choosing between them by formula misses the point. The column has a trend: 1.42, 1.45, 1.38, then 1.61, 1.63, 1.66. Something changed around 2021 — development is genuinely slower to emerge, or case reserving weakened, or the mix shifted toward longer-reporting claims.

Any all-year average blends two regimes and lands between them, describing neither. The defensible selection is from the latest three years (\approx 1.63), provided the actuary can identify what changed and satisfy themselves it will persist. If it cannot be explained, the factor selection is not the real problem — the unexplained change is, and it needs the operational enquiry Friedland's Chapter 4 describes.

The 2020 year deserves separate handling in any case: at 900 against a normal 4{,}000+, it is a partial year whose factor carries little information, which is precisely the case for a medial or volume-weighted approach rather than a simple average.