age-to-age (link ratio) factor

Look at one row of a paid triangle for an old accident year: it paid 5 million by month 12 and 8 million by month 24. The losses grew by a factor of 8/5 = 1.60 from age 12 to age 24. That ratio — how much cumulative losses multiply as you step from one development age to the next — is an age-to-age factor, also called a link ratio. It is the basic building block from which whole projections are made: chain together the right ratios and you can grow an immature year all the way to ultimate.

Precisely, the age-to-age factor from age t to age t+1 is the ratio of cumulative losses at t+1 to cumulative losses at t. Because you have several old rows, you get several such ratios for each step, and you must pick a representative one to use going forward — often a volume-weighted (all-year) average, sometimes a simple average, a recent-years average, or one that excludes an outlier. For instance, if three mature years show 12-to-24 ratios of 1.55, 1.62 and 1.58, you might select 1.58 as the factor to apply to the latest year's age-12 value. The selected factors at each step are the heart of the chain-ladder method.

Age-to-age factors matter because selecting them is where judgment enters reserving — the arithmetic is trivial, but deciding which years are representative, whether to smooth a noisy step, and how to treat a one-off shock is the actual skill. Two cautions: a factor estimated from very few or very volatile observations is unreliable, and the further out the column, the thinner the data — which is exactly why the long-run 'tail' beyond the triangle must often be handled by a separate tail factor rather than by reading a ratio that does not really exist yet.