Development Triangle

Study guides: Exam 5, Exam GI-101

Development Triangle is the fundamental data structure of reserving: cumulative amounts arranged with cohorts as rows (usually accident years), maturities as columns (age in months since the start of the cohort), and valuation dates as diagonals.

f_{k \to k+1} = \frac{\sum_{\text{AY}} C_{\text{AY},\,k+1}}{\sum_{\text{AY}} C_{\text{AY},\,k}}

• Cell C_{\text{AY},k} holds the cumulative amount for that accident year at age k. The triangle is triangular because a 2024 accident year cannot yet have a 36-month valuation.
• The three directions carry three different signals, and reading them apart is the core diagnostic skill:
  • Along a row — how one cohort develops.
  • Down a column — how successive cohorts compare at the same maturity. A trend here is a change in cost level, mix or rate adequacy.
  • Along a diagonal — everything that happened in one calendar period. A diagonal effect is inflation, a reserve strengthening, or a settlement push, and it hits every cohort at once.
• Build the same triangle on several bases — paid, reported, counts (reported, closed, open), ALAE, and recoveries. The comparison between them is where the diagnostics live: paid-to-reported ratios, average case outstanding, closure rates.
• Triangles may be annual, semi-annual or quarterly. Finer periods are more responsive and show seasonality; they also thin the data and multiply the factors to select.
• Assumptions being made implicitly by using one at all: each cohort develops in a similar pattern, the pattern is stable over time, and the mix inside each cohort is comparable. Every reserving distortion on the syllabus is one of these three failing.

Reading a Triangle in Three Directions Cumulative reported losses (000s), valued 12/31/2024$:

AY	12 mo	24 mo	36 mo	48 mo
2021	2{,}000	3{,}000	3{,}450	3{,}588
2022	2{,}200	3{,}300	3{,}795
2023	2{,}600	3{,}900
2024	3{,}100

Read the triangle in each direction.

Answer Along rows — AY 2021 grew 2{,}000 \to 3{,}000 \to 3{,}450 \to 3{,}588: factors 1.500, 1.150, 1.040. Development decelerates with maturity, as expected.

Down columns — the 12-month column runs 2{,}000, 2{,}200, 2{,}600, 3{,}100: growth of 10\%, 18\%, 19\%. Something is increasing — exposure volume, cost level, or both. Without exposure counts the triangle cannot tell which, which is why counts and premium are analyzed alongside it.

Along diagonals — the 12/31/2024 diagonal is 3{,}588, 3{,}795, 3{,}900, 3{,}100. Its age-to-age factors (1.040, 1.150, 1.500) match the corresponding factors from earlier years exactly, so there is no calendar year effect: nothing unusual happened in 2024.

A clean triangle like this one is the case in which the chain ladder is at its most defensible.

Spotting a Diagonal Effect Age-to-age factors from a reported triangle, with the latest diagonal in bold:

AY	12–24	24–36	36–48
2020	1.48	1.15	1.19
2021	1.51	1.29
2022	1.67

Prior column averages: 1.49, 1.16, 1.06. What should the actuary conclude?

Answer Every bolded factor exceeds its column history, and the excess is roughly proportional at each maturity (+12\%, +11\%, +12\%). That pattern — a uniform lift across all maturities on one diagonal — is the signature of a calendar year effect, not of a change in any cohort.

The candidates are a case reserve strengthening (Case Adequacy), an inflation or legal shock, or a change in claims handling. Distinguishing them requires the companion triangles:

• Average case outstanding jumping on the same diagonal → case strengthening.
• Paid triangle showing the same lift → inflation or faster settlement, since payments are not a reserving judgment.
• Closure rates moving → settlement rate change.

What must not happen is selecting factors that average the elevated diagonal in with the history. That applies a one-time level shift as though it were an ongoing pattern, and over-states every future year's development. The remedies are on Berquist-Sherman Method.