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Diagnostic Triangles

Note Section 2.2 • Reading time: ~5 mins

Diagnostic triangles are analytical tools constructed from raw claims and premium data. Actuaries use them to identify operational shifts in claims handling and reserving practices that would otherwise violate the basic assumptions of traditional development methods (such as the Chain Ladder method).

Source Data Inputs

To construct diagnostic triangles, the following basic datasets are required (typically organized by Accident Year/Report Year and maturity):

  • Reported Losses (Incurred Losses): Cumulative paid losses plus case outstanding reserves.
  • Paid Losses: Cumulative claim payments.
  • Reported Claim Counts: Total number of claims reported.
  • Closed Claim Counts: Total number of claims closed (both with and without payment).
  • Open Claim Counts (or Case Outstanding Counts): Cumulative reported claims minus closed claims.
  • Earned Premiums (and On-Level Earned Premiums): Used to analyze loss ratios.

Primary Diagnostic Triangles and Formulas

Actuaries analyze ratios and averages across maturities to diagnose operational shifts:

1. Count-Based Triangles

  • Closed-to-Reported Claim Count Ratio: Ratio=Closed Claim CountsReported Claim Counts\text{Ratio} = \dfrac{\text{Closed Claim Counts}}{\text{Reported Claim Counts}} Use: Measures the Claim Settlement Rate (CSR). An upward trend down columns indicates that claims are closing faster than in the past.

2. Loss-to-Loss Triangles

  • Paid-to-Reported Loss Ratio: Ratio=Cumulative Paid LossesCumulative Reported Losses\text{Ratio} = \dfrac{\text{Cumulative Paid Losses}}{\text{Cumulative Reported Losses}} Use: Detects changes in settlement speed (CSR) or Case Reserve Adequacy (CRA). A downward trend down columns suggests either a slowdown in claim payments or an increase in case reserving strength.

3. Average Severity Triangles

  • Average Reported Claim Size: Average Reported Size=Cumulative Reported LossesReported Claim Counts\text{Average Reported Size} = \dfrac{\text{Cumulative Reported Losses}}{\text{Reported Claim Counts}} Use: Detects severity trends and shifts in reserve strength.
  • Average Paid Claim Size: Average Paid Size=Cumulative Paid LossesClosed Claim Counts\text{Average Paid Size} = \dfrac{\text{Cumulative Paid Losses}}{\text{Closed Claim Counts}} Use: Identifies severity trends. If small claims are being settled faster (CSR speedup), this average will temporarily decrease.
  • Average Case Outstanding (Average Case Reserve): Average Case Outstanding=Reported Losses−Paid LossesReported Claim Counts−Closed Claim Counts=Total Case ReservesOpen Claim Counts\text{Average Case Outstanding} = \dfrac{\text{Reported Losses} - \text{Paid Losses}}{\text{Reported Claim Counts} - \text{Closed Claim Counts}} = \dfrac{\text{Total Case Reserves}}{\text{Open Claim Counts}} Use: Directly measures Case Reserve Adequacy (CRA). An increase down columns indicates reserve strengthening, assuming the mix of open claims remains constant.
  • Reported Losses / Earned Premium: Tracks the development of the reported loss ratio.
  • Paid Losses / Earned Premium: Tracks the development of the paid loss ratio.
  • Reported Losses / On-Level Earned Premium: Adjusts the premium base to current rate levels to isolate actual loss ratio trends.

Summary Interpretation Matrix

When diagnostic trends appear in the triangles, actuaries use the following relationships to isolate the root cause:

Observed PatternCSR SpeedupCRA Strengthening
Closed / Reported Counts📈 Increasing➡️ Stable
Average Case Outstanding📈 Increasing (small claims close first)📈 Increasing
Average Paid Claim Size📉 Decreasing (small claims close first)➡️ Stable
Paid Losses / Reported Losses📈 Increasing📉 Decreasing
Ultimate Loss Projections⚠️ Paid LDF overestimates⚠️ Reported LDF overestimates