Demo 3 · Monthly health paid-claims triangles

r2 distinguishes the 72×36 estimation basis, the traditional 36×36 display, and the synthetic full runoff through age 48. A 36×36 triangle is familiar, but at one valuation date it contributes only one observation to link 34→35. The 72-origin basis retains 37 observations for that link.

Precise result name

A paid-only aggregate triangle supports an estimated unpaid claim liability. It does not split pure IBNR, RBNS and IBNER without report dates, case reserves/incurred data and claim status.

The visible projection ends at age 35. The generator knows synthetic runoff through age 48 and applies the explicit tail

\[ f_{35\rightarrow48}=C_{48}^{simulated}/C_{35}^{simulated}\approx1.00524958. \]

Without a supported tail, the result should be called projected cumulative at terminal age, not final cost.

For origin \(i\):

\[ \widehat{L}_{i,unpaid}=\widehat{C}_{i,35}\times f_{35\rightarrow48}-P_{i,k}. \]

No zero floor is imposed. Negative residuals remain visible for review.

Data needed beyond a paid triangle

More realistic work should add report/notice date, case reserve or reported incurred, claim status and reopenings, exposure and an independent prior, homogeneous segmentation, gross/net and recoveries, large-claim flags, later runoff and documented operational changes. These fields determine what can be identified and how credible the selected pattern is.

r2 files

File	Meaning
monthly_paid_cumulative_triangle.csv	full 72×36 estimation basis
traditional_36x36_view.csv	latest 36 origins for presentation
monthly_age_to_age_factors.csv	factors to age 35 and counts
unpaid_claim_liability_projection_results.csv	terminal projection, tail, final cost and unpaid liability
monthly_bornhuetter_ferguson_prior.csv	synthetic prior for Demo 6

Run with:

python scripts/generate_demo_monthly_triangles.py --language en

This is synthetic educational evidence, not a booked reserve or a universal recommendation to use 72×36.