Loss Triangle Labyrinth

How it works

You are handed a run-off triangle of cumulative paid claims in thousands: five origin years (2021 to 2025) as rows, development years as columns. 2021 is fully developed through five columns (1000, 1500, 1750, 1875, 1900); each later year has one fewer entry, down to 2025 with a single figure of 1400. The dashes in the lower-right are money you owe but have not paid, and the game is about estimating it.

Phase 1 asks for the four volume-weighted link ratios, one per development step: for each step, sum the later column across every origin year where both columns are known, sum the earlier column over the same years, and divide. Each ratio is checked within 0.01. Phase 2 asks for the total chain ladder reserve: project each origin year's latest figure to ultimate using the product of remaining link ratios, subtract what is already paid, and sum across years (tolerance 40).

Phase 3 focuses on 2025, the year with one data point and a cumulative development factor near 1.9. You choose a method - stay with chain ladder or switch to Bornhuetter-Ferguson - and then compute the 2025 reserve given a premium of 2,600 and an expected loss ratio of 72%. The value is checked against the BF answer within 40. All answers lock after one check; you can rerun the triangle at the end.

How scoring works

Phase 1: +1 point per link ratio within 0.01, four available. Phase 2: +2 points for the total reserve within 40. Phase 3: +1 point for choosing Bornhuetter-Ferguson (the only choice scored as correct) and +2 points for the reserve value within 40 of the BF figure. Maximum 9 per run.

The value check in phase 3 is against the BF reserve regardless of which method you selected, so picking chain ladder and entering the chain ladder number scores zero on both counts.

Checks record skill attempts under distributions (link ratios), expected-value (both reserve figures), and selection-bias (the method choice).

Volume-weighted link ratios, not simple averages

The chain ladder estimator for each development step is LDF_j = sum of C_(i,j+1) over sum of C_(i,j), taken only across origin years where both columns exist. It is a ratio of sums, not an average of ratios: a large mature year carries more weight than a tiny immature one, which is the standard estimator precisely because it weights each year by its exposure.

Mechanically, for step 0 to 1 you use four origin years (2021 through 2024 all have both columns), for step 1 to 2 three years, and so on down to a single year for the last step. Write the two column sums before dividing - the discipline prevents the classic error of including an origin year whose later column does not exist.

This triangle is built to develop cleanly: the ratios come out at 1.5, about 1.1667, about 1.0713, and about 1.0133. Ratios should shrink toward 1 as development matures; if yours do not decline monotonically, recheck which rows you summed.

From ratios to reserves via CDFs

The cumulative development factor for an origin year is the product of every link ratio from its current development age to ultimate. 2021 is at ultimate (CDF 1, reserve zero); 2022 needs only the last ratio; 2025 needs all four multiplied together, about 1.90.

Each year's ultimate is its latest cumulative figure times its CDF, and its reserve is ultimate minus latest - the reserve is only what has not been paid yet, never the whole ultimate. Sum the per-year reserves for the total. A useful check: the reserve should be dominated by the youngest years, since they have the most development remaining.

The 40-tolerance on the total is tight enough that you should carry the ratios to four decimals through the CDF products rather than rounding each ratio to two places first.

Why the thin year breaks chain ladder, and what BF does instead

2025 has exactly one payment, 1,400, and a CDF near 1.9. Chain ladder multiplies that single noisy figure by almost 1.9, so any randomness in the first payment is amplified straight into the reserve - this is the leverage problem, and it is the one situation where chain ladder is the fragile choice.

Bornhuetter-Ferguson sidesteps it by never scaling the actual claims up. The BF reserve is (1 - 1/CDF) x premium x expected loss ratio: apply the undeveloped share of the pattern to an independent a-priori expectation instead of to the observed figure. An erratic early payment cannot dominate the estimate.

BF is exactly a credibility blend: Z x chain ladder + (1 - Z) x a-priori, with Z = 1/CDF, the share of losses you would expect to have paid by now. It always sits between the two estimates and leans on the a-priori whenever the year is barely developed - which is the phase 3 situation by construction.

A worked example

Phase 1. Step 0 to 1: (1500 + 1650 + 1800 + 1950) / (1000 + 1100 + 1200 + 1300) = 6900 / 4600 = 1.5. Step 1 to 2: (1750 + 1925 + 2100) / (1500 + 1650 + 1800) = 5775 / 4950 = 1.1667. Step 2 to 3: (1875 + 2062) / (1750 + 1925) = 3937 / 3675 = 1.0713. Step 3 to 4: 1900 / 1875 = 1.0133.

Phase 2. CDFs: 2021 fully developed, 1.0000. 2022: 1.0133. 2023: 1.0713 x 1.0133 = 1.0856. 2024: 1.1667 x 1.0856 = 1.2665. 2025: 1.5 x 1.2665 = 1.8998.

Reserves per year: 2021: 0. 2022: 2062 x 1.0133 - 2062 = about 27. 2023: 2100 x 1.0856 - 2100 = about 180. 2024: 1950 x 1.2665 - 1950 = about 520. 2025: 1400 x 1.8998 - 1400 = about 1,260. Total: about 1,987 - enter that (tolerance 40). Note nearly two thirds of the whole reserve sits in the one year with a single data point.

Phase 3. Choose Bornhuetter-Ferguson. BF reserve = (1 - 1/1.8998) x 2600 x 0.72 = 0.4736 x 1872 = about 887, versus chain ladder's roughly 1,260. Enter about 887. The gap between the two is exactly how much the chain ladder answer depends on one noisy first payment.

Common mistakes

• Averaging the individual year-over-year ratios instead of taking the ratio of column sums. The volume-weighted estimator is sums first, then divide; the simple average lets a small year distort the factor.

• Including an origin year in a link ratio when its later column does not exist yet. Only rows with both columns known enter the sums.

• Reporting ultimates instead of reserves. The reserve is ultimate minus latest paid - forgetting the subtraction roughly doubles your phase 2 answer.

• Rounding link ratios to two decimals before multiplying into CDFs. The compounding pushes the total outside the 40 tolerance; keep four decimals.

• Entering the chain ladder figure in phase 3. The value check is against the BF reserve of about 887, and staying with chain ladder also forfeits the method point.

Why interviews test this

Every P&C reserving interview starts with a triangle, and this is the standard sequence: compute volume-weighted link ratios, chain the CDFs, state the reserve as ultimate minus paid. Getting the ratio-of-sums estimator right (and saying why it beats a simple average) is the baseline; the differentiator is knowing the one situation where chain ladder blows up.

"When would you not use chain ladder?" is the classic follow-up, and the answer this game drills is the one interviewers want: an immature year with high leverage, where you switch to Bornhuetter-Ferguson and can explain it as a credibility weighting between the development-based estimate and an a-priori expectation, with Z = 1/CDF. This material maps directly to the reserving portions of the actuarial exam track and to case questions at consulting and insurance employers.