Estimating Outstanding Claims

An insurer's auto liability book shows $42 million paid for accident year 2024 as of year-end 2025. Claims are still being reported and settled. The actuary must estimate the ultimate cost, and that estimate drives the reserve, the surplus, and solvency. Three methods (expected loss ratio, Chain-Ladder, and Bornhuetter-Ferguson) attack this from different angles. The exam expects you to apply all three from a loss development triangle.

HIGH-FREQUENCY: The three reserving methods are tested heavily. You must apply each from a development triangle.

A triangle organizes cumulative losses by accident year (rows) and development period (columns). The lower-right is missing, that is what you estimate.

Every formula below is built from these six symbols. Fix them now and the three methods reduce to rearrangements of the same pieces.

$$\text{Ultimate}=\text{ELR}\times \text{Earned Premium}$$

Ignores actual emergence. Useful for immature years or new lines.

NOTE: ELR is the simplest method but the least responsive. It is appropriate only when there is too little data for development patterns to be meaningful (brand new lines or...

Common mistakes

• Confusing percent developed with percent unreported. Percent developed = $1/\text{CDF}$. Percent unreported = $1-1/\text{CDF}$. BF uses percent unreported. With CDF = 2.3337: developed = 0.4285, unreported = 0.5715. Using 0.4285 in BF gives IBNR of 977 instead of 1,300. Trap: 977.
• Using the wrong denominator in age-to-age factors. Only accident years with data at both periods enter the calculation. Including AY 2024 in $f_{12\to 24}$ (with 0 in the numerator) gives 3,150/2,700 = 1.167 instead of 1.853. Trap: an unrealistically low factor.
• Forgetting the tail factor. Omitting the 1.05 tail gives CDF = 2.2235 instead of 2.3337. Ultimate drops from 2,334 to 2,224. Trap: answer about 5% too low.

Bottom line

• Reserve = ultimate losses - paid losses; IBNR = ultimate losses - incurred losses
• ELR: Ultimate = ELR $\times$ EP. Ignores actual reported data entirely
• Chain-Ladder: Ultimate = $C_{i,k}\times {\text{CDF}}_k$, with CDF from chained age-to-age factors. Fully data-driven
• BF: Ultimate = $C_{i,k}+\text{ELR}\times \text{EP}\times (1-1/{\text{CDF}}_k)$. Blends actual reported with a priori expected unreported

Exam shortcut

When given a triangle, immediately compute age-to-age factors column by column below the triangle, then build CDFs from right to left. For BF, write the formula with blanks: Ultimate = __ + __ $\times$ __ $\times$ (1 - 1/__) and fill in four values. Remember: "ELR-CL-BF" in order of data dependence. Expected only, Completely from Losses, Blend of Factors. BF mnemonic: "Paid Plus Expected Unreported."

The full lesson (about 2,560 words, 17 min read) adds 4 worked examples, all 5 common mistakes, a self-check, free in the app.

Learning objectives

• 5a

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