Loss reserving Study Guide

📖 Core Concepts Loss Reserving – Estimating the money an insurer must set aside now to pay all future claims from a given block of business. Claims Reserves – Funds held to meet reported‑but‑not‑settled (RBNS) and incurred‑but‑not‑reported (IBNR) liabilities. General‑Insurance vs. Life/Pension/Health – General‑insurance contracts are short‑term (≈ 1 yr) and premium is paid upfront; reserves are based on past loss development, not future premium cash‑flows. Chain‑Ladder (Development) Method – Uses observed loss development patterns to project future development, assuming the past repeats. Bornhuetter–Ferguson (BF) Method – Blends the chain‑ladder projection with an a priori estimate of ultimate loss, tempering noisy data. Stochastic Reserving Models – Add a probability distribution to reserve estimates (e.g., Mack’s distribution‑free CL, over‑dispersed Poisson, log‑normal CL, PIC, bootstrap, Bayesian). Prediction Uncertainty – Variability around the reserve estimate; under Solvency II the one‑year claims development result is a key risk metric. --- 📌 Must Remember Reserve purpose: meet all future claim payments for policies already written. Short‑duration contracts: reserve = forecast of future losses only; no premium‑earning period to consider. Chain‑Ladder assumption: “development factors are stable over time.” BF weighting: \( \text{BF}{i} = \text{Prior}i + \text{Development Factor}i \times (\text{Actual}{\text{latest}} - \text{Prior}{\text{latest}}) \). Mack CL variance: \( \operatorname{Var}(\hat{C}{i,n}) = \hat{C}{i,n}^2 \sum{j=i}^{n-1} \frac{\hat{\sigma}j^2}{\hat{f}j^2} \). Over‑Dispersed Poisson (ODP): variance = \(\phi \times\) mean ( \(\phi>1\) indicates extra‑Poisson variation). IBNR definition: losses incurred but not yet reported; part of outstanding claims reserves. --- 🔄 Key Processes Chain‑Ladder Reserve Calculation Assemble cumulative loss triangle (paid or incurred). Compute age‑to‑age development factors \(fj = \frac{\sumi C{i,j+1}}{\sumi C{i,j}}\). Project to ultimate: \(\hat{C}{i,n} = C{i,k} \times \prod{j=k}^{n-1} fj\). Reserve = \(\hat{C}{i,n} - C{i,k}\). Bornhuetter–Ferguson Reserve Calculation Obtain prior ultimate loss estimate (actuarial judgment, exposure‑based). Compute cumulative development factors \(Fk = \prod{j=k}^{n-1} fj\). BF reserve = Prior × (1 – 1/Fk) + (Observed – Prior/Fk). Bootstrap Variability Assessment Fit a deterministic CL model. Resample residuals (with replacement) to create pseudo‑triangles. Re‑apply CL to each pseudo‑triangle → distribution of reserve estimates. Bayesian Reserving (conceptual) Specify prior distribution for ultimate losses (e.g., log‑normal). Update with observed triangle via Bayes’ theorem → posterior predictive distribution. --- 🔍 Key Comparisons Chain‑Ladder vs. Bornhuetter–Ferguson Data reliance: CL relies entirely on observed development; BF blends data with a prior. Stability: BF is more stable when early‑development data are volatile or sparse. Distribution‑Free CL (Mack) vs. Over‑Dispersed Poisson Assumptions: Mack requires only mean‑square consistency, no distribution; ODP assumes a Poisson‑type count with extra dispersion. Variance formula: Mack gives analytic variance; ODP adds a dispersion parameter \(\phi\). Log‑Normal CL vs. Standard CL Distribution: Log‑normal imposes a specific shape on development factors; standard CL is non‑parametric. Paid‑Incurred Chain (PIC) vs. Paid‑Only CL Data: PIC uses both paid and incurred triangles, improving tail estimates; paid‑only ignores incurred information. --- ⚠️ Common Misunderstandings “Reserves = Future Premiums” – Wrong for short‑term general insurance; reserves are future losses, not premium income. “Chain‑Ladder works for any line of business” – It performs poorly with sparse data or long‑tail lines; BF or Bayesian methods are preferred. “IBNR is the same as RBNS” – IBNR = losses not yet reported; RBNS = reported but not yet settled. They are distinct components of outstanding reserves. “Stochastic models give a single ‘correct’ reserve” – They provide a distribution (mean, variance, quantiles); the point estimate is only one summary. --- 🧠 Mental Models / Intuition Development Triangle as a “Growth Chart” – Think of each accident year as a plant; the development factors are the growth rates you expect the plant to follow based on past seasons. BF as “Anchoring” – Your prior ultimate loss is an anchor; the observed data can tug the estimate, but the anchor prevents wild swings when data are thin. Bootstrap as “What‑If Scenarios” – By reshuffling residuals you ask, “If the random noise had been slightly different, how would my reserve change?” --- 🚩 Exceptions & Edge Cases Sparse data (few years or low claim counts): Prefer BF, frequency‑severity, or Bayesian methods; CL can be unstable. Heavy‑tailed lines (e.g., liability): Log‑normal CL or Bayesian heavy‑tailed priors may capture extreme loss development better. Regulatory Solvency II one‑year horizon: Focus on the claims development result rather than the full ultimate reserve distribution. --- 📍 When to Use Which | Situation | Preferred Method | |-----------|------------------| | Well‑populated paid/incurred triangle, stable development | Chain‑Ladder (Mack) | | Early years noisy, need stability | Bornhuetter–Ferguson | | Very few observations or only exposure data | Frequency–Severity or Average Cost per Claim | | Need full predictive distribution & prior knowledge | Bayesian | | Want to quantify reserve variability without strong distributional assumptions | Bootstrap | | Tail risk important, data suggest over‑dispersion | Over‑Dispersed Poisson | | Want to incorporate both paid and incurred data for better tail | Paid‑Incurred Chain (PIC) | --- 👀 Patterns to Recognize Flattening development factors (e.g., 1.05, 1.02, 1.00) → indicates nearing ultimate losses. Large jumps in early development followed by stabilization → classic CL pattern; beware of outliers. Consistently > 1 % IBNR proportion across accident years → suggests reporting lag is material; may need separate IBNR model. Increasing dispersion (variance growing faster than mean) → signals over‑dispersed Poisson or need for log‑normal modeling. --- 🗂️ Exam Traps Choosing CL when data are sparse – The exam may present a triangle with only 2‑3 years; the correct answer is often BF or a Bayesian approach. Confusing RBNS with IBNR – Remember RBNS = reported, not settled; IBNR = not reported at all. Using the development factor of the latest calendar year – Development factors should be calculated across all available accident years, not just the most recent. Assuming variance = 0 for deterministic methods – Even CL has an analytic variance (Mack); ignoring it leads to under‑estimating risk. Treating “average cost per claim” as a full reserve – It only estimates the IBNR component; RBNS must still be added. ---