Reserve Variability (Mack) and an Excess-of-Loss Retention Decision

Background

You are the reserving and reinsurance actuary for a workers' compensation writer. Two questions land on your desk in the same week:

1. The appointed actuary wants a reserve range and a 75th-percentile risk margin for the paid workers' compensation triangle below, rather than a single chain-ladder point estimate.
2. Underwriting is quoting a large manufacturing account and wants to know whether the company should keep a $250,000 or$500,000 per-occurrence retention and buy excess-of-loss cover above it up to $1,000,000.

Data — cumulative paid losses ($000)

Real cumulative paid workers' compensation losses for one large national carrier group, accident years 1988–1997, taken from the CAS Loss Reserve Database (NAIC Schedule P, Part 3):

Part 1 — Chain ladder with Mack standard errors

Volume-weighted age-to-age factors are

with no tail beyond 120 months. Projecting each accident year to ultimate gives a **total paid reserve of $304,882K**, of which accident year 1997 alone contributes$103,885K.

Mack's distribution-free model treats the chain ladder as a sequence of weighted regressions through the origin, $C_{i,k+1} = f_k C_{i,k} + \sigma_k \sqrt{C_{i,k}}\,\varepsilon_{i,k}$, and estimates the variance parameters as

$\hat\sigma_k^2 = \frac{1}{n-k-1}\sum_{i=1}^{n-k} C_{i,k}\left(\frac{C_{i,k+1}}{C_{i,k}} - \hat f_k\right)^2 .$

The fitted $\hat\sigma_k^2$ fall from 3,158.9 at 12–24 months to 1.41 at the last two development periods, and the mean-squared-error formula

$\widehat{\mathrm{mse}}(\hat R_i) = \hat C_{i,n}^2 \sum_{k=n+1-i}^{n-1} \frac{\hat\sigma_k^2}{\hat f_k^2}\left(\frac{1}{\hat C_{i,k}} + \frac{1}{\sum_{j=1}^{n-k} C_{j,k}}\right)$

yields the following prediction standard errors by accident year (process plus parameter error):

The total standard error shown is the square root of the sum of the individual accident-year MSEs; the full Mack total also adds a positive covariance term for the shared factor estimates, so 6.4% is a slight understatement of the true coefficient of variation.

Fitting a lognormal to the total reserve with mean $304,882K and CV 6.4$\sigma^2=\ln(1+0.064^2)=0.00408$, so the **75th percentile is about$317.7M** and the 99.5th percentile about $358.7M.

Part 2 — Excess-of-loss retention for a large account

The account is expected to generate 180 claims per year. Its per-occurrence severity is modelled as lognormal with $\mu = 9.2$ and $\sigma = 1.6$ (mean severity $35,596; median$9,897). The limited expected value of a lognormal at limit $x$ is

$\mathrm{LEV}(x) = e^{\mu+\sigma^2/2}\,\Phi\!\left(\frac{\ln x - \mu - \sigma^2}{\sigma}\right) + x\left[1-\Phi\!\left(\frac{\ln x-\mu}{\sigma}\right)\right],$

which gives

Expected annual losses in the two candidate layers are therefore

The reinsurer prices either layer at expected ceded loss × 1.30 (a 30% load for expenses, risk and profit): about **$1,170K** for the$250K retention programme and **$441K** for the$500K retention programme. The company's stated risk appetite is that no single occurrence may consume more than 1% of surplus, and surplus is $45 million.

Your task

Answer the questions below using the figures above as given. Show formulas where asked; numerical answers within rounding of the tables are acceptable.