Actuarial Notes

Unpaid Claim Distribution

Study guides: Exam 7, Exam GI-301

An unpaid claim distribution is the probability distribution of the total amount R that will be paid after the valuation date on claims that have already occurred. Its mean is the central estimate of unpaid claims, and its spread, measured by the prediction error and percentiles, measures how uncertain the reserve is.

R = \sum_{w + d \,>\, n + 1} q_{w,d}

\sigma^2 = \ln\!\left(1 + \mathrm{CV}^2\right)

\mu = \ln E[R] - \tfrac{1}{2}\sigma^2

  • R sums the future incremental payments q_{w,d} below the diagonal of an n-year triangle, plus any tail. The second and third blocks fit a lognormal to a mean and a CV. Mack suggested this for his chain ladder moments. Its p-th percentile is then R_p = e^{\mu + z_p\sigma}.

  • Estimating the parameters. One route is to take moments from a model (Mack's mean and standard error, or Clark's maximum likelihood fit with its variance) and attach a distribution. The other is to simulate: an ODP bootstrap, a GLM or MCMC posterior draws produce thousands of outcomes, and moments and percentiles are read off directly. See Stochastic Reserving.

  • Moments and shape. Report the mean, standard error, CoV and percentiles such as the 50th, 75th, 95th and 99th. Unpaid claims are right-skewed, so a normal approximation understates the upper tail. At a modest percentile such as the 75th, however, a normal puts the value slightly above a lognormal with the same mean and CV.

  • Simulating percentiles. Each iteration draws the parameters (the bootstrap pseudo-triangle, or the posterior draw) and then the outcome. The 95th percentile of the total is not the sum of the accident years' 95th percentiles, because years diversify. See Parameter Risk.

  • Reasonableness of the output (Shapland):

    • the standard error should rise from the oldest to the most recent years;
    • the CoV should generally fall, though it may turn up again for the latest years as parameter uncertainty grows;
    • the total's standard error should exceed any single year's, and its CoV should be below any single year's;
    • the minimum and maximum iterations must be plausible.

    Meyers adds a retrospective test: the percentiles of actual outcomes should be uniform. See Reasonableness Testing.

  • The distribution drives risk margins, capital and reinsurance decisions. It is not a range of reasonable estimates: its percentiles describe outcomes, not alternative estimates of the mean.

Percentiles from a Mean and Standard Error A Mack analysis gives a total reserve of 5{,}000 with standard error 800 ($000s). Assuming a lognormal, find the 75th and 95th percentiles and compare them with a normal.

Answer

\begin{align*}
\mathrm{CV} &= 800/5{,}000 \\
&= 0.16 \\
\sigma^2 &= \ln(1.0256) \\
&= 0.02528 \\
\sigma &= 0.1590 \\
\mu &= \ln 5{,}000 - 0.01264 \\
&= 8.50455
\end{align*}
\begin{align*}
R_{0.75} &= e^{8.50455 + 0.6745(0.1590)} \\
&= 5{,}496 \\
R_{0.95} &= e^{8.50455 + 1.645(0.1590)} \\
&= 6{,}413
\end{align*}

The normal gives 5{,}000 + 0.6745(800) = 5{,}540 and 5{,}000 + 1.645(800) = 6{,}316. The lognormal is lower at the 75th percentile and higher at the 95th. The choice of distribution matters most exactly where capital and reinsurance decisions are made. The mean and standard error alone do not settle it.

Reviewing Bootstrap Output An ODP bootstrap on paid data (10,000 iterations, $000s) returns:

AY Mean Std error CoV
2016 120 95 79\%
2017 410 160 39\%
2018 900 250 28\%
2019 1{,}650 330 20\%
2020 2{,}800 450 16\%
2021 4{,}100 1{,}350 33\%
Total 9{,}980 1{,}520 15\%

The minimum iteration for 2016 is -60. Assess the output.

Answer

  • Standard errors rise from 95 to 1{,}350, and the total's 1{,}520 exceeds every year's. ✓
  • The total's CoV (15.2\%) is below every year's, the lowest being 2020's 16.1\%. ✓
  • CoVs fall from 79\% to 16\% through 2020, as they should. A small, nearly settled reserve for 2016 has most of its uncertainty relative to its size. ✓
  • 2021 jumps to 33\%. A chain ladder on the latest year multiplies one thin diagonal by the full CDF, so parameter uncertainty swamps the decline. Shapland's remedy is to consider a BF or Cape Cod model for the recent years, which borrows an a priori loss ratio.
  • The negative minimum for 2016 comes from negative incremental values in a few iterations. It is acceptable only if negative development is genuinely possible, for example from salvage or case-reserve takedowns. Otherwise the simulation should be constrained.

The distribution is usable for 2016 to 2020, but 2021 should be re-modelled before any percentile is quoted for the total.