Actuarial Notes

Loss Cost

Study guides: Exam FAM, Exam ASTAM, Exam 8, Exam 9

Loss cost is the expected cost of claims, before any allowance for expenses or profit, for a unit of exposure, a rating class, a layer or a reinsurance contract. It includes loss adjustment expense only where that is stated. For primary business it is the same thing as the pure premium. In reinsurance it is usually quoted as a rate on the cedant's subject premium.

\text{Loss Cost} = \frac{E[\text{Losses}]}{\text{Exposure}}

\text{LC}_{(a,\,b]} = \text{LC} \times \frac{E[X \wedge b] - E[X \wedge a]}{E[X]}

  • Cost of a layer from the overall loss cost (Exam 8). The second block works this out. The severity distribution gives the share of ground-up loss dollars that falls in (a, b], and the layer's loss cost is that share of the total. Frequency cancels out because every claim contributes to the ground-up mean. Increased Limits factors do the same calculation with a basic limit as the base.
  • By class (Exam 8). A GLM estimates each class's loss cost. It is either a Tweedie model of pure premium or separate frequency and severity models multiplied together (Frequency-Severity Models). The standard errors of the fitted relativities show whether a class really differs from the base or the gap is just noise (Statistical Significance).
  • Reinsurance contracts (Exam 9).
    • A quota share cedes a fixed share of every loss, so its loss cost is that share of the subject losses.
    • A surplus share cedes a different share of each risk, set by the line size.
    • An excess of loss layer gets its loss cost from experience rating, the burning cost: trended, developed layer losses over on-level subject premium. It can also come from exposure rating, which applies exposure curves or ILFs to the risk profile. The two are credibility-weighted.
    • An aggregate cover needs the whole aggregate loss distribution, not only its mean.
  • Contract provisions. A loss corridor hands a band of ceded loss back to the cedant, which lowers the reinsurer's loss cost. Reinstatement premiums, sliding-scale commissions and profit commissions change what the reinsurer is paid as losses vary. Evaluating them takes the loss distribution; the mean alone isn't enough.

Cost of a Layer from the Overall Loss Cost The ground-up loss cost is \$400 per exposure. Severity is Pareto with \alpha = 3 and \theta = 50{,}000, so the share of expected loss below u is 1 - \big(\theta / (\theta + u)\big)^2.

Split the loss cost among the first \$25{,}000, the layer \$50{,}000 excess of \$25{,}000, and the amount above \$75{,}000.

Answer

\begin{align*}
\Big(\tfrac{50}{75}\Big)^2 &= \tfrac{4}{9} \\
\Big(\tfrac{50}{125}\Big)^2 &= 0.1600
\end{align*}
\begin{align*}
\text{LC}_{(0,\,25\text{K}]} &= 400 \times \big(1 - \tfrac{4}{9}\big) \\
&= \$222.22 \\[4pt]
\text{LC}_{(25\text{K},\,75\text{K}]} &= 400 \times \big(\tfrac{4}{9} - 0.16\big) \\
&= \$113.78 \\[4pt]
\text{LC}_{(75\text{K},\,\infty)} &= 400 \times 0.1600 \\
&= \$64.00
\end{align*}

The pieces sum to \$400. A \$25{,}000 deductible would remove 55.6\% of the loss cost, which is its loss elimination ratio.

Quota Share Versus Excess of Loss A cedant expects \$20{,}000{,}000 of subject premium at a 65\% loss ratio. It is considering:

  • a 25\% quota share;
  • a \$500{,}000 excess of \$500{,}000 per-risk layer. Five years of trended, developed layer losses total \$5{,}400{,}000 against \$90{,}000{,}000 of on-level subject premium.

Find each contract's expected loss cost.

Answer

\begin{align*}
\text{QS loss cost} &= 0.25 \times 0.65 \times 20{,}000{,}000 \\
&= \$3{,}250{,}000
\end{align*}
\begin{align*}
\text{Burning cost} &= \frac{5{,}400{,}000}{90{,}000{,}000} \\
&= 6.0\% \\[4pt]
\text{XL loss cost} &= 0.060 \times 20{,}000{,}000 \\
&= \$1{,}200{,}000
\end{align*}

The quota share reinsurer gets the cedant's own 65\% loss ratio on \$5{,}000{,}000 of ceded premium. The excess layer's loss cost is a small rate on subject premium but far more volatile, since five years may hold only a handful of layer losses. That is why the burning cost gets credibility-weighted against an exposure rate. Each loss must also be trended before the retention is applied, because the layer trends faster than ground-up losses.