Loss Sensitive Rating

Study guide: Exam 8

Loss Sensitive Rating covers any plan under which what the insured ultimately pays — premium, or premium plus retained losses — depends on its own losses during the policy period. Retrospective rating is the classic form. Large deductibles, self-insured retentions and loss-sensitive dividend plans are the others.

L = \min\left(\max(A,\ r_H E),\ r_G E\right)

\mathrm{E}[L] = E\left(1 - \phi(r_G) + \psi(r_H)\right)

B = e - (c - 1)E + cI

• A is the actual aggregate (ratable) loss, E = \mathrm{E}[A] its expected value, and r = A/E the entry ratio. r_G and r_H are the entry ratios at the maximum and minimum. The Table M charge \phi(r) = \mathrm{E}[\max(A/E - r,\,0)] is the expected loss above rE per unit of E, and the savings \psi(r) = \mathrm{E}[\max(r - A/E,\,0)] is the expected shortfall below it. Always \psi(r) = \phi(r) + r - 1.
• Retro premium is R = (B + cL)\,T, with loss conversion factor c (loads LAE) and tax multiplier T = 1/(1 - \text{tax rate}). Setting \mathrm{E}[R]/T = e + E gives the basic premium B above. Here e is total expense before taxes, including LAE and profit, and I = E[\phi(r_G) - \psi(r_H)] is the net insurance charge. Fisher et al. note that this balanced premium is no longer required or desirable for most plans, because the plans transfer less risk than guaranteed cost; the expected losses must still balance.
• Per-occurrence limits. A loss limit adds an excess-loss charge. The aggregate charge is then computed on limited losses (Table M_D), or both are combined in one charge (Table L). Table M is entered by risk size, because small risks have volatile entry ratios and therefore larger charges.
• The other plans (large deductibles are now more common than retro plans):
  • Large deductible: the insurer pays every claim and bills the deductible back. It must have a per-occurrence limit, may have an aggregate, and has no minimum. Reimbursements are not premium, so they escape premium tax.
  • SIR: the insured handles its own claims, and the insurer's limit sits above the retention.
  • Dividend plans: return money when losses are low but collect nothing when they are high, so they are unbalanced.
• Analysing a plan means looking at credit risk on future premium or reimbursements (managed with collateral, loss development factors and holdbacks). The insured should keep the predictable working layer, and retentions should rise with trend. Cash flow differs between paid and incurred retros. The aggregate charge is highly sensitive to the loss pick.

Table M Charges from Aggregate Outcomes Five similar accounts, equally likely, finish at entry ratios 0.4, 0.7, 0.9, 1.2 and 1.8. Compute \phi(1.5) and \psi(0.6), and verify the charge–savings identity at r = 0.6.

Answer

\begin{align*}
\phi(1.5) &= \frac{0 + 0 + 0 + 0 + 0.3}{5} \\
&= 0.06 \\[4pt]
\psi(0.6) &= \frac{0.2 + 0 + 0 + 0 + 0}{5} \\
&= 0.04 \\[4pt]
\phi(0.6) &= \frac{0 + 0.1 + 0.3 + 0.6 + 1.2}{5} \\
&= 0.44
\end{align*}

Check: \phi(0.6) + 0.6 - 1 = 0.04 = \psi(0.6). A maximum at 1.5E transfers 6\% of expected loss to the insurer, and a minimum at 0.6E hands 4\% back. The net insurance charge is 2\% of E.

Pricing a Balanced Retro Plan Using those charges, let E = \$600{,}000, e = \$250{,}000, c = 1.10, T = 1.03, with the maximum and minimum at entry ratios 1.5 and 0.6. Find B, the maximum and minimum premiums, and confirm the plan balances.

Answer

\begin{align*}
I &= 600{,}000 \times (0.06 - 0.04) \\
&= 12{,}000 \\[4pt]
B &= 250{,}000 - 0.10(600{,}000) + 1.10(12{,}000) \\
&= \$203{,}200 \\[4pt]
G &= \left[203{,}200 + 1.10(900{,}000)\right] \times 1.03 \\
&= \$1{,}228{,}996 \\[4pt]
H &= \left[203{,}200 + 1.10(360{,}000)\right] \times 1.03 \\
&= \$617{,}176
\end{align*}

The five outcomes produce retro premiums of \$617{,}176, \$685{,}156, \$821{,}116, \$1{,}025{,}056 and \$1{,}228{,}996. Their average is \$875{,}500 = (250{,}000 + 600{,}000) \times 1.03 — exactly the guaranteed-cost premium. The good year is floored and the bad one capped. The \$13{,}200 loaded into B is what pays for that asymmetry.