Actuarial Notes

Deductible Rating

Study guides: Exam 5, Exam GI-301

Deductible Rating is the pricing of policies under which the insured retains the first d dollars of each loss. The rate credit is derived from the loss elimination ratio — the share of expected losses the deductible removes.

\text{LER}(d) = \frac{E[X \wedge d]}{E[X]}

\text{Deductible Relativity} = 1 - \text{LER}(d)

  • Werner derives the relativity (1 - \text{LER}) assuming all expenses are variable and the variable expense and profit provisions are a constant percentage of premium, so under that assumption it applies to the whole rate. Fixed expenses do not fall because the insured took a deductible: where they are material and are not charged as a separate fee, Werner's general rule for relativities is to include the fixed-expense loading, which flattens the indicated relativity — in effect the credit applies to the loss portion only.
  • LERs are computed from the size-of-loss distribution — empirically when ground-up losses are known, or from a fitted continuous distribution. When the data hold only losses net of the deductible, policies with a deductible higher than the one being priced cannot be used.
  • Deductible credits must satisfy a consistency requirement. Werner writes the losses eliminated by a deductible D as \int_0^D x f(x)\,dx + D\,[1 - F(D)], so each extra dollar of deductible eliminates 1 - F(D) per claim — the chance a loss exceeds D — which can only fall as D rises. The credit per dollar of deductible must therefore decrease as the deductible rises; a schedule that violates this is not consistent with any size-of-loss distribution.
  • The LER approach assumes claiming behaviour is the same at every deductible, and Werner notes two ways it is not: an insured with a higher deductible may not report a claim whose small net payment is not worth a claim surcharge, and where insureds choose freely, lower-risk insureds tend to choose higher deductibles — so higher-deductible policies may end up more profitable than lower ones. A classification analysis of the historical data reflects these differences, and the actuary may look at both sets of indications before selecting.
  • Deductibles erode with inflation: a fixed d eliminates a falling share of a growing loss, so relativities must be refreshed or higher-deductible business becomes under-priced.
  • On a large deductible policy the insurer may pay the whole claim and seek reimbursement below the deductible from the insured — the deductible recovery — so the premium must cover invoicing and monitoring and a provision for the credit risk that the insured cannot pay, which collateral rarely covers fully.

Pricing a Deductible Option A homeowners programme has a full-coverage loss cost of \$620 per house-year. LERs from the fitted severity distribution are:

Deductible LER
\$500 0.145
\$1{,}000 0.225
\$2{,}500 0.360

Fixed expenses are \$70 per policy, variable expenses 24\%, target profit 5\%. Compute the rate at each deductible and check consistency.

Answer Apply the relativity to the loss cost only, then load expenses:

Deductible Loss cost Rate = (\text{LC} + 70)/0.71
\$500 620 \times 0.855 = \$530.10 \$845.21
\$1{,}000 620 \times 0.775 = \$480.50 \$775.35
\$2{,}500 620 \times 0.640 = \$396.80 \$657.46

Consistency check — the extra credit per extra dollar of deductible must fall:

\begin{align*}
\$500 \to \$1{,}000: \; \frac{845.21 - 775.35}{500} &= \$0.140 \text{ per \$1} \\[6pt]
\$1{,}000 \to \$2{,}500: \; \frac{775.35 - 657.46}{1{,}500} &= \$0.079 \text{ per \$1}
\end{align*}

The marginal credit falls from 14.0 cents to 7.9 cents per dollar of retention, as it must — the second, wider layer contains proportionally less expected loss.

Note also that the rate does not fall by the full LER: raising the deductible from \$500 to \$2{,}500 eliminates 25\% of losses but only 22\% of premium, because the \$70 of fixed expense rides on regardless — the variable expense and profit loads scale with the loss cost, so they fall in proportion.

An Inconsistent Credit Schedule A commercial property filing proposes these deductible credits against a \$1{,}000-deductible base:

Deductible Proposed credit
\$1{,}000 —
\$2{,}500 8\%
\$5{,}000 14\%
\$10{,}000 28\%

Is the schedule sound?

Answer Marginal credit per dollar of additional retention:

\begin{align*}
\$1{,}000 \to \$2{,}500: \; \frac{8\%}{1{,}500} &= 0.53\% \text{ per \$100} \\[6pt]
\$2{,}500 \to \$5{,}000: \; \frac{6\%}{2{,}500} &= 0.24\% \text{ per \$100} \\[6pt]
\$5{,}000 \to \$10{,}000: \; \frac{14\%}{5{,}000} &= 0.28\% \text{ per \$100}
\end{align*}

The marginal credit falls, then rises at the top step. That is inconsistent: the \$5{,}000–\$10{,}000 layer cannot contain more expected loss per dollar than the \$2{,}500–\$5{,}000 layer beneath it.

The practical consequence is arbitrage. An insured whose expected loss in the top layer is below average buys the \$10{,}000 deductible and captures a credit exceeding the loss it retains; the insurer loses on every such sale, and the insureds who take the option are precisely the ones for whom it is mispriced.

The fix is to re-derive the credits from a single consistent size-of-loss distribution rather than setting them competitively step by step — the same consistency test applied to increased limits factors, run in the opposite direction.