Deductible Mix Change vs. True Pure Premium Trend in Homeowners
A homeowners insurer writes in a state where deductibles are rising. Five years ago, 80% of policies had a $500 deductible and 20% had $1,000. Today, 25% have $500, 45% have $1,000, 25% have $2,500, and 5% have $5,000. The actuarial analyst computed a pure premium trend of −4.5% per year from the raw average paid loss per exposure and concluded that rates should decrease. The chief actuary is skeptical. The company's deductible credits are: $500 = 1.00, $1,000 = 0.92, $2,500 = 0.80, $5,000 = 0.70. Losses are being inflated by 6% per year in severity, and frequency is flat. The current rate level is 1.00, and the average deductible is now $1,450 versus $600 five years ago.
1. Compute the average deductible credit at the start and end and the implied annual effect of the deductible mix change.
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Start: 0.80 × 1.00 + 0.20 × 0.92 = 0.984. End: 0.25 × 1.00 + 0.45 × 0.92 + 0.25 × 0.80 + 0.05 × 0.70 = 0.25 + 0.414 + 0.20 + 0.035 = 0.899. The ratio is 0.899/0.984 = 0.9136 over five years. Annualized: 0.9136^(1/5) − 1 = −0.0180, or −1.8% per year. The shift to higher deductibles reduced the average loss cost by about 1.8% per year, all else equal. If the raw average paid loss per exposure fell 4.5% per year, the trend at a constant deductible mix would be roughly −4.5% − (−1.8%) = −2.7% per year. That is still negative, but much less than the raw figure. This calculation assumes the deductible credits are correct; if the credits are too generous, the mix effect is misstated. It also assumes the distribution of losses within each deductible band is unchanged, which may not hold if the company changed underwriting or policy forms.
2. Given flat frequency and 6% severity inflation, what is the indicated pure premium trend? What does the inconsistency imply about the analyst's calculation?
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At a constant deductible, pure premium trend ≈ (1 + severity trend) × (1 + frequency trend) − 1 = 1.06 × 1.00 − 1 = +6.0% per year. Adding the deductible mix effect of −1.8% gives an expected raw trend of about 1.06 × 0.982 − 1 = +4.1% per year, not −4.5%. The analyst's −4.5% is inconsistent with the stated severity and frequency trends by about 8.6 percentage points. That gap is too large to be noise. Likely causes: the analyst did not adjust for rate changes, so the raw premium is not on-level; the exposure base changed, for example more policies with low coverage; the loss data are not developed to ultimate; or the severity trend is wrong. Before using any trend, recompute on-level earned premium and use a frequency-severity method: fit frequency trend and severity trend separately, then combine. Do not file a negative trend based on a raw average that is contaminated by mix change. If the true trend is +4% to +6%, filing a decrease would seriously underprice the book.
3. How should the deductible credits themselves be tested? What risk do stale credits create?
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Deductible credits should be derived from excess ratios: for each deductible D, the credit is approximately the ratio of expected insurer losses with deductible D to expected losses with the base deductible, using a fitted severity distribution and reflecting claim-handling expense savings. Test the credits by comparing loss ratios by deductible band on a level premium basis: if higher-deductible policies show materially higher loss ratios, the credits are too large and the company is undercharging them. Also check that the credits do not exceed the maximum possible; they should approach the proportion of losses below the deductible, not zero. The company has not updated credits in six years; during that time severity has increased 6% per year, which changes the excess ratio because a fixed deductible becomes relatively smaller. For example, a $500 deductible is a smaller share of a $10,000 claim than of a $6,000 claim, so the credit for a given deductible should decrease as severity rises. Stale credits likely overcredit high deductibles, causing adverse selection. Re-estimate the credits, file them as a revamp, and test revenue neutrality. Separate the loss credit from the expense credit, because a higher deductible reduces claim handling.