Credibility for a Large Workers' Compensation Account
A workers' compensation insurer has insured a large construction account for three years. The account's annual loss ratios, on a common exposure base, were 0.80, 0.60 and 0.70. Exposure has been stable, so the three years can be treated as three observations of equal weight. Collateral data from 20 comparable accounts produce an estimated within-account process variance of v = 0.04 and an estimated between-account variance of a = 0.01, both on the loss-ratio scale. The manual loss ratio for this class is 0.65. The underwriter wants to use the account's three-year average of 0.70 for next year's renewal and is arguing for a 7.7% rate increase. The account has just implemented a formal safety program, which the underwriter believes will reduce future losses but which is not reflected in the three-year history. The chief actuary must decide how much credibility to give the account's own experience and how to incorporate the safety program.
1. Compute the Bühlmann credibility factor for the account and the credibility-weighted loss ratio.
Read the worked answer
The Bühlmann credibility factor is Z = n / (n + K), where K = v / a. Here v = 0.04 and a = 0.01, so K = 4. With n = 3 years, Z = 3 / (3 + 4) = 3/7 = 0.4286.
The account's own average loss ratio is 0.70 and the manual complement is 0.65. The credibility-weighted estimate is 0.4286 x 0.70 + 0.5714 x 0.65 = 0.3000 + 0.3714 = 0.6714.
So the actuarial estimate for next year, before any safety-program adjustment, is about 0.671, not 0.70. The account's experience gets only 43% weight; the manual rate gets 57%. The underwriter's proposed 7.7% increase is based on the raw average and overstates the indicated change.
2. How many years of stable experience would be needed for Z at least 0.90? What does that imply for the underwriter's proposed increase?
Read the worked answer
Set n / (n + 4) >= 0.90. Then n >= 0.90(n + 4), so n >= 0.90n + 3.6 and 0.10n >= 3.6. Therefore n >= 36 years.
Three years is far from full credibility under these variance components. The credible estimate is 0.671, only 2.1 loss-ratio points above the manual 0.65, which is about a 3.3% increase rather than 7.7%. Moreover, the safety program may reduce future losses, so the historical 0.70 is not a clean estimate of next year's expected loss ratio. The underwriter should not apply the full 7.7% increase based on three years alone; a partial credibility increase plus a separate safety-program adjustment is more defensible.
3. Suppose the insurer believes the safety program will reduce the next-year loss ratio by 10%, but only with probability 0.6; otherwise it has no effect. How could this be incorporated, and what is the resulting estimate using the credibility weight from part 1?
Read the worked answer
The expected multiplicative effect of the program is 0.6 x 0.90 + 0.4 x 1.00 = 0.54 + 0.40 = 0.94. Equivalently, the expected reduction is 0.6 x 10% = 6%. Apply this factor after the credibility weighting, because the credibility estimate represents the expected loss ratio before the new program: 0.6714 x 0.94 = 0.6311.
The resulting estimate is about 0.631, which is below the manual 0.65. This would suggest no rate increase, or even a decrease, if the safety program is believed. However, the 10% effect is uncertain and may be optimistic. A prudent approach is to present a range: without the program, 0.671; with the expected program effect, 0.631. The insurer should track the account's actual losses after the program before committing to a large decrease, and should avoid double-counting the program if the manual rate already includes a safety discount.