Actuarial Notes

Complement of Credibility

Study guides: Exam MAS-II, Exam 5

Complement of Credibility is the estimate that receives the weight (1-Z) when observed experience receives credibility Z. When Z is low — a small class, a new territory, an excess layer — the complement, not the data, determines the answer.

\text{Estimate} = Z \times \text{Experience} + (1 - Z) \times \text{Complement}

Boor's desirable qualities of a complement. It should be (1) accurate — as low an error variance as possible around the future expected losses; (2) unbiased — not routinely higher or lower than the observed experience; (3) statistically independent from the base statistic, or any error in the base statistic can be compounded; (4) available — the data needed to compute it readily obtainable; (5) easy to compute — easy to perform and understand, which matters most when the actuary must justify it to a third party such as a regulator; and (6) logically related to the base statistic, so that its use is easy to support.

Werner & Modlin's six first-dollar complements:

  1. Loss costs of a larger group that includes the group being rated — e.g. all states in the region for the state under review. It can be built to include or exclude the subject experience; if included, the subject must not dominate the group.
  2. Loss costs of a larger related group — a separate but similar group, e.g. owners-form contents experience for the condo form. It is generally biased, but because it does not contain the subject experience it may be a better choice than method 1 on independence.
  3. Rate change from the larger group applied to present rates — the larger group's indicated change applied to the subject's current loss cost, rather than the larger group's loss costs used directly.
  4. Harwayne's method — other states' data adjusted to remove the overall differences between states before it is used, so that a state with a higher cost level does not import that level into the class complement.
  5. Trended present rates — the current rate adjusted to what was last indicated rather than implemented, and trended from the original target effective date of the current rates to that of the new rates.
  6. Competitors' rates.

Excess (increased-limits) complements: increased limits analysis, lower limits analysis, limits analysis, and the fitted-curve approach — all built on the fact that excess layers have too few claims ever to be credible on their own.

Further points:

  • The loss costs of a larger related group may be biased. Method 3 mitigates this by transferring the larger group's indicated change rather than its loss cost, so even when the subject and the larger group have different overall loss costs the complement is largely unbiased; its independence depends on how large the subject is relative to the larger group.
  • ASOP No. 25, as Werner quotes it, requires related experience whose frequency, severity or other determinable characteristics may reasonably be expected to be similar to the subject experience; related experience that does not, or cannot be adjusted to, meet that criterion should not be used. Trended present rates are the worked case: the current rate is brought forward for loss trend before it is blended.
  • In reserving, the BF method is a credibility weighting of the development method (credibility Z = 1/\text{CDF}) with the expected claims method, which receives the complement (1-Z); the Cape Cod technique differs from BF chiefly in deriving the expected claim ratio from the reported claims experience rather than from an independent, often judgmental selection.

The Complement Drives a Low-Credibility Indication A territory's experience indicates a rate change of +25\% but earns only Z = 0.20. The complement is the statewide indicated change applied to the territory's present rates (method 3): +6\%.

Compute the weighted change, and assess how much the answer depends on the complement.

Answer

\begin{align*}
\text{Weighted} &= 0.20(25\%) + 0.80(6\%) \\
&= 5.0\% + 4.8\% \\
&= +9.8\%
\end{align*}

Nearly half the answer (4.8 of 9.8 points) comes from the complement, and the result moves 0.8 points for every point the complement moves: had the complement been +12\%, the result would be +14.6\%.

This is Werner's point that the complement can often be more important than the observed data: with low credibility, most of the rate is driven by the complement. Method 3 suits the case because it transfers the statewide change rather than the statewide loss cost, so it stays largely unbiased even if the territory's cost level differs from the state's; the territory being a small part of the state keeps it largely independent too.

Harwayne's Method Class 1 in State X has thin data. States Y and Z write the same two classes but have different overall cost levels and different class mixes:

State Class Exposures Losses Pure premium
X 1 200 \$600 \$3.00
X 2 300 \$1{,}500 \$5.00
Y 1 400 \$1{,}440 \$3.60
Y 2 200 \$1{,}200 \$6.00
Z 1 300 \$750 \$2.50
Z 2 600 \$2{,}400 \$4.00

Compute the Harwayne complement for class 1 in State X.

Answer Step 1 — State X's average pure premium:

L_X = \frac{200(3.00) + 300(5.00)}{500} = \$4.20

Step 2 — Y's and Z's average pure premiums on State X's class mix (X's exposures, each state's own class pure premiums):

\begin{align*}
\hat{L}_Y &= \frac{200(3.60) + 300(6.00)}{500} = \$5.04 \\[4pt]
\hat{L}_Z &= \frac{200(2.50) + 300(4.00)}{500} = \$3.40
\end{align*}

Step 3 — adjustment factors, applied to each state's class 1 pure premium:

\begin{align*}
F_Y &= \frac{4.20}{5.04} = 0.833, & \hat{L}_{1,Y} &= 3.60(0.833) = \$3.00 \\[4pt]
F_Z &= \frac{4.20}{3.40} = 1.235, & \hat{L}_{1,Z} &= 2.50(1.235) = \$3.09
\end{align*}

Step 4 — combine, weighted by each state's class 1 exposures:

C = \frac{400(3.00) + 300(3.09)}{400 + 300} = \$3.04

Step 2 is the step that makes this Harwayne's method. State Y's own average is \$2{,}640 / 600 = \$4.40, but that figure carries Y's class mix, which is heavier in the cheap class 1 than X's. Scaling by it (3.60 \times 4.20/4.40 = \$3.44) would treat part of a mix difference as a level difference. Re-weighting both states to X's exposure distribution leaves only the difference in cost level — legal environment, population density — for the factor to remove.

Werner's evaluation: the complement is unbiased because it adjusts for the distributional differences, reasonably accurate when there is enough multi-state data, and mostly independent because it uses other states' data; the data is usually available, but the computation is time-consuming and harder to explain.