Limited Fluctuation Credibility Calculations | Free Exam FAM Lesson

Your company insures 3,000 taxicabs. Last year: 412 claims. Is that enough to let the fleet's experience stand on its own, or must you blend with the class rate? The answer is the full credibility standard, and it depends on whether you are measuring claim counts, average severity, or aggregate losses. Get the target wrong and your standard is off by a factor of 2 or more.

HIGH-FREQUENCY: The Poisson full credibility formula is the most commonly tested credibility calculation.

Tolerance criterion: observed claims within $100 r %$ of expected with probability $p$ :

$n_{0} = (\frac{z _{p}}{r})^{2} \cdot \frac{Var ( N )}{E [ N ]}$

For Poisson counts ( $Var (N) = E [N]$ ):

$n_{0} = (\frac{z _{p}}{r})^{2}$

Standard exam values ( $r = 0.05$ ):

$p = 0.90$ : $z_{p} = 1.645$ , $n_{0} = (32.9)^{2} = 1, 083$
$p = 0.95$ : $z_{p} = 1.960$ , $n_{0} = (39.2)^{2} = 1, 537$

Always round up to the next integer.

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Common mistakes

Forgetting the +1 for aggregate losses. Aggregate uses $1 + C V^{2}$ , not just $C V^{2}$ . For exponential severity, the aggregate standard is $2 \times (z_{p} / r)^{2}$ . Omitting the 1 gives the severity-only standard, exactly half the correct answer. Trap: 1,083 instead of 2,166.
Using the wrong $z_{p}$ . $z_{p}$ is the two-sided critical value: $z_{0.90} = 1.645$ , $z_{0.95} = 1.960$ . Using the one-sided 1.645 when the problem says $p = 0.95$ gives 1,083 instead of 1,537. Trap: wrong base factor.
Computing $C V^{2}$ for the wrong distribution. Pareto( $α = 3$ ): $C V^{2} = 3$ . Gamma( $α = 3$ ): $C V^{2} = 1/3$ . A factor of 9 difference. Trap: using the gamma formula for a Pareto or vice versa.

Bottom line

Poisson frequency standard: $n_{0} = (z_{p} / r)^{2}$ ; memorize 1,083 ( $p = 0.90$ ) and 1,537 ( $p = 0.95$ ) at $r = 0.05$ .
Severity-only standard: $n_{0} = (z_{p} / r)^{2} \cdot C V^{2}$ , no "+1".
Aggregate standard: $n_{0} = (z_{p} / r)^{2} \cdot (1 + C V^{2})$ ; aggregate = frequency + severity.
Partial credibility: $Z = n / n_{0}$ , capped at 1.

Exam shortcut

Before computing, identify two things: (1) is the target claim counts, average severity, or aggregate losses? This determines whether you multiply by 1, $C V^{2}$ , or $1 + C V^{2}$ . (2) What are $p$ and $r$ ? Memorize the two base factors: 1,083 ( $p = 0.90$ ) and 1,537 ( $p = 0.95$ ) for $r = 0.05$ . Remember: "1-CV-1+CV", frequency uses 1, severity uses $C V^{2}$ , aggregate uses $1 + C V^{2}$ . The Pareto always needs more data than the exponential; the gamma always needs less.

The full lesson (about 2,093 words, 14 min read) adds 3 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

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