Actuarial Pure Premium (Frequency x Severity) & Credibility Factor Z

Separate observed loss cost from frequency credibility

Observed frequency is claims divided by exposure; observed severity is losses divided by claims. For positive claim counts their product equals losses per exposure. These are historical summaries, not automatically expected future losses or a price for a policy. Exposure units must be defined consistently, such as car-years.

Dividing observed loss cost by one minus a proportional load is a simplified gross-up. It omits fixed expenses, trend, development, coverage changes and other pricing components. A loss-free sample produces zero under this formula, not a recommendation to charge zero.

The illustrative threshold of 1,082 relates to claim frequency under a homogeneous independent Poisson model and an approximate 90% probability of being within 5% of the mean. It is not a universal standard for loss cost: severity variation and non-Poisson frequency require different treatment. B6 is a supplied model parameter, not a generally applicable pricing rule.

The square-root credibility factor is capped at one. This article does not apply that frequency weight to a statewide premium rate. A full-credibility label is a property of the selected model and threshold, not certainty or a guarantee about future claims.

Illustrative gross-up of observed loss cost

B5 is a proportional load in the interval [0,1). The calculation assumes this is the only loading and does not include a fixed cost per exposure.

=IF(COUNT(B2:B6)<>5,"Invalid inputs",IF(OR(B2<=0,B3<0,B3<>INT(B3),B4<0,B5<0,B5>=1,B6<=0,AND(B3=0,B4<>0)),"Invalid inputs",ROUND((B4 / B2) / (1 - B5), 2)))

Illustrative frequency credibility weight

The full-precision weight is MIN(1,SQRT(B3/B6)); round only its display. The four-decimal formula below is for reporting, not an intermediate weight for further calculations.

=IF(COUNT(B2:B6)<>5,"Invalid inputs",IF(OR(B2<=0,B3<0,B3<>INT(B3),B4<0,B5<0,B5>=1,B6<=0,AND(B3=0,B4<>0)),"Invalid inputs",MIN(1, ROUND(SQRT(B3 / B6), 4))))

Observed losses per exposure

Historical losses divided by exposure; display rounded to cents.

=IF(COUNT(B2:B6)<>5,"Invalid inputs",IF(OR(B2<=0,B3<0,B3<>INT(B3),B4<0,B5<0,B5>=1,B6<=0,AND(B3=0,B4<>0)),"Invalid inputs",ROUND(B4 / B2, 2)))

Observed claim frequency

Claims per exposure can exceed one when an exposure has multiple claims.

=IF(COUNT(B2:B6)<>5,"Invalid inputs",IF(OR(B2<=0,B3<0,B3<>INT(B3),B4<0,B5<0,B5>=1,B6<=0,AND(B3=0,B4<>0)),"Invalid inputs",B3/B2))

Observed average severity

No claims gives an undefined average, displayed as N/A. This example requires zero loss when there are zero claims; reserve-only adjustments need a different data model.

=IF(COUNT(B2:B6)<>5,"Invalid inputs",IF(OR(B2<=0,B3<0,B3<>INT(B3),B4<0,B5<0,B5>=1,B6<=0,AND(B3=0,B4<>0)),"Invalid inputs",IF(B3=0,"N/A",B4/B3)))

Full-precision credibility weight

Use this unrounded value for subsequent calculations under the chosen frequency model.

=IF(COUNT(B2:B6)<>5,"Invalid inputs",IF(OR(B2<=0,B3<0,B3<>INT(B3),B4<0,B5<0,B5>=1,B6<=0,AND(B3=0,B4<>0)),"Invalid inputs",MIN(1,SQRT(B3/B6))))

Further reading

• Microsoft Support — SQRT function
• Casualty Actuarial Society — Full credibility for estimating claim frequency
• Actuarial Standards Board — Credibility procedures