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Pure Premium Engine

Calculate the pure premium — the expected loss per exposure unit — from claim frequency and severity. The foundation of all insurance pricing before loading for expenses, profit, and contingencies.

Calculate the strict mathematical severity and frequency of a risk pool to determine the absolute breakeven dollar amount required per exposure.

Cohort Statistical Data

$

Actuarial Outputs

Claim Frequency

0.05
5.00% probability per exposure

Average Claim Severity

$5,000.00
Average cost when a loss occurs

Pure Premium

$250.00
Expected Loss per Exposure
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What is Pure Premium: The Actuarial Foundation of Insurance Pricing?

The pure premium (also called loss cost or risk premium) is the expected loss per exposure unit — the minimum amount an insurer must charge just to break even on claims. It is calculated as Frequency × Severity, or equivalently, Total Losses ÷ Total Exposures. Every insurance premium starts with the pure premium, then adds loadings for expenses (commissions, overhead), profit margin, and contingency reserves. Rating bureaus like NCCI and ISO publish pure premiums that individual insurers modify with their own loss cost multipliers.

Mathematical Foundation

Pure Premium (Loss Cost) Formula
Pure Premium=Frequency×Severity=Total Incurred LossesTotal Exposures\text{Pure Premium} = \text{Frequency} \times \text{Severity} = \frac{\text{Total Incurred Losses}}{\text{Total Exposures}}
Pure Premium\text{Pure Premium}= Expected loss dollars per exposure unit — the cost of risk before any insurer markup.
Frequency\text{Frequency}= Number of claims per exposure unit. Example: 0.05 = 5 claims per 100 exposures.
Severity\text{Severity}= Average cost per claim. Example: $15,000 mean claim payment.
Exposures\text{Exposures}= Denominator units: earned car-years, payroll in $100s, revenue in $1,000s, etc.

Laws & Principles

  • Frequency × Severity Decomposition: Pure Premium = Frequency × Severity. Knowing both components separately is crucial: a line with high frequency/low severity (auto glass claims) is managed differently than low frequency/high severity (product liability). The same $750 pure premium could come from 15% frequency × $5,000 severity or 0.1% frequency × $750,000 severity.
  • Exposure Base Must Match the Risk: Workers comp uses payroll ($100s), auto uses car-years, property uses insured value ($1,000s). Using the wrong exposure base produces misleading loss costs. A restaurant with 50 employees and $2M payroll needs payroll-based exposures, not headcount.
  • Development and Trend: Raw historical losses must be developed to ultimate (estimating IBNR — incurred but not reported claims) and trended to the future policy period. A $500 pure premium from 2020 data may be $575 after 3% annual loss trend for a 2025 policy.

Step-by-Step Example Walkthrough

" An auto insurer had 500 claims totaling $7.5 million in losses across 10,000 earned car-years. "

  1. 1. Pure Premium = Total Losses / Exposures = $7,500,000 / 10,000 = $750 per car-year.
  2. 2. Frequency = Claims / Exposures = 500 / 10,000 = 0.05 (5%).
  3. 3. Severity = Total Losses / Claims = $7,500,000 / 500 = $15,000 per claim.
  4. 4. Verify: $750 = 0.05 × $15,000. ✓
Final Result: Pure Premium = $750 per car-year. Frequency = 5%. Severity = $15,000. After a 25% expense loading, the gross premium = $750 / (1 - 0.25) = $1,000 per car-year.

Quick Answer: How does the Pure Premium Calculator work?

Enter total incurred losses, number of claims, and total exposures. The calculator computes pure premium = losses/exposures, then decomposes it into frequency (claims/exposures) and severity (losses/claims).

Core Formula

Pure Premium = Frequency × Severity = Total Losses / Total Exposures

Where Frequency = claims per exposure, Severity = average cost per claim. Gross premium adds expense and profit loadings on top.

Industry Applications

Workers' Compensation

NCCI publishes pure premium rates by class code. Class 8810 (clerical) has a pure premium around $0.15 per $100 payroll, while Class 5403 (carpentry) runs $8-12 per $100 payroll — an 80× difference reflecting the true cost of risk for each occupation.

Auto Insurance Rating

Auto pure premiums vary dramatically by territory, age, and coverage. Urban ZIP codes can have 3× the pure premium of rural areas due to higher frequency (congestion) AND higher severity (repair costs). A 19-year-old male driver may have a pure premium 4× higher than a 45-year-old female.

Pure Premium by Line of Business

Line Typical Frequency Typical Severity Exposure Base
Auto Physical Damage5-8%$3,000-$8,000Car-years
Workers' Comp2-6%$10,000-$40,000Payroll ($100s)
General Liability0.5-2%$15,000-$100,000Revenue ($1,000s)
Product Liability0.01-0.1%$100,000-$5MUnits sold
Homeowners3-7%$5,000-$25,000House-years

Actuarial Best Practices (Pro Tips)

Do This

  • ✓Always decompose into Frequency × Severity. A rising pure premium could mean more claims (frequency-driven) or more expensive claims (severity-driven) — each requires a completely different response. Frequency is controlled by loss prevention; severity is controlled by claims management.

Avoid This

  • ✗Don't use undeveloped losses. Raw reported losses understate ultimate costs because many claims are still open (reserves not finalized) and some haven't been reported yet (IBNR). Always apply loss development factors before calculating pure premiums for pricing.

Frequently Asked Questions

What is the difference between pure premium and gross premium?

Pure premium covers only expected losses. Gross premium = pure premium / (1 - expense ratio). If expenses (commissions, overhead, profit) are 30% of premium, gross premium = pure premium / 0.70. An $800 pure premium becomes a $1,143 gross premium.

What is a loss cost multiplier (LCM)?

An LCM is a single multiplier applied to bureau-published pure premiums to produce an insurer's final rate. LCM = 1 / (1 - expense provision). An LCM of 1.45 means the insurer loads 45% on top of the loss cost for expenses and profit. Each insurer files its own LCM with regulators.

Why do actuaries separate frequency from severity?

Each component is driven by different factors and trends differently. Frequency is influenced by safety programs and economic conditions (fewer drivers in recessions). Severity is influenced by medical cost inflation, litigation trends, and repair costs. Trend factors must be applied separately for accurate projections.

What is IBNR and why does it matter?

IBNR (Incurred But Not Reported) represents claims that have occurred but haven't been filed yet. Workers' comp has long IBNR tails — an asbestos exposure today may not generate a claim for 20 years. Ignoring IBNR grossly understates pure premiums and threatens insurer solvency.

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