Interactive tools

Actuarial & Premium Sandbox

Build intuition for how insurance is priced. Start from the pure risk premium — probability × severity — then add loadings, test deductibles, and split risk between retention and transfer.

1 · Pure Risk Premium & Rate

The expected cost of claims, then loaded for expenses and profit to reach the charged premium.

Claim frequency — expected claims per year (λ)

Average severity — mean cost per claim

Expense ratio — acquisition + admin, % of premium

Profit & contingency load — % of premium

Pure premium (expected loss)

Pure premium = frequency × severity

Gross premium (charged)

• Expenses

• Profit & contingency

Permissible loss ratio

Gross = pure ÷ (1 − expense − profit)

2 · Deductible Sensitivity

How a per-claim deductible lowers the insurer's expected payout and the premium. Model: severity ~ Exponential(mean θ)

Claim frequency — claims per year (λ)

Mean severity θ — average ground-up claim

Deductible — retained by the insured per claim

Insurer expected annual loss

E[(X−d)₊] = θ·e^(−d/θ) · ×λ per year

Ground-up expected loss

Loss eliminated by deductible

Loss Elimination Ratio (LER)

Deductible	Insurer loss/yr	Savings

3 · Risk Retention & Transfer (Layering)

Split expected loss between a self-insured retention and a transferred layer up to a limit. Model: severity ~ Exponential(mean θ)

Claim frequency — claims per year (λ)

Mean severity θ

Self-insured retention (SIR) — you keep losses below this

Transferred limit — insurer covers this layer above the SIR

Expected annual loss, split

■ Retained (you)

■ Transferred (insurer)

Excess above limit (uninsured)

Total ground-up expected loss

Layer (d, d+L) = θ·(e^(−d/θ) − e^(−(d+L)/θ)) · λ

About the models. Calculator 1 uses the identity expected loss = frequency × severity and the standard rate-making loading gross = pure ÷ permissible loss ratio. Calculators 2 and 3 assume claim severity follows an exponential distribution with mean θ, which gives clean closed forms for deductibles and layers and is a common teaching model. Real pricing uses fitted severity distributions, credibility, trend, and reinsurance structure. These tools are for education only — not actuarial advice or a rate filing.