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Actuarial & Insurance

Pricing risk: premiums, loss ratios, annuities, life expectancy and expected claims

Pure Premium

NEWIntermediate
PP=f×SPP = f \times S

Expected claim cost per policy: frequency times severity.

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Gross Premium

NEWIntermediate
GP=PP1eGP = \frac{PP}{1 - e}

Premium that covers claims plus an expense/profit load.

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Loss Ratio

NEWBasic
LR=CPLR = \frac{C}{P}

Claims paid divided by premiums earned.

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Combined Ratio

NEWIntermediate
CR=LR+ERCR = LR + ER

Total cost ratio: losses plus expenses vs premium.

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Expected Value of a Claim

NEWBasic
E[X]=pixiE[X] = \sum p_i x_i

Probability-weighted average of possible claim amounts.

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Present Value of an Annuity

NEWIntermediate
PV=P1(1+r)nrPV = P \frac{1 - (1+r)^{-n}}{r}

Value today of a stream of future payments.

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Future Value of an Annuity

NEWIntermediate
FV=P(1+r)n1rFV = P \frac{(1+r)^n - 1}{r}

Accumulated value of regular contributions.

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Curtate Life Expectancy

NEWAdvanced
ex=k=1kpxe_x = \sum_{k=1}^{\infty} {}_k p_x

Expected remaining whole years of life from survival probabilities.

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Mortality Rate

NEWIntermediate
qx=dxlxq_x = \frac{d_x}{l_x}

Probability that a life aged x dies within the year.

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Loss Reserve

NEWAdvanced
R=C^CpaidR = \hat{C} - C_{paid}

Money set aside for claims incurred but not yet fully paid.

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Credibility Weighting

NEWAdvanced
P=Zxˉ+(1Z)μP = Z\bar{x} + (1-Z)\mu

Blend own experience with the wider average by a credibility factor.

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Risk Premium (Variance Load)

NEWAdvanced
P=E[X]+θVar(X)P = E[X] + \theta\,\text{Var}(X)

Premium that adds a load proportional to risk variance.

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Claim Frequency

NEWBasic
f=NEf = \frac{N}{E}

Number of claims per unit of exposure.

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Simple Term Insurance Cost

NEWBasic
C=qx×BC = q_x \times B

Expected one-year cost of a term policy.

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Solvency Ratio

NEWIntermediate
SR=ALLSR = \frac{A - L}{L}

Insurer's capital buffer relative to its liabilities.

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