Actuarial & Insurance

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

Actuarial & Insurance formulas help you price uncertainty, measure risk, and understand how insurers estimate premiums, claims, reserves, and life-contingent payments. From pure and gross premiums to loss ratios, annuities, and mortality measures, this category is useful for students, exam candidates, and professionals working in insurance, pensions, and risk management, where accurate calculations support fair pricing and sound financial decisions.

Pure Premium

NEWIntermediate

PP = f \times S

Expected claim cost per policy: frequency times severity.

Gross Premium

NEWIntermediate

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

Premium that covers claims plus an expense/profit load.

Loss Ratio

LR = \frac{C}{P}

Claims paid divided by premiums earned.

Combined Ratio

NEWIntermediate

CR = LR + ER

Total cost ratio: losses plus expenses vs premium.

Expected Value of a Claim

E[X] = \sum p_i x_i

Probability-weighted average of possible claim amounts.

Present Value of an Annuity

NEWIntermediate

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

Value today of a stream of future payments.

Future Value of an Annuity

NEWIntermediate

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

Accumulated value of regular contributions.

Curtate Life Expectancy

e_x = \sum_{k=1}^{\infty} {}_k p_x

Expected remaining whole years of life from survival probabilities.

Mortality Rate

NEWIntermediate

q_x = \frac{d_x}{l_x}

Probability that a life aged x dies within the year.

Loss Reserve

R = \hat{C} - C_{paid}

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

Credibility Weighting

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

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

Risk Premium (Variance Load)

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

Premium that adds a load proportional to risk variance.

Claim Frequency

f = \frac{N}{E}

Number of claims per unit of exposure.

Simple Term Insurance Cost

C = q_x \times B

Expected one-year cost of a term policy.

Solvency Ratio

NEWIntermediate

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

Insurer's capital buffer relative to its liabilities.