The Nature of Risk and Classification

Any economic activity is accompanied by uncertainty: a fire may destroy a warehouse, a borrower may default on a loan, the price of oil may collapse, or a pandemic may paralyze entire industries. Actuarial science turns “fears” into numbers: it formalizes risk, quantitatively assesses it, and offers management instruments. Without this apparatus, modern insurance companies, pension funds, banks, and regulators would not exist—in short, the entire financial infrastructure of the 21st century.

Definition and Types of Risk

Risk is the possibility of an undesirable event occurring with some probability, leading to financial losses. Formally: a random variable $X$ describing loss (or return with a negative sign), plus a probability measure on the space of elementary outcomes.

Pure risk vs. speculative.

• Pure risk: only the possibility of loss (fire, illness, accident). The ideal sphere for insurance.
• Speculative risk: possibility of both loss and gain (investments, exchange rate). Managed with hedging, diversification.

Classification by nature:

• Insurable risk: random, measurable, there is a large number of homogeneous units, loss is non-catastrophic.
• Financial risk: market (stock prices, rates), credit (default), liquidity (inability to sell quickly).
• Operational risk: process errors, systems, personnel, external events.
• Systemic risk: correlated shocks affecting the entire system (2008 crisis, COVID-2020).

Insurability criteria: randomness (does not depend on the will of the insured), definiteness (the loss can be measured), independence (between insured parties), a large number of homogeneous risks (for the law of large numbers), acceptable premium. Not all risks are insurable: nuclear catastrophe, war, reputational risk, risk of changes in consumer tastes are often excluded.

Frequency and Severity of Losses

Compound model: Total annual loss $S = X_1 + X_2 + ... + X_N$, where $N$ is a random number of losses (frequency), $X_i$ is the amount of the $i$-th loss (severity). The standard assumption is the independence of $N$ and ${X_i}$, and independence among the $X_i$.

Distributions of frequency $N$:

• Poisson: $P(N = k) = e^{−λ}·λ^k/k!$, $E[N] = λ$, $Var[N] = λ$. Ideal for rare independent events (fires, accidents).
• Binomial: fixed $n$ risks, each defaults with probability $p$. $E[N] = n·p$, $Var[N] = n·p·(1−p)$.
• Negative binomial: heterogeneous portfolio—$Var > E$ (overdispersion).

Distributions of severity $X$:

• Exponential: $f(x) = λ·e^{−λ·x}$, $E[X] = 1/λ$. Light tail, few catastrophes.
• Lognormal: $\ln X \sim N(\mu, \sigma^2)$. Moderate tail, classic for insurance losses.
• Pareto: $P(X > x) = (θ/(θ+x))^α$. Heavy tail—catastrophes are possible. Used for fires, floods, cyber-attacks.

Formulas for $S$: $E[S] = E[N]·E[X]$, $Var[S] = E[N]·Var[X] + Var[N]·E[X]^2$ (Wald-style formula).

Law of Large Numbers in Insurance

This is the mathematical foundation of the entire insurance business. $n$ insured parties with independent losses $X_i$. The average loss $\bar{X}_n = (1/n)·\Sigma X_i$. By the LLN: $\bar{X}_n \rightarrow E[X]$ almost surely as $n \rightarrow \infty$. Variance of the average: $Var[\bar{X}_n] = Var[X]/n \rightarrow 0$.

Safety loading. The insurance premium $P$ is set to cover the expected loss plus a "loading" for risk:

• Expected value principle: $P = (1 + θ)·E[X]$, $θ ≈ 0.1–0.5$.
• Standard deviation principle: $P = E[X] + λ·σ(X)$.
• Variance principle: $P = E[X] + α·Var[X]$.
• Exponential (Esscher) principle: $P = (1/α)·\ln E[e^{α·X}]$—connected to utility theory.

Numerical Example: Motor Insurance Portfolio

1000 policyholders. Claim frequency $N_i \sim Poisson(λ = 0.05)$ per year. Severity $X \sim Lognormal(μ = 8, σ = 1)$ (median $e^8 ≈ 2981$ rubles, mean $≈ 4915$ rubles).

$E[S_i] = 0.05·4915 ≈ 246$ rubles. $Var[X] = (e^{σ^2} − 1)·e^{2μ+σ^2} = (e − 1)·e^{17} ≈ 4.18·10^7$. $E[S_i^2] = E[N]·E[X^2] + (E[N]·E[X])^2$ (for Compound Poisson: $Var[S_i] = λ·E[X^2]$). $Var[S_i] = 0.05·E[X^2] = 0.05·(Var[X] + E[X]^2) ≈ 0.05·(4.18·10^7 + 2.42·10^7) = 3.30·10^6$. $σ(S_i) ≈ 1817$ rubles.

For 1000 policyholders, total loss $S = \Sigma S_i$: $E[S] = 246,000$, $σ(S) = \sqrt{1000}·1817 ≈ 57,460$. By normal approximation $P(S > 1.1·E[S]) = P(Z > (24600/57460)) = P(Z > 0.428) ≈ 33.4%$. Premium $P = E[S] + 1.96·σ(S) ≈ 246,000 + 112,600 = 358,600$ rubles, ~358 rubles per policy, provides a 97.5% safety level.

Real-World Applications

• Automobile insurance (comprehensive, compulsory MTPL). Russia: market of 250+ billion rubles per year. Tariff coefficients are constructed on compound models with discrimination by age, experience, region, engine power.
• Healthcare. Bismarck systems (Germany, Japan) and market-based systems (USA) use actuarial models to calculate premiums considering age and lifestyle.
• Corporate insurance. D&O (directors) policies, cyber insurance, business interruption insurance—exotic models with heavy tails.
• Reinsurance (Munich Re, Swiss Re). Global reinsurers manage portfolios of catastrophic risks worth tens of billions of dollars—our frequency-severity apparatus with adjustments for correlations.

Exercise. For a portfolio of 1000 motor insurance policies: claim frequency $N \sim Poisson(λ = 0.05)$, severity $X \sim Lognormal(μ = 8, σ = 1)$. (a) Calculate $E[S], Var[S]$ analytically. (b) Approximate $P(S > 1.1·E[S])$ using the normal approximation. (c) How many policyholders $n$ are needed for $σ(\bar{X}_n) < 100$ rubles? (d) Use Monte Carlo simulation (10,000 iterations) to find the empirical distribution of $S$—how close is it to normal (Shapiro-Wilk test)?