A discrete random variable \(X\) takes a finite or countable set of values; the PMF \(p(x) = P(X = x)\) gives the probability of each value. Total probability is \(\sum p(x) = 1\).
Important discrete distributions in actuarial work:
| Distribution | Use in insurance |
|---|---|
| Bernoulli(p) | Whether a single insured suffers a loss in a year (1) or not (0) |
| Binomial(n, p) | Number of policy-holders out of n who suffer a loss |
| Poisson(λ) | Number of claims in a fixed period (e.g., a year) |
| Negative Binomial(r, p) | Number of claims when there is over-dispersion compared to Poisson |
| Geometric(p) | Number of policies issued before the first claim |
A continuous random variable has a density function \(f(x)\) with \(\int f(x)\,dx = 1\) and \(P(a \le X \le b) = \int_a^b f(x)\,dx\). Useful in modelling claim size.
| Distribution | Use |
|---|---|
| Exponential(λ) | Inter-arrival times of claims; lifetime of a simple device |
| Gamma(α, λ) | Sum of independent exponential claim sizes; positively-skewed losses |
| Lognormal(μ, σ²) | Right-skewed claim severities (typical for property/auto claims) |
| Pareto(α, β) | Heavy-tailed loss distributions (large catastrophic claims) |
| Weibull(α, β) | Failure-time / human-lifetime modelling |
| Normal(μ, σ²) | Approximation of aggregate-claims (via CLT) |
A mixed distribution has both a discrete component (point masses) and a continuous component (density on an interval). In insurance this arises naturally:
40 % of policies have no claim. For policies with a claim, claim size is Exponential with mean ₹10 000.
\(P(X = 0) = 0.4;\;\; f(x) = 0.6 \cdot (1/10000) e^{-x/10000}\) for \(x > 0\).
Pure premium = \(E(X) = 0 \cdot 0.4 + 0.6 \cdot 10000 = \) ₹6 000.
Aggregate annual loss \(S\) is continuous on \((0, \infty)\). A stop-loss treaty pays \(\max(S - d, 0)\) — this transformed claim is 0 with positive probability and continuous beyond \(d\). It is a mixed r.v.
Insurance is a contract in which the insurer (insurance company) agrees, in exchange for a premium, to compensate the insured for specified random losses. By pooling many independent risks, the insurer transforms each individual's uncertain loss into a (nearly) certain expected loss — the foundation of the business.
For random variables \(X_1, X_2, \ldots, X_n\) with means \(\mu_i\) and variances \(\sigma_i^2\):
If the \(X_i\) are independent: \(\text{Var}(S) = \sum \sigma_i^2\).
| If each \(X_i\) is … | Then \(S = \sum X_i\) is … | Parameters |
|---|---|---|
| Bernoulli(p) (iid) | Binomial(n, p) | n trials |
| Binomial(n_i, p) | Binomial(Σn_i, p) | Same p |
| Poisson(λ_i) | Poisson(Σλ_i) | Sum of rates |
| Exponential(λ) (iid) | Gamma(n, λ) | — |
| Gamma(α_i, λ) | Gamma(Σα_i, λ) | Same scale λ |
| Normal(μ_i, σ_i²) | Normal(Σμ_i, Σσ_i²) | Independent |
If individual claim sizes \(X_i\) have finite variance, then for large \(n\):
where \(\mu = E(X_i)\), \(\sigma^2 = \text{Var}(X_i)\). This justifies normal approximations used throughout insurance practice.
An insurer covers 400 independent policy-holders. Each policy has claim with mean ₹500 and SD ₹200. Find approximate probability that total claims exceed ₹2 10 000.
\(E(S) = 400 \cdot 500 = 2,00,000\); SD\((S) = \sqrt{400} \cdot 200 = 4{,}000\).
By CLT: \(P(S > 210000) \approx P\!\left(Z > \frac{210000 - 200000}{4000}\right) = P(Z > 2.5) = 0.0062\).
Two independent product lines have annual claim counts \(N_1 \sim\) Poisson(15) and \(N_2 \sim\) Poisson(20). Total \(N = N_1 + N_2 \sim\) Poisson(35). \(E(N) = 35,\; \text{Var}(N) = 35\).
Utility theory describes how individuals (or organisations) make decisions in the presence of risk. Instead of comparing monetary outcomes \(X\), the decision-maker assigns a utility \(u(X)\) to each outcome and chooses the action that maximises expected utility \(E[u(X)]\).
A utility function \(u(w)\) representing preferences over wealth \(w\) typically satisfies:
Two summary measures of risk aversion:
| Type | Form \(u(w)\) | Risk attitude |
|---|---|---|
| Linear | \(a + bw\) | Risk-neutral (\(u''=0\)) |
| Quadratic | \(w - \dfrac{a}{2}w^2\) (for \(w < 1/a\)) | Risk-averse on its domain |
| Logarithmic | \(\ln w\) (for \(w > 0\)) | Risk-averse, constant relative risk aversion (CRRA = 1) |
| Power (CRRA) | \(\dfrac{w^{1-\gamma}}{1-\gamma}\) (\(\gamma \ne 1\)) | Constant relative risk aversion = \(\gamma\) |
| Exponential (CARA) | \(-\dfrac{1}{a}e^{-aw}\) (\(a > 0\)) | Constant absolute risk aversion = \(a\) |
| Quadratic (HARA) | Hyperbolic Absolute Risk Aversion family | General family |
Among alternatives \(A_1, A_2, \ldots\) each producing a random outcome \(X_i\), the decision-maker chooses the alternative that maximises \(E[u(X_i)]\). This is the foundation of rational decision-making under risk (von Neumann–Morgenstern axioms).
Insurance exists because both parties have different utility functions for the same risk.
A risk-averse individual is willing to pay a premium greater than the expected loss in exchange for certainty. A risk-neutral or less risk-averse insurer is willing to accept that premium because, by pooling many independent risks, it faces effectively no risk on aggregate (law of large numbers).
Consider an individual with initial wealth \(w\) facing a random loss \(X\). They are willing to pay a premium \(G\) for full insurance if:
The maximum acceptable premium \(G^*\) is the value satisfying equality:
\[ u(w - G^*) \;=\; E\bigl[u(w - X)\bigr]. \]Similarly, the insurer is willing to sell at minimum premium \(H^*\) satisfying:
where \(u_I\) is the insurer's utility and \(w_I\) its initial wealth. A trade can occur if \(G^* \ge H^*\) — i.e., the buyer's maximum is at least the seller's minimum.
The certainty equivalent \(CE\) of a random outcome \(X\) is the certain amount that yields the same utility:
\[ u(CE) = E[u(X)]. \]For a risk-averse individual, \(CE < E(X)\). The difference \(E(X) - CE\) is the risk premium — the discount on the expected value required to make the random outcome acceptable.
An individual has utility \(u(w) = -e^{-0.000001\, w}\) (constant absolute risk aversion \(a = 10^{-6}\)) and initial wealth ₹1 00 000. They face a loss of ₹50 000 with probability 0.10.
Expected loss: \(E(X) = 50000 \cdot 0.10 = \text{₹}5{,}000\) (pure premium).
Expected utility without insurance:
\(E[u(w - X)] = 0.9 \cdot u(100000) + 0.1 \cdot u(50000)\)
\(= 0.9 \cdot (-e^{-0.1}) + 0.1 \cdot (-e^{-0.05}) = 0.9(-0.9048) + 0.1(-0.9512) \approx -0.9095\).
Maximum premium \(G^*\): for CARA utility the premium is wealth-independent and solves \(e^{aG^*} = E[e^{aX}] = 0.1\,e^{0.05} + 0.9 = 1.00513\), so \(G^* = \dfrac{\ln 1.00513}{10^{-6}} \approx \text{₹}5{,}114\).
The risk loading \(G^* - E(X) \approx \text{₹}114\) is small — with \(a = 10^{-6}\) the individual is only mildly risk-averse at this wealth, so the premium sits just above the pure premium. (A much larger \(a\) would push \(G^*\) toward the maximum loss of ₹50 000.)
An individual with \(u(w) = \ln w\) and wealth ₹1 00 000 faces a loss \(X\) that is ₹40 000 with probability 0.25 and 0 otherwise.
\(E[u(w - X)] = 0.25 \ln(60000) + 0.75 \ln(100000)\)
\(= 0.25 (11.002) + 0.75(11.513) = 2.7505 + 8.6347 = 11.3852\).
\(u(100000 - G^*) = 11.3852 \Rightarrow 100000 - G^* = e^{11.3852} = 88{,}011\) (equivalently \(0.6^{0.25}\times 10^5\)).
Hence \(G^* = \text{₹}11{,}989\). Expected loss = 0.25 × 40000 = ₹10 000.
Risk premium = \(G^* - E(X) = \text{₹}1{,}989\) — the individual is willing to pay ₹1 989 more than the pure premium for the safety of full coverage.