Role of insurance in reducing or removing risk: formula sheet

Key formulas

Mean loss per policy for n independent identical risks

E(S/n) = μ, where S = X₁ + X₂ + … + Xₙ and each Xᵢ has mean μ

Expected average loss does not change with pooling.

Variance of average loss (independent, identical risks)

Var(S/n) = σ² ÷ n, so SD(S/n) = σ ÷ √n

This is why diversifiable risk falls as the pool grows. It needs independence.

Variance of average loss with common correlation ρ

Var(S/n) = σ²/n + ((n − 1)/n) × ρσ²

As n grows, this tends to ρσ². Correlated (systematic) risk is not removed by pooling.

Classification rules

Pure risk: loss or no loss. Speculative risk: loss, no change or gain.

Diversifiable: removed by pooling. Systematic: remains.

Checklist of insurable features

Insurable interest; definable and measurable loss; uncertain event; estimable probability; large pool of similar independent risks; no moral hazard; not catastrophic; affordable premium

Use as a checklist, not as strict pass or fail rules.

Net (pure) premium

Pure premium = E[S] = E[N] × E[X]

Expected claim cost. N is number of claims and X is claim size, assuming N and claim sizes are independent. Actual premiums add loadings for expenses, risk and profit.

Premium with loading

P = (1 + θ) × E[S]

θ is the relative safety loading. This is one simple premium principle. θ > 0 is needed for the insurer to be profitable on average.

Risk-averse person (Jensen's inequality)

E[U(W)] < U(E[W]) for strictly concave U

Risk-averse utility is concave. This is why a certain outcome is preferred to a gamble with the same mean.

Maximum premium the individual will pay

E[U(W − X)] = U(W − P_max)

W is initial wealth and X the random loss. P_max is where the person is indifferent between insuring and not insuring. For a risk-averse person P_max > E[X].

Variance of average claim from n independent identical risks

Var(S̄) = σ² ÷ n

Standard deviation of the average falls as σ ÷ √n. This holds for independent risks with the same variance σ². Correlated risks (e.g. a flood hitting a region) reduce the benefit.

Total claim

S = X₁ + X₂ + ... + Xₙ

Sum of the claims on n policies.

Mean and variance of total claim

E(S) = nμ; Var(S) = nσ²

Requires the Xᵢ to be independent (or uncorrelated) with common mean μ and variance σ².

Average claim per policy

E(S ÷ n) = μ; Var(S ÷ n) = σ² ÷ n

Variance per policy falls in proportion to 1 ÷ n.

Standard deviation of average claim

SD(S ÷ n) = σ ÷ √n

Falls with the square root of n.

Coefficient of variation of total claim

CV(S) = √(nσ²) ÷ (nμ) = σ ÷ (μ√n)

Measures relative risk. It falls as n rises.

Variance with common correlation ρ

Var(S ÷ n) = σ² ÷ n + ((n − 1) ÷ n) ρσ²

As n → ∞ this tends to ρσ². Correlated risk cannot be pooled away.

Weak law of large numbers

P(|S ÷ n − μ| > ε) → 0 as n → ∞, for any ε > 0

Holds for independent identically distributed claims with finite mean.

Expected utility criterion

Choose the option with highest E[U(W)]

W is final wealth, a random variable. Compare options by their expected utility, not by their utility of expected wealth.

Risk aversion condition

U'(w) > 0 and U''(w) < 0

Positive marginal utility means more is preferred. Negative second derivative means concave, so risk averse.

Jensen's inequality

E[U(W)] ≤ U(E[W]) for concave U

Strict if U is strictly concave and W is not constant. Reverses for convex U.

Maximum premium for full insurance

U(w − P_max) = E[U(w − X)]

w is initial wealth, X is the random loss. Solve for P_max.

Risk premium

Risk premium = P_max − E[X]

Positive for risk-averse individuals, zero for risk neutral.

Certainty equivalent

U(CE) = E[U(W)]

The certain wealth giving the same utility as the gamble. Then P_max = w − CE.

Absolute risk aversion

R(w) = −U''(w) ÷ U'(w)

Higher R means more risk averse and a larger risk premium.

Insurance purchase condition

Buy if P ≤ P_max

Equivalent to U(w − P) ≥ E[U(w − X)].

Claim paid with a deductible (fixed amount d)

Payment = max(X − d, 0)

X is the loss. The policyholder bears min(X, d). Excess and deductible are used for this in IAI material; state which meaning you use.

Claim paid with co-insurance share α

Payment = α × X, where 0 < α < 1

The policyholder bears (1 − α) × X of every loss.

Deductible with a policy limit L

Payment = min(max(X − d, 0), L)

Use when cover has both a deductible and a maximum payout.

Fair premium (no expenses or profit)

Premium = E[payment]

Loadings are added on top for expenses, profit and risk.

Adverse selection pricing with mixed groups

Average premium = Σ (proportion in group i × expected claim of group i)

Valid only if the mix of groups stays fixed. If low risks leave, the proportions change and the premium must be recalculated.

Quota share, retained claim

Retained claim = a × X, where 0 < a ≤ 1

The insurer keeps proportion a of every claim X. The reinsurer pays (1 − a)X. Premium is also shared in the same proportion, before any reinsurance commission.

Quota share moments

E(aX) = a E(X); Var(aX) = a² Var(X); SD(aX) = a SD(X)

Variance falls by a², so the coefficient of variation stays the same.

Individual excess of loss, retained claim

Retained = min(X, M); Reinsurer pays max(X − M, 0)

M is the retention. The insurer pays all of a claim up to M and the reinsurer pays the excess.

Stop loss, retained aggregate claim

Retained = min(S, d); Reinsurer pays max(S − d, 0)

S is total claims in the period and d is the aggregate retention.

Expected reinsurer payment, excess of loss

E[max(X − M, 0)] = ∫ from M to ∞ of (1 − F(x)) dx

For a continuous or general claim X ≥ 0. For Exp(λ), this equals e^(−λM) ÷ λ.

Reinsurer's premium

Reinsurance premium = (1 + θ) × expected reinsurer payment

θ is the reinsurer's loading. If θ is above the insurer's own loading, reinsurance reduces expected profit.

Quick revision

• Risk means uncertainty about outcomes; insurance deals with the financial effect of loss.
• An insurable risk needs a definable loss, and the chance of loss should be estimable and not under the insured's control.
• Insurance transfers the financial loss from the policyholder to the insurer in return for a premium.
• Pooling works best with many similar, independent risks; correlated risks such as a widespread disaster weaken it.
• Pooling reduces the variability of the average loss per policy; it does not remove risk from the whole pool.
• A risk-averse person has a concave utility function and prefers a certain wealth to a gamble with the same expected value.
• With full insurance at a fair premium, a risk-averse person is better off than without insurance.
• The maximum premium a person will pay is the one that gives the same utility as the uninsured gamble, and it exceeds the expected loss.
• Moral hazard is a change in behaviour after cover is bought; adverse selection is higher-risk people buying cover more often.
• Deductibles, co-payments and policy limits reduce moral hazard; underwriting and risk-based pricing reduce adverse selection.
• Reinsurance lets an insurer pass part of its risk to another insurer, reducing its own variability and capital need.
• Always state your assumptions, such as independence and the form of the utility function.

Common mistakes

• Saying insurers cover any risk if the premium is high enough. Fix: Say that a high premium cannot fix lack of data, catastrophic correlation or severe moral hazard.
• Treating pure and speculative as the same as diversifiable and systematic. Fix: Remember they are separate classifications. Pure/speculative is about possible outcomes. Diversifiable/systematic is about correlation across exposures.
• Saying insurance removes risk altogether. Fix: Say risk is transferred to the insurer and reduced per policy by pooling. Total risk in the economy remains.
• Claiming a risk-averse person will pay only up to the expected loss. Fix: Remember P_max > E[X] for a strictly risk-averse person. The fair premium E[X] is the minimum an insurer breaking even needs, before loadings.
• Saying pooling reduces the total variance of claims. Fix: Var(S) = nσ² rises with n. Only Var(S ÷ n) and the coefficient of variation fall.
• Writing Var(S ÷ n) = σ² ÷ √n or SD = σ ÷ n. Fix: Variance is σ² ÷ n. Standard deviation is σ ÷ √n. Check which one you are writing.
• Comparing U(E[W]) with U(w − P) instead of E[U(W)] with U(w − P). Fix: Apply U to each outcome first, then take the expectation. Expectation goes outside U.
• Writing the maximum premium as E[U(w − X)] instead of solving for P. Fix: E[U] is in utility units. Invert U to convert it back to wealth, then subtract from w.
• Mixing up moral hazard and adverse selection. Fix: Use timing. Hidden type before purchase is adverse selection. Changed behaviour after purchase is moral hazard.
• Saying deductibles solve adverse selection. Fix: Deductibles mainly address moral hazard. They can also help sort risk types, since high risks prefer low deductibles, but state that as a secondary point.

Exam tips

• In written answers, always apply each insurable feature to the scenario. A bare list earns few marks.
• Show the variance formula for the average loss when numbers are given, and state the independence assumption.
• Keep the two classification pairs separate and use both in your answer: pure or speculative, then diversifiable or systematic.
• For MCQs on systematic risk, look for events hitting many policyholders at once, such as pandemics, recessions or widespread floods.
• End scenario answers with a clear conclusion and one practical remedy such as reinsurance, deductibles or exclusions.
• Write definitions in one line, then explain the mechanism. Examiners reward the link between risk aversion, pooling and premium loading.
• In utility questions, define final wealth in each state before applying U, and show the check that P_max exceeds the expected loss.
• When you mention pooling, state the assumptions: independent, similar risks and a large number of policies.