Risk Aversion and Insurance Demand: Utility, Certainty Equivalent and Risk Premium

Understand Risk Aversion and Insurance Demand

Key rules to remember

Expected utility

E[U(W)] = Σ p_i × U(w_i)

Use the probabilities of each wealth outcome. Wealth is final wealth, not just the gain or loss.

Risk attitude by curvature

U''(w) < 0: risk-averse; U''(w) = 0: risk-neutral; U''(w) > 0: risk-seeking

Assume U'(w) > 0 (more wealth is preferred).

Jensen's inequality for risk aversion

E[U(W)] < U(E[W]) for concave U and non-constant W

Reverses for convex U. Equality holds for linear U.

Certainty equivalent

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

Solve by inverting U. For U = ln w, CE = exp(E[ln W]).

Risk premium

π = E[W] − CE

Positive for risk-averse, zero for risk-neutral, negative for risk-seeking.

Maximum insurance premium

U(w − P) = E[U(w − X)], so P_max = w − CE = E[X] + π

X is the random loss and w is initial wealth. P_max exceeds E[X] for a risk-averse person.

Absolute risk aversion (Arrow-Pratt)

A(w) = −U''(w) ÷ U'(w)

Larger A means more risk-averse at wealth w.

Relative risk aversion

R(w) = w × A(w) = −w U''(w) ÷ U'(w)

Measures aversion to proportional risk.

Approximate risk premium

π ≈ ½ × A(w) × Var(W)

An approximation for small risks around mean wealth w. Not exact.

How to solve Risk Aversion and Insurance Demand questions

1. 1Write down the utility function U(w) and check U'(w) > 0.
2. 2Find U''(w). Its sign gives the risk attitude. Compute A(w) = −U''/U' if asked.
3. 3List the possible final wealth outcomes and their probabilities. Include initial wealth in each outcome.
4. 4Compute E[W] and E[U(W)] = Σ p × U(w).
5. 5Find the certainty equivalent by solving U(CE) = E[U(W)].
6. 6Compute the risk premium π = E[W] − CE.
7. 7For insurance, set U(w − P) = E[U(W without insurance)] and solve for the maximum premium P. Compare it with the actual premium and the expected loss.
8. 8State the conclusion in words: buy or not, and why.

Quickest way: Shortcut with inverse utility

Common mistakes in Risk Aversion and Insurance Demand

• Taking U(E[W]) as the expected utility.

  The notation looks similar and averaging wealth first feels natural.

  Fix: Compute U for each outcome first, then weight by probability. Remember E[U(W)] ≠ U(E[W]) unless U is linear.

• Confusing certainty equivalent with risk premium.

  Both come from the same calculation and both are amounts of money.

  Fix: CE is a level of sure wealth. The risk premium is E[W] − CE, the amount given up. Always label which one you have found.

• Using the gain or loss instead of final wealth in U.

  The question lists a gamble of ±₹ amounts.

  Fix: Add the gain or loss to initial wealth before applying U. For log or square root utility this changes the answer.

• Deciding risk attitude from U' instead of U''.

  Students see an increasing function and call it risk-seeking.

  Fix: Every sensible U is increasing. Only the sign of U'' tells you the attitude.

• Forgetting the sign convention in A(w).

  The minus sign is dropped, giving negative A for risk-averse people.

  Fix: A(w) = −U''/U'. For a concave U it is positive.

• Saying a risk-averse person will pay any premium for full cover.

  The idea of preferring certainty is overextended.

  Fix: The person buys only if the premium is at most P_max = E[X] + π. Above that, they prefer to stay uninsured.

Worked examples