Economics Tutor Geneva Lausanne Zurich and Zug regions- Switzerland

Insurance looks like a losing bet. You pay a premium every year, and a fair insurer pays back, on average, a little less than it collects. So why are you better off buying it? Because you do not value wealth in a straight line — and that one fact is the whole of this topic, the model second-years most want to pin down with an economics tutor in Switzerland.

1 · Expected value is not expected utility

There are two ways to score a risky prospect. Its expected value is the probability-weighted average of the money outcomes. Its expected utility is the probability-weighted average of the utility those outcomes deliver — and that is the one that predicts choice. The St Petersburg gamble makes the point: a pot that doubles until a coin lands heads has infinite expected value, yet nobody stakes more than a few pounds on it. You rank utility, not money — and it rises ever more slowly with wealth. Score a lottery paying w_L with probability p and w_H otherwise by its expected utility:

2 · Concavity is risk aversion

Take a concrete utility function, u(w) = √w. The first pound buys plenty of utility; the ten-thousandth buys almost none. Marginal utility falls, so the curve rises and flattens — it is concave, and that has a sharp consequence. Compare the utility of average wealth, u(E[w]), with the average of the utilities, E[u(w)]. For a concave curve the first always wins:

This is Jensen’s inequality — the curve sitting above its own chord. Preferring the certain average to the gamble that only averages it is risk aversion. Risk-neutral (straight-line) utility makes the two sides equal; concavity makes risk costly.

3 · The certainty equivalent and the risk premium

Now price that cost. The certainty equivalent CE is the guaranteed wealth that leaves you exactly as well off as the gamble — the wealth whose utility equals the gamble’s expected utility:

The inversion is special to u = √w: wealth is utility squared, so squaring the expected utility reads CE straight off. Because the curve sits above its chord, CE falls below E[w], and the gap is the risk premium:

risk premium = E[w] − CE

It is the most a risk-averse person pays to shed the risk. The chord below is the whole geometry.

4 · Why this makes insurance mutually profitable

An actuarially fair premium equals the insurer’s expected payout — the expected loss, nothing added. Charged that, a risk-averse buyer insures — and pays strictly more: her ceiling is the fair premium plus her own risk premium, since even then her certain wealth equals her certainty equivalent.

A large insurer holds thousands of roughly independent policies, so by the law of large numbers its average payout is nearly certain; it acts as if risk-neutral and prices near fair. The buyer will go up to fair-plus-risk-premium; the insurer is content at fair; every premium in that band leaves both ahead. That overlap — the risk premium — is why insurance is positive-sum, not zero-sum.

At a fair premium the buyer takes full cover; loaded above fair, she balances loading against risk and buys partial cover with a deductible. Three topics sit next door. Pooling independent losses is not portfolio diversification, whose leftover systematic risk is priced on the security market line. The full-insurance result is formally a Lagrangian optimisation. And two frictions — moral hazard (a hidden action) and adverse selection (a hidden type) — have their own lessons.

Worked example — a stylised 50–50 risk

You hold £400. A mishap would cost £300, and it strikes on a coin toss: probability ½ you keep £400, probability ½ you are left with £100. Utility is u(w) = √w.

Step 1 — Expected wealth. E[w] = ½(400) + ½(100) = £250.

Step 2 — Expected utility. E[u(w)] = ½√400 + ½√100 = ½(20) + ½(10) = 15. Beside it, the utility of the average is u(250) = √250 ≈ 15.81. Concavity gives 15.81 > 15: the sure £250 beats the gamble — Jensen’s inequality in numbers.

Step 3 — Certainty equivalent and risk premium. Square the expected utility: CE = 15² = £225. You would swap the gamble for £225 certain, though it averages £250. The gap, 250 − 225 = £25, is your risk premium.

Step 4 — Turn it into insurance. An insurer covers the £300 loss for a premium π paid up front, making your wealth £(400 − π) for certain. You buy while √(400 − π) ≥ 15, i.e. 400 − π ≥ 225, so π_max = £175.

Step 5 — Split the ceiling. The fair premium is the expected payout, ½ × £300 = £150. Your £175 ceiling is exactly that plus your £25 risk premium — because £175 = £400 − CE.

Step 6 — Read the trade. The insurer prices near £150; you pay up to £175; any premium in between helps both, and the band’s width is your £25 risk premium. A risk-neutral person (CE = £250) would not pay a penny above £150.

Given a utility function and a gamble, could you find the certainty equivalent and price the risk premium — and say when insurance is worth buying? That chain, from Jensen’s inequality to the premium ceiling, is exactly what risk questions reward. A one-on-one economics tutor works the concave-curve geometry with you until it comes out cleanly under exam pressure. Book a trial session.

Practice

Q1. Same utility and the same £400, but the £300 loss now strikes with probability 0.1. Find E[w], E[u(w)], the certainty equivalent, the risk premium and the fair premium. What is the most you would pay for full cover, and would you buy at £42?

Q2. A risk-averse student (u = √w) faces a 50–50 gamble between £196 and £324. A second gamble keeps the same mean but widens the outcomes to £36 or £484 — a mean-preserving spread. Find each risk premium. What does the spread do?

Answers. Q1: E[w] = 0.9(400) + 0.1(100) = £370; E[u] = 0.9(20) + 0.1(10) = 19; CE = 19² = £361; risk premium = 370 − 361 = £9; fair premium = 0.1 × 300 = £30. The most you would pay is 400 − 361 = £39 = fair (£30) + risk premium (£9). At £42 you decline — the £12 loading exceeds the £9 of risk you wanted rid of. Q2: the narrow gamble averages £260 with CE = 256, a risk premium of £4; the spread still averages £260 but CE = 196, a risk premium of £64. Same mean, wider outcomes: the premium jumps £4 → £64. A mean-preserving spread always raises it.

Key takeaways

• Rank by expected utility, not expected value. E[u(w)] predicts choice under risk; E[w] does not.
• Concavity is risk aversion. It puts the curve above its chord, so u(E[w]) > E[u(w)] — Jensen’s inequality.
• The certainty equivalent prices the risk. CE solves u(CE) = E[u(w)]; the risk premium E[w] − CE is what you would pay to be rid of the gamble.
• Insurance is positive-sum. A risk-averse buyer pays up to fair-plus-risk-premium; a diversified insurer prices near fair. The overlap — the risk premium — is the mutual gain.

Why Swiss students choose our economics tutoring

• Models rebuilt from the utility function, not memorised: you derive the certainty equivalent and risk premium from u(w) itself, so you can reconstruct the picture under exam pressure instead of quoting a half-remembered formula.
• The distinctions examiners reward, drilled: expected value versus expected utility, fair versus loaded premiums, full versus partial cover, and the two information frictions — the differences that separate a first from a 2:1.
• In person across the region, or online: sessions run in Geneva, Lausanne, Zurich and Zug, or online wherever your course is taught.
• One-on-one, matched to your syllabus: tutors work from your own notation and past papers, whether your micro module follows Varian, Nicholson and Snyder, or Mas-Colell.

FAQ

Q: What is the difference between expected value and expected utility?
A: Expected value averages the money outcomes; expected utility averages the satisfaction they give, through a utility function u(w). A risk-averse person ranks prospects by expected utility, so two gambles with equal expected value can rank differently.

Q: What is the certainty equivalent?
A: The guaranteed wealth that makes you exactly as well off as the gamble — it solves u(CE) = E[u(w)]. For a risk-averse person it lies below the gamble’s expected value, and the shortfall is the risk premium.

Q: Why does the risk premium matter?
A: It measures, in pounds, what the risk itself costs you — the most you would pay above a fair premium to insure it away.

Q: If insurance costs more than the expected loss, why buy it?
A: Because certainty has value. While the premium stays below the fair premium plus your risk premium, the peace of mind outweighs the markup.

Q: Does more risk always mean a bigger premium?
A: Holding the average outcome fixed, yes. A mean-preserving spread — same mean, wider outcomes — lowers the certainty equivalent and raises the risk premium.

Book an economics tutor in Switzerland or online

Expected utility rewards the student who can build the picture — the concave curve, the chord, the certainty equivalent, the risk premium — not just recite a definition. One-on-one sessions drill the derivations, the examiner-tested distinctions and the insurance logic that follows. Tell us your module and exam date, and we will match you with the right tutor this week.