Introduction

🔁 Where We Came From

So far everything has been certain: the consumer knew the prices, the firm knew its costs.

But almost no financial decision works like that. Today uncertainty and hidden information enter the picture. 🎲

This is the session closest to what you will actually do in banking, insurance and risk management. 🛡️

🗺️ Today’s Map

  1. 🎲 Expected utility and risk aversion
  2. 💶 Certainty equivalent and risk premium
  3. 🌫️ Asymmetric information: adverse selection and moral hazard
  4. 🛡️ How insurance works
  5. 🧰 How insurers respond to asymmetric information

Part 1 · Choice Under Uncertainty

🎲 A Gamble

I offer you a choice:

(A) €52 thousand for certain.

(B) A coin toss: €4 thousand on tails, €100 thousand on heads.

The two are worth the same on average. Most people choose (A). Why? 🤔

🧮 Expected Utility

The answer: people do not maximize expected value, they maximize expected utility.

\[ E[u(w)] = \sum_i p_i \, u(w_i) \]

Each outcome \(w_i\) occurs with probability \(p_i\), and what counts is the utility it delivers, not the euro itself.

📉 Risk Aversion Is Concavity

If \(u\) is concave, each additional euro is worth less than the one before (\(u'' < 0\)).

By Jensen’s inequality, for any nondegenerate gamble: \[ E[u(w)] < u(E[w]). \]

Risk aversion: preferring the expected value for certain over the gamble. It is exactly the concavity of \(u\). 🎯

📊 Seeing Risk Aversion

The chord (the gamble) lies below the curve (certainty). That is concavity.

💶 Certainty Equivalent and Risk Premium

Certainty equivalent (CE): the sure amount that gives the same utility as the gamble, \(u(CE) = E[u(w)]\).

Risk premium: \(E[w] - CE\). It is what you are willing to pay not to bear the risk.

And this is exactly where willingness to pay for insurance comes from. 🛡️

📏 How Risk Averse, Exactly?

“Concave” is qualitative. The standard quantitative measure is Arrow-Pratt:

\[A(w) = -\frac{u''(w)}{u'(w)}\]

Why the ratio?

Because \(u\) is ordinal up to an increasing affine transformation, and \(u''\) alone is not invariant. Dividing by \(u'\) makes \(A(w)\) invariant, so it measures the curvature and not the scale.

A second order expansion around \(E[w]\) gives the risk premium directly: \[\pi \approx \tfrac{1}{2}\,A(w)\,\sigma^{2}\]

So the price of risk is curvature times variance. It is why variance is the risk measure you will meet everywhere in finance. 📊

📈 Diversification

If risk aversion is expensive, how can we reduce risk without giving up return?

Two assets with the same mean and variance \(\sigma^{2}\), held half and half. If \(\operatorname{Cov}(X_1, X_2) = 0\): \[ \operatorname{Var}\!\left(\tfrac{1}{2}X_1 + \tfrac{1}{2}X_2\right) = \tfrac{1}{4}\sigma^2 + \tfrac{1}{4}\sigma^2 + 2 \cdot \tfrac{1}{4}\underbrace{\operatorname{Cov}}_{= \,0} = \tfrac{\sigma^2}{2}. \]

The same expected return, half the variance. With \(n\) independent assets, \(\sigma^{2}/n \to 0\). 📊

⚠️ The Covariance Term Is the Whole Story

Independence was doing all the work above. In general, for \(n\) equally weighted assets:

\[\operatorname{Var}(\bar{X}) = \frac{1}{n}\,\overline{\sigma^{2}} + \frac{n-1}{n}\,\overline{\operatorname{Cov}}\]

As \(n \to \infty\) the first term vanishes and the second converges to \(\overline{\operatorname{Cov}}\). Individual variance can be diversified away; average covariance cannot.

That residual is systematic risk: what remains when everything moves together. It is why a portfolio of 500 European bank stocks is still a bet on European banking.

Diversification is therefore not “hold many things”. It is “hold things that do not move together”. 🎯

📝 Review · Part 1

Two multiple choice questions and one exercise. ✍️

❓ Multiple Choice 1

An agent with \(u(w) = \sqrt{w}\) is offered a fair gamble. They:

A. Accept it, because the expected value is unchanged.

B. Are indifferent, because \(E[w]\) is the same either way.

C. Refuse it, because \(\sqrt{\cdot}\) is concave and Jensen gives \(E[u(w)] < u(E[w])\).

D. Cannot decide without knowing their initial wealth.

✅ C. A fair gamble leaves the mean unchanged and adds variance, and a concave \(u\) dislikes variance. Note that D is wrong here for a specific reason: the sign of the answer does not depend on wealth, only its size does.

❓ Multiple Choice 2

The risk premium is:

A. The difference between the expected value and the certainty equivalent.

B. The price charged by the insurance company.

C. Always equal to the expected loss.

D. Zero for any risk averse agent.

✅ A. It is \(E[w] - CE\): how much the agent will give up, on average, to be rid of the risk. It is what makes an insurer’s margin possible, but it is not the premium itself.

🧮 Numerical Exercise

Utility \(u(w) = \sqrt{w}\), and a 50/50 gamble between 4 and 100 thousand euros.

a) Find \(E[w]\) and \(E[u(w)]\).
b) Find the certainty equivalent.
c) Find the risk premium.
d) Compute \(A(w)\) at \(w = E[w]\) and compare \(\tfrac{1}{2}A\sigma^{2}\) with your answer to (c).

✅ Solution

a) \(E[w] = 0.5(4) + 0.5(100) = 52\), and \(E[u] = 0.5(2) + 0.5(10) = 6\) thousand euros.

b) \(\sqrt{CE} = 6\), so \(CE = 36\) thousand euros.

c) Risk premium \(= 52 - 36 = 16\) thousand euros. Almost a third of the expected value: this is a very large gamble.

d) \(u' = \tfrac{1}{2}w^{-1/2}\) and \(u'' = -\tfrac{1}{4}w^{-3/2}\), so \(A(w) = 1/(2w) = 1/104\). With \(\sigma^{2} = 48^{2} = 2304\): \(\tfrac{1}{2}(2304)/104 \approx 11\). The right order of magnitude, and short of 16 because the approximation is local and this gamble is anything but. ✅

Part 2 · Asymmetric Information

🌫️ Imperfect Information

We rarely decide with perfect information.

You buy a used phone without knowing the real state of the battery.

Asymmetric information: when one side of the transaction knows more than the other.

🍋 The Used Car Market

A classic example (Akerlof). There are good cars and “lemons” (bad ones). Only the seller knows which is which.

The buyer, not knowing, will only pay an average price.

At that average price, whoever has a good car will not sell (it is worth more). Only the lemons remain. 🍋

📉 Adverse Selection

Adverse selection: when hidden information causes the “worst” types to dominate the market.

The average price drives out the good ones, which lowers quality and price further still. The market can collapse.

Hold on to this idea: it is exactly the central problem of insurance. 🔁

🎭 Moral Hazard

A different problem, arising after the contract is signed.

Moral hazard: when someone shielded from a consequence changes their behavior and takes on more risk.

With your car fully insured against theft, you might leave it unlocked more often. 🚗

🔀 Adverse Selection vs Moral Hazard

When it occurs The problem
Adverse selection before the contract you hide your type
Moral hazard after the contract you change your behavior

Both arise from asymmetric information, but at different moments. ⏱️

🎓 Signaling

How do you solve hidden information? One side can signal its quality.

A degree signals competence. A warranty signals that the car is good.

The signal only works if it is expensive to imitate for whoever lacks the quality. 🎯

📝 Review · Part 2

Two multiple choice questions and one conceptual exercise. ✍️

❓ Multiple Choice 3

An insurer offers one policy at a single price to everyone. Low risk clients leave, so the insurer raises the price, so more clients leave. This is:

A. Moral hazard, since behavior changed after the contract.

B. Risk pooling working exactly as intended.

C. A signaling equilibrium.

D. Adverse selection, because the price cannot separate hidden types.

✅ D. A single price averages over types the insurer cannot observe. It is too expensive for the good risks and a bargain for the bad ones, so the pool deteriorates at every round. Insurers call it the death spiral.

❓ Multiple Choice 4

A risk averse agent is offered insurance at an actuarially fair premium (premium equals expected loss). They will buy:

A. No insurance, since the premium equals the expected loss and there is no gain.

B. Full insurance, eliminating all the risk.

C. Partial cover, to balance premium against risk.

D. An amount depending on how large the loss is.

✅ B. At a fair price, buying cover has zero expected cost and strictly cuts variance, so a concave \(u\) takes all of it. Partial cover (C) is what happens once the premium carries a loading above fair, which is the real world case.

🧮 Applied Exercise

At a dealership, half the cars are worth 10,000 euros (good) and half 4,000 euros (lemons). Buyers cannot tell them apart, and are risk neutral.

a) What is the most a buyer will pay?
b) What happens to the owners of good cars?
c) What is the new market price once they have left?
d) Name one institution that stops this happening in practice.

✅ Solution

a) \(0.5(10{,}000) + 0.5(4{,}000) = 7{,}000\) euros.

b) At 7,000 they will not sell: the car is worth more to them than the market offers. They withdraw.

c) Only lemons are left, so buyers work that out and pay 4,000. The good half of the market has disappeared, and nobody behaved irrationally at any step. 🍋

d) A warranty, an independent inspection, a dealer’s reputation, a manufacturer certification. Each one is a signal that is costly to fake, which is why it works. ✅

Part 3 · Insurance

🛡️ What Insurance Is

Insurance trades an uncertain, large loss for a certain, small cost: the premium.

You pay €300 a year so as not to risk a €20,000 loss in an accident.

Why are you willing to do this?

Because you are risk averse: you prefer certainty. 🎯

🎲 The Idea of Risk Pooling

The insurer brings many clients together. Individually, the risk is unpredictable.

But across thousands of policies, the fraction of claims is stable and predictable.

Risk pooling: bringing together many independent risks makes the total predictable.

📊 The Law of Large Numbers

With more policies, the fraction of claims approaches the true value (10%). The red line is the true risk.

💶 How the Premium Is Set

The premium covers the expected loss plus costs and margin.

If the probability of a claim is 2% and the loss is €20,000, the expected loss is 0.02 × 20,000 = €400.

The premium will sit a little above €400, to cover administrative costs and margin. 🧮

⚠️ The Two Enemies of Insurance

The problems from Part 1 return, now from the insurer’s side:

🍋 Adverse selection: whoever knows they are high risk is the one who most wants coverage.

🎭 Moral hazard: whoever is insured may become less careful.

🧰 How Insurers Respond

🔍 Screening: medical questionnaires, driving history, age of the car.

💶 Deductibles: the insured pays part of the loss, which keeps them careful (this fights moral hazard).

📊 Differentiated premiums: prices by risk profile, to fight adverse selection.

🏦 Why This Matters for You

Managing asymmetric information is the heart of the insurance and banking business.

Assessing credit risk, setting premiums, designing deductibles: all of that is applying what we saw today. 💼

📝 Review · Part 3

Two multiple choice questions and one exercise. ✍️

❓ Multiple Choice 5

An insurer covers a region against flooding. Risk pooling works poorly here because:

A. The losses are too large to insure at any price.

B. Policyholders will behave less carefully once covered.

C. The risks are strongly correlated, so the covariance term does not vanish.

D. The law of large numbers does not apply to natural events.

✅ C. Pooling needs risks that are close to independent. When one flood hits every policy at once, \(\overline{\operatorname{Cov}}\) stays large no matter how many policies are written. This is exactly the systematic risk term from Part 1, and it is why catastrophe cover is reinsured or state backed.

❓ Multiple Choice 6

A deductible, where the insured pays the first part of any claim, works mainly by:

A. Leaving the insured exposed enough to stay careful.

B. Raising the insurer’s revenue per policy.

C. Removing the need to price by risk profile.

D. Making the policy compulsory.

✅ A. It targets moral hazard, by keeping some of the loss with the person whose behavior affects it. Screening and risk-based pricing are the tools aimed at adverse selection instead.

🧮 Numerical Exercise

The probability of a claim is 5 percent and the loss is 30,000 euros.

a) Compute the expected loss.
b) The insurer loads 20 percent for costs and margin. What is the premium?
c) A client with \(u(w) = \sqrt{w}\) and wealth 100,000 euros faces this risk. What is the most they would pay for full cover?
d) Do they buy the policy at the price in (b)?

✅ Solution

a) \(0.05 \times 30{,}000 = 1{,}500\) euros.

b) \(1{,}500 \times 1.20 = 1{,}800\) euros. The 300 euros above fair is the loading.

c) \(E[u] = 0.95\sqrt{100{,}000} + 0.05\sqrt{70{,}000} \approx 313.65\), so \(CE \approx 313.65^{2} \approx 98{,}373\). The most they would pay is \(100{,}000 - CE \approx 1{,}627\) euros.

d) No. 1,800 is above their 1,627 ceiling. But at the fair premium of 1,500 they would buy, and buy in full, which is question 4 in numbers: the loading is what makes cover refusable. ✅

Wrap-Up

🎯 What to Take From This Session

🌫️ Asymmetric information generates adverse selection (before) and moral hazard (after the contract).

🛡️ Insurance trades an uncertain loss for a certain cost, resting on risk pooling.

🧰 Insurers fight asymmetric information with screening, deductibles and differentiated premiums.

👋 End of Block 1

Next comes the Midterm Test, and after that we move into macroeconomics: GDP, inflation, money and economic policy. 🌍

Good luck with the studying. 🙌