Lecture 1 - Expected Utility and Risk Aversion

Question 1

Besides assuming that people are maximizing expected utility functions, we usually assume that their utilities make them risk averse.
▪ An investor’s attitude towards risk is characterised by his/her preference between a simple gamble 𝛿̃ with expected value 𝑔, denoted 𝛿̃𝑔, and a certain outcome being equal to 𝑔.

Knowing the following, please define the notations for risk averse, risk neutral and risk lover

Answer 1

Risk averse = 𝑔≻𝛿̃𝑔, ∀𝑔,𝛿̃𝑔.

Risk neutral = 𝑔∽𝛿̃𝑔, ∀𝑔,𝛿̃𝑔

Risk lover = 𝑔≺𝛿̃𝑔, ∀𝑔,𝛿̃𝑔

Question 2

Please explain this graph and also explain why a risk averse investor has concave utility curve, risk neutral - linear and risk lover - convex

Answer 2

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Question 3

What is the markowitz risk premium?

Answer 3

For risk averse investors, this is the maximum amount they would be willing to pay for insurance that gets the payoff rather than have to bear the risk of the gamble. This will be the difference between expected wealth and the level of certain wealth at which she is indifferent between it and the gamble (called the certainty equivalent wealth, CEW).

Question 4

If 𝑈 = √𝑊 and the gamble is (0, 100: 0.5, 0.5) what is the Markowitz risk premium?

Answer 4

It is the difference between Expected wealth from the gamble and the CEW of the gamble ⇒ (a) 𝐸(𝐺𝑎𝑚𝑝𝑙𝑒𝑝𝑎𝑦𝑜𝑢𝑡) = 𝐸(0, 100: 0.5, 0.5) = 50; (b)𝐸(𝑈(𝐺𝑎𝑚𝑏𝑙𝑒)) =
𝐸(𝑈(0, 100: 0.5, 0.5)) = 0.5 × √0 + 0.5 × √100 = 5.

At what certain wealth W is 𝑈(𝑊) = 5? ⇒ √𝑊 = 5 ⇒ 𝑊 = 25. So CEW is 25. This implies that Risk premium is 𝐸(𝐺𝑎𝑚𝑝𝑙𝑒𝑝𝑎𝑦𝑜𝑢𝑡) − 𝐶𝐸𝑊 = 50 − 25=25.

Question 5

Consider a utility function in the form U(W) = lnW. Is this utility function consistent with risk aversion and non-satiation (prefer more to less)?

Answer 5

A utility function satisfies non-satiation if its first derivative is postive, meaning that increasing wealth increases utility.

First derivative: U’(W) shows the gradient of the curve. U’(W) = 1/W. Since U(W) > 0 for all W>0, this is positive, meaning more wealth = more utility.

Second derivative: U’‘(W) = -W^-2 = -(1/W^2). Since U’‘(W) < 0 for W > 0, the function is concave meaning it represents a risk-averse individual.

Question 6

Please explain how the second derivative of the utility function U’’( ) explains risk aversion

Answer 6

Question 7

Please explain ARA(W) and define its formula

Answer 7

Question 8

Explain RRA(W) and provide its formula

Answer 8

Question 9

Please explain decreasing, constant and increasing risk aversion

Answer 9

Question 10

Please define the quadratic utility function and its first and second derivative. Please

Answer 10