Lecture 5

Question 1

How is uncertainty modelled? What is it?

Answer 1

It is modelled through a state space, S. it is a set of states where only one is true but it is uncertain which

Question 2

Special name for states under uncertainty and why?

Answer 2

States of nature since the decision maker has no influence on which of the states is true

Question 3

What are events?

Answer 3

Sunsets of the state space

Question 4

What is E^c?

Answer 4

The complementary event to E (the negation of E)

Question 5

How are outcomes denoted?

Answer 5

Either Greek letters or x’s

Question 6

How is each state denoted?

Answer 6

Each state of S is denoted s1, s2….

Question 7

See rest of notation in notes

Answer 7

Now

Question 8

What are prospects?

Answer 8

Prospects designate state-contingent outcomes - they are courses of action for the decision maker

Question 9

Define decision under uncertainty?

Answer 9

In decision under uncertainty, S is a finite or infinite state space, and R is the outcome set. Prospects map states to outcomes. The domain of preference is the set of all prospects. >= is a preference relation on the set of prospects

Question 10

What is a certainty equivalent?

Answer 10

A certainty equivalent of a prospect x is an outcome α such that α~x

Question 11

Explain first way to make decisions under uncertainty?

Answer 11

By choosing probabilities for uncertain events (subjective) and then maximising EV wrt those probabilities

Question 12

What is an EV maximiser? And what if not?

Answer 12

If your preference towards all decisions can be represented by EV we say you are an EV maximiser

If not, we say EV doesn’t hold for you

Question 13

With EV maximin method what are the subjective parameters?

Answer 13

The subjective probabilities - they characterise the decision maker

Question 14

See

Answer 14

Slides 19-24

Question 15

See

Answer 15

Top of 2nd side of notes on steps 1-3

Question 16

What is the axiom of additivity?

Answer 16

[x>=y -> x+z>=y+z] for all prospects x, y and z

Question 17

Define arbitrage/a Dutch book?

Answer 17

When preferences x(j)>=y(j) for j=1,…,m, such that the preferred prospects x(j), when COMBINED, always (for each state S) yield less than the non-preferred prospects y(j)

See notes

Question 18

Define risk neutral?

Answer 18

When a party is neither risk averse or risk seeking. Tf given 2 events with equal EVs, they will be indifferent even if one is much riskier!

Question 19

Fundamental point in finance?

Answer 19

Idea that no arbitrage implies an as-if risk neutral (ie. EV) evaluation of financial assets

Question 20

Define de finetti’s theorem?

Answer 20

The person’s preferences follow the preference relation >= as a weak order; for each prospect there is a CE, and additivity, transitivity and monotonicity all hold

Question 21

What does de finetti’s theorem indicate?

Answer 21

That the EV maximiser will not be arbitraged

Question 22

Read

Answer 22

In finance, arbitrage means that you can combine a number of portfolios in such a way that you always (in every state of the world) make a profit. In such a case, you can make money from the market without contributing anything, which you will of course do as much as possible. In practice, arbitrage opportunities disappear within fractions of seconds because they are immediately exploited by the largest and quickest market participants, i.e. investment banks that use automated computer programs. A fundamental result in finance is that no-arbitrage implies an as-if risk neutral (= EV) evaluation of financial assets. The same result was discovered before by de Finetti for individual choice, where arbitrage is called a Dutch book.

Question 23

A person maximises EV is the same as saying?

Answer 23

The person’s preferences follow the preference relation ≽as a weak order; for each prospect there exists a certainty equivalent; and additivity, transitivityand monotonicity are satisfied.

ie. De Finetti’s Theorem!

Question 24

What does a dutch book show?

Answer 24

It shows that if someone is not an EV maximiser (eg. they follow a maximin rule) then arbitrage can be made against them. This arbitrage is what a Dutch Book shows!