Week 3 - Utility Theory

• What is the nature of future wealth in a risky environment?

  It is a random variable
• Do investors only care about expected wealth?

  No, they care about other characteristics
• What is the first action in Euler's game, the St Petersburg Paradox?

  Toss a coin until it shows heads
• How much would you pay to enter the St Petersburg game?

  Depends on the number of throws
• What is the payoff if a head does not appear until the nth throw?

  $2^{(n-1)}$
• What is the expected payoff from the St Petersburg game?

  It is infinitely large
• How is expected return calculated in the St Petersburg game?

  By summing payoffs multiplied by probabilities
• What does the utility function represent?

  Preferences of an individual or group
• What does the utility function not quantify?

  Amount of utility from consumption
• How can we learn about an individual's utility function?

  By observing their choices
• What is the utility of having £100 with the given utility function?

  20 (utils)
• What is the expected value of the lottery with given pay-offs?

  £1,705
• What is the expected utility of the lottery?

  29.12
• What do we prefer if the lottery costs £1,000 to enter?

  £1,000 over the lottery
• What is the certainty equivalent of the lottery?

  £848
• What is the utility of the lottery under the new utility function?

  112,406
• What is the certainty equivalent under the new utility function?

  £2,329
• What characterizes a risk-averse investor?

  Diminishing marginal utility of wealth
• What characterizes a risk-loving investor?

  Increasing marginal utility of wealth
• What defines a fair gamble?

  Expected wealth equals available choice
• What will a risk-averse individual do with a fair gamble?

  Never accept it
• What will a risk-loving individual do with a fair gamble?

  Always accept it
• What is the expected value of the lottery with a £2,000 win?

  £1,000
• What probability makes an individual indifferent between a lottery and a certain payoff?

  $p =$$0.707$
• What does an affine transformation of the utility function do?

  Does not alter preference ordering
• What are the axioms of rational choice?

  Comparability, transitivity, independence
• What is an indifference curve?

  Locus of all indifferent assets
• How is variance calculated for a random variable?

  Using expected values of outcomes
• What is the risk-averse utility function expressed as?

  $U(R) =$$200r_x - 100r_x^2$
• What is the expected utility of asset X?

  18
• What does it mean if two assets lie on the same indifference curve?

  They yield the same expected utility
• What is the expected utility of asset Z?

  20
• What does a higher utility curve indicate?

  Higher utility than lower curves
• What does increasing risk aversion imply?

  Preference for lower risk investments