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Principles of Finance Grzegorz Trojanowski Lecture 2: Theory of investor’s choice October 12, 2004 Principles of Finance - Lecture 2 1 Lecture 2 material • Required reading: Elton et al., Chapters 1, 10 • Supplementary reading: Sharpe et al., Chapter 6 Any intermediate or advanced microeconomics textbook October 12, 2004 Principles of Finance - Lecture 2 2 1 Lecture 2: Checklist • By the end of this lecture you should: Be familiar with the notions of preference relation, utility function, certainty equivalent, and risk premium Be familiar with different measures of risk aversion Be able to compute risk premium coefficients for a given utility function Be familiar with the properties of the most commonly used utility functions Be aware of the corresponding empirical evidence October 12, 2004 Principles of Finance - Lecture 2 3 Example: Choice under certainty (1) • Two periods: Year 1 and Year 2 • Investor receives an income of ₤ 10,000 each period (with certainty) • Interest rate for borrowing/lending: 5% • Questions: What are the options open to an investor? Which of them should be chosen? October 12, 2004 Principles of Finance - Lecture 2 4 2 Example: Choice under certainty (2) • Scenario A: Save all the income until Year 2; the amounts consumed equal to ₤ 0 in Year 1 and ₤ 20,500 (i.e. ₤ 10,000 + ₤ 10,000 · 1.05) in Year 2 • Scenario B: Consume the Year 1 income in Year 1 and the Year 2 income in Year 2; the amounts consumed equal to ₤ 10,000 in each period • Scenario C: Consume as much as possible in Year 1 and borrow against Year 2 income; the amounts consumed in Year 1 and 2 equal to ₤ 19,524 and ₤ 0, respectively 10,000 X + 0.05 X = 10,000 ⇒ X= = 9,524 1.05 October 12, 2004 Principles of Finance - Lecture 2 5 Example: Choice under certainty (3) • Intermediate scenarios also possible, e.g. the investor can consume just ₤ 5,000 in Year 1 and save the remainder of the Year 1 income until Year 2 • All the available options (the opportunity set) can be described by the following equation: ⎡Year 2 ⎤ ⎡Year 2 ⎤ ⎡Amount ⎤ ⎢consumption ⎥ = ⎢income⎥ + ⎢saved in Year 1⎥ (1 + 0.05) ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ C2 = £ 10,000 + (£ 10 ,000-C1 ) ⋅1.05 C2 = £ 20,500 + 1.05 ⋅ C1 October 12, 2004 Principles of Finance - Lecture 2 6 3 Example: Choice under certainty (4) October 12, 2004 Principles of Finance - Lecture 2 7 Example: Choice under certainty (5) • Investor is likely to have some preferences over the possible options (consumption bundles) • A useful tool for the analysis of preferences are the indifference curves October 12, 2004 Principles of Finance - Lecture 2 8 4 Example: Choice under certainty (6) October 12, 2004 Principles of Finance - Lecture 2 9 Example: Choice under certainty (7) October 12, 2004 Principles of Finance - Lecture 2 10 5 Preference relation • Consider a consumption bundle x ∈ {x1, x2, …, xn} from some feasible set, where the elements of x are sure things • For pairs (x, y): If x ≥ y, then x is weakly preferred to y Strict preference: x > y if x ≥ y but not y ≥ x Indifference: x ~ y, if x ≥ y and y ≥ x • A preference relation is a binary relation with the properties of completeness and transitivity October 12, 2004 Principles of Finance - Lecture 2 11 Preference relation (aside) • Completeness (or comparability) means that given any two bundles x and y, the agent is able to state whether x ≥ y, y ≥ x, or x ~ y • Transitivity means that the choices are consistent, i.e. if x ≥ y and y ≥ z, then x ≥ z October 12, 2004 Principles of Finance - Lecture 2 12 6 Ordinal utility functions • A preference relation ≥ can be represented by an ordinal utility function Φ: Φ(x) > Φ(y) ⇔ x > y Φ(x) = Φ(y) ⇔ x = y • The function Φ is not unique • It can only be used to rank bundles • It does not quantify exactly how much better one bundle is compared to another October 12, 2004 Principles of Finance - Lecture 2 13 Risk and preference • Suppose x is not a sure thing, but a lottery Example: The agent can receive £ 20, £ 12, or £ 4 with equal probability (i.e. ⅓ for each outcome) • How much is the lottery worth? • The certainty equivalent of a lottery (i.e. the certain payment which makes the agent indifferent between this payment and the lottery) depends on the agent’s preference towards risk • Certainty equivalent for such a lottery would be very small if the agent is very risk averse October 12, 2004 Principles of Finance - Lecture 2 14 7 State-contingent payoffs (1) • Let’s think of x as an asset with payoffs depending on the state of the world next period State ω1 ω2 … ωn Payoff x1 x2 … xn • Suppose that the probability for state ω1 is π1, the probability for state ω2 is π2, etc. • Then, x is like a lottery October 12, 2004 Principles of Finance - Lecture 2 15 State-contingent payoffs (2) • Can we define a preference relation on lotteries? • We can use ordinal utility function to rank these lotteries • However, if there are many possible states (sometimes infinitely many), x may be a large vector (sometimes of an infinite length) • We need a more convenient solution • Let’s construct a measure to replace ordinal ranking of states with cardinal ranking of certain prospects October 12, 2004 Principles of Finance - Lecture 2 16 8 Expected utility • Let U(·) be a cardinal utility measure on sure things n with the expected value Q = ∑ π iU ( xi ) i =1 • U(·) has the property that n n x ≥ y ⇔ ∑ π iU ( xi ) ≥ ∑ π iU ( yi ) i =1 i =1 • Q can be thought of as an expected utility function October 12, 2004 Principles of Finance - Lecture 2 17 Risk aversion • At a given wealth level, an investor is risk averse if and only if he refuses every fair and immediately resolved gamble which has consequences only for wealth • An investor who is risk averse at all relevant wealth levels is globally risk averse • An investor is risk averse if and only if his utility function is strictly concave • Let w be the wealth level. For risk averse investors Ε[U ( w)] < U [Ε( w)] October 12, 2004 Principles of Finance - Lecture 2 18 9 Markowitz risk premium • Let ε be a lottery where E(ε) = 0; ε is a fair gamble • Let U(·) be the utility function of an investor who is only concerned about his wealth level • The Markowitz risk premium is the solution πM to the equation Ε[U ( w + ε )] = U ( w − π M ) • It is the difference between the investor’s expected wealth (given the lottery) and the certainty equivalent wealth • Interpretation: It is the maximum the investor would pay to avoid the gamble October 12, 2004 Principles of Finance - Lecture 2 19 Pratt-Arrow risk premium (1) • For small and fair gambles ε, expand U(w+ε) as a Taylor series to obtain the Pratt-Arrow risk premium U (w + ε ) = = U ( w) + U ' ( w)[( w + ε ) − w] + U ' ' ( w)[( w + ε ) 2 − w] + K 1 2 • Take expectation on both sides and ignore higher order terms Ε[U ( w + ε )] ≈ Ε[U ( w)] + U ' ( w)Ε[ε ] + U ' ' ( w)Ε[ε 2 ] 1 2 • Note that U(w) is constant, E[ε] = 0, and Ε[ε 2 ] = σ ε2 • So 1 Ε[U ( w + ε )] ≈ U ( w) + U ' ' ( w)σ ε2 2 October 12, 2004 Principles of Finance - Lecture 2 20 10 Pratt-Arrow risk premium (2) • The Taylor series expansion for U(w - π ) is U ( w − π ) = U ( w) − πU ' ( w) + K • Using equation for E[U(w+ε)] we have 1 U (ε ) + U ' ' ( w)σ ε2 = U ( w) − πU ' ( w) 2 • Solving for π renders the Pratt-Arrow risk premium 1 U ' ' ( w) 2 π =− σε 2 U ' ( w) October 12, 2004 Principles of Finance - Lecture 2 21 Absolute risk aversion • The Pratt-Arrow coefficient of absolute risk aversion U ' ' ( w) (ARA) is defined as A( w) = − U ' ( w) • Sometimes, risk tolerance function is used instead 1 • Risk tolerance function is defined as T ( w) = A( w) October 12, 2004 Principles of Finance - Lecture 2 22 11 Absolute vs. relative risk aversion • ARA is a measure of local absolute risk aversion • The coefficient of relative risk aversion (RRA) measures aversion to gambles that are proportional to wealth levels R(w) = w · A(w) • The more risk averse the investor, the higher the risk premium required from the risky asset in order to persuade the investor all of his wealth in it October 12, 2004 Principles of Finance - Lecture 2 23 Risk aversion and wealth (1) • How does the investor preference change when his wealth increases? If the investor increases the amount of invested in risky assets as his wealth increases, he has a decreasing absolute risk aversion: A’(w) < 0 If the investor holds the same amount of money in risky assets as his wealth increases, he has a constant absolute risk aversion: A’(w) = 0 If the investor invests more money in risky assets as his wealth increases, he has an increasing absolute risk aversion: A’(w) > 0 October 12, 2004 Principles of Finance - Lecture 2 24 12 Risk aversion and wealth (2) • How does the percentage of wealth invested in risky assets change as the wealth increases? If the investor increases the percentage of wealth invested in risky assets, he has a decreasing relative risk aversion: R’(w) < 0 If the investor invests the same percentage of wealth in risky assets, he has a constant relative risk aversion: R’(w) = 0 If the investor decreases the percentage of wealth invested in risky assets, he has an increasing relative risk aversion: R’(w) > 0 October 12, 2004 Principles of Finance - Lecture 2 25 Preferences and risk aversion: Empirical evidence • Sources of evidence: Experiments from simple choice situations Survey data on investor’s asset choices • Major findings: Investors prefer more to less Vast majority of individuals can be considered risk averse Investors are usually found to have decreasing absolute risk aversion The evidence on the coefficient of relative risk aversion appears inconclusive October 12, 2004 Principles of Finance - Lecture 2 26 13 Quadratic utility function (1) b 2 • Quadratic utility function is given by U ( w) = w − w 2 • The first derivative: U’(w) = 1 – bw • The second derivative: U’’(w) = -b b • Absolute risk aversion is A( w) = 1 − bw b2 • Thus, A' ( w) = >0 (1 − bw) 2 • The quadratic utility function exhibits increasing absolute risk aversion October 12, 2004 Principles of Finance - Lecture 2 27 Quadratic utility function (2) • The relative risk aversion coefficient equals to bw R ( w) = wA( w) = 1 − bw b • Thus, R ' ( w) = >0 (1 − bw) 2 • The quadratic utility function exhibits increasing relative risk aversion • The assumption of a quadratic utility function leads to mean variance analysis being optimum • However, if the investor is assumed to prefer more to less, the quadratic utility function is valid only for a restricted range of wealth October 12, 2004 Principles of Finance - Lecture 2 28 14 Log utility function (1) • The log utility function is given by U(w) = ln(w) 1 • The first derivative: U ' ( w) = w 1 • The second derivative: U ' ' ( w) = − w2 • An investor with a log utility function prefers more to less and is risk averse October 12, 2004 Principles of Finance - Lecture 2 29 Log utility function (2) 1 • Absolute risk aversion: A( w) = w 1 • Hence, A' ( w) = − w2 • The log utility function exhibits decreasing absolute risk aversion • The relative risk aversion coefficient equals R(w) = wA(w) = 1 • Therefore, R’(w) = 0 • The log utility function exhibits constant relative risk aversion October 12, 2004 Principles of Finance - Lecture 2 30 15 HARA utility function (1) • The hyperbolic absolute risk-aversion (HARA) or linear risk-tolerance (LRT) γ 1 − γ ⎛ αw ⎞ utility is given by: U ( w) = ⎜ +β⎟ γ ⎝1 − γ ⎠ αw • It is properly defined for +β >0 1− γ October 12, 2004 Principles of Finance - Lecture 2 31 HARA utility function (2) • For γ < 1 there is a lower bound on the domain • For γ > 1 there is an upper bound on the domain • The risk tolerance is linear in w: 1 w β T ( w) = = + A( w) 1 − γ α October 12, 2004 Principles of Finance - Lecture 2 32 16 HARA utility function (3) • For γ → 1, HARA reduces to a linear function (characterised by risk neutrality) • For γ = 2, one obtains quadratic utility function • For γ → -∞ and β = 1 we obtain (negative) exponential utility (with constant absolute risk aversion) • For γ < 1 and β = 0, we obtain power utility • For γ → 0 and β = 0, we obtain log utility October 12, 2004 Principles of Finance - Lecture 2 33 17 Principles of Finance Week 3: October 19, 2004 Tutorial problems Problem 1 • ECBG Chapter 1, Exercise 4, p. 9 Problem 2 • EGBG Chapter 1, Exercise 12, p. 10 Problem 3 • EGBG Chapter 10, Exercises 3 and 4, p. 229 Problem 4 • EGBG Chapter 10, Exercise 7, p. 229 Problem 5 • EGBG Chapter 10, Exercise 10-11, p. 229-230 Problem 6 (optional) HARA utility function is defined as: γ 1 − γ ⎛ αw ⎞ U ( w) = ⎜ +β⎟ γ ⎝1 − γ ⎠ where α, β, and γ are the parameters, while w is the level of wealth. As discussed during the lecture, this representation renders many commonly used utility functions as the special cases. In particular: • For γ → 1, HARA reduces to a linear function (characterised by risk neutrality) • For γ = 2, one obtains quadratic utility function • For γ → -∞ and β = 1 we obtain (negative) exponential utility (with constant absolute risk aversion) • For γ < 1 and β = 0, we obtain power utility • For γ → 0 and β = 0, we obtain log utility A. Prove that you can indeed obtain linear, quadratic, exponential, power, and log utility functions, respectively, for the values of parameters α, β, and γ given above. B. Derive the measures of ARA and RRA for each of those special cases.