Principles of Finance Lecture 5: Utility Theory Under Uncertainty PDF

Introduction Rational decision making when uncertainty exists Principles of Finance Lecture 5: Utility theory under uncertainty (CWS ch. 3) Rikke Sejer Ni...

Introduction Rational decision making when uncertainty exists Principles of Finance Lecture 5: Utility theory under uncertainty (CWS ch. 3) Rikke Sejer Nielsen 1 / 32  Introduction Rational decision making when uncertainty exists Uncertainty In earlier lectures: Consumption and investment decision under certainty The market interest rate is nonstochastic - known with certainty in all time periods (not necessarily constant!), All payoffs from current investment decisions are known with certainty. BUT rarely the case We don’t know the future market interest rate Future payoffs from current investment decisions are not known. ⇒ Uncertainty exists! 3 / 32  Introduction Rational decision making when uncertainty exists Uncertainty Today: The theory of rational decision making in case of uncertainty Theory of investor choice ▶ How we choose between timeless risky alternatives ▶ Behavior of individuals is central ∗ Degrees of risk aversion ∗ Time preferences of consumption. The object of choice ▶ The objects that the individual’s choice is based on. Later, the theory of optimal decision-making under uncertainty. 4 / 32  Introduction Rational decision making when uncertainty exists Indifference curves 5 / 32  Introduction Rational decision making when uncertainty exists Theory of investor choice The object of choice Axioms (assumptions) of Choice under Uncertainty Five Axioms of Choice under Uncertainty Axiom 1 Comparability (sometimes called completeness). ⇒ x ≻ y, y ≻ x, or x ∽ y Axiom 2 Transitivity (sometimes called consistency). ⇒ If x ≻ y and y ≻ z, then x ≻ z ⇒ If x ∼ y and y ∼ z, then x ∼ z Axiom 3 Strong independence. ⇒ Gamble G(x, z : α), where Prob(x) = α and Prob(z) = 1 − α ⇒ If x ∼ y, then G(x, z : α) ∼ G(y, z : α) Axiom 4 Measurability. ⇒ If x ≻ y ⪰ z or x ⪰ y ≻ z , then a unique α exists, such that y ∼ G(x, z : α) Axiom 5 Ranking. ⇒ If x ⪰ y ⪰ z and x ⪰ u ⪰ z, and y ∼ G(x, z : α1 ) and u ∼ G(x, z : α2 ), then y ≻ u if α1 > α2 , or y ∼ u if α1 = α2. If all axioms hold ⇒ consistent and rational behavior. 8 / 32  Introduction Rational decision making when uncertainty exists Theory of investor choice The object of choice Utility functions Mapping preferences for risky alternatives using measurable utilities Two properties of utility functions: 1 Order preserving: If x ≻ y, then U(x) > U(y ) Proof: From Axiom 4 & 5, interpret α(x) and α(y ) as numerical utilities that uniquely rank outcome x and y. 2 Combinations of risky alternatives can be ranked by expected utility U[G(x, y : α)] = αU(x) + (1 − α)U(y) Proof on the board! ⇒ Ranking function for risky alternatives is expected utility. 9 / 32  Introduction Rational decision making when uncertainty exists Theory of investor choice The object of choice Expected utility The objective function of a greedy investor (U ′ (W ) > 0) is to maximize expected utility of end-of-period wealth, defined as: S X E[U(W )] = pi U(Wi ), i where S is the entire set of uncertain alternatives, i refers to a specific uncertain alternative, and pi and Wi is the probability and end-of-period wealth of a specific uncertain alternative, respectively. 10 / 32  Introduction Rational decision making when uncertainty exists Theory of investor choice The object of choice Expected utility Note: Utility functions are specific to individuals Changes in utility/Marginal utility is comparable across individuals U1 (x) − U1 (y ) = constant U2 (x) − U2 (y ) Example on the board 11 / 32  Introduction Rational decision making when uncertainty exists Theory of investor choice The object of choice Expected utility Expected utility of end-of-period wealth: S X E[U(W )] = pi U(Wi ), i Thus, it reflects both Expected wealth at the end of the period Risk profile of the investor ⇒ Investor’s risk attitude towards variation in the end-of-period wealth 12 / 32  Introduction Rational decision making when uncertainty exists Theory of investor choice The object of choice Risk profile of the investor Three types of risk profiles: Risk averse: Investor dislikes risk and wants to be compensated for taking risk Risk neutral: Investor doesn’t care about risk, and hence decisions only based on expected end-of-period wealth Risk lover: Investor likes risk and wants to be compensated for avoiding risk 13 / 32  Introduction Rational decision making when uncertainty exists Theory of investor choice The object of choice Risk profile of the investor Example of a fair gamble: Assume no discounting, k = 0 Investor has initial wealth of 1,000 to invest or not: Invest Keep (Not invest) End-of-period wealth Prob. End-of-period wealth Prob. 2000 50% 1000 100% 0 50% - - ⇒ Expected end-of-period wealth = 1000 for both alternatives ⇒ BUT expected utility is not necessarily equal What would you choose? Invest or not? How do attitudes towards risk affect the investment choice? ⇒ Solved on the board. 14 / 32  Introduction Rational decision making when uncertainty exists Theory of investor choice The object of choice Risk profile of the investor by Markowitz (1959) Note that this is the framwwork of Markowitz (1959) Risk loving investor Risk neutral investor Risk averse investor U ′ (W ) ≥ 0 & U ′′ (W ) ≥ 0 U ′ (W ) ≥ 0 & U ′′ (W ) = 0 U ′ (W ) ≥ 0 & U ′′ (W ) ≤ 0 (convex utility function) (linear utility function) (concave utility function) U[E(W )] < E[U(W )] U[E(W ))] = E[U(W )] U[E(W )] > E[U(W )] 15 / 32  Introduction Rational decision making when uncertainty exists Theory of investor choice The object of choice Certainty equivalent wealth and risk premium by Markowitz (1959) Certainty equivalent wealth, CE The level of wealth the investor would accept with certainty if the gamble were removed. CE is defined as U(CE) = E[U(W )] Risk premium, π: π = E(W ) − CE Graphical illustration on the board. 16 / 32  Introduction Rational decision making when uncertainty exists Theory of investor choice The object of choice Exercise - Risk profile of the investor Assume no discounting, k = 0. An investor has initial wealth of 2,000 and two risky investment opportunities: Investment 1 Investment 2 End-of-period Wealth Prob. End-of-period Wealth Prob. 3000 50% 4000 50% 1000 50% 0 50% Which of the investments do you prefer, if your utility function is defined as 1 U(W ) = W 2 ? 2 U(W ) = ln(W )? Illustrate your investment decisions graphically. 17 / 32  Introduction Rational decision making when uncertainty exists Theory of investor choice The object of choice Level of risk aversion by Pratt (1964) and Arrow (1971) Empirically, it is shown that investors are typically risk averse. Two key definitions of risk aversion 1 Absolute risk aversion 2 Relative risk aversion ⇒ Introduced by Pratt (1964) and Arrow (1971) 18 / 32  Introduction Rational decision making when uncertainty exists Theory of investor choice The object of choice Level of risk aversion by Pratt (1964) and Arrow (1971) Assume, an investor with Current wealth, W Actuarially neutral gamble of Z e dollars, with E(Z e) = 0 Two examples: z1 z gamble 1 gamble 2 (p1 = p2 = 0.5) z2 −z where E(z1 + z2 ) = 0 e ) = 0.5(z − z) = 0 E(Z End-of-period wealth of W + Z e: Risk premium, π(W , Z e) 19 / 32  Introduction Rational decision making when uncertainty exists Theory of investor choice The object of choice Level of risk aversion by Pratt (1964) and Arrow (1971) We know from before that π(W , Z e ) − CE e ) = E(W + Z ⇔ CE = E(W + Z e ) − π(W , Z e) e ) − π(W , Z = W + E(Z e) CE is defined as: E[U(W + Z e )] = U[CE] ⇔ E[U(W + Z e ) − π(W , Z e )] = U[W + E(Z e )] Since E(Z e) = 0 ⇔ E[U(W + Z e )] = U[W − π(W , Z e )] 20 / 32  Introduction Rational decision making when uncertainty exists Theory of investor choice The object of choice Level of risk aversion by Pratt (1964) and Arrow (1971) Using Taylor series expansion, On the board 1 2 ′′ ⇔ U(W ) + σ U (W ) + · · · = U(W ) − πU ′ (W ) + · · · 2 Z... ! 1 2 U ′′ (W ) ⇔ π = σZ − ′ 2 U (W ) ⇒ Pratt-Arrow measure of a local risk premium. 21 / 32  Introduction Rational decision making when uncertainty exists Theory of investor choice The object of choice Level of risk aversion by Pratt (1964) and Arrow (1971) Pratt-Arrow measure of a local risk premium ! 1 U ′′ (W ) π = σZ2 − ′ 2 U (W ) Absolute risk aversion, ARA U ′′ (W ) ARA = − U ′ (W ) Relative risk aversion, RRA U ′′ (W ) RRA = −W U ′ (W ) 22 / 32  Introduction Rational decision making when uncertainty exists Theory of investor choice The object of choice Example - Level of risk aversion An investor with a power utility function 1 U(W ) = − W Determine ARA and RRA. Are the ARA and RRA constant/increasing/decreasing in W? 23 / 32  Introduction Rational decision making when uncertainty exists Theory of investor choice The object of choice Exercise - Level of risk aversion An investor with a logarithmic utility function U(W ) = ln(W ) Determine ARA and RRA. Are the ARA and RRA constant/increasing/decreasing in W? 24 / 32  Introduction Rational decision making when uncertainty exists Theory of investor choice The object of choice Exercise - Differences in measures of risk premiums Assume no discounting, k = 0. An investor with a logarithmic utility function, U(W ) = ln(W ), has an initial wealth of 2,000 and two investment opp. with 1 a 50/50 chance of gaining or losing $10 2 75% chance of losing $1,000 and a 25% chance of winning $500 Investment 1 Investment 2 Prob. Z e end-of-period wealth Prob. Z e end-of-period wealth 50% 10 2010 75% -1000 1000 50% -10 1990 25% 500 2500 What is the two investments’ 1 Pratt-Arrow risk premium? 2 Risk premium based on the concept by Markowitz? Is there a difference? If yes, why? 25 / 32  Introduction Rational decision making when uncertainty exists Theory of investor choice The object of choice The object of choice Recall that the investors’ expected utility of stochastic end-of-period wealth (W f ) depends on Average end-of-period wealth, E[W f] 2 Variance of end-of-period wealth, σW and higher moments of end-of-period wealth. Object of choice: Stochastic dominance ⇒ Rational decision making (maximizing expected utility of end-of-period wealth) requires stochastic dominance (read it by yourself!) Alternative object of choice: mean and variance, when assuming normal distributed end-of-period wealth ⇒ Computationally much simpler 27 / 32  Introduction Rational decision making when uncertainty exists Theory of investor choice The object of choice The object of choice Under normality assumption Normality assumption for end-of-period wealth. f ∼ N(W̄ , σ 2 ), W W f ) and σ 2 = E W f) 2 , 
 where W̄ = E(W f − E(W W ⇒ Normal distributions completely described by mean and variance ⇒ Rational decision making based solely on the mean and variance 28 / 32  Introduction Rational decision making when uncertainty exists Theory of investor choice The object of choice The object of choice Under normality assumption Focus is typically on the asset returns f − W0 W R= W0 with expectation and variance: " # f − W0 W f ) − W0 E[W E[W f) R̄ = E(R) = E = = −1 W0 W0 W0 " # f−W W 0 f ) − W0 2 E(W σR2 = E[(R − E(R)) ] = E 2 − W0 W0 " # f − E(W W f ) 2 2 1 h f f ) 2 = σW
i =E = E W − E(W W0 W02 W02 29 / 32  Introduction Rational decision making when uncertainty exists Theory of investor choice The object of choice The object of choice Under normality assumption Equivalent to the normally distributed end-of-period wealth, we thus have normally distributed returns: R ∼ N(R̄, σR2 ), 2 E[W e) σW where R̄ = W0 − 1 and σR2 = W02. ⇒ Rational decision making based solely on mean and variance of the returns (referred to as return and risk) ⇒ Investment criteria when uncertainty exist: Return and risk! 30 / 32  Introduction Rational decision making when uncertainty exists Theory of investor choice The object of choice The object of choice Under normality assumption Indifference-curves as a function of return-variance (risk) and expected return: ⇒ Derivation on the board. 31 / 32  Introduction Rational decision making when uncertainty exists Theory of investor choice The object of choice References CWS, ch. 3 32 / 32