Lecture 4: Risk Preferences & Expected Utility Theory

FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (1)

Markus K. Brunnermeier LECTURE 4: RISK PREFERENCES & EXPECTED UTILITY THEORY FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (2)

specify observe/specify Preferences & existing Technology Asset Prices

• evolution of states NAC/LOOP NAC/LOOP • risk preferences • aggregation

State Prices q absolute (or stochastic discount relative asset pricing factor/Martingale measure) asset pricing

LOOP

derive derive Asset Prices Price for (new) asset

Only works as long as market completeness doesn’t change FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (3) Overview: Risk Preferences

1. State-by-state dominance 2. Stochastic dominance [DD4] 3. vNM expected utility theory a) Intuition [L4] b) Axiomatic foundations [DD3] 4. Risk aversion coefficients and portfolio choice [DD5,L4] 5. Uncertainty/ambiguity aversion 6. Prudence coefficient and precautionary savings [DD5] 7. Mean-variance preferences [L4.6] FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (4) State-by-state Dominance

- State-by-state dominance is an incomplete ranking

풕 = ퟎ 풕 = ퟏ

Cost Payoff 휋1 = 휋2 = 1/2 푠 = 1 푠 = 2

Investment 1 -1000 1050 1200 Investment 2 -1000 500 1600 Investment 3 -1000 1050 1600

- Investment 3 state-by-state dominates investment 1 - Recall: 푦, 푥 ∈ ℝ푆

. 푦 ≥ 푥 ⇔ 푦푠 ≥ 푥푠 for each 푠 = 1, … , 푆 . 푦 > 푥 ⇔ 푦 ≥ 푥, 푦 ≠ 푥

. 푦 ≫ 푥 ⇔ 푦푠 > 푥푠 for each 푠 = 1, … , 푆 FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (5) State-by-state Dominance (ctd.)

풕 = ퟎ 풕 = ퟏ

Cost Return 휋1 = 휋2 = 1/2 E[Return] 휎 푠 = 1 푠 = 2

Investment 1 -1000 + 5% + 20% 12.5% 7.5% Investment 2 -1000 - 50% + 60% 5% 55% Investment 3 -1000 + 5% + 60% 32.5% 27.5%

• Investment 1 mean-variance dominates 2 • But, investment 3 does not mean-variance dominate 1 FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (6) State-by-state Dominance (ctd.)

풕 = ퟎ 풕 = ퟏ

Cost Return 휋1 = 휋2 = 1/2 E[Return] 휎 푠 = 1 푠 = 2

Investment 4 -1000 + 3% + 5% 4.0% 1.0% Investment 5 -1000 + 3% + 8% 5.5% 2.5%

• What is the trade-off between risk and expected return? 퐸 푟 −푟 • Investment 4 has a higher Sharpe ratio 푓 than investment 5 for 푟 = 0 𝜎 푓 FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (7) Overview: Risk Preferences

1. State-by-state dominance 2. Stochastic dominance [DD4] 3. vNM expected utility theory a) Intuition [L4] b) Axiomatic foundations [DD3] 4. Risk aversion coefficients and portfolio choice [DD5,L4] 5. Prudence coefficient and precautionary savings [DD5] 6. Mean-variance preferences [L4.6] FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (8) Stochastic Dominance

• No state-space – probabilities are not assigned specific states . Only applicable for final payoff gamble • Not for stocks/lotteries that form a portfolio (whose payoff is final) . Random variables before introduction of (Ω, ℱ, 푃) • Still incomplete ordering . “More complete” than state-by-state ordering . State-by-state dominance ⇒ stochastic dominance . Risk preference not needed for ranking! • independently of the specific trade-offs (between return, risk and other characteristics of probability distributions) represented by an agent's utility function. (“risk-preference-free”) • Next Section: . Complete preference ordering and utility representations FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (9) From payoffs per state to probability

state 풔ퟏ 풔ퟐ 풔ퟑ 풔ퟒ 풔ퟓ

probability 휋1 휋2 휋3 휋4 휋5 Payoff x 10 10 20 20 20 Payoff y 10 20 20 20 30

Expressed in “probability lotteries” – only useful for final payoffs (since some cross correlation information ins lost) payoff 10 20 30

Prob x 푝10 = 휋1 + 휋2 푝20 = 휋3 + 휋4 + 휋5 푝30 = 0

Prob y 푞10 = 휋1 푞20 = 휋2 + 휋3 + 휋4 푞30 = 휋5

푆 Preference 푥 ≻ 푦 ∈ ℝ expressed in probabilities 푝푥 ≻ 푞푦 FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (10)

Expected Standard Payoff Deviation Payoffs 10 100 2000 Probability 1 .4 .6 0 64 44 Probability 2 .4 .4 .2 444 779

Pr obability F1 1.0

0.9 F2 0.8

0.7

0.6

0.5 F1 and F 2 0.4

0.3 Note: Payoff of $10 is 0.2 equally likely, but can be 0.1 in different states of the world

Payoff 0 10 100 2000 FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (11) First Order Stochastic Dominance

• Definition: Let 퐹퐴 푥 , 퐹퐵 푥 , respectively, represent the cumulative distribution functions of two random variables (cash payoffs) that, without loss of generality assume values in the interval 푎, 푏 . We say that 퐹퐴 푥 first order stochastically dominates (FSD) 퐹퐵 푥 if and only if for all 푥 ∈ 푎, 푏 퐹퐴 푥 ≤ 퐹퐵 푥

Homework: Provide an example which can be ranked according to FSD, but not according to state dominance. FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (12) First Order Stochastic Dominance

0.9

0.8 FB

0.7 FA

0.6

0.5

0.4

0.3

0.2

0.1

0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 X FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (13)

Payoff 1 4 5 6 8 12 Probability 3 0 .25 0.50 0 0 .25 Probability 4 .33 0 0 .33 .33 0

0.9

0.8

0.7 0.6 investment 4 0.5

0.4

0.3 0.2 investment 3 0.1

0 0 1 2 3 4 5 6 7 8 9 10 11 12 13

CDFs of investment 3 and 4 FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (14) Second Order Stochastic Dominance

• Definition: Let 퐹퐴 푥 , 퐹퐵 푥 be two cumulative probability distribution for random payoffs in 푎, 푏 . We say that 퐹퐴 푥 second order stochastically dominates (SSD) 퐹퐵 푥 if and only if for any 푥 ∈ 푎, 푏 푥 퐹퐵 푡 − 퐹퐴 푡 푑푡 ≥ 0 −∞ (with strict inequality for some meaningful interval of values of t). FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (15) Mean Preserving Spread

푥퐵 = 푥퐴 + 푧 (where 푧 is independent and has zero mean)

Mean Preserving Spread: f x (for normal distributions) A

fB x

~x, Payoff     x fA xdx  x fBxdx FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (16) Mean Preserving Spread & SSD

• Theorem: Let 퐹퐴 푥 and 퐹퐵 푥 be two distribution functions defined on the same state space with identical means. Then the following statements are equivalent :

. 퐹퐴 푥 SSD 퐹퐵 푥 . 퐹퐵 푥 is a mean-preserving spread of 퐹퐴 푥 FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (17) Overview: Risk Preferences

1. State-by-state dominance 2. Stochastic dominance [DD4] 3. vNM expected utility theory a) Intuition [L4] b) Axiomatic foundations [DD3] 4. Risk aversion coefficients and portfolio choice [DD5,L4] 5. Prudence coefficient and precautionary savings [DD5] 6. Mean-variance preferences [L4.6] FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (18) A Hypothetical Gamble

• Suppose someone offers you this gamble: . "I have a fair coin here. I'll flip it, and if it's tails I pay you $1 and the gamble is over. If it's heads, I'll flip again. If it's tails then, I pay you $2, if not I'll flip again. With every round, I double the amount I will pay to you if it turns up tails." • Sounds like a good deal. After all, you can't lose. So here's the question: . How much are you willing to pay to take this gamble? FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (19) Proposal 1: Expected Value

1 1 1 • With probability you get $1, × 20 2 2 1 1 2 • With probability you get $2, × 21 4 2 1 1 3 • With probability you get $4, × 22 8 2 • …

. The expected payoff is given by the sum of all these terms, i.e. ∞ 푡 ∞ 1 1 × 2푡−1 = = ∞ 2 2 푡=1 푡=1 FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (20) St. Petersburg Paradox

• You should pay everything you own and more to purchase the 0.5 right to take this gamble! • Yet, in practice, no one is 0.4 prepared to pay such a high price. Why? 0.3 • Even though the expected 0.2 payoff is infinite, the distribution of payoffs is not 0.1 attractive… 0 . with 93% probability 0 20 40 60 we get $8 or less; . with 99% probability we get $64 or less FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (21) Proposal 2

• Bernoulli suggests that large gains should be weighted less. He suggests to use the natural logarithm. [Cremer - another great mathematician of the time - suggests the square root.]

∞ 푡 1 × ln 2푡−1 = ln 2 < ∞ 2 푡=1 According to this Bernoulli would pay at most 푒ln 2 = 2 to participate in this gamble FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (22) Representation of Preferences

A preference ordering is (i) complete, (ii) transitive, (iii) continuous and [(iv) relatively stable] can be represented by a utility function, i.e. ′ ′ ′ 푐0, 푐1, … , 푐푆 ≻ 푐0, 푐1, … , 푐푆 ′ ′ ′ ⇔ 푈 푐0, 푐1, … , 푐푆 > 푈 푐0, 푐1, … , 푐푆

(preference ordering over lotteries – (푆 + 1)-dimensional space) FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (23) Indifference curves

in ℝ2 (for 푆 = 2) 푐2

흅

푧

45° 푐1 푧 FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (24) Preferences over Prob. Distributions

• Consider 푐0 fixed, 푐1 is a random variable • Preference ordering over probability distributions • Let . 푃 be a set of probability distributions with a finite support over a set 푋, . ≽ preference ordering over 푃 (that is, a subset of 푃 × 푃) FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (25) Prob. Distributions

• S states of the world • Set of all possible lotteries

푃 = 푝 ∈ ℝ푆 푝 푐 ≥ 0, 푝 푐 = 1

• Space with S dimensions

• Can we simplify the utility representation of preferences over lotteries? • Space with one dimension – income • We need to assume further axioms FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (26) Expected Utility Theory

• A binary relation that satisfies the following three axioms if and only if there exists a function 푢 ⋅ such that 푝 ≻ 푞 ⇔ 푝 푐 푢 푐 > 푞 푐 푢 푐

i.e. preferences correspond to expected utility. FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (27) vNM Expected Utility Theory

• Axiom 1 (Completeness and Transitivity): . Agents have preference relation over 푃 (repeated) • Axiom 2 (Substitution/Independence) . For all lotteries 푝, 푞, 푟 ∈ 푃 and 훼 ∈ 0,1 , 푝 ≽ 푞 ⇔ 훼푝 + 1 − 훼 푟 ≽ 훼푞 + 1 − 훼 푟 • Axiom 3 (Archimedian/Continuity) . For all lotteries 푝, 푞, 푟 ∈ 푃 if 푝 ≻ 푞 ≻ 푟 then there exists 훼, 훽 ∈ 0,1 such that, 훼푝 + 1 − 훼 푟 ≻ 푞 ≻ 훽푝 + 1 − 훽 푟 Problem: 푝 you get $100 for sure, 푞 you get $ 10 for sure, 푟 you are killed FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (28) Independence Axiom

• Independence of irrelevant alternatives:

푝 푞 휋 휋

푝 ≽ 푞 ⇔ ≽

푟 푟 FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (29) Allais Paradox – Violation of Independence Axiom

10’ 15’ 10% 9% ≺

0 0 FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (30) Allais Paradox – Violation of Independence Axiom

10’ 15’ 10% 9% ≺

0 0

10’ 15’ 100% 90% ≻

0 0 FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (31) Allais Paradox – Violation of Independence Axiom

10’ 15’ 10% 9% ≺

0 0

10’ 15’ 100% 90% ≻ 10% 10% 0 0

0 0 FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (32) vNM EU Theorem

• A binary relation that satisfies the axioms 1-3 if and only if there exists a function 푢 ⋅ such that 푝 ≻ 푞 ⇔ 푝 푐 푢 푐 > 푞 푐 푢 푐

i.e. preferences correspond to expected utility. FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (33) Risk-Aversion and Concavity 푢 푐 • The shape of the von Neumann Morgenstern (NM) utility function reflects risk

preference 푢 푐2 • Consider lottery with final wealth of 푐1 or 푐2 퐸 푢 푐

푢 푐1

푐1 퐸 푐 푐2 푐 FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (34) Risk-aversion and concavity 푢 푥 • Risk-aversion means that the certainty equivalent is smaller than the expected prize. 푢 푐2 . We conclude that a risk- 푢 퐸 푐 averse vNM utility function must be concave. 퐸 푢 푐

푢 푐1

푐1 퐸 푐 푐2 푥 푢−1 퐸 푢 푐 FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (35) Jensen’s Inequality Theorem: • Let 푔 ⋅ be a concave function on the interval 푎, 푏 , and 푥 be a random variable such that 푃 푥 ∈ 푎, 푏 = 1 • Suppose the expectations 퐸 푥 and 퐸 푔 푥 exist; then 퐸 푔 푥 ≤ 푔 퐸 푥 Furthermore, if 푔 ⋅ is strictly concave, then the inequality is strict. FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (36) Expected Utility & Stochastic Dominance

• Theorem: Let 퐹퐴 푥 , 퐹퐵 푥 be two cumulative probability distribution for random payoffs 푥 ∈ 푎, 푏 . Then 퐹퐴 푥 FSD 퐹퐵 푥 if and only if 퐸퐴[푢 푥 ] ≥ 퐸퐵[푢 푥 ] for all non decreasing utility functions 푈 ⋅ .

• Theorem: Let 퐹퐴 푥 , 퐹퐵 푥 be two cumulative probability distribution for random payoffs 푥 ∈ 푎, 푏 . Then 퐹퐴 푥 SSD 퐹퐵 푥 if and only if 퐸퐴[푢 푥 ] ≥ 퐸퐵[푢 푥 ] for all non decreasing concave utility functions 푈 ⋅ . FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (37) Certainty Equivalent and Risk Premium

퐸 푢 푐 + 푍 = 푢 푐 + 퐶퐸 푐, 푍

퐸 푢 푐 + 푍 = 푢 푐 + 퐸 푍 − Π 푐, 푍 FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (38) Certainty Equivalent and Risk Premium

U(Y)

U(Y0 + Z2 ) ~ U(Y0 + E(Z)) ~ EU(Y0 + Z)

U(Y0 + Z1)

~ CE(Z) P

~ ~ Y Y + Z CE(Y + Z) Y + E(Z) Y + Z 0 0 1 0 0 0 2 Y

Certainty Equivalent and Risk Premium FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (39) Utility Transformations

• General utility function: ′ ′ ′ . Suppose 푈 푐0, 푐1, … , 푐푆 > 푈 푐0, 푐1, … , 푐푆 represents complete, transitive,… preference ordering, . then 푉 ⋅ = 푓 푈 ⋅ , where 푓 ⋅ is strictly increasing represents the same preference ordering • vNM utility function . Suppose 퐸 푢 푐 represents preference ordering satisfying vNM axioms, . then 푣 푐 = 푎 + 푏푢 푐 represents the same. “affine transformation” FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (40) Overview: Risk Preferences

U(W) tangent lines

U(Y+h)

U[0.5(Y+h)+0.5(Y-h)]

0.5U(Y+h)+0.5U(Y-h)

U(Y-h)

Y-h Y Y+h W A Strictly Concave Utility Function FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (42) Arrow-Pratt Measures of Risk aversion

푢′′ 푐 • absolute risk aversion = − ≡ 푅 푐 푢′ 푐 퐴

푐푢′′ 푐 • relative risk aversion = − ≡ 푅 푐 푢′ 푐 푅

1 • risk tolerance = 푅퐴 FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (43) Absolute risk aversion coefficient

푢′′ 푐 푅퐴 = − ′ 푢 푐 휋 푐 + Δ

푐

1 − 휋 푐 − Δ

1 1 휋 푐, Δ = + Δ푅 푐 + 퐻푂푇 2 4 퐴 FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (44) Relative risk aversion coefficient

푢′′ 푐 푅 = − 푐 푅 푢′ 푐 푐(1 + 훿) 휋

푐

1 − 휋 푐(1 − 훿)

1 1 휋 푐, 휃 = + 훿푅 푐 + 퐻푂푇 2 4 푅 Homework: Derive this result. FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (45) CARA and CRRA-utility functions

• Constant Absolute RA utility function 푢 푐 = −푒−𝜌푐

• Constant Relative RA utility function 푐1−훾 푢 푐 = , 훾 ≠ 1 1 − 훾 푢 푐 = ln 푐 , 훾 = 1 FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (46) Level of Relative Risk Aversion

1 1 푌 + 퐶퐸 1−훾 푌 + 50000 1−훾 푌 + 100000 1−훾 = 2 + 2 1 − 훾 1 − 훾 1 − 훾

푌 = 0  = 0 CE = 75,000 (risk neutrality)  = 1 CE = 70,711  = 2 CE = 66,246  = 5 CE = 58,566  = 10 CE = 53,991  = 20 CE = 51,858  = 30 CE = 51,209

푌 = 100000  = 5 CE = 66,530 FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (47) Risk aversion and Portfolio Allocation

• No savings decision (consumption occurs only at t=1) • Asset structure

. One risk free bond with net return 푟푓 . One risky asset with random net return 푟 (a =quantity of risky assets)

max 퐸 푢 푌0 1 + 푟푓 + 푎 푟 − 푟푓 푎

′ FOC ⇒ 퐸 푢 푌0 1 + 푟푓 + 푎 푟 − 푟푓 푟 − 푟푓 = 0 FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (48) Risk aversion and Portfolio Allocation

• Theorem 4.1: Assume 푈′ > 0, 푈′′ < 0 and let 푎 denote the solution to above problem. Then 푎 > 0 ⇔ 퐸 푟 > 푟푓 푎 = 0 ⇔ 퐸 푟 = 푟푓 푎 < 0 ⇔ 퐸 푟 < 푟푓

• Define 푊 푎 = 퐸 푢 푌0 1 + 푟푓 + 푎 푟 − 푟푓 . The FOC can then be ′ ′ written 푊 푎 = 퐸 푢 푌0 1 + 푟푓 + 푎 푟 − 푟푓 푟 − 푟푓 = 0. ′′ ′′ 2 • By risk aversion 푊 푎 = 퐸 푢 푌0 1 + 푟푓 + 푎 푟 − 푟푓 푟 − 푟푓 < 0, that is, 푊′ 푎 is everywhere decreasing . It follows that 푎 will be positive ⇔ 푊′ 0 > 0 ′ • Since 푢 > 0 this implies that 푎 > 0 ⇔ 퐸 푟 − 푟푓 > 0 W’(a) . The other assertion follows similarly a 0 FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (49) Portfolio as wealth changes

• Theorem (Arrow, 1971): Let 푎 = 푎 푌0 be the solution to max-problem above; then: 휕푅 휕푎 i. 퐴 < 0 (DARA) ⇒ > 0 휕푌 휕푌0 휕푅 휕푎 ii. 퐴 = 0 (CARA) ⇒ = 0 휕푌 휕푌0 휕푅 휕푎 iii. 퐴 > 0 (IARA) ⇒ < 0 휕푌 휕푌0 FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (50) Portfolio as wealth changes

• Theorem (Arrow 1971): If, for all wealth levels Y,

휕푅 i. 푅 = 0 (CRRA) ⇒ 휂 = 1 휕푌 휕푅 ii. 푅 < 0 (DRRA) ⇒ 휂 > 1 휕푌 휕푅 iii. 푅 > 0 (IRRA) ⇒ 휂 < 1 휕푌 푑푎 푎 where 휂 = 푑푌 푌 FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (51) Log utility & Portfolio Allocation

푢 푌 = ln 푌

푟 − 푟푓 퐸 = 0 푌0 1 + 푟푓 + 푎 푟 − 푟푓

푎 1 + 푟푓 퐸 푟 − 푟푓 = > 0 푌0 − 푟1 − 푟푓 푟2 − 푟푓

2 states, where 푟2 > 푟푓 > 푟1

Constant fraction of wealth is invested in risky asset! Homework: show that this result holds for • any CRRA utility function • any distribution of r FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (52) Risk aversion and Portfolio Allocation

푎 1 푌0 • Theorem (Cass and Stiglitz,1970): Let the vector ⋮ denote the 푎 퐽 푌0

amount optimally invested in the 퐽 risky assets if the wealth level is 푌0.

푎 1 푌0 푎1 Then ⋮ = ⋮ 푓 푌0 if and only if either 푎 퐽 푌0 푎퐽

′ Δ i. 푢 (푌0) = 퐵푌0 + 퐶 or ′ −𝜌푌0 ii. 푢 푌0 = 휉푒

• In words, it is sufficient to offer a mutual fund. FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (53) LRT/HARA-utility functions

• Linear Risk Tolerance/hyperbolic absolute risk aversion 푢′′ 푐 1 − = 푢′ 푐 퐴 + 퐵푐 • Special Cases 1 퐵−1 . 퐵 = 0, 퐴 > 0 CARA 푢 푐 = 퐴 + 퐵푐 퐵 퐵−1 . 퐵 ≠ 0, ≠ 1 Generalized Power • 퐵 = 1 Log utility 푢 푐 = ln 퐴 + 퐵푐 • 퐵 = −1 Quadratic Utility 푢 푐 = − 퐴 − 푐 2 1 퐵−1 • 퐵 ≠ 1, 퐴 = 0 CRRA Utility function 푢 푐 = 퐵푐 퐵 퐵−1 FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (54) Overview: Risk Preferences

Digression: Subjective EU Theory

• Derive perceived probability from preferences! . Set 푆 of prizes/consequences . Set 푍 of states . Set of functions 푓 푠 ∈ 푍, called acts (consumption plans) • Seven SAVAGE Axioms . Goes beyond scope of this course. FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (56)

Digression: Ellsberg Paradox • 10 balls in an urn Lottery 1: win $100 if you draw a red ball Lottery 2: win $100 if you draw a blue ball • Uncertainty: Probability distribution is not known • Risk: Probability distribution is known (5 balls are red, 5 balls are blue)

• Individuals are “uncertainty/ambiguity averse” (non-additive probability approach) FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (57)

Digression: Prospect Theory • Value function (over gains and losses)

• Overweight low probability events • Experimental evidence FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (58) Overview: Risk Preferences

• Introduce savings decision: Consumption at 푡 = 0 and 푡 = 1 • Asset structure 1: – risk free bond 푅푓 – NO risky asset with random return – Increase 푅푓: – Substitution effect: shift consumption from 푡 = 0 to 푡 = 1 ⇒ save more – Income effect: agent is “effectively richer” and wants to consume some of the additional richness at 푡 = 0 ⇒ save less – For log-utility (훾 = 1) both effects cancel each other FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (60) Savings: Euler Equation 푐1−훾 for CRRA: 푢 푐 = 1−훾

• max 푢 푐0 + 훿푢(푐1) 푐0,푐1 푓 . s.t. 푐1 = 푅 (푒0−푐0) + 푒1 푓 • max 푢 푐0 + 훿푢(푅 푒0 + 푒1) 푐0 ′ −훾 푢 푐1 푓 푐1 푓 • FOC: 1 = 훿 ′ 푅 1 = 훿 푅 푢 푐0 푐0 ′ 푓 푓 푢 푐1 . 핣 ≈ ln 푅 = − ln ′ − 푙푛훿 푢 푐0 for log: 푢 푐 = ln 푐 & 푒1 = 0 1 1 푐0 = 훿(훿+1)[푒0 + 푅푒1] 1 푐1 = 1 − 훿 훿+1 [푅푒0 + 푒_1]

푓 푐0 푓 for e1 = 0 saving does not depend on (risk of) 푅 : 1 = 훿 푓 푅 푅 (푒0−푐0) FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (61) Intertemporal Elasticity of Substitution

푐 푐1 휕 ln 1 휕 ln 푐0 푐0 • 퐼퐸푆 ≔ = − 푢′(푐 ) 휕푟 휕 ln 1 푢′(푐0)

푐1−훾 1 • For CRRA 푢 푐 = 1−훾 퐼퐸푆 = 훾 FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (62) Investment Risk

• Savings decision: Consumption at 푡 = 0 and 푡 = 1 • No endowment risk at 푡 = 1 • Asset structure 2: (no portfolio choice yet) . Single risky asset only . No risk-free asset

• Theorem (Rothschild and Stiglitz, 1971): For 푅퐵 = 푅퐴 + ε, where E 휀 = 0 and 휀 ⊥ 푅퐴, then 퐴 퐵 respective savings 푠 , 푠 out of initial wealth level 푊0 are . If 휕푅푅 ≤ 0 and 푅 > 1, then 푠퐴 < 푠퐵. 휕푊0 푅 . If 휕푅푅 ≥ 0 and 푅 < 1, then 푠퐴 > 푠퐵. 휕푊0 푅 FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (63) Investment Risk with Portfolio and Savings Decision • Savings decision: Consumption at 푡 = 0 and 푡 = 1

• No endowment risk at 푡 = 1, 푒1 = 0 • Asset structure 3: portfolio shares 훼푗

• max푐0,푐1,휶ퟎ 푢 푐0 + 훿퐸0[푢(푐1)] 푗 푗 s.t. 푊1 = 훼0푅1(푊0 − 푐0)

′ ′ 푗 푢 푐0 = 퐸0 훿푢 푐1 푅 ∀푗 FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (64) Investment Risk: Excess Return

1 = 퐸 훿푢(푐1)푅푗 ∀푗 0 푢(푐0) −훾 • For CRRA 1 = 훿퐸 푐1 푅푗 0 푐0 푗 푗 • In “log-notation”: 핔푡 ≡ log 푐푡, 핣푡 ≡ log 푅푡 푗 −훾 핔1−핔0 +핣 1 = 훿퐸0 푒 푗 • Assume 핔푡, 핣푡 ~풩 1 −훾퐸 Δ핔 +퐸 핣푗 + 푉푎푟 −훾Δ핔 +핣푗 1 = 훿[푒 0 1 0 2 0 1 ] 훾2 1 0 = ln 훿 − 훾퐸 Δ핔 + 퐸 핣푗 + 푉푎푟 Δ핔 + 푉푎푟 핣푗 − 훾퐶표푣 [Δ핔 , 핣푗] 0 1 0 2 0 1 2 0 0 1 • For risk free asset: 훾2 핣푓 = − ln 훿 + 훾퐸 Δ핔 − 푉푎푟 Δ핔 0 1 2 0 1 • Excess return of any asset: 1 퐸 핣푗 + 푉푎푟 핣푗 − 핣푓 = 훾퐶표푣 [Δ핔 , 핣푗] 0 2 0 0 1 FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (65) Investment Risk: Portfolio Shares

• Excess return 1 퐸 핣푗 + 푉푎푟 핣푗 − 핣푓 = 훾퐶표푣 [Δ핔 , 핣푗] 0 2 0 0 1 • If consumption growth Δ핔1 = Δ함1 wealth growth 푗 푗 푗 푗 푗 푗 • 퐶표푣0 Δ함1, 핣 = 퐶표푣0 훼0핣 , 핣 = 훼0푉푎푟0[핣 ] • Hence, optimal portfolio share 1 퐸 핣푗 + 푉푎푟 핣푗 −핣푓 푗 0 2 0 훼0 = 푗 훾푉푎푟0[핣 ] FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (66)

Making Δ함1 Linear in 핔0 − 함0

푗 푗 • 푊1 = 훼0푅1(푊0 − 푐0) recall 푒1 = 0 • 푊1 = 훼푗푅푗(1 − 푐0 ) let 푅푝 = 훼푗푅푗 푊0 0 1 푊0 1 0 1 푗 핔0−함0 • In “log-notation”: Δ함1 = 핣1 + log(1 − 푒 )

nonlinear • Linearize using Taylor expansion around 핔 − 함

푗 1 • Δ함1 = 핣1 + 푘 + 1 − 휌 핔0 − 함0 1−휌 핔−함 . Where 푘 ≡ log 휌 + 1 − 휌 log 휌 , 휌 = 1 − 푒

Hint: in continuous time this approximation is precise FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (67) Endowment Risk: Prudence and Pre-cautionary Savings • Savings decision Consumption at 푡 = 0 and 푡 = 1 • Asset structure 2: . No investment risk: riskfree bond . Endowment at 푡 = 1 is random (background risk)

• 2 effects: Tomorrow consumption is more volatile . consume more today, since it’s not risky . save more for precautionary reasons FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (68) Prudence and Pre-cautionary Savings

• Risk aversion is about the willingness to insure … • … but not about its comparative statics. • How does the behavior of an agent change when we marginally increase his exposure to risk? • An old hypothesis (J.M. Keynes) is that . people save more when they face greater uncertainty . precautionary saving • Two forms: . Shape of utility function 푢′′′

. Borrowing constraint 푎푡 ≥ −푏 FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (69) Precautionary Savings 1: Prudence

• Utility maximization 푢 푐0 + 훿푢퐸0[푢(푐1)] . Budget constraint: 푐1 = 푒1 + 1 + 푟 (푒0 − 푐0) ′ ′ . Standard Euler equation: 푢 푐푡 = 훿 1 + 푟 퐸푡 푢 푐푡+1 • If 푢′′′ > 0, then Jensen’s inequality implies: ′ ′ 1 퐸푡 푢 푐푡+1 푢 퐸푡 푐푡+1 • = ′ > ′ 훿 1+푟 푢 푐푡 푢 푐푡 • Increase variance of 푒1 (mean preserving spread) ′ • Numerator 퐸푡[푢 푐푡+1 ] increases with variance of 푐푡+1 • For equality to hold, denominator has to increase 푐푡 has to decrease, i.e. savings has to increase precautionary savings 푢′′′ • Prudence refers to curvature of 푢′, i.e. 푃 = − 푢′′ FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (70) Precautionary Savings 1: Prudence

• Does not directly follow from risk aversion, involves 푢′′′ . Leland (1968) • Kimball (1990) defines absolute prudence as 푢′′′ 푐 푃 푐 ≔ − 푢′′ 푐 • Precautionary saving if any only if prudent. . important for comparative statics of interest rates. • DARA ⇒ Prudence 푢′′ 휕 − ′′′ 푢′ 푢 푢′′ < 0, − > − 휕푐 푢′′ 푢′ FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (71) Precautionary Savings 2: Future Borrowing Constraint • Agent might be concerned that he faces borrowing constraints in some state in the future • agents engage in precautionary savings (self-insurance) • In Bewley (1977) idiosyncratic income shocks, mean asset holdings mean 푎 (across individuals) result from individual optimization

r 휌

mean[a]

-b FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (76) Precautionary Savings 1: Prudence

• Asset structure 3: – No risk free bond – One risky asset with random gross return 푅

• Theorem (Rothschild and Stiglitz,1971) : Let 푅 퐴, 푅 퐵 be two return distributions with identical means such that 푅 퐵 = 푅 퐴 + 푒, where 푒 is white noise, and let 푠퐴, 푠퐵 be the savings out of 푌0 corresponding to the return distributions 푅 퐴, 푅 퐵 respectively.

′ . If 푅푅 푌 ≤ 0 and 푅푅 푌 > 1, then 푠퐴 < 푠퐵 ′ . If 푅푅 푌 ≥ 0 and 푅푅 푌 < 1, then 푠퐴 > 푠퐵 FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (77) Precautionary Savings 1: Prudence

푢′′′ 푐 푃 푐 = − 푢′′ 푐 푐푢′′′ 푐 푃 푐 푐 = − 푢′′ 푐

• Theorem: Let 푅 퐴, 푅 퐵 be two return distributions such that 푅 퐴 SSD 푅 퐵, let 푠퐴 and 푠퐵 be, respectively, the savings out of 푌0. Then,

. 푠퐴 ≥ 푠퐵 ⇔ 푐푃 푐 ≤ 2 and conversely, . 푠퐴 < 푠퐵 ⇔ 푐푃 푐 > 2 FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (78) Overview: Risk Preferences

• Early research (e.g. Markowitz and Sharpe) simply used mean and variance of return • Mean-variance utility often easier than vNM utility function • … but is it compatible with vNM theory? • The answer is yes … approximately … under some conditions. FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (80) Mean-Variance: quadratic utility

Suppose utility is quadratic, 푢 푐 = 푎푐 − 푏푐2 Expected utility is then 퐸 푢 푐 = 푎퐸 푐 − 푏퐸 푐2 = 푎퐸 푐 − 푏 퐸 푐 2 + var 푐

Thus, expected utility is a function of the mean 퐸 푐 , and the variance var 푐 ,only. FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (81) Mean-Variance: joint normals

• Suppose all lotteries in the domain have normally distributed prized. (independence is not needed). . This requires an infinite state space. • Any linear combination of jointly normals is also normal. • The normal distribution is completely described by its first two moments. • Hence, expected utility can be expressed as a function of just these two numbers as well. FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (86) Mean-Variance: small risks

• Let 푓: ℝ → ℝ be a smooth function. The Taylor approximation is 푓 푥 푥 − 푥 1 ≈ 푓 푥 + 푓′ 푥 0 0 0 1! 푥 − 푥 2 + 푓′′ 푥 0 + ⋯ 0 2!

• Use the Taylor approximation for 퐸 푢 푥 FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (87) Mean-Variance: small risks

• Since 퐸 푢 푤 + 푥 = 푢 푐퐶퐸 , this simplifies var 푥 to 푤 − 푐 ≈ 푅 푤 퐶퐸 퐴 2 . 푤 − 푐퐶퐸 is the risk premium . We see here that the risk premium is approximately a linear function of the variance of the additive risk, with the slope of the effect equal to half the coefficient of absolute risk. FIN501 Asset Pricing Lecture 04 Risk Prefs & EU (88) Mean-Variance: small risks

• Same exercise can be done with a multiplicative risk. • Let 푦 = 푔푤, where 푔 is a positive random variable with unit mean. • Doing the same steps as before leads to var 푔 1 − 휅 ≈ 푅 푤 푅 2 . where 휅 is the certainty equivalent growth rate, 푢 휅푤 = 퐸[푢 푔푤 ]. . The coefficient of relative risk aversion is relevant for multiplicative risk, absolute risk aversion for additive risk.