LECTURE 06: SHARPE RATIO, BONDS, & THE EQUITY PREMIUM PUZZLE

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1 Lecture 06 Equity Premium Puzzle (1) Markus K. Brunnermeier LECTURE 06: SHARPE RATIO, BONDS, & THE EQUITY PREMIUM PUZZLE

2 FIN501 Asset Pricing Lecture 06 Equity Premium Puzzle (2) Money, Bonds vs. Stocks $50 $45 $40 $35 $30 $25 $20 $15 $10 $5 $0 SP500 US 1y Tsy US 10Y Tsy

3 Lecture 06 Equity Premium Puzzle (3) Sharpe Ratios and Bounds Consider a one period security available at date t with payo x t+1. We have p t = E t m t+1 x t+1 or p t = E t m t+1 E t x t+1 + cov m t+1, x t+1 For a given m t+1 we let R t+1 E t m t+1 Note that R t+1 will depend on the choice o m t+1 unless there exists a riskless portolio R t+1 is the return rom t to t + 1, typically measurable w.r.t. F t+1. (An exception is R, which is measurable w.r.t. F t, but we stick with subscript t + 1.) = 1

4 Lecture 06 Equity Premium Puzzle (4) Sharpe Ratios and Bounds (ctd.) Hence p t = 1 E t x t+1 R t+1 Expected PV + cov m t+1, x t+1 Risk Adjustment Positive correlation with the discount actor adds value, i.e. decreases required return

5 Lecture 06 Equity Premium Puzzle (5) in Returns E t m t+1 x t+1 = p t Divide both sides by p t and note that x t+1 p t Using R t+1 E t m t+1 E t m t+1 R t+1 = 1 = 1/E t m t+1, we obtain R t+1 R t+1 = 0 = R t+1 m-discounted expected excess return or all assets is zero.

6 Lecture 06 Equity Premium Puzzle (6) in Returns Since E t m t+1 R t+1 R t+1 cov t m t+1, R t+1 R t+1 = 0 = E t m t+1 E t R t+1 R t+1 That is, risk premium or expected excess return E t R t+1 R t+1 = cov t m t+1, R t+1 E t m t+1 is determined by its covariance with the stochastic discount actor

7 Lecture 06 Equity Premium Puzzle (7) Sharpe Ratio Multiply both sides with portolio h E t E t R t+1 R t+1 R t+1 R t+1 h = cov t m t+1, R t+1 h E t m t+1 h = ρ m t+1, R t+1 h σ R t+1 h σ m t+1 E t m t+1 NB: All results also hold or unconditional expectations E Rewritten in terms o Sharpe Ratio =... σ m t+1 ρ m E m t+1, R t+1 h = E R t+1 R t+1 t+1 σ R t+1 h h

8 Lecture 06 Equity Premium Puzzle (8) Hansen-Jagannathan Bound Since ρ 1,1 we have σ m t+1 E m t+1 sup h E R t+1 R t+1 σ R t+1 h Theorem (Hansen-Jagannathan Bound): The ratio o the standard deviation o a stochastic discount actor to its mean exceeds the Sharpe Ratio attained by any portolio. h

9 Lecture 06 Equity Premium Puzzle (9) Hansen-Jagannathan Bound Theorem (Hansen-Jagannathan Bound): The ratio o the standard deviation o a stochastic discount actor to its mean exceeds the Sharpe Ratio attained by any portolio. Can be used to easy check the viability o a proposed discount actor Given a discount actor, this inequality bounds the available risk-return possibilities The result also holds conditional on date t ino

10 Lecture 06 Equity Premium Puzzle (10) Hansen-Jagannathan Bound expected return slope s (m) / E[m] R available portolios s

11 Lecture 06 Equity Premium Puzzle (11) Assuming Expected Utility c 0 R, c 1 R S U c 0, c 1 = s π s u c 0, c 1,s U(c 0,c 1 ) 0 u = u c 0, c 1,1,, u c 0, c 1,S c 0 c 0 1 u = u c 0, c 1,1,, u c 0, c 1,S c 1,1 c 1,S Stochastic discount actor m = MRS π = 1u E 0 u RS

12 Lecture 06 Equity Premium Puzzle (12) Time-Separable Digression: i utility is in addition time-separable u c 0, c 1 = v c 0 + v c 1 Then 0 u = v c 0,, v c 0 c 0 c 0 And 1 u = v c 1,1 c 1,1 m s = 1 π s π s v c 1,s,, v c 1,S c 1,S v c 0 = v c 1,s v c 0

13 Lecture 06 Equity Premium Puzzle (13) A simple example S = 2, π 1 = securities with x 1 = 1,0, x 2 = 1,0, x 3 = 1,1 Let m = 1 2, 1, σ = 1 4 = Hence, p 1 = 1 4, p2 = 1 2 = p3 = 3 4 and R 1 = 4,0, R 2 = 0,2, R 3 = 4 3, 4 3 E R 1 = 2, E R 2 = 1, E R 3 = 4 3

14 Lecture 06 Equity Premium Puzzle (14) Example: Where does SDF come rom? Representative agent with Endowment: 1 in date 0, (2,1) in date 1 Utility EU c 0, c 1, c 2 = s π s ln c 0 + ln c 1,s i.e. u c 0, c 1,s = ln c 0 + ln c 1,s (additive) time separable u- unction m = 1u 1,2,1 E 0 u 1,2,1 = c 0 c 1,1, c 0 c 1,2 = 1 2, 1 1 = 1 2, 1 since endowment=consumption Low consumption states are high m-states Risk-neutral probabilities combine true probabilities and marginal utilities.

15 Lecture 06 Equity Premium Puzzle (15) Equity Premium Puzzle Recall E R i R = R cov m, R i Now: E R j R = R cov 1 u,r j E 0 u Recall Hansen-Jaganathan bound σ m E R R ; E m = 1 E m σ R R σ m 1 E R R R σ R

16 Lecture 06 Equity Premium Puzzle (16) Equity Premium Puzzle (ctd.) σ 1 u E 0 u 1 R E R R σ R u u c 1 c 2 c 1 c 2 Equity Premium Puzzle high observed Sharpe ratio o stock market indices low volatility o consumption ) (unrealistically) high level o risk aversion

17 Equity Premium or Low Risk-ree Rate Puzzle? Suppose we allow or suiciently high risk aversion s.t. σ 1 u E 0 u 1 R FIN501 Asset Pricing Lecture 06 Equity Premium Puzzle (17) E R R σ R New problem emerges: Strong orce to consumption smooth over time (low intertemporal elasticity o consumption (IES)) due to concavity o utility unction vnm utility unction smoothing over states = smoothing over time CRRA gamma = 1/ IES Model predicts much higher risk-ree rate

18 Solution: Equity Premium or Low Risk-ree Rate Puzzle? FIN501 Asset Pricing Lecture 06 Equity Premium Puzzle (18) depart rom vnm utility preerence representation Found preerence representation that allows split Risk-aversion Intertemporal elasticity o substitution Kreps-Porteus (special case: vnm) Epstein-Zin (special case: CRRA vnm)

19 Lecture 06 Equity Premium Puzzle (19) Digression: Preerence or the timing o uncertainty resolution p $100 $150 $100 $ 25 0 p $150 $100 Kreps-Porteus $ 25 U 0 x 1, x 2 s = W x 1, E U 1 x 1, x 2 s Early (late) resolution i W(P 1, ) is convex (concave) Do you want to know whether you will get cancer at the age o 55 now?