Lecture 03: Sharpe Ratio, Bounds and the Equity Premium Puzzle

Transcription

1 Lecture 03: Sharpe Ratio, Bounds and the Equity Premium Puzzle Prof. Markus K. Brunnermeier Slide 03-1

2 $1 invested in show graph! Slide 03-2

3 Sharpe Ratios and Bounds Consider a one period security available at date t with payoff 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 f t+1 = 1/ E t [m t+1 ] Note that R f t will depend on the choice of m t+1 unless there exists a riskless portfolio Slide 03-3

4 Sharpe Ratios and Bounds (ctd( ctd.) Hence p t = (1/R f t+1) E t [x t+1 ] + Cov[m t+1, x t+1 ] price = expected PV + Risk adjustment positive correlation with the discount factor adds value Slide 03-4

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 = R t+1 E t [m t+1 R t+1 ] = 1 (vector) using R f t+1 = 1/ E t [m t+1 ], we get E t [m t+1 (R t+1 R f t+1 )] = 0 m-discounted expected excess return for all assets is zero. Slide 03-5

6 in Returns Since E t [m t+1 (R t+1 R f t+1 )] = 0 Cov t [m t+1,r t+1 -R tf ] = E t [m t+1 (R t+1 R t+1f )] E t [m t+1 ]E t [R t+1 R t+1f ] = - E t [m t+1 ] E t [R t+1 R t+1f ] That is, risk premium or expected excess return E t [R t+1 -R tf ] = - Cov t [m t+1,r t+1 ] / E[m t+1 ] is determined by its covariance with the stochastic discount factor Slide 03-6

7 Sharpe Ratio Multiply both sides with portfolio h E t [(R t+1 -R tf )h] = - Cov t [m t+1,r t+1 h] / E[m t+1 ] NB: All results also hold for unconditional expectations E[ ] Rewritten in terms of Sharpe Ratio =... Slide 03-7

8 Hansen-Jagannathan Bound Since ρ [-1,1] we have Theorem (Hansen-Jagannathan Bound): The ratio of the standard deviation of a stochastic discount factor to its mean exceeds the Sharpe Ratio attained by any portfolio. Slide 03-8

9 Hansen-Jagannathan Bound Theorem (Hansen-Jagannathan Bound): The ratio of the standard deviation of a stochastic discount factor to its mean exceeds the Sharpe Ratio attained by any portfolio. Can be used to easy check the viability of a proposed discount factor Given a discount factor, this inequality bounds the available risk-return possibilities The result also holds conditional on date t info Slide 03-9

10 Hansen-Jagannathan Bound expected return slope σ (m) / E[m] R f available portfolios σ Slide 03-10

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 ) - Stochastic discount factor Slide 03-11

12 Digression: if utility is in addition time-separable u(c 0,c 1 ) = v(c 0 ) + v(c 1 ) Then and Slide 03-12

13 Equity Premium Puzzle Recall E[R j ]-R f = -R f Cov[m,R j ] Now: E[R j ]-R f = -R f Cov[ 1 u,r j ]/E[ 0 u] Recall Hansen-Jaganathan bound Slide 03-13

14 Equity Premium Puzzle (ctd( ctd.) u u c 1 c 2 c 1 c 2 Equity Premium Puzzle: high observed Sharpe ratio of stock market indices low volatility of consumption (unrealistically) high level of risk aversion Slide 03-14

15 A simple example S=2, π 1 = ½, 3 securities with x 1 = (1,0), x 2 =(0,1), x 3 = (1,1) Let m=(½,1), σ= ¼ =[½(½ - ¾) 2 + ½(1 - ¾) 2 ] ½ Hence, p 1 =¼, p 2 = 1/2, p 3 = ¾ 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 Slide 03-15

16 Example: Where does SDF come from? 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-function m= 1 u (1,2,1) / E[ 0 u(1,2,1)]=(c 0 /c 1,1, c 0 /c 1,2 )=(1/2, 1/1) m=(½,1) since endowment=consumption Low consumption states are high m-states Risk-neutral probabilities combine true probabilities and marginal utilities. Slide 03-16