Fin 501: Asset Pricing 00:45 Lecture 04Bounds and Equity Premium Puzzle1 Lecture 04: Sharpe Ratio, Bounds and the Equity Premium Puzzle Equity Premium Puzzle Prof. Markus K. Brunnermeier

Fin 501: Asset Pricing 00:45 Lecture 04Bounds and Equity Premium Puzzle2 $1 invested in show graph!

Fin 501: Asset Pricing 00:45 Lecture 04Bounds and Equity Premium Puzzle3 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

Fin 501: Asset Pricing 00:45 Lecture 04Bounds and Equity Premium Puzzle4 Sharpe Ratios and Bounds (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

Fin 501: Asset Pricing 00:45 Lecture 04Bounds and Equity Premium Puzzle5 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.

Fin 501: Asset Pricing 00:45 Lecture 04Bounds and Equity Premium Puzzle6 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 t f ] = E t [m t+1 (R t+1 – R t+1 f )] – E t [m t+1 ]E t [R t+1 – R t+1 f ] = - E t [m t+1 ] E t [R t+1 – R t+1 f ] That is, risk premium or expected excess return E t [R t+1 -R t f ] = - Cov t [m t+1,R t+1 ] / E[m t+1 ] is determined by its covariance with the stochastic discount factor

Fin 501: Asset Pricing 00:45 Lecture 04Bounds and Equity Premium Puzzle7 Sharpe Ratio Multiply both sides with portfolio h E t [(R t+1 -R t+1 f )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 =...

Fin 501: Asset Pricing 00:45 Lecture 04Bounds and Equity Premium Puzzle8 Hansen-Jagannathan Bound –Since  2 [-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.

Fin 501: Asset Pricing 00:45 Lecture 04Bounds and Equity Premium Puzzle9 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

Fin 501: Asset Pricing 00:45 Lecture 04Bounds and Equity Premium Puzzle10 Hansen-Jagannathan Bound expected return RfRf  available portfolios slope  (m) / E[m]

Fin 501: Asset Pricing 00:45 Lecture 04Bounds and Equity Premium Puzzle11 Assuming Expected Utility c 0 2 R, c 1 2 R S U(c 0,c 1 )=  s  s u(c 0,c 1,s ) - Stochastic discount factor

Fin 501: Asset Pricing 00:45 Lecture 04Bounds and Equity Premium Puzzle12 Digression: if utility is in addition time-separable u(c 0,c 1 ) = v(c 0 ) + v(c 1 ) Then and

Fin 501: Asset Pricing 00:45 Lecture 04Bounds and Equity Premium Puzzle13 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

Fin 501: Asset Pricing 00:45 Lecture 04Bounds and Equity Premium Puzzle14 Equity Premium Puzzle (ctd.) uu c1c1 c2c2 Equity Premium Puzzle: high observed Sharpe ratio of stock market indices low volatility of consumption ) (unrealistically) high level of risk aversion c1c1 c2c2

Fin 501: Asset Pricing 00:45 Lecture 04Bounds and Equity Premium Puzzle15 A simple example S=2,  1 = ½, 3 securities with x 1 = (1,0), x 2 =(0,1), x 3 = (1,1) Let m=(½,1),  = 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

Fin 501: Asset Pricing 00:45 Lecture 04Bounds and Equity Premium Puzzle16 Example: Where does SDF come from? “representative agent” with –endowment: 1 in date 0, (2,1) in date 1 –utility U(c 0, c 1, c 2 )=ln c 0 +  s  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.