Lecture 22. The Equity Premium Puzzle

Transcription

1 Lecture 22 The Equity Premium Puzzle 1 The Equity Premium Puzzle by Mehra and Prescott (1985) was considered one of the most influential paper in Macroeconomics Again, it is meant to be an exercise that takes a simple dynamic general equilibrium macro model to the data Not surprisingly, the simple model is decisively rejected by the data 2 The puzzle attracted so much attention and led to a great deal of follow-up work This is why Lawrence Summers in his 1991 paper gives so much credit to The Equity Premium Puzzle In this sense, Mehra and Prescott (1985) is more influential than Hansen and Singleton (1982), the winner of the best paper prize (Frisch Medal) in Econometrica 3 Lawrence Summers 1991 paper The Scientific Illusion in Empirical Macroeconomics contains some interesting and provocative views 4 The rest of the presentation is taken from a note typeset by Greenwood

2 43 The Equity Premium: A Puzzle 431 The Problem à la Mehra and Prescott (1985) Facts: From the average return on equity from the Standard and Poor 500 index as 7%

3 CHAPTER 4 MANUSCRIPT The average yield on short term debt was less than 1% Can such a differential be explained in a frictionless Arrow-Debreu-McKenzie economy? Finding: For the class of economies studied the average real return on equity is at a maximum 04 percentage points higher than on short-term debt Puzzle: To get a low risk free interest rate in a growing economy you need a high elasticity of intertemporal substitution To get a large equity premium, you need ahighcoefficient of intertemporal substitution But one is the reciprocal of the other 432 The Environment Tastes U(c, α) = c1 α 1 1 α, 0 < α < Endowments n-state Markov chain in growth rates y 0 = x 0 y, where x {λ 1,, λ n } and φ ij =Pr[x t+1 = λ j x t = λ i ] 97

4 433 Asset Pricing CHAPTER 4 MANUSCRIPT or p t = βe{ U 1(y t+1 ) U 1 (y t ) [y s + p t+1 } (41) Since U 1 (y) =y α then p t = E{ X s=t+1 p t = P (y t,x t )=E[ β s t U 1(y s ) U 1 (y t ) y s} X s=t+1 β s t yt α y ys α s x t,y t ] Note that (y t,x t ) are legitimate state variables for the pricing function since y s = y t x t+1 x s Clearly, then P (y, x) is homogeneous of degree one in y From (41) P (y, i) =β nx φ ij (λ j y) α [yλ j + P (λ j y, j)]y α (42) j=1 Now, using the fact that P (y, i) is homogeneous of degree one in y, conjecture a solution of the form P (y, i) =w i y, where the constant w i will have to be determined Substituting this solution into (42) yields w i = β nx j=1 φ ij λ 1 α j (1 + w j ), for i = 1, 2,,n (43) 98

5 CHAPTER 4 MANUSCRIPT Therefore, w = βλw + γ, where w = w 1 w n, Λ = φ 11 λ 1 α 1 φ 1n λ 1 α n φ n1 λ 1 α 1 φ nn λ 1 α n, γ = β P j φ 1jλ 1 α j β P j φ njλ 1 α j Thus, w =[I βλ] 1 γ, assuming that I βλ 6=0 What is the expected return from holding equity The realized return, r ij, from moving from state (y, i) to (λ j y, j) is r ij = P (λ jy, j)+λ j y P (y, i) P (y, i) = λ j(w j + 1) w i 1 Expected Returns, Conditional on State: The expected return on equity, conditional on that the current state is i, is R i = nx φ ij r ij j=1 Next consider the price of one-period discount bond in state i, orp f i Clearly, p f i = P f (c, i) = βe[u 1(λ j y)] U 1 (y) = β P n j=1 φ iju 1 (λ j y) U 1 (y) = β nx j=1 φ ij λ α j 99

6 The return on this risk free asset is CHAPTER 4 MANUSCRIPT R f i = 1/p f 1 Expected Returns, Unconditional: To calculate the expected return on either equity or bonds one needs to know the unconditional probability of being in a particular state, say i This comes from the matrix equation π = πφ, where π =(π 1,, π n ) and Φ =[φ ij ] Therefore, the unconditional return on equity and bonds is R e = X π i R e i, and R f = X π i R f i The risk premium is R e R f 434 Findings Two-State Markov Chain λ 1 = 1 + Growth z} { µ + δ, 100

7 CHAPTER 4 MANUSCRIPT λ 2 = 1 + µ {z} δ, St Dev Calibration φ 11 = φ 22 φ and φ {z} 12 = φ 21 =(1 φ) Autocorrelation: 2φ 1 For the US economy the mean growth rate in consumption was 0018 Its standard deviation and autocorrelation were 0036 and -014 Matching these facts necessitated setting µ =0018, δ =0036, andφ =043 Now, clearly 0 < β < 1, andlet0 < α < 10 Let X = {(α, β) :0< β < 1, 0 < α < 10, and I βλ 6=0} This defines two functions, so to speak, where R f = R(α, β) and R e R f = P (α, β) As can be seen the model can t simultaneously generate an equity premium of 698% and risk-free return of 08% 435 Conclusions Within the context of a frictionless Arrow-Debreu-McKenzie world it is difficult to rationalize why the average return on equity was so high while the risk-free return was so low 101

8 CHAPTER 4 MANUSCRIPT Figure 41: Equity Premium and Risk-free Rate Combinations Source: Mehra and Prescott (1985), pg 155