The Equity Premium. Blake LeBaron Reading: Cochrane(chap 21, 2017), Campbell(2017/2003) October Fin305f, LeBaron

Transcription

1 The Equity Premium Blake LeBaron Reading: Cochrane(chap 21, 2017), Campbell(2017/2003) October 2017 Fin305f, LeBaron

2 History Asset markets and real business cycle like models Macro asset pricing Measure macro risk Key early papers: Lucas(78), Econometrica Brenon Mehra/Prescott(JME 85) Fin305f, LeBaron

3 Goals Incorporate asset markets into RBC-like macro models Often highly simplified (exchange economies) Sometimes this doesn t matter Early big assumptions Complete markets or at some form of representative agent Zero transaction costs Fin305f, LeBaron

4 Facts: asset markets LeBaron merged data ( ) Nominal equity return 10.5% Real equity return 7.9% Std equity return 18.1% Variance of equity return Standard deviation of mean real ret: 1.59 Risk free rate, 3.8 Real risk free rate, 1.2% Std. risk free rate, 0.7% Var risk free rate, Monthly dynamics Autocorrelations of stock returns: 0.07 Autocorrelation of risk free: 0.95 Fin305f, LeBaron

5 Facts: asset markets Predictability facts Price/Div ratios Price/Earnings ratios Consumption/wealth ratio (CAY) Term premium Short term nominal rates VIX All technically forecast (significantly) future returns Some questions Risk Data snooping Fin305f, LeBaron

6 Facts: consumption Campbell paper Growth 2-3 percent Volatility about 2 percent (US postwar 1 percent) Covariance with stocks (very low) (table 5) divide by In the Kocherlakota data set this is , correlation 0.03 Fin305f, LeBaron

7 How well does this line up with rep like models? Kocherlakota, JEL, 1996 Fin305f, LeBaron

8 General model Representative St. max E 0 β t u(c t ) t=0 W t = C t + P t,e S t+1 + P t,b B t+1 = (P t,e + D t )S t + B t + Y t Fin305f, LeBaron

9 Euler equations 1 = E t {β u (C t+1 ) u (C t ) (1 + R t+1,j)} 1 = E t {β u (C t+1 ) u (C t ) (1 + R t+1,s)} 1 = E t {β u (C t+1 ) u (C t ) (1 + R t+1,f )} 1 = (1 + R t+1,f )E t {β u (C t+1 ) u (C t ) } 0 = E t {β u (C t+1 ) u (C t ) (R t+1,s R t+1,f )} j (Market return) (Risk free) Fin305f, LeBaron

10 Power utility u(c) = 1 1 γ c1 γ u (c) = c γ Standard power utility State/time separable Constant relative risk aversion, CRRA = γ Log preferences, γ = 1 Risk neutral, γ = 0 Fin305f, LeBaron

11 Kocherlakota (JEL, 1996) Estimate first order condition directly from macro/fin data Try different γ e t+1 = β( C t+1 C t ) γ (R t+1,s R t+1,f ) e t+1 = ( C t+1 C t ) γ (R t+1,s R t+1,f ) E(e t+1 ) = E(E t (e t+1 )) = 0 Report mean for e t, and t-test for equality with zero Data is Fin305f, LeBaron

12 Kocherlakota (JEL, 1996) γ Mean(e t ) t-test Fin305f, LeBaron

13 Another version of this 0 = E{( C t+1 C t ) γ (R t+1,s R t+1,f )} cov(x, y) = E(xy) E(x)E(y) cov(( C t+1 ) γ, R t+1,s R t+1,f ) + E(( C t+1 ) γ )E(R t+1,s R t+1,f ) = 0 C t C t 1 E(R t+1,s R t+1,f ) = E(( Ct+1 C t ) γ ) cov((c t+1 ) γ, R t+1,s R t+1,f ) C t Returns depend on covariance with consumption growth Payoff during good times, high returns Pays when you don t really need it Consumption CAPM (C-CAPM) Fin305f, LeBaron

14 Kocherlakota (JEL, 1996) γ r t+1,s r t+1,f % % % % % % Fin305f, LeBaron

15 Risk free rate puzzle 1 = (1 + R t+1,f )E t { βu (C t+1 ) u } (C t ) 1 = β(1 + R t+1,f )E t {( C t+1 C t ) γ } 0 = β(1 + R t+1,f )E t {( C t+1 C t ) γ } 1 e t+1 = β(1 + R t+1,f )( C t+1 C t ) γ 1 Fin305f, LeBaron

16 Risk free rate puzzle Assume β = 0.99 γ Mean(e t ) t-test Fin305f, LeBaron

17 EIS and CRRA for power utility σ = log(c t+1/c t ) log(1 + R f ) 1 = β( C t+1 ) γ 1 + R f C t log(1 + R f ) = log(β) γ log(c t+1 /C t ) log(c t+1 /C t ) = (1/γ)(log(β) + log(1 + R f )) σ = (1/γ) This measures response of consumption growth to interest rates (prices) As γ increases it takes a bigger interest rate to make consumers happy with a given c growth rate At actual (low) rates they will want to smooth by borrowing Fin305f, LeBaron

18 Another (common) statement pricing and the puzzle The basic Euler equation 1 = E t {β ( Ct+1 C t ) γ (1 + R t+1,j)} j (7.3.1) A feature of logs and expectations for X lognormal log(e t X ) = E t log(x ) + (1/2)var t (log(x )) Now take logs 0 = log[e t {β 0 = E t log[β ( Ct+1 C t ( Ct+1 C t (1/2)var t log[β ) γ (1 + R t+1,j)}] ) γ (1 + R t+1,j)]+ ( Ct+1 C t ) γ (1 + R t+1,j)] Fin305f, LeBaron

19 Another (common) statement pricing and the puzzle The two log parts are log[β E t log[β var t {log[β ( Ct+1 C t ( Ct+1 C t ( Ct+1 C t σ jc is the cov( c t+1, r j,t+1 ) ) γ (1 + R t+1,j)] = log β γ(c t+1 c t ) + r t+1,j = log β γ c t+1 + r t+1,j ) γ (1 + R t+1,j)] = log β γe t c t+1 + E t (r t+1,j) ) γ (1 + R t+1,j)]} = σ 2 j + γ 2 σ 2 c 2γσ jc Fin305f, LeBaron

20 Putting this together gives For all 0 = log β γe t c t+1 +E t (r t+1,j )+(1/2)(σ 2 j +γ 2 σ 2 c 2γσ ic ) (7.3.2) Risky Risk free E t (r t+1,j ) = log(β) + γe t c t+1 σ2 j + γ 2 σ 2 c 2γσ 2 jc 2 r t+1,f = log(β) + γe t c t+1 γ2 σ 2 c 2 Equity Premium σ 2 j E t (r t+1,j r t+1,f ) = γσ 2 jc σ2 j 2 is a term adjusting for log returns (known as a Jensen s inequality term) Otherwise premium depends directly on γ and σ 2 jc Fin305f, LeBaron

21 Calibrating Equity premium Using the Kocherlakota data set σ 2 jc = and σj 2 = gives γ = 83 Risk free High γ, σc 2 = , E c t+1 = 0.02 r t+1,f = log(β) + γe t c t+1 γ2 σ 2 c 2 For β = 0.99, γ = 83 this gives a rf = 139% For β = 0.99, γ = 10 this gives a rf = 21% For β = 0.99, γ = 1 this gives a rf = 3% For β = 1.2, γ = 10 this gives a r f = 1.3% Fin305f, LeBaron

22 More thoughts on the risk free puzzle High γ, σ 2 c = , E c t+1 = 0.02 r t+1,f = log(β) + γe t c t+1 γ2 σ 2 c 2 Since σ 2 c is low, much of this drives from the first term As EIS, 1/γ, falls then really high rates are necessary to make consumer happy with changes (increases) in consumption stream As γ gets really big, quadratic term kicks in and real rate can fall Fin305f, LeBaron

23 More results on using SDF Campbell derives some useful features for the SDF 0 = E t r t+1,i + E t m t [σ2 i + σ 2 m + 2σ im ] r t+1,f = E t m t+1 σ2 m 2 r t+1,i r t+1,f = σ2 i 2 σ im Key role of the SDF in terms of the equity premium We may need to use these later Fin305f, LeBaron

24 Hansen/Jagannathan bounds Much empirical work testing rep agent models All write down various Euler equations and then reject HJ want a more general approach 0 = E t (M t+1 (R t+1,j R t+1,f )) Idea is to get properties for M t+1 Any model gives an M t+1 and must meet these properties Reject large classes of models right away Fin305f, LeBaron

25 Deriving inequality Start with the Euler equation: 0 = E t (M t+1 (R t+1,j R t+1,f )) 0 = E(M t+1 (R t+1,j R t+1,f )) 0 = cov(m t+1, R t+1,j R t+1,f ) + E(M t+1 cov(m t+1, R t+1,j R t+1,f ) = E(M t+1 )E(R t+1,j R t+1,f ) Cauchy/Schwartz inequality: cov(x, y) σ x σ y σ M σ Rj E(M t+1 ) E(R t+1,j R t+1,f ) E(M t+1 > 0) σ M E(M t+1 ) E(R t+1,j R t+1,f ) σ Rj Fin305f, LeBaron

26 Evidence Kocherlakota data range ( ) β = 0.99, M t+1 = β( C t+1 C t ) γ σ M E(M t+1 ) E(R SP R f ) = 0.34 σ SP γ σ M E(M) Fin305f, LeBaron

27 HJ figure σ M E(M) Fin305f, LeBaron

28 More general (multi-asset) bound For vector of assets Ω is the variance/cov matrix σm 2 (1 E(M)E(1 + R)) Ω 1 (1 E(M)E(1 + R)) This formula maximizes right side value Optimal mean/variance portfolio Fin305f, LeBaron

29 The price volatility puzzle Prices move too much to be explained by dividend or discount movements Shiller,Grossman/Grossman, Campbell Related to the dividend discount model, and lots of stuff Fin305f, LeBaron

30 Dividend discount model(gordon) A very simple model for price volatility P t = j=1 D t+j (1 + R t,t+j ) Constant dividend growth, and discount rates P t = j=1 (1 + g) j D t (1 + R) j P t = D t(1 + g) R g P t (1 + g) = D t R g What about small changes in beliefs about R and g? Fin305f, LeBaron

31 More generally A very simple model for price volatility D t+j = D t P t D t = E t j (1 + g t+k ) k=1 j k=1 (1 + g t+k) j=1 (1 + R t,t+j ) Changes in P/D must come from either changes in perceived growth rates (cash/flow), or changes in discounts (returns) Dividends (cash/flows) are difficult to forecast Most likely something with R drives P/D volatility Fin305f, LeBaron

32 Bounds with SDF (Grossman/Shiller), (Campbell/Shiller) Can write with the SDF Put in your favorite M t,t+j P t = E t P t = E t P t D t = E t j=1 j=1 j=1 M t,t+j D t+j β j u (C t+j ) u (C t ) D t+j β j u (C t+j ) D t+j u (C t ) D t To get price/dividend ratio to move need either Movement in SDF Beliefs about dividend growth rates Connected to predictability As important as equity premium itself Fin305f, LeBaron