The Cross-Section and Time-Series of Stock and Bond Returns

Transcription

1 The Cross-Section and Time-Series of Ralph S.J. Koijen, Hanno Lustig, and Stijn Van Nieuwerburgh University of Chicago, UCLA & NBER, and NYU, NBER & CEPR UC Berkeley, September 10, 2009

2 Unified Stochastic Discount Factor Model Models that price bonds do not explain expected stock returns nor stock return predictability Models that price stocks do not explain expected bond returns nor bond return predictability

3 Unified Stochastic Discount Factor Model Models that price bonds do not explain expected stock returns nor stock return predictability Models that price stocks do not explain expected bond returns nor bond return predictability Puzzling: unless bond and stock markets are completely segmented, the same pricing kernel (marginal utility of the bond/stock trader) should price both!

4 Unified Stochastic Discount Factor Model Models that price bonds do not explain expected stock returns nor stock return predictability Models that price stocks do not explain expected bond returns nor bond return predictability Puzzling: unless bond and stock markets are completely segmented, the same pricing kernel (marginal utility of the bond/stock trader) should price both! Main contribution: Develop a parsimonious SDF model that:

5 Unified Stochastic Discount Factor Model Models that price bonds do not explain expected stock returns nor stock return predictability Models that price stocks do not explain expected bond returns nor bond return predictability Puzzling: unless bond and stock markets are completely segmented, the same pricing kernel (marginal utility of the bond/stock trader) should price both! Main contribution: Develop a parsimonious SDF model that: 1 Explains the cross-section of expected stock and bond returns

6 Unified Stochastic Discount Factor Model Models that price bonds do not explain expected stock returns nor stock return predictability Models that price stocks do not explain expected bond returns nor bond return predictability Puzzling: unless bond and stock markets are completely segmented, the same pricing kernel (marginal utility of the bond/stock trader) should price both! Main contribution: Develop a parsimonious SDF model that: 1 Explains the cross-section of expected stock and bond returns 2 Captures the predictability of bond returns, dynamics of bond yields, and stock return predictability

7 Unified Stochastic Discount Factor Model Models that price bonds do not explain expected stock returns nor stock return predictability Models that price stocks do not explain expected bond returns nor bond return predictability Puzzling: unless bond and stock markets are completely segmented, the same pricing kernel (marginal utility of the bond/stock trader) should price both! Main contribution: Develop a parsimonious SDF model that: 1 Explains the cross-section of expected stock and bond returns 2 Captures the predictability of bond returns, dynamics of bond yields, and stock return predictability 3 Improves our understanding of connection between stock and bond returns

8 Unified Stochastic Discount Factor Model Models that price bonds do not explain expected stock returns nor stock return predictability Models that price stocks do not explain expected bond returns nor bond return predictability Puzzling: unless bond and stock markets are completely segmented, the same pricing kernel (marginal utility of the bond/stock trader) should price both! Main contribution: Develop a parsimonious SDF model that: 1 Explains the cross-section of expected stock and bond returns 2 Captures the predictability of bond returns, dynamics of bond yields, and stock return predictability 3 Improves our understanding of connection between stock and bond returns Ultimate goal: understanding of (macroeconomic) sources of risk agents demand compensation for when holding financial assets

9 Developing a Unified Stochastic Discount Factor Our SDF model contains three priced risk factors

10 Developing a Unified Stochastic Discount Factor Our SDF model contains three priced risk factors Motivated by a temporal decomposition of risk Alvarez and Jermann (2005) and Hansen and Scheinkman (2008) Shocks to the dividend yield permanent shocks to SDF

11 Developing a Unified Stochastic Discount Factor Our SDF model contains three priced risk factors Motivated by a temporal decomposition of risk Alvarez and Jermann (2005) and Hansen and Scheinkman (2008) Shocks to the dividend yield permanent shocks to SDF Shocks to the level factor of the yield curve transitory shocks to SDF

12 Developing a Unified Stochastic Discount Factor Our SDF model contains three priced risk factors Motivated by a temporal decomposition of risk Alvarez and Jermann (2005) and Hansen and Scheinkman (2008) Shocks to the dividend yield permanent shocks to SDF Shocks to the level factor of the yield curve transitory shocks to SDF Shocks to the Cochrane-Piazzesi (2005) factor relative importance of the two components

13 Interpreting the Value Premium Differential exposure to CP shocks enables the model to account for the spread on average returns between value and growth stocks Value stocks have a positive exposure to CP shocks, while growth stocks have a small, negative exposure

14 Interpreting the Value Premium Differential exposure to CP shocks enables the model to account for the spread on average returns between value and growth stocks Value stocks have a positive exposure to CP shocks, while growth stocks have a small, negative exposure CP turns out to be a strong forecaster of future economic activity Value stocks pay off when economic activity is expected to increase

15 Interpreting the Value Premium Differential exposure to CP shocks enables the model to account for the spread on average returns between value and growth stocks Value stocks have a positive exposure to CP shocks, while growth stocks have a small, negative exposure CP turns out to be a strong forecaster of future economic activity Value stocks pay off when economic activity is expected to increase This explains why value stocks are riskier and why the price of CP risk is expected to be positive

16 Related Literature Large, separately developed literatures on pricing bonds and on pricing stocks

17 Related Literature Large, separately developed literatures on pricing bonds and on pricing stocks Recent term structure models are able to: Explain the cross-section of expected bond returns across maturities E.g., Duffie and Kan (1996), Dai and Singleton (2000, 2002), Duffee (2002), and Cochrane and Piazzesi (2008) Capture the predictability of bond returns across maturities E.g., Stambaugh (1988), Campbell and Shiller (1991) and Cochrane and Piazzesi (2005, 2008)

18 Related Literature Large, separately developed literatures on pricing bonds and on pricing stocks Recent equity valuation models are able to: Explain the cross-section of expected stock returns E.g., Fama and French (1992) Capture the predictability of returns on market and on cross-section of stocks E.g., Petkova (2005), Lettau and Van Nieuwerburgh (2008), Cochrane (2008), Pastor and Stambaugh (2009), Binsbergen and Koijen (2009)

19 Related Literature Large, separately developed literatures on pricing bonds and on pricing stocks Empirical work on joint properties of stocks and bonds E.g., Chen, Roll, and Ross (1986), Ferson and Harvey (1991), Fama and French (1993), Brennan, Wang, and Xia (2004), Petkova (2006), Baker and Wurgler (2007), Lustig, Van Nieuwerburgh, and Verdelhan (2008)

20 Related Literature Large, separately developed literatures on pricing bonds and on pricing stocks Empirical work on joint properties of stocks and bonds Theoretical work on joint pricing of stocks and bonds Habit and long-run risk models Most closely related: Bekaert, Engstrom, and Xing (2008), Lettau and Wachter (2009) and Gabaix (2009)

21 Outline Affine valuation model for stocks and bonds Decomposition of SDF into permanent and transitory component Estimation from cross-section and time series of expected stock and bond returns What is CP and why does it help account for value premium Extensions

22 Affine Valuation Models We show how to decompose the SDF in any affine valuation model The SDF is given by: SDF t+1 = M t+1 /M t ( = exp y t 1 ) 2 Λ t ΣΛ t Λ t ε t+1 Short rate, risk prices, and state dynamics: y t = δ 0 + δ 1 X t, Λ t = Λ 0 + Λ 1 X t, X t+1 = γ 0 + ΓX t + ε t+1

23 Bond Pricing The model implies: P t (τ) = exp ( A τ + B τx ) t, A(τ) and B(τ) follow from the recursions: A(τ) = δ 0 + A(τ 1) + B (τ 1)γ 0 Λ 0ΣB(τ 1) B (τ 1)ΣB(τ 1), B(τ) = δ 1 + (Γ ΣΛ 1 ) B(τ 1), initiated at A(0) = 0 and B(0) = 0 1 N.

24 Bond Pricing The model implies: P t (τ) = exp ( A τ + B τx ) t, A(τ) and B(τ) follow from the recursions: A(τ) = δ 0 + A(τ 1) + B (τ 1)γ 0 Λ 0ΣB(τ 1) B (τ 1)ΣB(τ 1), B(τ) = δ 1 + (Γ ΣΛ 1 ) B(τ 1), initiated at A(0) = 0 and B(0) = 0 1 N. Bond yields: y t (τ) = log (P t (τ)) /τ

25 Bond Pricing The model implies: P t (τ) = exp ( A τ + B τx ) t, A(τ) and B(τ) follow from the recursions: A(τ) = δ 0 + A(τ 1) + B (τ 1)γ 0 Λ 0ΣB(τ 1) B (τ 1)ΣB(τ 1), B(τ) = δ 1 + (Γ ΣΛ 1 ) B(τ 1), initiated at A(0) = 0 and B(0) = 0 1 N. Bond yields: y t (τ) = log (P t (τ)) /τ Transposed risk-neutral companion matrix Θ = (Γ ΣΛ 1 )

26 Bond Pricing The model implies: P t (τ) = exp ( A τ + B τx t ), A(τ) and B(τ) follow from the recursions: A(τ) = δ 0 + A(τ 1) + B (τ 1)γ 0 Λ 0ΣB(τ 1) B (τ 1)ΣB(τ 1), B(τ) = δ 1 + (Γ ΣΛ 1 ) B(τ 1), initiated at A(0) = 0 and B(0) = 0 1 N. Bond yields: y t (τ) = log (P t (τ)) /τ Transposed risk-neutral companion matrix Θ = (Γ ΣΛ 1 ) B(τ) = (I Θ τ )(I Θ) 1 δ 1. B lim τ B(τ) = (I Θ) 1 δ 1.

27 No-Arbitrage Condition for Risky Assets For each risky asset j: ] log E t [SDF t+1 R j t+1 = 0,

28 No-Arbitrage Condition for Risky Assets For each risky asset j: ] log E t [SDF t+1 R j t+1 = 0, Let r j t+1 = E t[r j t+1 ] + ηj t+1, and Σ Xj = Cov(ε, η j )

29 No-Arbitrage Condition for Risky Assets For each risky asset j: ] log E t [SDF t+1 R j t+1 = 0, Let r j t+1 = E t[r j t+1 ] + ηj t+1, and Σ Xj = Cov(ε, η j ) ] Define rx j t+1 rj t+1 y t [η Var j t+1

30 No-Arbitrage Condition for Risky Assets For each risky asset j: ] log E t [SDF t+1 R j t+1 = 0, Let r j t+1 = E t[r j t+1 ] + ηj t+1, and Σ Xj = Cov(ε, η j ) ] Define rx j t+1 rj t+1 y t [η Var j t+1 No arbitrage condition implies: [ ] E rx j t+1 = Σ Xj (Λ 0 + Λ 1 E [X t ]) = Σ Xj ˆΛ 0.

31 No-Arbitrage Condition for Risky Assets For each risky asset j: ] log E t [SDF t+1 R j t+1 = 0, Let r j t+1 = E t[r j t+1 ] + ηj t+1, and Σ Xj = Cov(ε, η j ) ] Define rx j t+1 rj t+1 y t [η Var j t+1 No arbitrage condition implies: [ ] E rx j t+1 = Σ Xj (Λ 0 + Λ 1 E [X t ]) = Σ Xj ˆΛ 0. Estimate ˆΛ 0 to match average excess returns on test assets

32 Decomposing Affine Valuation Models Following Alverez and Jermann (2005), we decompose the pricing kernel as M t = M P t MT t, with: M T t = lim τ β t+τ /P t (τ), E t [M P t+1 ] = MP t

33 Decomposing Affine Valuation Models Following Alverez and Jermann (2005), we decompose the pricing kernel as M t = M P t MT t, with: M T t = lim τ β t+τ /P t (τ), E t [M P t+1 ] = MP t Proposition The SDF of any affine model can be decomposed into: M T t+1 M T t Mt+1 P Mt P = β exp ( B γ 0 + B (I Γ)X t B ε t+1), ( = β 1 exp δ 0 + B γ 0 [ B (I Γ) + δ 1] Xt 1 2 Λ tσλ t ( ) Λ t B ) εt+1, with constant β given by β = exp ( δ 0 + ( γ 0 Λ 0 Σ) B + 1 ) 2 B ΣB.

34 Conditional Variance Ratio AJ use metric L t (SDF t+1 ) = log E t [SDF t+1 ] E t [log SDF t+1 ] In Gaussian models: L t (SDF t+1 ) = 1 2 V t[sdf t+1 ] The conditional variance of the SDF that comes from the permanent component ω t = L t(m P t+1 /MP t ) L t (SDF t+1 ) = 1 B ΣΛ t 2 1B ΣB 1 2 Λ t ΣΛ, t = 1 E t[r b t+1 ( ) y t] max j E t [r j t+1 y t].

35 Link with Stocks and Bonds Variation in the transitory component of SDF linked to bond market Mt+1 T = (R Mt T t+1 ( )) 1 Without transitory component, yield curve is constant and bond risk premia are zero (e.g., C-CAPM) Shocks to Level factor capture shocks to the transitory component

36 Link with Stocks and Bonds Variation in the transitory component of SDF linked to bond market Variation in the permanent component of SDF linked to stock market Without permanent component, highest risk premium is that on long bond AJ use stocks and bonds to find E[ω] 1 Shocks to DP factor capture shocks to the permanent component

37 Link with Stocks and Bonds Variation in the transitory component of SDF linked to bond market Variation in the permanent component of SDF linked to stock market Return predictability shows ω t cannot be constant Bond risk premia move over time, with CP, generating time variation in conditional variance of transitory component Stock risk premia move over time, with DP, generating time variation in conditional variance of permanent component Shocks to CP & DP factor capture shocks to the conditional variance ratio ω t

38 Example of Structural Model Preferences are CRRA

39 Example of Structural Model Preferences are CRRA Aggregate consumption has two components: C t+1 = C P t+1 CT t+1 Transitory component follows AR(1) in logs Permanent component is random walk in levels Variance of shock to permanent component, s 2 t, varies over time (time-varying economic uncertainty)

40 Example of Structural Model Preferences are CRRA Aggregate consumption has two components: C t+1 = C P t+1 CT t+1 Transitory component follows AR(1) in logs Permanent component is random walk in levels Variance of shock to permanent component, s 2 t, varies over time (time-varying economic uncertainty) Conditional variance ratio ω t varies positively with s 2 t

41 Estimation Strategy 1 Estimate risk-neutral companion matrix Θ and short-rate parameters in δ 1 to match dynamics of forward rates (20); estimate δ 0 to match average level of interest rate (1), as in Cochrane and Piazzesi (2008)

42 Estimation Strategy 1 Estimate risk-neutral companion matrix Θ and short-rate parameters in δ 1 to match dynamics of forward rates (20); estimate δ 0 to match average level of interest rate (1), as in Cochrane and Piazzesi (2008) 2 Estimate average prices of Level, DP, and CP risk in ˆΛ 0 to match unconditional expected returns on stock and bond portfolios (3)

43 Estimation Strategy 1 Estimate risk-neutral companion matrix Θ and short-rate parameters in δ 1 to match dynamics of forward rates (20); estimate δ 0 to match average level of interest rate (1), as in Cochrane and Piazzesi (2008) 2 Estimate average prices of Level, DP, and CP risk in ˆΛ 0 to match unconditional expected returns on stock and bond portfolios (3) 3 Estimate the time-variation in risk prices in Λ 1 to match conditional expected returns on stock and bond portfolios (3)

44 Model Specification and Test Assets The state vector X t contains: CP factor (as in Cochrane and Piazzesi 2005) Level factor (first principal comp. of Fama-Bliss yields) Slope factor (unpriced) Curvature factor (unpriced) DP factor (log dividend yield on stock market)

45 Model Specification and Test Assets The state vector X t contains: CP factor (as in Cochrane and Piazzesi 2005) Level factor (first principal comp. of Fama-Bliss yields) Slope factor (unpriced) Curvature factor (unpriced) DP factor (log dividend yield on stock market) Dividend yield is not a priced risk factor in the bond market by setting Γ (1:4,5) =

46 Data Main set of 16 test assets CRSP value-weighted market portfolio (AMEX/NASDAQ/NYSE) 10 Fama-French portfolios sorted along book-to-market (BM) 5 CRSP bond portfolios with maturities 1y, 2y, 5y, 7y, and 10y

47 Data Main set of 16 test assets CRSP value-weighted market portfolio (AMEX/NASDAQ/NYSE) 10 Fama-French portfolios sorted along book-to-market (BM) 5 CRSP bond portfolios with maturities 1y, 2y, 5y, 7y, and 10y Fama-Bliss yield/forward rate data with maturities 1y, 2y,..., 5y Fama 1-month risk-free rate

48 Data Main set of 16 test assets CRSP value-weighted market portfolio (AMEX/NASDAQ/NYSE) 10 Fama-French portfolios sorted along book-to-market (BM) 5 CRSP bond portfolios with maturities 1y, 2y, 5y, 7y, and 10y Fama-Bliss yield/forward rate data with maturities 1y, 2y,..., 5y Fama 1-month risk-free rate Data at monthly frequency Sample period June December 2008

49 Data Main set of 16 test assets CRSP value-weighted market portfolio (AMEX/NASDAQ/NYSE) 10 Fama-French portfolios sorted along book-to-market (BM) 5 CRSP bond portfolios with maturities 1y, 2y, 5y, 7y, and 10y Fama-Bliss yield/forward rate data with maturities 1y, 2y,..., 5y Fama 1-month risk-free rate Data at monthly frequency Sample period June December 2008 Robustness analysis: corporate bonds, other equity portfolios

50 1. Cross-Section of Unconditional Expected Returns RN Our Model Level Level-only bonds DP Level + DP Panel A: Pricing Errors (in % per year) 1-yr yr yr yr yr Mkt BM BM BM BM BM BM BM BM BM BM MAPE Panel B: Prices of Risk Estimates CP Level DP

51 Decomposing Risk Premia Decomposition of the market and bond risk premia DP Level CP Decomposition of risk premia on value and growth

52 2. Time Series of Conditional Expected Returns Recall r j t+1 = E t[r j t+1 ] + ηj t+1 Return predictability governed by Λ 1 matrix

53 2. Time Series of Conditional Expected Returns Recall r j t+1 = E t[r j t+1 ] + ηj t+1 Return predictability governed by Λ 1 matrix Bond return predictability: Price of level risk depends on CP: Λ 1(2,1) = 0 Chosen to exactly match predictive coefficient of equally-weighted portfolio of CRSP bonds on lagged CP

54 2. Time Series of Conditional Expected Returns Recall r j t+1 = E t[r j t+1 ] + ηj t+1 Return predictability governed by Λ 1 matrix Bond return predictability: Price of level risk depends on CP: Λ 1(2,1) = 0 Chosen to exactly match predictive coefficient of equally-weighted portfolio of CRSP bonds on lagged CP Implies predictability of annual 2-yr through 5-yr returns, constructed from Fama-Bliss yields, on lagged CP

55 2. Time Series of Conditional Expected Returns Recall r j t+1 = E t[r j t+1 ] + ηj t+1 Return predictability governed by Λ 1 matrix Bond return predictability: Stock return predictability Price of DP risk depends on DP: Λ 1(5,5) = 0 Chosen to exactly match predictive coefficient of the aggregate stock market on lagged DP Λ 1(5,1) = 0 to make sure CP does not predict aggregate stock return

56 2. Time Series of Conditional Expected Returns Recall r j t+1 = E t[r j t+1 ] + ηj t+1 Return predictability governed by Λ 1 matrix Bond return predictability: Stock return predictability Price of DP risk depends on DP: Λ 1(5,5) = 0 Chosen to exactly match predictive coefficient of the aggregate stock market on lagged DP Λ 1(5,1) = 0 to make sure CP does not predict aggregate stock return Implies predictive coefficients for 10 book-to-market portfolios on lagged DP

57 Conditional Risk Pricing: Bond Return Predictability Rerun the CP (2005) bond return predictability regressions using simulations to form yields, forward rates, and bond returns Model Data

58 Conditional Risk Pricing: Stock Return Predictability Regress 10 BM portfolio excess returns on lagged DP Predictability of value and growth portfolios Data Model

59 Time Series of Expected Two frequencies in risk premia equity risk premium has 83 mo. half-life, bond risk premium only 3 mo Equity risk premium Bond risk premium Annual Risk Premium

60 3. Term Structure of Interest Rates Short Rate: y t (1) = δ 0 + δ 1 X t. Recall: Bond yields of maturity τ affine in X: y t (τ) = A(τ) τ B (τ) X t τ Forward rates also affine Estimate Θ and δ 1 to minimize pricing errors on 1-yr through 5-yr Fama-Bliss demeaned forward rates Estimate δ 0 to match average yield level of Fama-Bliss bonds

61 Implied Fit for Bond Yields 0.2 Model Data Volatility of yield pricing errors: 12bp, 7bp, 6bp, 9bp, 3bp per year

62 Summary and Remaining Questions Decomposing the SDF suggests three priced factors: 1 CP innovations to explain the cross-section of expected stock returns 2 Level innovations to explain the cross-section of expected bond returns, price of Level risk varies with CP 3 DP innovations to capture the level of expected stock returns, price of DP risk varies with DP

63 Summary and Remaining Questions Decomposing the SDF suggests three priced factors: 1 CP innovations to explain the cross-section of expected stock returns 2 Level innovations to explain the cross-section of expected bond returns, price of Level risk varies with CP 3 DP innovations to capture the level of expected stock returns, price of DP risk varies with DP What is CP factor? Why are value stocks riskier than growth stocks? Why is the price of CP risk positive?

64 Revisiting Conditional Variance Ratio 1.1 Contribution of Permanent Component to Conditional Variance of SDF Cochrane Piazzesi Factor Most the of the variation in ω is due to CP: correlation is -99.2% When CP is high, importance of the transitory shocks is high, and persistence of pricing kernel is low This happens when economic activity is expected to increase

65 Interpreting the CP Factor When CP is high, economic activity is expected to increase Predicting economic activity k = 1,..., 36 months ahead with CP: CFNAI t+k = c k + β k CP t + ε t+k, where CFNAI is the Chicago FED National Activity Indicator Shocks to CP are good news for the economy, hence positive price of risk

66 Interpreting the CP Factor 40 β k Months of CP lag 5 T statistic R squared

67 Interpreting the CP Factor and the Value Premium High values of CP forecast an increase in future economic activity Returns on value stocks are positively correlated with CP shocks (while growth stocks have zero or even negative correlation) Value stocks are riskier because they pay off exactly when economic activity is expected to increase

68 Stocks Also Predicted by CP rx j t+1 = rx 0 + ξ j s1 DP t + ξ j s2 CP t + η j t Predictability of value and growth portfolios by DP Data Model Predictability of value and growth portfolios by CP 0.2 Data Model

69 Comparison to Fama-French Three-Factor Model RN Our Model FF Panel A: Pricing Errors (in % per year) 1-yr yr yr yr yr Mkt BM BM BM BM BM BM BM BM BM BM MAPE Panel B: Prices of Risk Estimates CP MKT 5.08 Level SMB DP HML 6.93

70 Pricing Corporate Bonds RN SDF Our SDF Our SDF FF SDF not re-estimated re-estimated 1-yr yr yr yr yr Market BM BM BM BM BM BM BM BM BM BM Credit Credit Credit Credit MAPE CP/Market Level/SMB DP/HML

71 Pricing Other Test Assets 10 FF Size 10 FF E-P 25 FF Size/Value RN KLN FF RN KLN FF RN KLN FF MAPE Market Prices of Risk CP/Market Level/SMB Market/HML

72 Different Sample Periods RN kernel KLN kernel FF kernel RN kernel KLN kernel FF kernel MAPE Panel B: Market Prices of Risk CP/Market Level/SMB DP/HML

73 Maximum Sharpe Ratio Affine term structure models commonly imply maximum Sharpe ratios that are very high Maximum SR achieved through large long-short positions in bonds Impose constraints on positions: α w i 1, with α = 0,.5, 1 Model Data mean st.dev. mean st.dev. Panel A: Annual Sharpe ratios on individual bonds 1-yr 0.81 (0.28) 0.71 (0.28) 2-yr 0.71 (0.31) 0.48 (0.32) 5-yr 0.45 (0.33) 0.33 (0.30) 7-yr 0.33 (0.33) 0.33 (0.29) 10-yr 0.24 (0.32) 0.22 (0.30) Panel B: Annual Sharpe ratios on bond portfolios unconstr (0.07) α = (0.30) α = (0.30) α = (0.30)

74 Conclusions Main contribution: develop a parsimonious SDF model which 1 Explains the cross-section of expected stock and bond returns 2 Captures the predictability of bond and stock returns 3 Captures the dynamics of bond yields

75 Conclusions Main contribution: develop a parsimonious SDF model which Requires only three priced factors: 1 CP innovations to explain the cross-section of expected stock returns 2 Level innovations to explain the cross-section of expected bond returns 3 DP innovations to capture the level of expected stock returns

76 Conclusions Main contribution: develop a parsimonious SDF model which Requires only three priced factors: Main economic insights: CP factor captures the variation in importance of transitory component of pricing kernel This transitory component becomes more important 1-2 years before economic activity peaks

77 Conclusions Main contribution: develop a parsimonious SDF model which Requires only three priced factors: Main economic insights: CP factor captures the variation in importance of transitory component of pricing kernel This transitory component becomes more important 1-2 years before economic activity peaks Values stocks have cash-flows that are more sensitive to this cyclical component, hence the value premium

78 Model with Temporary and Permanent Component in Consumption Endowment: C t+1 = C P t+1 CT t+1 with ct+1 T = µ c + ρct T + σε t+1, ct+1 P = ct P 1 2 s2 t + s t η t+1, st+1 2 s2 = ν(st 2 s 2 ) + σ w w t+1, Preferences are CRRA: sdf t+1 = log(β) γ c t+1, = log(β) γµ c + γ(1 ρ)c T t + γ 2 s2 t γσε t+1 γs t η t+1,

79 Model with Temporary and Permanent Component in Consumption Term structure is affine: { ( P t (τ) = exp A(τ) + B(τ)ct T + C(τ) st 2 s 2)}. Decomposition sdf t+1 = sdft+1 T + sdf t+1 P follows general model Conditional variance ratio ] 1 2 V t [sdf t+1 P ω t = 1 2 V = 1 B γσ B2 σ C2 σw 2 ( 1 t [sdf t+1 ] 2 γ 2 σ 2 + γ 2 st 2 ). = 1- bond risk premium / maximum risk premium When economic uncertainty st 2 SDF becomes more important decreases, transitory component of

80 The CP Factor and NBER Recessions 1 CP and Recessions CP factor

81 Consistent Risk Pricing Across Stocks and Bonds GMM estimation where we allow for = risk price on CP and Level for stocks and bonds Point estimates in Λ 0 : 91.98/88.16, /-24.03, -2.12/-1.96 Incremental risk prices for bonds (GMM s.e.): (38.57) and (37.77) Cannot reject null hypothesis that risk prices are same for stocks and bonds

82 Internal consistency with CP Factor Consistency between first VAR element and implied CP? Simulate SDF model for 100,000 months Calculate model-implied nominal yields, forwards, and bond returns from simulation Re-estimate CP factor on model-implied bond returns and yields Does ĈP equal the first element of the state vector? Their correlation is 80%

83 Does the Slope Factor Also Work? No: Innovations of slope factor have correlation with CP innovations of only 17% Estimate model where slope is priced instead of CP: Λ 0(1) = 0 and Λ 0(3) = 0 and price of level risk depends on slope instead of CP: Λ 1(2,1) = 0 and Λ 1(2,3) = 0. MAPE deteriorates substantially from 40bp to 105bp per year Pricing errors contain a value spread, and a spread between shortand long-horizon bonds Slope factor plays no meaningful role in pricing either stock or bond portfolios

84 Why Are Value Stocks Riskier? Role of Cash-Flow News Decompose returns r t+1 E t [r t+1 ] = (E t+1 E t ) ρ j d t+1+j (E t+1 E t ) ρ j r t+1+j j=0 j=1 Cov(ε R,i t+1, εcp,i t+1 ) = Cov(εNCFDG,i t+1, ε CP,i t+1 ) Cov(εNFR,i t+1, εcp,i t+1 ) 0.5 Covariance of Unexpected Stock Returns with CP Innovations ICR NCFDG NFR (Growth) (Value) Market CP future bond returns high NCFDG on value stocks is good