International Asset Pricing and Risk Sharing with Recursive Preferences

p. 1/3 International Asset Pricing and Risk Sharing with Recursive Preferences Riccardo Colacito Prepared for Tom Sargent s PhD class (Part 1)

Roadmap p. 2/3 Today International asset pricing (exchange rates, co-movements of int l stock market returns,...) Agents have recursive preferences No equilibrium trade

Roadmap p. 2/3 Today International asset pricing (exchange rates, co-movements of int l stock market returns,...) Agents have recursive preferences No equilibrium trade Tomorrow Agents consume bundles of domestic and foreign goods Trade arises as an equilibrium outcome Efficient risk-sharing with recursive preferences

Set the stage p. 3/3 Study the link between international stochastic discount factors and the depreciation of the US dollar: M uk t+1 M us t+1 = e t+1 e t where [ E t M i t+1 Rt+1 i ] = 1, i {us,uk}

Set the stage p. 3/3 Study the link between international stochastic discount factors and the depreciation of the US dollar: M uk t+1 M us t+1 = e t+1 e t where [ E t M i t+1 Rt+1 i ] = 1, i {us,uk} How to evaluate Mt+1? i 1. From prices: σ ( ) Mt+1 i E[R i t+1 Rf t+1] σ(rt+1 i Rf t+1) e.g. if E [ R i R f] 7%, σ ( R i R f) 17% then σ ( M i) 40%

The International equity premium puzzle p. 4/3 By no arbitrage E t [M f t+1 Rf t+1 ] [ = 1 = E t M h t+1 Rt+1 h ]

The International equity premium puzzle p. 4/3 By no arbitrage E t [M f t+1 Rf t+1 ] = 1 = E t [M h t+1 ] e t+1 R f t+1 e t

The International equity premium puzzle p. 4/3 By no arbitrage logm f t+1 logmh t+1 = log e t+1 e t

The International equity premium puzzle p. 4/3 By no arbitrage logm f t+1 logmh t+1 = log e t+1 e t Correlation of stochastic discount factors σ 2 m f +σ 2 m h 2ρ mf,m hσ m fσ m h = σ2 π

The International equity premium puzzle p. 4/3 By no arbitrage logm f t+1 logmh t+1 = log e t+1 e t Correlation of stochastic discount factors σ 2 m f }{{} 20% +σ 2 m h }{{} 20% 2ρ mf,m h }{{} 0.96 σ m fσ m h }{{} 20% = σ 2 π }{{} 1.5% Assuming identical CRRA preferences: logm i t+1 = γδc i t+1 γ 2 σ 2 Δc f +γ 2 σ 2 Δc h 2ρ Δc f,δc hγ2 σ Δc fσ Δc h = σ 2 π

The International equity premium puzzle p. 4/3 By no arbitrage logm f t+1 logmh t+1 = log e t+1 e t Correlation of stochastic discount factors σ 2 m f }{{} 20% +σ 2 m h }{{} 20% 2ρ mf,m h }{{} 0.96 σ m fσ m h }{{} 20% = σ 2 π }{{} 1.5% Assuming identical CRRA preferences: logm i t+1 = γδc i t+1 γ 2 σ 2 Δc f }{{} 20% +γ 2 σ 2 Δc h }{{} 20% 2ρ Δc f,δc h γ2 σ Δc fσ Δc h }{{} 20% = σ 2 π

The International equity premium puzzle p. 4/3 By no arbitrage logm f t+1 logmh t+1 = log e t+1 e t Correlation of stochastic discount factors σ 2 m f }{{} 20% +σ 2 m h }{{} 20% 2ρ mf,m h }{{} 0.96 σ m fσ m h }{{} 20% = σ 2 π }{{} 1.5% Assuming identical CRRA preferences: logm i t+1 = γδc i t+1 γ 2 σ 2 Δc f }{{} 20% +γ 2 σ 2 Δc h }{{} 20% 2ρ Δc f,δc h }{{} 0.3 γ 2 σ Δc fσ Δc h }{{} 20% = σ 2 π }{{} 28%

The puzzle in a cross section of countries HJ bound and depreciation rate: high correlation of SDF and low variance of depreciation rate. 1.0 0.9 0.85 Prices UK France Germany Netherlands Sweden Correlations 0.35 0.3 0.15 0 0 2 4 17 18 21 24 27 30 p. 5/3

The puzzle in a cross section of countries Consumption data and CRRA preferences: low correlation of SDF and high variance of depreciation rate. 1.0 0.9 0.85 Prices UK France Germany Netherlands Sweden Correlations 0.35 0.3 Quantities 0.15 0 0 2 4 17 18 21 24 27 30 p. 5/3

The puzzle in a cross section of countries This paper: high correlation of SDF, low correlation of Δc and low variance of depreciation rate. 1.0 0.9 0.85 corr(m h,m f ) UK France Germany Netherlands Sweden Correlations 0.35 0.3 corr( c h, c f ) 0.15 0 0 2 4 17 18 21 24 27 30 p. 5/3

The plan p. 6/3 1. Asset pricing within each country The long-run risk model by Bansal and Yaron 2. Asset pricing across countries The international long-run risks model by Colacito and Croce

Comparison of Utilities p. 7/3 Expected Utility U t = (1 δ)logc t +δe t U t+1 Risk Sensitive preferences U t = (1 δ)logc t +δθloge t exp { Ut+1 θ }

Comparison of Utilities p. 7/3 Expected Utility U t = (1 δ)logc t +δe t U t+1 Risk Sensitive preferences U t = (1 δ)logc t +δθloge t exp { Ut+1 (1 δ)logc t +δe t U t+1 + δ 2θ V tu t+1 θ }

Comparison of Utilities p. 7/3 Expected Utility U t = (1 δ)logc t +δe t U t+1 Risk Sensitive preferences What about SDF s? U t = (1 δ)logc t +δθloge t exp { Ut+1 (1 δ)logc t +δe t U t+1 + δ 2θ V tu t+1 θ }

Comparison of SDF s p. 8/3 SDF s can be calculated as m t+1 = log U t/ C t+1 U t / C t

Comparison of SDF s p. 8/3 SDF s can be calculated as m t+1 = log U t/ C t+1 U t / C t 1. Expected Utility m t+1 = δ Δc t+1

Comparison of SDF s p. 8/3 SDF s can be calculated as m t+1 = log U t/ C t+1 U t / C t 1. Expected Utility m t+1 = δ Δc t+1 2. Risk Sensitive Preferences m t+1 = δ Δc t+1 + U t+1 θ loge t exp { Ut+1 θ }

Comparison of SDF s p. 8/3 SDF s can be calculated as m t+1 = log U t/ C t+1 U t / C t 1. Expected Utility m t+1 = δ Δc t+1 2. Risk Sensitive Preferences m t+1 = δ Δc t+1 + U t+1 θ loge t exp { Ut+1 θ } Hansen-Jagannathan: SDF s should be highly volatile

Two states examples p. 9/3 2.2 Log Consumption 2 c t+1 = 2 1.8 1.6 1.4 1.2 1 c t+1 = 1 0.8 0 10 20 30 40 Time

Two states examples p. 9/3 Log Consumption 2.2 2 c t+1 = 2 + 0.1 c t+1 = 2 1.8 1.6 1.4 1.2 1 0.8 c t+1 = 1 c t+1 = 1 0.1 0 10 20 30 40 Time

Two states examples p. 9/3 Log Consumption 2.2 c t+1 = 2 + 0.1 2 c t+1 = 2 1.8 2.2 2 1.8 Utility 1.6 1.6 1.4 1.4 1.2 1.2 1 c t+1 = 1 c t+1 = 1 0.1 0.8 0 10 20 30 40 Time 1 0.8 0 10 20 30 Time

Two states examples p. 9/3 Log Consumption 2.2 c t+1 = 2 + 0.1 2 c t+1 = 2 1.8 2.2 2 1.8 Utility 1.6 1.4 1.6 1.4 E 0 [U 1 ] = 1.5, V 0 [U 1 ] = (0.5) 2 E 0 [U 1 ] = 1.5, V 0 [U 1 ] = (0.6) 2 1.2 1.2 1 c t+1 = 1 c t+1 = 1 0.1 0.8 0 10 20 30 40 Time 1 0.8 0 10 20 30 Time

Two states examples p. 9/3 Log Consumption 2.2 2 1.8 c t+1 = 2 + c t+1 = 2 t j=0 ρ j x 2.2 2 1.8 Utility 1.6 1.6 1.4 1.4 1.2 1.2 1 c t+1 = 1 t 0.8 c t+1 = 1 0 10 20 30 40 Time j=0 ρ j x 1 0.8 0 10 20 30 Time

p. 10/3 Exercise Let log-consumption growth evolve according to Δc t+1 = c 0 + t j=0 ρ j x 0 where c 0 N(0,σc) 2 and x 0 N(0,σx) 2 Let preferences be described by U t = (1 δ)logc t +δθloge t exp { Ut+1 θ } Show that the premium of a claim to consumption in excess of the risk-free rate is increasing in 1. σ 2 x 2. ρ

Exercise: hints p. 11/3 By definition the gross return of a claim to consumption is R c t+1 = Pc t+1 +C t+1 P c t = 1+vc t+1 v c t exp{δc t+1 } where v t = P t /C t is the price-consumption ratio. Define the log-stochastic discount factor as m t+1 = log U t/ C t = δ Δc t+1 + U t+1 U t / C t+1 θ loge t exp { Ut+1 θ } [ Solve for v t using E t exp{mt+1 }Rt+1] c = 1. (Hint: vt is constant). Solve for the risk-free rate, R f ] t+1, using E t [exp{m t+1 }R f t+1 = 1. (Hint: R f t+1 is known at date t). Compute the premium.

Summary p. 12/3 The equity premium puzzle: consumption is too smooth to make the stochastic discount factor volatile enough in the HJ sense. The Bansal and Yaron recipe: 1. small, but long-lasting shocks to consumption (ρ is large) 2. that investors care about (1/θ = 0) can increase the volatility of the stochastic discount factor. How does this extend to international asset pricing?

International Asset Pricing p. 13/3 Facts: 1. Equity Sharpe ratios are large in the cross-section of major industrialized countries International SDF s must be very volatile! 2. Correlation of equity markets returns is large ( 0.6) Puzzling because fundamentals have low correlations (i.e. cash flows are poorly correlated across countries) 3. The volatility of real exchange rates movements is in the 10 15%. The fix for 2+3: the int l correlation of SDF s must be high!

Rules of the game and outline p. 14/3 1. We want: to reconcile the correlation of SDF from (a) prices (b) quantities

Rules of the game and outline p. 14/3 1. We want: to reconcile the correlation of SDF from (a) prices (b) quantities low volatility of depreciation rate low volatility of consumption growth low correlation of consumption growths

Setup of the economy p. 15/3 Endowment economy. Two country specific goods. Complete home bias in consumption.

Setup of the economy p. 15/3 Endowment economy. Two country specific goods. Complete home bias in consumption. Epstein, Zin and Weil preferences: U i t = {(1 δ)(ct) i 1 γ θ +δ [ E t (Ut+1) i 1 γ] 1} θ 1 γ θ, i {h,f} where θ = 1 γ 1 1/ψ

Stochastic discount factors p. 16/3 Assume ψ = 1: U i t = (1 δ)logct i + δ 1 γ loge [ ] t exp(1 γ)u i t+1 The stochastic discount factors are logm i t+1 = log Ui / C i t+1 U i / C i t = logδ +log Ci t C i t+1 +log exp{(1 γ)u i t+1} E t [ exp{(1 γ)u i t+1 } ] Brandt, Cochrane and Santa-Clara use: logm i t+1 = logδ +log Ci t C i t+1

Remainder of the economy p. 17/3 Home country Δc h t = μ c +x h t 1 +σε h c,t x h t = ρx h t 1 +σφ e ε h x,t Foreign country Δc f t = μ c +x f t 1 +σεf c,t x f t = ρx f t 1 +σφ eε f x,t Shocks are i.i.d. within each country Shocks are correlated across countries ρ c = corr(ε h c,t,ε f c,t) ρ x = corr(ε h x,t,ε f x,t)

p. 18/3 Calibration δ γ ψ θ μ c σ ρ φ e ρ x ρ c.998 4.25 2 6.5.0015.0068.987.048 1.3 m i t+1 = θlogδ θ ψ Δci t +(θ 1)logR i c,t+1 Δc i t = μ c +x i t 1 +σε i c,t x i t = ρx i t 1 +σφ e ε i x,t Preferences: Low risk aversion (γ) IES from Bansal, Gallant and Tauchen (2004) Monthly model: high discounting

Calibration p. 18/3 δ γ ψ θ μ c σ ρ φ e ρ x ρ c.998 4.25 2 6.5.0015.0068.987.048 1.3 m i t+1 = θlogδ θ ψ Δci t +(θ 1)logR i c,t+1 Δc i t x i t = μ c +x i t 1 +σε i c,t = ρx i t 1 +σφ e ε i x,t Consumption process: Average consumption growth 2% Standard deviation of consumption growth 2.5% Variance explained by long run risk 7 8%

p. 18/3 Calibration δ γ ψ θ μ c σ ρ φ e ρ x ρ c.998 4.25 2 6.5.0015.0068.987.048 1.3 m i t+1 = θlogδ θ ψ Δci t +(θ 1)logR i c,t+1 Δc i t = μ c +x i t 1 +σε i c,t x i t = ρx i t 1 +σφ e ε i x,t Cross correlations of shocks: Correlation of consumption growths 0.3

Three ingredients p. 19/3 We can solve the puzzle by appropriately combining three ingredients:

Three ingredients p. 19/3 We can solve the puzzle by appropriately combining three ingredients: 1. Use Epstein and Zin preferences: BCSC (2005): m i t+1 = E [ g(δc i t+1) I t+1 ] = γδc i t+1 This paper: m i t+1 = E [ g(δc i t+1,δc i t+2,δc i t+3,...) I t+1 ] Alter the conditional distribution of ( Δc h,δc f) : Δc i t+1 = μ c +x i t +σε i c,t+1 x i t+1 = ρ i x i t +σφ e ε i x,t+1 by assuming 2. High persistence ρ i 3. High cross country correlation corr ( ) ε h x,t+1,ε f x,t+1

Stochastic discount factors p. 20/3 m i t+1 = θlogδ 1 ψ xi t γσε i c,t+1 + δ(1 γψ) ψ(1 ρδ) σφ eε i x,t+1

Stochastic discount factors p. 20/3 m i t+1 = θlogδ 1 ψ xi t γσε i c,t+1 + δ(1 γψ) ψ(1 ρδ) σφ eε i x,t+1 1 0.9 Correlation of stochastic discount factors 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.5 1 1.5 2 2.5 Intertemporal Elasticity of substitution (ψ)

Stochastic discount factors p. 20/3 m i t+1 = θlogδ 1 ψ xi t γσε i c,t+1 + δ(1 γψ) ψ(1 ρδ) σφ eε i x,t+1 1 0.9 Correlation of stochastic discount factors 0.8 0.7 0.6 0.5 0.4 0.3 ρ=0.987 ρ=0.9 0.2 0.1 0 0 0.5 1 1.5 2 2.5 1/γ Intertemporal Elasticity of substitution (ψ)

Stochastic discount factors p. 20/3 m i t+1 = θlogδ 1 ψ xi t γσε i c,t+1 + δ(1 γψ) ψ(1 ρδ) σφ eε i x,t+1 1 0.9 Correlation of stochastic discount factors 0.8 0.7 0.6 0.5 0.4 0.3 ρ x =1 ρ x =0.75 0.2 0.1 0 0 0.5 1 1.5 2 2.5 1/γ Intertemporal Elasticity of substitution (ψ)

Exchange rate depreciation p. 21/3 Var( et+1 e t ) { = 2(1 ρ x) ψ 2 1 1 ρ 2 + [ δ(1 γψ) (1 ρδ) ] } 2 φ 2 eσ 2 +2γ 2 (1 ρ c )σ 2

Exchange rate depreciation p. 21/3 Var( et+1 e t ) { = 2(1 ρ x) ψ 2 1 1 ρ 2 + [ δ(1 γψ) (1 ρδ) ] } 2 φ 2 eσ 2 +2γ 2 (1 ρ c )σ 2 70 Volatility of depreciation rate (Annualized %) 30 20 15 13 11 8 6 4 2 0 0 1.5 3 4.25 6 7.5 10 15 Risk aversion (γ)

Exchange rate depreciation p. 21/3 Var( et+1 e t ) { = 2(1 ρ x) ψ 2 1 1 ρ 2 + [ δ(1 γψ) (1 ρδ) ] } 2 φ 2 eσ 2 +2γ 2 (1 ρ c )σ 2 70 Volatility of depreciation rate (Annualized %) 30 20 15 13 11 ρ x =0.9 ρ x =1 8 6 4 2 0 0 1.5 3 4.25 6 7.5 10 15 Risk aversion (γ)

Exchange rate depreciation p. 21/3 Var( et+1 e t ) { = 2(1 ρ x) ψ 2 1 1 ρ 2 + [ δ(1 γψ) (1 ρδ) ] } 2 φ 2 eσ 2 +2γ 2 (1 ρ c )σ 2 70 Volatility of depreciation rate (Annualized %) 30 20 15 13 11 ρ x =0.9, ρ=.98 ρ x =1, ρ=.987 8 6 4 2 0 0 1.5 3 4.25 6 7.5 10 15 Risk aversion (γ)

Every assumption counts p. 22/3 Ingredients needed to solve the puzzle: 1. Disentangle elasticity of substitution from risk aversion 2. Highly persistent predictable component 3. Highly correlated predictable components

Introducing dividends p. 23/3 The system becomes Δc i t = μ c +x i t 1 +σε i c,t Δd i t = μ d +λx i t 1 +σφ d ε i d,t x i t = ρx i t 1 +σφ e ε i x,t sssssssssssssssssssssssssssssssssssss i {h, f} Shocks are i.i.d. within each country Shocks are correlated across countries

Introducing dividends p. 23/3 The system becomes Δc i t = μ c +x i t 1 +σε i c,t Δd i t =.0007+3 x i t 1 +σ 5 ε i d,t x i t = ρx i t 1 +σφ e ε i x,t sssssssssssssssssssssssssssssssssssss i {h, f} Shocks are i.i.d. within each country Shocks are correlated across countries Calibrate coefficients of dividend growth to match: Average dividend growth 1% Standard deviation of dividend growth 12% Leverage is 5 Small correlation of dividend growths: corr ( ) ε h d,t,εf d,t 0

Introducing dividends: results p. 24/3 US UK Model ρ ( m h,m f) - - 0.93 σ( et+1 e t ) 11.21 11.83 E(r d r f ) 7.02 9.17 7.01 σ(r ( d r f ) ) 17.13 22.83 19.60 ρ rd h rh f,rf d rf f 0.60 0.58 E(r f ) 1.47 1.62 1.33 σ(r ( f ) ) 1.53 2.92 1.19 ρ rf h,rf f 0.65 1.00

Estimating long run risks p. 25/3 Can we estimate this model? Δch t Δc f t xh t x f t = = 1 0 0 1 ρh 0 0 ρ f xh t 1 x f t 1 xh t 1 x f t 1 + σ 0 0 σ + 1 ρ x 1 εh c,t ε f c,t 1 2 σφ e 0 0 σφ e ρ x εh x,t ε f x,t

Estimating long run risks p. 25/3 Can we estimate this model? Δch t Δc f t xh t x f t = = 1 0 0 1 ρh 0 0 ρ f xh t 1 x f t 1 xh t 1 x f t 1 + σ 0 0 σ + 1 ρ x 1 εh c,t ε f c,t 1 2 σφ e 0 0 σφ e ρ x εh x,t ε f x,t Roadmap: 1. Use consumption data only Use Kalman filter to get a recursive representation of the likelihood function Multi-country provide inconclusive evidence

Likelihood ratio tests p. 26/3 0.9 0.8 0.01 0.7 0.05 0.025 0.6 0.10 x US = x UK 0.5 0.4 0.3 0.2 0.10 0.1 0.05 0.025 0.01 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 corr(x US,x UK )

A simulation exercise p. 27/3 Sample Size ρ(x,x ) # of c s # of x s 50 80 100 200 1 1 1 (0.000, 2.107) (0.000, 1.478) (0.000, 1.234) (0.000, 0.746) 2 1 (0.000, 0.695) (0.066, 0.667) (0.093, 0.637) (0.178, 0.527) 5 1 (0.135, 0.553) (0.182, 0.500) (0.204, 0.476) (0.242, 0.435) 0.95 2 2 (0.000, 1.108) (0.000, 0.740) (0.000, 0.672) (0.099, 0.552) 5 5 (0.000, 0.583) (0.115, 0.559) (0.149, 0.525) (0.218, 0.448) 0.9 2 2 (0.000, 1.159) (0.000, 0.780) (0.000, 0.683) (0.138, 0.580) 5 5 (0.000, 0.610) (0.089, 0.564) (0.131, 0.531) (0.217, 0.457) Notes - Each column reports the 95% confidence interval for the estimated ϕ e parameter for simulated samples of increasing size. The true value of ϕ e is 0.34.

Predictive regressions p. 28/3 Regress consumption growth on lagged values of price-dividend ratio risk-free rate consumption-output ratio consumption growth Fitted consumption growth is predictive component (x) Use annual data from 1929 to 2006 Repeat analysis for US and UK separately

Predictive regressions: results p. 29/3 F stat R 2 ρ x US UK US UK US UK corr ( x US,x UK) Pd and risk-free 6.641 3.476 0.145 0.086 0.768 0.787 0.579 (0.002) (0.036) (0.076) (0.074) [0.137, 0.896] All predictive variables 6.585 6.852 0.315 0.278 0.672 0.759 0.758 (0.000) (0.000) (0.074) (0.074) [0.531, 0.922] Pd only 7.299 6.013 0.086 0.076 0.885 0.726 0.849 (0.008) (0.016) (0.065) (0.099) [0.762, 0.941]

FX volatility and correlation of long-run risks p. 30/3 0.02 UK 0 0.02 US 0.04 0.06 ρ(x us,x uk )=0.22 ρ(x us,x uk )=0.68 ρ(x us,x uk )=0.73 ρ(x us,x uk )=0.87 0.08 σ π =17.55 σ π =7.67 σ π =13.81 σ π =11.02 1930 1940 1950 1960 1970 1980 1990 2000

FX volatility and correlation of long-run risks: post 1970 p. 31/3 14 13.5 13 R² = 0.9238 e 12.5 12 11.5 11 10.5 10 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 corr(x US, x UK )

FX volatility and correlation of long-run risks: table p. 32/3 R 2 β Var(x US x UK ) Var( e) Pd and risk-free 0.929-5.02 0.007 [0.013] All predictive variables 0.610-8.59 0.004 [0.077] Pd only 0.792-12.34 0.003 [0.014]

Estimating preference parameters p. 33/3 Use Euler equation restrictions (each country has two domestic and two foreign assets) and first two moments of FX change to estimate preference parameters Two exercises: 1. use OLS estimates of previous tables 2. jointly estimate OLS parameters and preference parameters

GMM estimation p. 34/3 Conditional Estimation Joint Estimation P/D P/D, R f All P/D P/D, R f All ψ 4.094 1.371 1.276 2.781 2.719 2.001 [ 0.398, 6.739 ] [ 0.391, 1.792 ] [ 0.404, 1.992 ] [ 0.243, 2.971 ] [ 0.288, 3.030 ] [ 0.274, 3.099 ] γ 4.359 3.402 2.936 3.387 3.225 4.147 [ 3.181, 9.575 ] [ 2.282, 7.376 ] [ 2.144, 6.339 ] [ 1.014, 10.267 ] [ 1.090, 9.509 ] [ 1.167, 10.802 ] ρ x - - - 0.997 0.997 0.988 [ 0.427, 0.999 ] [ 0.413, 0.999 ] [ 0.662, 0.999 ] ρ ( x US,x UK) - - - 0.879 0.867 0.830 Wald-stat p-value γ 1/ψ p-value J-stat p-value [ 0.618, 0.936 ] [ 0.582, 0.939 ] [ 0.471, 0.963 ] - - - 0.000 0.000 0.000 0.000 0.002 0.008 0.000 0.001 0.000 0.000 0.000 0.000 0.035 0.000 0.011

Concluding remarks p. 35/3 Key ingredients Separate elasticity of substitution from risk aversion Highly persistent predictable component Highly correlated predictable components It is possible to explain low volatility of the depreciation of the US dollar high equity premium high persistence of the risk free rate high correlation of int l financial markets correlation of bonds low correlation of consumption growths low persistence of consumption growths