Leads, Lags, and Logs: Asset Prices in Business Cycle Analysis David Backus (NYU), Bryan Routledge (CMU), and Stanley Zin (CMU) Zicklin School of Business, Baruch College October 24, 2007 This version: October 23, 2007 Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 1 / 46
Overview of recursive preferences Time preference Risk Preference ◮ Chew-Dekel ◮ Risk premiums Applications of Kreps-Porteus preferences ◮ Pricing kernels ◮ Risk sharing ◮ Business cycles (the paper in the title) Extensions? Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 1 / 46
Time preference Time preference Time aggregator V = V ( u t , U t +1 ) U t Additive preferences ∞ � β j u t + j U t = (1 − β ) u t + β U t +1 = (1 − β ) j =0 Why don’t we care about this? Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 2 / 46
Risk preference Risk preference overview Certainty equivalent functions Chew-Dekel preferences Small, lognormal, and extreme risks Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 3 / 46
Risk preference Risk preference Basics: states s ∈ { 1 , . . . , S } , consumption c ( s ), probabilities p ( s ) Certainty equivalent function: µ satisfying U ( µ, . . . , µ ) = U [ c (1) , . . . , c ( S )] Risk aversion: µ ( c ) ≤ E ( c ) Chew-Dekel preferences (risk aggregator M ) � µ = p ( s ) M [ c ( s ) , µ ] s Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 4 / 46
Risk preference Chew-Dekel examples Expected utility c α m 1 − α /α + m (1 − 1 /α ) M ( c , m ) = Weighted utility ( c / m ) γ c α m 1 − α /α + m [1 − ( c / m ) γ /α ] . M ( c , m ) = Disappointment aversion c α m 1 − α /α + m (1 − 1 /α ) M ( c , m ) = + δ I ( m − c )( c α m 1 − α − m ) /α I ( x ) = 1 if x > 0 , 0 otherwise Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 5 / 46
Risk preference Chew-Dekel as adjusted probabilities Expected utility �� � 1 /α p ( s ) c ( s ) α µ = s Weighted utility: ditto with p ( s ) c ( s ) γ p ( s ) ˆ = � u p ( u ) c ( u ) γ , Disappointment aversion: ditto with p ( s )(1 + δ I [ µ − c ( s )]) ˆ p ( s ) = � u p ( s )(1 + δ I [ µ − c ( s )]) , Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 6 / 46
Risk preference Small risks Two states (1 + σ, 1 − σ ), equal probs, Taylor series around σ = 0 Expected utility µ (EU) ≈ 1 − (1 − α ) σ 2 / 2 Weighted utility µ (WU) ≈ 1 − [1 − ( α + 2 γ )] σ 2 / 2 Disappointment aversion � � � � δ 4 + 4 δ σ 2 / 2 µ (DA) ≈ 1 − σ − (1 − α ) 4 + 4 δ + δ 2 2 + δ Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 7 / 46
Risk preference Lognormal risks Let: log c ∼ N( κ 1 , κ 2 ), rp = log[ E ( c ) /µ ] Expected utility rp(EU) = (1 − α ) κ 2 / 2 Weighted utility rp(WU) = [1 − ( α + 2 γ )] κ 2 / 2 Disappointment aversion rp(WU) = E2C2E Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 8 / 46
Risk preference Extreme risks Let: log E exp(log c ) = κ 1 + κ 2 / 2! + κ 3 / 3! + κ 4 / 4! Expected utility (1 − α ) κ 2 / 2 + (1 − α 2 ) κ 3 / 3! + (1 − α 3 ) κ 4 / 4! rp(EU) = Weighted utility [1 − ( α + 2 γ )] κ 2 / 2 + [1 − ( α + 2 γ ) 2 + γ ( α + γ )] κ 3 / 3! rp(WU) = + [1 − ( α + 2 γ ) 3 + 2 γ ( α + γ )( α + 2 γ )] κ 4 / 4! Disappointment aversion rp(DA) = Another E2C2E Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 9 / 46
Kreps-Porteus preferences Kreps-Porteus preferences Recursive preferences U t = V [ u t , µ t ( U t +1 )] Kreps-Porteus/Epstein-Zin-Weil [(1 − β ) u ρ t + βµ ρ t ] 1 /ρ V ( u t , µ t ) = � � 1 /α E t U α µ t ( U t +1 ) = t +1 IES = 1 / (1 − ρ ) CRRA = 1 − α α = ρ ⇒ additive preferences U t = U ρ Invariant to monotonic transformations: eg, � t /ρ Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 10 / 46
Kreps-Porteus preferences Kreps-Porteus overview Pricing kernels Risk sharing Business cycles Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 11 / 46
Kreps-Porteus preferences Kreps-Porteus pricing kernels Marginal rate of substitution β ( c t +1 / c t ) ρ − 1 [ U t +1 /µ t ( U t +1 )] α − ρ m t +1 = Note role of future utility Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 12 / 46
Kreps-Porteus preferences Kreps-Porteus pricing kernels (continued) Example: let consumption growth follow ∞ � log x t = log x + χ j w t − j j =0 Pricing kernel log m t +1 = constant + [( ρ − 1) χ 0 + ( α − ρ )( χ 0 + X 1 )] w t +1 ∞ � + ( ρ − 1) χ j +1 w t − j j =0 ∞ � β j χ j = (“Bansal-Yaron” term) X 1 j =1 Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 13 / 46
Kreps-Porteus preferences Kreps-Porteus risk sharing Pareto problem with two recursive agents Bryan did this a few weeks ago Issues ◮ Time-varying pareto weights ◮ Representative agent may look different from individuals ◮ Possible nonstationary consumption distribution Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 14 / 46
Kreps-Porteus preferences Kreps-Porteus business cycle overview Pictures: leads and lags in US data Equations: the usual suspects + bells & whistles Computations: loglinear approximation More pictures: leads and lags in the model Extensions Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 15 / 46
Leads and lags in data Leads and lags in US data Cross-correlation functions of GDP with ◮ Stock price indexes ◮ Interest rates and spreads ◮ Consumption and employment US data, quarterly, 1960 to present Quarterly growth rates ( log x t − log x t − 1 ) except ◮ Interest rates and spreads (used as is) ◮ Occasional year-on-year comparisons (log x t +2 − log x t − 2 ) Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 16 / 46
Leads and lags in data Stock prices and GDP S&P 500 1.00 1.00 Leads GDP Lags GDP Cross−Correlation with GDP 0.50 0.50 0.00 0.00 −0.50 −0.50 −1.00 −1.00 −10 −5 0 5 10 Lag Relative to GDP Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 17 / 46
Leads and lags in data Stock prices and GDP (year-on-year) S&P 500 (yoy) 1.00 1.00 Cross−Correlation with GDP 0.50 0.50 0.00 0.00 −0.50 −0.50 −1.00 −1.00 −10 −5 0 5 10 Lag Relative to GDP Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 18 / 46
Backus, Routledge, & Zin (NYU & CMU) Stock prices and GDP Cross−Correlation with GDP Cross−Correlation with GDP −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 −10 −10 Leads GDP NYSE Composite −5 −5 Lag Relative to GDP Lag Relative to GDP S&P 500 0 0 Leads and lags in data Lags GDP 5 5 Leads, lags, and logs 10 10 −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 Cross−Correlation with GDP Cross−Correlation with GDP −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 −10 −10 S&P 500 minus Short Rate Nasdaq Composite −5 −5 Lag Relative to GDP Lag Relative to GDP 0 0 5 5 10 10 −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 19 / 46
Backus, Routledge, & Zin (NYU & CMU) Interest rates and GDP Cross−Correlation with GDP Cross−Correlation with GDP −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 −10 −10 Yield Spread (10y−3m) Short Rate (3m) −5 −5 Lag Relative to GDP Lag Relative to GDP 0 0 Leads and lags in data 5 5 Leads, lags, and logs 10 10 −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 Cross−Correlation with GDP Cross−Correlation with GDP −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 −10 −10 Yield Spread (GDP yoy) −5 −5 Lag Relative to GDP Lag Relative to GDP Real Rate 0 0 5 5 10 10 −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 20 / 46
Backus, Routledge, & Zin (NYU & CMU) Consumption and GDP Cross−Correlation with GDP Cross−Correlation with GDP −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 −10 −10 −5 −5 Lag Relative to GDP Lag Relative to GDP Consumption Nondurables 0 0 Leads and lags in data 5 5 Leads, lags, and logs 10 10 −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 Cross−Correlation with GDP Cross−Correlation with GDP −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 −10 −10 −5 −5 Lag Relative to GDP Lag Relative to GDP Durables Services 0 0 5 5 10 10 −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 21 / 46
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