Exotic (Recursive) Preferences & Cyclical Properties of US Asset - PowerPoint PPT Presentation

“Exotic” (Recursive) Preferences & Cyclical Properties of US Asset Returns David Backus (NYU), Bryan Routledge (CMU), and Stanley Zin (CMU) Birkbeck College | November 18, 2008 This version: December 10, 2008 Backus, Routledge, & Zin (NYU & CMU) Exotic + Cyclical Returns 1 / 37

Overview Time preference Risk preference ◮ Chew-Dekel ◮ Risk premiums Recursive preferences Applications of recursive preferences ◮ Pricing kernels ◮ Risk sharing ◮ Asset returns Backus, Routledge, & Zin (NYU & CMU) Exotic + Cyclical Returns 1 / 37

Time preference Time preference Additive preferences ∞ � β j u t + j = (1 − β ) u t + β U t +1 = (1 − β ) U t j =0 Time aggregator V = V ( u t , U t +1 ) U t (discounting built into V 2 ) Why don’t we care about this? Backus, Routledge, & Zin (NYU & CMU) Exotic + Cyclical Returns 2 / 37

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 )] Properties: ◮ Sure things: µ ( c ) = E ( c ) = c ◮ FOSD: µ ( c + a ) ≥ µ ( c ) for constant a > 0 ◮ SOSD: µ ( c + a ) ≤ µ ( c ) for mean preserving spread a ◮ ⇒ µ ( c ) ≤ E ( c ) Backus, Routledge, & Zin (NYU & CMU) Exotic + Cyclical Returns 3 / 37

Risk preference Chew-Dekel preferences Certainty equivalent function defined by risk aggregator M � µ = p ( s ) M [ c ( s ) , µ ] s Recursive definition unavoidable (you’ll see why) Generalization of expected utility (weaker independence axiom) Backus, Routledge, & Zin (NYU & CMU) Exotic + Cyclical Returns 4 / 37

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) Exotic + Cyclical Returns 5 / 37

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) Exotic + Cyclical Returns 6 / 37

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) Exotic + Cyclical Returns 7 / 37

Risk preference Lognormal risks Let: log c ∼ N( κ 1 , κ 2 ), rp = log[ E ( c ) /µ ( c )] Expected utility rp(EU) = (1 − α ) κ 2 / 2 Weighted utility rp(WU) = [1 − ( α + 2 γ )] κ 2 / 2 Disappointment aversion rp(DA) = E2C2E Backus, Routledge, & Zin (NYU & CMU) Exotic + Cyclical Returns 8 / 37

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) Exotic + Cyclical Returns 9 / 37

Recursive preferences Recursive preferences General form U t = V [ u t , µ t ( U t +1 )] Kreps-Porteus/Epstein-Zin/Weil t ] 1 /ρ [(1 − β ) u ρ t + βµ 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 Backus, Routledge, & Zin (NYU & CMU) Exotic + Cyclical Returns 10 / 37

Recursive preferences Applications Pricing kernels Risk sharing Cyclical properties of US asset returns Backus, Routledge, & Zin (NYU & CMU) Exotic + Cyclical Returns 11 / 37

Recursive preferences Kreps-Porteus pricing kernel 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 ◮ Allows role for predictable future consumption growth ◮ Ditto volatility Backus, Routledge, & Zin (NYU & CMU) Exotic + Cyclical Returns 12 / 37

Recursive preferences Kreps-Porteus pricing kernel (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) Exotic + Cyclical Returns 13 / 37

Recursive preferences Kreps-Porteus risk sharing Pareto problem with two (different) recursive agents Issues ◮ Time-varying pareto weights ◮ Representative agent may look different from individuals ◮ Possible nonstationary consumption distribution Backus, Routledge, & Zin (NYU & CMU) Exotic + Cyclical Returns 14 / 37

US asset returns Cyclical properties of US asset returns Data: cyclical properties of US asset prices and returns Theory: numerical example [“Bansal-Yaron plus”] Backus, Routledge, & Zin (NYU & CMU) Exotic + Cyclical Returns 15 / 37

Data Cyclical properties of US asset prices and returns Cross correlations for financial indicators and economic growth ◮ Returns: logs of gross returns ◮ Excess returns: differences in logs of gross returns Economic growth ◮ Monthly: log x t − log x t − 1 ◮ Or year-on-year: log x t +6 − log x t − 6 ◮ Computed from: industrial production, consumption, employment US data, monthly, 1960 to present Backus, Routledge, & Zin (NYU & CMU) Exotic + Cyclical Returns 16 / 37

Data Equity returns (monthly growth) 1.00 1.00 Leads Lags 0.50 0.50 Cross Correlation 0.00 0.00 −0.50 −0.50 −1.00 −1.00 −24 −18 −12 −6 0 6 12 18 24 Lead or Lag in Months Backus, Routledge, & Zin (NYU & CMU) Exotic + Cyclical Returns 17 / 37

Data Equity returns (yoy growth) 1.00 1.00 Leads Lags 0.50 0.50 Cross Correlation 0.00 0.00 −0.50 −0.50 −1.00 −1.00 −24 −18 −12 −6 0 6 12 18 24 Lead or Lag in Months Backus, Routledge, & Zin (NYU & CMU) Exotic + Cyclical Returns 18 / 37

Data Equity returns (variations) Year−on−Year Growth Real −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 Leads Lags Leads Lags Cross Correlation Cross Correlation −24 −18 −12 −6 0 6 12 18 24 −24 −18 −12 −6 0 6 12 18 24 Lead or Lag in Months Lead or Lag in Months Monthly 1990 and After −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 Leads Lags Leads Lags Cross Correlation Cross Correlation −24 −18 −12 −6 0 6 12 18 24 −24 −18 −12 −6 0 6 12 18 24 Lead or Lag in Months Lead or Lag in Months Backus, Routledge, & Zin (NYU & CMU) Exotic + Cyclical Returns 19 / 37

Data Term spread (monthly growth) 1.00 1.00 Leads Lags 0.50 0.50 Cross Correlation 0.00 0.00 −0.50 −0.50 −1.00 −1.00 −24 −18 −12 −6 0 6 12 18 24 Lead or Lag in Months Backus, Routledge, & Zin (NYU & CMU) Exotic + Cyclical Returns 20 / 37

Data Term spread (variations) Term Spread Short Rate −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 Leads Lags Leads Lags Cross Correlation Cross Correlation −24 −18 −12 −6 0 6 12 18 24 −24 −18 −12 −6 0 6 12 18 24 Lead or Lag in Months Lead or Lag in Months Year−on−Year Growth 1990 and After −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 Leads Lags Leads Lags Cross Correlation Cross Correlation −24 −18 −12 −6 0 6 12 18 24 −24 −18 −12 −6 0 6 12 18 24 Lead or Lag in Months Lead or Lag in Months Backus, Routledge, & Zin (NYU & CMU) Exotic + Cyclical Returns 21 / 37

Data Think about this for a minute... Backus, Routledge, & Zin (NYU & CMU) Exotic + Cyclical Returns 22 / 37

Data Excess returns: equity (yoy) 1.00 1.00 Leads Lags 0.50 0.50 Cross Correlation 0.00 0.00 −0.50 −0.50 −1.00 −1.00 −24 −18 −12 −6 0 6 12 18 24 Lead or Lag in Months Backus, Routledge, & Zin (NYU & CMU) Exotic + Cyclical Returns 23 / 37