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

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“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

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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

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Time preference

Additive preferences Ut = (1 − β)ut + βUt+1 = (1 − β)

∞

j=0

βjut+j Time aggregator V Ut = V (ut, Ut+1) (discounting built into V2) Why don’t we care about this?

Backus, Routledge, & Zin (NYU & CMU) Exotic + Cyclical Returns 2 / 37

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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

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Risk preference

Chew-Dekel preferences

Certainty equivalent function defined by risk aggregator M µ =

s

p(s)M[c(s), µ] Recursive definition unavoidable (you’ll see why) Generalization of expected utility (weaker independence axiom)

Backus, Routledge, & Zin (NYU & CMU) Exotic + Cyclical Returns 4 / 37

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Risk preference

Chew-Dekel examples

Expected utility M(c, m) = cαm1−α/α + m(1 − 1/α) Weighted utility M(c, m) = (c/m)γcαm1−α/α + m[1 − (c/m)γ/α]. Disappointment aversion M(c, m) = cαm1−α/α + m(1 − 1/α) + δI(m − c)(cαm1−α − m)/α I(x) = 1 if x > 0, 0 otherwise

Backus, Routledge, & Zin (NYU & CMU) Exotic + Cyclical Returns 5 / 37

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Risk preference

Chew-Dekel as adjusted probabilities

Expected utility µ =

s

p(s)c(s)α 1/α Weighted utility: ditto with ˆ p(s) = p(s)c(s)γ

u p(u)c(u)γ ,

Disappointment aversion: ditto with ˆ p(s) = p(s)(1 + δI[µ − c(s)])

u p(s)(1 + δI[µ − c(s)]),

Backus, Routledge, & Zin (NYU & CMU) Exotic + Cyclical Returns 6 / 37

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Risk preference

Small risks

Two states (1 + σ, 1 − σ), equal probs, Taylor series around σ = 0 Expected utility µ(EU) ≈ 1 − (1 − α)σ2/2 Weighted utility µ(WU) ≈ 1 − [1 − (α + 2γ)]σ2/2 Disappointment aversion µ(DA) ≈ 1 −

δ

2 + δ

σ − (1 − α)
4 + 4δ

4 + 4δ + δ2

σ2/2

Backus, Routledge, & Zin (NYU & CMU) Exotic + Cyclical Returns 7 / 37

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Risk preference

Lognormal risks

Let: log c ∼ N(κ1, κ2), rp = log[E(c)/µ(c)] Expected utility rp(EU) = (1 − α)κ2/2 Weighted utility rp(WU) = [1 − (α + 2γ)]κ2/2 Disappointment aversion rp(DA) = E2C2E

Backus, Routledge, & Zin (NYU & CMU) Exotic + Cyclical Returns 8 / 37

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Risk preference

Extreme risks

Let: log E exp(log c) = κ1 + κ2/2! + κ3/3! + κ4/4! Expected utility rp(EU) = (1 − α)κ2/2 + (1 − α2)κ3/3! + (1 − α3)κ4/4! Weighted utility rp(WU) = [1 − (α + 2γ)]κ2/2 + [1 − (α + 2γ)2 + γ(α + γ)]κ3/3! + [1 − (α + 2γ)3 + 2γ(α + γ)(α + 2γ)]κ4/4! Disappointment aversion rp(DA) = Another E2C2E

Backus, Routledge, & Zin (NYU & CMU) Exotic + Cyclical Returns 9 / 37

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Recursive preferences

General form Ut = V [ut, µt(Ut+1)] Kreps-Porteus/Epstein-Zin/Weil V (ut, µt) = [(1 − β)uρ

t + βµρ t ]1/ρ

µt(Ut+1) =

EtUα

t+1

1/α IES = 1/(1 − ρ) CRRA = 1 − α α = ρ ⇒ additive preferences

Backus, Routledge, & Zin (NYU & CMU) Exotic + Cyclical Returns 10 / 37

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Recursive preferences

Applications

Pricing kernels Risk sharing Cyclical properties of US asset returns

Backus, Routledge, & Zin (NYU & CMU) Exotic + Cyclical Returns 11 / 37

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Recursive preferences

Kreps-Porteus pricing kernel

Marginal rate of substitution mt+1 = β(ct+1/ct)ρ−1[Ut+1/µt(Ut+1)]α−ρ Note role of future utility

◮ Allows role for predictable future consumption growth ◮ Ditto volatility Backus, Routledge, & Zin (NYU & CMU) Exotic + Cyclical Returns 12 / 37

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Recursive preferences

Kreps-Porteus pricing kernel (continued)

Example: let consumption growth follow log xt = log x +

∞

j=0

χjwt−j Pricing kernel log mt+1 = constant + [(ρ − 1)χ0 + (α − ρ)(χ0 + X1)]wt+1 + (ρ − 1)

∞

j=0

χj+1wt−j X1 =

∞

j=1

βjχj (“Bansal-Yaron” term)

Backus, Routledge, & Zin (NYU & CMU) Exotic + Cyclical Returns 13 / 37

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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

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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

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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 xt − log xt−1 ◮ Or year-on-year: log xt+6 − log xt−6 ◮ Computed from: industrial production, consumption, employment

US data, monthly, 1960 to present

Backus, Routledge, & Zin (NYU & CMU) Exotic + Cyclical Returns 16 / 37

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Data

Equity returns (monthly growth)

Leads Lags −1.00 −0.50 0.00 0.50 1.00 −1.00 −0.50 0.00 0.50 1.00 Cross Correlation −24 −18 −12 −6 6 12 18 24 Lead or Lag in Months

Backus, Routledge, & Zin (NYU & CMU) Exotic + Cyclical Returns 17 / 37

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Data

Equity returns (yoy growth)

Leads Lags −1.00 −0.50 0.00 0.50 1.00 −1.00 −0.50 0.00 0.50 1.00 Cross Correlation −24 −18 −12 −6 6 12 18 24 Lead or Lag in Months

Backus, Routledge, & Zin (NYU & CMU) Exotic + Cyclical Returns 18 / 37

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Data

Equity returns (variations)

Leads Lags −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 Cross Correlation −24 −18 −12 −6 6 12 18 24 Lead or Lag in Months

Year−on−Year Growth

Leads Lags −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 Cross Correlation −24 −18 −12 −6 6 12 18 24 Lead or Lag in Months

Real

Leads Lags −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 Cross Correlation −24 −18 −12 −6 6 12 18 24 Lead or Lag in Months

Monthly

Leads Lags −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 Cross Correlation −24 −18 −12 −6 6 12 18 24 Lead or Lag in Months

1990 and After

Backus, Routledge, & Zin (NYU & CMU) Exotic + Cyclical Returns 19 / 37

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Data

Term spread (monthly growth)

Leads Lags −1.00 −0.50 0.00 0.50 1.00 −1.00 −0.50 0.00 0.50 1.00 Cross Correlation −24 −18 −12 −6 6 12 18 24 Lead or Lag in Months

Backus, Routledge, & Zin (NYU & CMU) Exotic + Cyclical Returns 20 / 37

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Data

Term spread (variations)

Leads Lags −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 Cross Correlation −24 −18 −12 −6 6 12 18 24 Lead or Lag in Months

Term Spread

Leads Lags −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 Cross Correlation −24 −18 −12 −6 6 12 18 24 Lead or Lag in Months

Short Rate

Leads Lags −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 Cross Correlation −24 −18 −12 −6 6 12 18 24 Lead or Lag in Months

Year−on−Year Growth

Leads Lags −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 Cross Correlation −24 −18 −12 −6 6 12 18 24 Lead or Lag in Months

1990 and After

Backus, Routledge, & Zin (NYU & CMU) Exotic + Cyclical Returns 21 / 37

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Data

Think about this for a minute...

Backus, Routledge, & Zin (NYU & CMU) Exotic + Cyclical Returns 22 / 37

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Data

Excess returns: equity (yoy)

Leads Lags −1.00 −0.50 0.00 0.50 1.00 −1.00 −0.50 0.00 0.50 1.00 Cross Correlation −24 −18 −12 −6 6 12 18 24 Lead or Lag in Months

Backus, Routledge, & Zin (NYU & CMU) Exotic + Cyclical Returns 23 / 37

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Data

Excess returns: Fama-French portfolios (yoy)

Leads Lags −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 Cross Correlation −24 −18 −12 −6 6 12 18 24 Lead or Lag in Months

Small

Leads Lags −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 Cross Correlation −24 −18 −12 −6 6 12 18 24 Lead or Lag in Months

Big

Leads Lags −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 Cross Correlation −24 −18 −12 −6 6 12 18 24 Lead or Lag in Months

Low Book−to−Market

Leads Lags −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 Cross Correlation −24 −18 −12 −6 6 12 18 24 Lead or Lag in Months

High Book−to−Market

Backus, Routledge, & Zin (NYU & CMU) Exotic + Cyclical Returns 24 / 37

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Data

Excess returns: industries (yoy)

Leads Lags −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 Cross Correlation −24 −18 −12 −6 6 12 18 24 Lead or Lag in Months

Machinery

Leads Lags −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 Cross Correlation −24 −18 −12 −6 6 12 18 24 Lead or Lag in Months

Automobiles

Leads Lags −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 Cross Correlation −24 −18 −12 −6 6 12 18 24 Lead or Lag in Months

Food

Leads Lags −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 Cross Correlation −24 −18 −12 −6 6 12 18 24 Lead or Lag in Months

Drugs

Backus, Routledge, & Zin (NYU & CMU) Exotic + Cyclical Returns 25 / 37

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Data

Excess returns: bonds (yoy)

Leads Lags −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 Cross Correlation −24 −18 −12 −6 6 12 18 24 Lead or Lag in Months

1−6 Months

Leads Lags −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 Cross Correlation −24 −18 −12 −6 6 12 18 24 Lead or Lag in Months

19−24 Months

Leads Lags −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 Cross Correlation −24 −18 −12 −6 6 12 18 24 Lead or Lag in Months

55−60 Months

Leads Lags −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 Cross Correlation −24 −18 −12 −6 6 12 18 24 Lead or Lag in Months

115−120 Months

Backus, Routledge, & Zin (NYU & CMU) Exotic + Cyclical Returns 26 / 37

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Theory

Theoretical economy

Take a breath What do we need?

◮ Variation in risk and/or price of risk ◮ ... tied to economic growth

Bansal-Yaron plus

◮ Representative agent exchange economy ◮ Recursive preferences (Kreps-Porteus/Epstein-Zin/Weil) ◮ Loglinear process for consumption growth ◮ Stochastic volatility ◮ Interaction between growth and volatility Backus, Routledge, & Zin (NYU & CMU) Exotic + Cyclical Returns 27 / 37

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Theory

Theoretical economy

Take a breath What do we need?

◮ Variation in risk and/or price of risk ◮ ... tied to economic growth

Bansal-Yaron plus

◮ Representative agent exchange economy ◮ Recursive preferences (Kreps-Porteus/Epstein-Zin/Weil) ◮ Loglinear process for consumption growth ◮ Stochastic volatility ◮ Interaction between growth and volatility Backus, Routledge, & Zin (NYU & CMU) Exotic + Cyclical Returns 27 / 37

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Theory

Kreps-Porteus preferences

Equations Ut = [(1 − β)cρ

t + βµt(Ut+1)ρ]1/ρ

µt(Ut+1) =

EtUα

t+1

1/α α, ρ ≤ 1 Interpretation IES = 1/(1 − ρ) CRRA = 1 − α α = ρ ⇒ additive preferences

Backus, Routledge, & Zin (NYU & CMU) Exotic + Cyclical Returns 28 / 37

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Theory

Kreps-Porteus pricing kernel

Marginal rate of substitution mt+1 = β ct+1 ct ρ−1 Ut+1 µt(Ut+1) α−ρ If α = ρ

◮ Second term disappears ◮ No roles for volatility or predictable consumption growth Backus, Routledge, & Zin (NYU & CMU) Exotic + Cyclical Returns 29 / 37

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Theory

Consumption growth

Consumption growth follows from log gt = g + e⊤xt xt+1 = Axt + a(vt − v) + v1/2

t

Bwt+1 vt+1 = (1 − ϕv)v + ϕvvt + bwt+1 Note

◮ A generates predictable component ◮ vt is stochastic ◮ a generates interaction Backus, Routledge, & Zin (NYU & CMU) Exotic + Cyclical Returns 30 / 37

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Theory

Theoretical excess returns

Transparent loglinear solution

◮ We love this, but won’t bore you with the details ◮ Still needs some work

Excess returns depend on

◮ Volatility (vt) ◮ Innovations in consumption growth and volatility ◮ Not expected future consumption growth (xt)! Backus, Routledge, & Zin (NYU & CMU) Exotic + Cyclical Returns 31 / 37

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Theory

Excess returns: numerical example

−20 −15 −10 −5 5 10 15 20 −1 −0.5 0.5 1 Cross Correlation Volatility −20 −15 −10 −5 5 10 15 20 −1 −0.5 0.5 1 Cross Correlation Lead or Lag in Months Excess return on equity Backus, Routledge, & Zin (NYU & CMU) Exotic + Cyclical Returns 32 / 37

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Summary and extensions

Summary

◮ Data: excess returns correlated with future growth ◮ Model: ditto via stochastic volatility

Fixups and extensions

◮ Model dividends explicitly ◮ Production economies: volatility acts like shock to discount factor,

affects consumption and labor supply

Backus, Routledge, & Zin (NYU & CMU) Exotic + Cyclical Returns 33 / 37

SLIDE 36

Extra slides

Related work (some of it)

Evidence on financial indicators of business cycles

◮ Ang-Piazzesi-Wei, Estrella-Hardouvelis, King-Watson, Rouwenhorst,

Stock-Watson

Kreps-Porteus pricing kernel

◮ Hansen-Heaton-Li, Weil

Stochastic volatility and returns

◮ Atkeson-Kehoe, Gallmeyer-Hollifield-Zin, Naik,

Primiceri-Schaumburg-Tambalotti

Backus, Routledge, & Zin (NYU & CMU) Exotic + Cyclical Returns 34 / 37

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Extra slides

Earnings and dividends (yoy)

Leads Lags −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 −20 −10 10 20 Lead or Lag in Months

Earnings (YOY)

Leads Lags −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 −20 −10 10 20 Lead or Lag in Months

Dividends (YOY)

Backus, Routledge, & Zin (NYU & CMU) Exotic + Cyclical Returns 35 / 37

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Extra slides

Approximation: two flavors

Problem: find decision rule ut = h(xt) satisfying EtF(xt, ut, wt+1) = 1, wt ∼ N(0, κ2) Judd + many others

◮ Taylor series expansion of F ◮ nth moment shows up in nth-order term

Us + much of modern finance

◮ Taylor series expansion of f = log F in

Et exp[f (xt, ut, wt+1)] = 1

◮ All moments show up even in linear approximation Backus, Routledge, & Zin (NYU & CMU) Exotic + Cyclical Returns 36 / 37

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Extra slides

Approximation: example

Linear “perturbation” method

◮ Linear approximation of F

F(xt, ut, wt+1) = F + Fx(xt − x) + Fu(ut − u) + Fwwt+1 EtF = 1 ⇒ ut − u = (1 − F)/Fu − (Fx/Fu)(xt − x)

◮ Decision rule doesn’t depend on variance of w (or higher moments)

“Affine” finance method

◮ Linear approximation of f = log F

f (xt, ut, wt+1) = f + fx(xt − x) + fu(ut − u) + fwwt+1 Et exp(f ) = 1 ⇒ ut − u = −(f + fwκ2/2)/fu − (fx/fu)(xt − x)

◮ Note impact of variance v (higher moments would show up, too) Backus, Routledge, & Zin (NYU & CMU) Exotic + Cyclical Returns 37 / 37