A MULTIPLIER APPROACH TO UNDERSTANDING THE MACRO IMPLICATIONS OF HOUSEHOLD FINANCE

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Ευτυχός Παπάζογλου
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1 A MULTIPLIER APPROACH TO UNDERSTANDING THE MACRO IMPLICATIONS OF HOUSEHOLD FINANCE YiLi Chien, Harold Cole and Hanno Lustig Roine Vestman, New York University October 30, 2007

2 OVERVIEW OF THE PAPER Taking heterogeneity in households trading technology as a given, what are the implications for the moments of asset prices? What role does non-participation play? 2....households portfolios? 3....consumption moments? 4....the wealth distribution? Methodological contribution A solution method based on multipliers, rather than moments of the wealth distribution Related papers Empirical household finance: Calvet, Campbell and Sodini (2007) Models with non-participation: Basak and Cuoco (1998), Guvenen (2005) Distributional implications: Krusell and Smith (1997), Favilukis (2007)

3 MODEL Notation Aggregate shock: z t Idiosyncratic shock: η t (iid across HH s) π(z t+1, η t+1 z t, η t ) = π(z t+1 z t )π(η t+1 z t+1, η t ) Successor nodes: z t+1 z t ; set of successor histories: {z τ z t } History from zero to t 1 contained in η t : η t 1 (η t ) Endowment Y (z t ) = exp {z t } Y (z t 1 ) Diversifiable income: (1 γ)y (z t ) Non-diversifiable income: γy (z t )η t Preferences } U(c) = E { t 1 βt π(z t, η t ) c(zt,η t ) 1 α 1 α

4 FOUR TRADING TECHNOLOGIES 4 types of HH s Active traders: Complete, z-complete Passive traders: Diversified, Non-participants Prices Aggregate state prices: q [( z t+1, η t+1), (z t, η t ) ] = π(η t+1 z t+1, η t )q(z t+1, z t ) Arrow-Debreu prices: P(z t, η t ) = π(z t, η t )P(z t, η t ) = q(z t, z t 1 )q(z t 1, z t 2 )..q(z 1, z 0 )q(z 0 ) m(z t+1 z t ) = P(z t+1 )/P(z t ) is one SDF

5 BUDGET CONSTRAINTS 1. Complete traders (c) γy (z t )η t + a t 1 (z t, η t ) + σ(z t 1, η t 1 ) [ (1 γ)y (z t ) + ω(z t ) ] c(z t, η t ) q(z t+1, z t ) a t (z t+1, η t+1 )π(η t+1 z t+1, η t ) + σ(z t, η t )ω(z t ) z t+1 z t η t+1 η t 2. z-complete traders (z) γy (z t )η t + a t 1 (z t, η t 1 ) + σ(z t 1, η t 1 ) [ (1 γ)y (z t ) + ω(z t ) ] c(z t, η t q(z t+1, z t )a t (z t+1, η t ) + σ(z t, η t )ω(z t ) z t+1 z t

6 BUDGET CONSTRAINTS AND MARKET CLEARING 3. Diversified traders (div) γy (z t )η t + σ(z t 1, η t 1 ) [ (1 γ)y (z t ) + ω(z t ) ] c(z t, η t ) σ(z t, η t )ω(z t ) 4. Non-participants (np) γy (z t )η t + a t 1 (z t 1, η t 1 ) c(z t, η t ) Market clearing η t [ µ1 at 1 c (zt, η t ) + µ 2 at 1 z (zt, η t 1 (η t )) +µ 4 a np t 1 (zt 1 (z t ), η t 1 (η t )) z t+1 z t q(z t+1, z t )a t (z t, η t ) ] π(η t z t ) = 0 [ ] µ 1 σ c (z t, η t ) + µ 2 σ z (z t, η t ) + µ 3 σ div (z t, η t ) π(η t z t ) = 1 η t

7 MEASURABILITY CONSTRAINTS Net wealth â t 1 (z t, η t ) a t 1 (z t, η t ) + σ(z t 1, η t 1 ) [ (1 γ)y (z t ) + ϖ(z t ) ] 2. z-complete traders ] ] â t 1 (z t, [η t 1, η t ) = â t 1 (z t, [η t 1, η t ) z t, η t 1 and η t, η t N 3. Diversified traders â t 1 ( [ z t 1 ] [, z t, η t 1 ] [, η t ) (1 γ)y (z t 1, z t ) + ϖ(z t 1, z t ) = â t 1 ( z t 1 ] [, z t, η t 1 ], η t ) (1 γ)y (z t 1, z t ) + ϖ(z t 1, z t ) 4. Non-participants z t 1, η t 1 and z t, z t Z and η t, η t N ] ] ] ] â t 1 ( [z t 1, z t, [η t 1, η t ) = â t 1 ( [z t 1, z t, [η t 1, η t ) z t 1, η t 1 and z t, z t Z and η t, η t N

8 THE HOUSEHOLD PROBLEM Lagrangian L = min max β t {χ,ν,ϕ}{c,â} t=1 u(c(z t, η t ))π(z t, η t ) (z t,η t ) +χ {present value budget constr.} + ν(z t, η t ) { measurability constr. in (z t, η t ) } t 1 (z t,η t ) + ϕ(z t, η t ) { borrowing constr. in (z t, η t ) } t 1 (z t,η t ) Lagrangian a la Marcet and Marimon (1999) ζ 0 = χ; ζ(z t, η t ) = ζ(z t 1, η t 1 ) + ν(z t, η t ) ϕ(z t, η t ) L z = min max β t u(c(z t, η t ))π(z t, η t ) {χ,ν,ϕ}{c,â} t=1 (z t,η t ) { + ζ(z t, η t ) ( γy (z t )η t c(z t, η t) ) } t 1 (z t,η t ) { ν(z + t, η t )â t 1 (z t, η t 1 } ) ϕ(z t, η t ) {borrowing constr. in (z t, η t )} t 1 (z t,η t )

9 THE AGGREGATE MULTIPLIER AND THE SDF FOC for consumption, irrespective of trading tech. Consumption sharing rule c(z t, η t ) C (z t ) β t u (c(z t, η t )) P(z t ) = ζ(zt, η t ) 1 α h(z t ) Stochastic discount factor m(z t+1 z t ) = β = ζ(z t, η t ), h(z t ) = ζ(z t, η t ) 1 α π(η t z t ) η t ( C (z t+1 ) C (z t ) ) α ( h(z t+1 ) α ) h(z t )

10 A NET SAVINGS FUNCTION THE PRESENT DISCOUNTED VALUE OF FUTURE SAVINGS HH s net savings functions, j {c, z, div, np}: ] S j (ζ(z t, η t ); z t, η t ) = [γη t ζ(zt, η t ) 1 α h(z t C (z t ) ) Ratio of savings to aggr. consumption + π(zt+1, ηt+1 )P(zt+1 ) π(z z t+1,η t, η t )P(z t ) t+1 S(ζ(z t+1, η t+1 ); z t+1, η t+1 ) S j (ζ(z t, η t ); z t, η t ) = Sj (ζ(z t, η t ); z t, η t ) Y (z t )

11 WITHOUT NON-PARTICIPANTS, THE (CONDITIONAL) EQUITY RISK PREMIUM IS THE BREEDEN-LUCAS ONE Conditions φ(z t+1 z t ) = φ(z t+1 ) π(η t+1, z t+1 η t, z t ) = ϕ(η t+1 η t )φ(z t+1 z t ) Given the conditions and without non-participants, ζ and the consumption share are independent of z t h t+1 h t non-random m(z t+1 z t ) = β ( C (z t+1 ) C (z t ) The role of non-participants ) α ( ) h(z t+1 ) α h(z t ) np s measurability condition in terms of net savings is given by: S np t+1 (ζ(zt+1, η t+1 ); z t+1, η t+1 np ) S t+1 e z = (ζ( zt+1, η t+1 ); z t+1, η t+1 ) t+1 e z t+1 np spill risk to other household types In the presence of np, {h t+1 /h t } depends on z t+1

12 SHIFTING AGGREGATE RISK TO ACTIVE TRADERS Div. do not bear any of the residual aggregated risk created by np Div s measurability constraint Sa div (z t, z t+1 ) [(1 γ)y (z t, z t+1 ) + ω(z t, z t+1 )] = Sa div (z t, z t+1 ) [(1 γ)y (z t, z t+1 ) + ω(z t, z t+1 )]

13 CALIBRATION Endowment process Y (z t ) as in Mehra and Prescott (1985) η as in STY (2003) (1 γ) = 0.1 Households µ np = 0.7 α = 5 β = 0.95

14 ASSET PRICING (TABLE 3)

15 CONSUMPTION (TABLE 5)

16 THE EQUITY SHARE ALONG THE WEALTH DISTRIBUTION (TABLE 9)

17 IS THERE A MAPPING BETWEEN CCL AND CALVET, CAMPBELL, AND SODINI (2007)? Is there no financial income in non-diversifiable income? Calvet et al (2007) focus on households financial portfolios and argue that many are subject to idiosyncratic risk Would Calvet et al (2007) agree that a diversified trader is somebody who is afraid of making investment mistakes (CCL p. 5)? How categorize households in the data? Portfolio holdings would be natural! np=easy div=well-diversified stock-holders? c=? z=? without c or z there is no equilibrium! What about HH s Sharpe ratios? What about HH s trading volume or trading pattern?

18 WHERE ARE ENTREPRENEURS AND THEIR CAPITAL? Where does an entrepreneur with little non-proprietary wealth belong? Moskowitz and Vissing-Jorgenssen (AER, 2004) Entrepreneurs as a group own a lot of capital They face a lot of uninsurable risk (they are not c or z) They are also poorly diversified (not div) Are they np? But np in the strict sense are typically poor!

19 IT S NICE TO BE AN ACTIVE TRADER - A LITTLE TOO NICE? c z div np E [ R W R f ] σ [ log(c)] Portfolio positions In model: c holds leveraged equity portfolios (160%), z: 93% Average returns are increasing in the degree of insurance Empirical research (Vissing-Jorgensen, Guvenen) reject the hypothesis of complete markets for stockholders, but not for non-stockholders Would their tests reject complete markets among active traders? Again, who are c and z? Trading frequency and volume Empirically, the evidence of households trading frequency and trading volumes is mixed std of fraction invested in stocks for c is 60% std of fraction invested in stocks for z is 30%

20 COMPETING SOLUTION METHODS Does the multiplier method outperform the Krusell-Smith method? Regressing log SDF on mean wealth R 2 << R 2 KS Is the solution method applicable in the presence of capital? Yes? Market clearing vs. finding SDF Krusell-Smith: Clear market by market CCL: Find sdf as a function of households multipliers (h(z t )) Many HH s and few assets easier to clear market? Few HH s and many asset easier to find h(z t )? Are there other model features that may matter?

21 HOUSEHOLD PORTFOLIOS (TABLE 4)

22 THE WEALTH DISTRIBUTION (TABLE 8) OUT OF SAMPLE

23 COMPUTATION (1) MARTINGALE CONDITIONS FOR CUMULATIVE MULTIPLIERS Complete z-complete ζ(z t, η t ) ζ(z t, η t ) = ζ(z t 1, η t 1 ) ϕ(z t, η t ) η t+1 η t ζ(z t+1, η t+1 )π(η t+1 z t+1, η t ) Diversified ζ(z t, η t ) ζ(z t+1, η t+1 m(z ) t+1 z t )R(z t+1 ) E {m(z t+1 z t )R(z t+1 ) z t } π(ηt+1 z t+1, η t ) z t+1 z t η t+1 η t R(z t+1 ) = return on tradable income claim Non-participants ζ(z t, η t ) ζ(z t+1, η t+1 m(z ) t+1 z t ) E {m(z z t+1 z t,η t+1 η t+1 z t ) z t } π(ηt+1 z t+1, η t ) t

24 COMPUTATION (2) RECURSIVE FORMULATION THROUGH PDV OF FUTURE SAVINGS HH s net savings functions, j {c, z, div, np}: ] S j (ζ(z t, η t ); z t, η t ) = [γη t ζ(zt, η t ) 1 α h(z t C (z t ) + ) π(zt+1, ηt+1 )P(zt+1 π(z z t+1,η t, η t )P(z t ) t+1 Operator S(ζ(z t+1, η t+1 ); z t+1, η t+1 ) T j (z t+1, η t+1 z t, η t )(ζ(z t, η t )) = ζ(z t+1, η t+1 ) j {c, z, div, np} Operator for the aggregate multiplier T h Joint distr of multipliers and endowments: Φ j t Recall h(z t ) = η t ζ(z t, η t ) 1 α π(η t z t ) Integrate [ T j ] 1 α over η t+1 η t and then over j to get fixed point of { h 1 t (z t ) } = T h { ht 0 (z t ) }

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