Consumption Based Asset Pricing Models: Theory

Transcription

1 Consumpion Based Asse Pricing Models: Theory Faih Guvenen UT-Ausin Hanno Lusig UCLA March 3, 2007 Absrac The essenial elemen in modern asse pricing heory is a posiive random variable called he sochasic discoun facor SDF. This objec allows one o price any payoff sream. Is exisence is implied by he absence of arbirage opporuniies. Consumpion-based asse pricing models link he SDF o he marginal uiliy growh of invesors and in urn o observable economic variables and in doing so, hey provide empirical conen o asse pricing heory. This enry discusses his class of models. JEL Classificaions: G12 1 Consumpion-Based Asse Pricing Models: Theory Consumpion-based asse pricing models sudy he pricing of risky payoff sreams using he covariance of hese payoffs wih he marginal uiliy growh of invesors. The cenral componen of a consumpion-based asse pricing model is he Euler equaion, which imposes resricions on he covariance beween asse reurns and he marginal uiliy growh of invesors. An easy and inuiive way o derive his equaion is by using a variaional argumen. Suppose ha he opimal consumpion pah of invesor i is given by {C} i T =0 where T is possibly infinie. Suppose furher ha an asse j is available wih a reurn R j,+1 beween periods and + 1, and he invesor is no facing a binding porfolio consrain wih respec o his asse. Then a feasible sraegy is o reduce consumpion a ime by a small amoun ε, inves i in asse j, and consume he proceeds, C i +1+εR j +1, in he nex period. Assuming a ime-separable uiliy funcion, wih he one-period feliciy funcion denoed by U and a ime discoun facor of β, his sraegy changes he invesors expeced lifeime uiliy by U c C i, X ε + E βuc C i +1, X +1 εr j +1], where E is he mahemaical condiional expecaion operaor; X represens he argumens of he uiliy funcion oher han consumpion; and U c denoes he parial derivaive wih respec o consumpion. The opimaliy of he original sequence implies ha his sraegy canno be 1 Elecronic copy of his paper is available a: hp://ssrn.com/absrac=968061

2 profiable for any amoun ε and any asse available. Seing his gain o zero and rearranging yields he Euler equaion: E M,+1 R j,+1 ] = 1 where M,+1 = β U c C i +1, X U c C, i X This Euler equaion was firs derived by Rubinsein 1976 and Lucas 1978 in discree ime, and by Breeden 1979 in coninuous ime. While his class of models can in principle be used o sudy a broad variey of asses, his enry will focus on socks and shor-erm bonds, which have received he greaes aenion in he consumpion-based asse pricing lieraure. In he case of a one-period discoun bond wih gross reurn R f,+1 = 1/P f a bond ha coss P f dollars oday and pays off 1 dollar omorrow he Euler equaion can be rewrien as Similarly, when he asse is a sock wih ex-dividend price P s equaion can be rearranged o read P s equaion yields: P s P f = E M,+1 ]. 2 and dividend paymen D, he Euler = E M,+1 P s +1 + D +1 ]. By forward subsiuion his ] = E M,+s D +s, 3 s=1 which deermines he price of a share of equiy as he value of all fuure dividends i eniles discouned by he pricing kernel. Lucas 1978 and Mehra and Presco 1985 used a represenaive-agen endowmen economy srucure in which he dividend sream, {D } =1, is exogenously produced by a ree. Furhermore, hese dividends are assumed o be perishable frui, so in equilibrium he price of equiy in he ree adjuss o he poin where he represenaive agen is willing o consume all available dividends: C = D. Subsiuing his condiion ino he expression for M in equaion 1, and hen using M in equaions 2 and 3 shows ha he price of his sock and ha of he one-period bond are enirely deermined by he sochasic process for D ogeher wih he funcional form for U ignoring X for now. Hansen and Singleon 1983 esed he represenaive agen s Euler equaion on US consumpion daa, and hey found he model was rejeced. Mehra and Presco 1985 showed ha when one chooses he properies of C o mach he momens of aggregae consumpion in he daa calibrae he model o daa, he equiy premium E R+1 s R f generaed by he model was abou 60 imes smaller han ha observed in he hisorical U.S. daa. This equiy premium puzzle has generaed enormous ineres and led o he developmen of a wide range of consumpion-based asse pricing models in an aemp o resolve i. For furher discussion of he empirical performance of hese models, see Consumpion-based asse pricing models: Empirical performance. 2 Elecronic copy of his paper is available a: hp://ssrn.com/absrac=968061

3 An alernaive way o explain he hurdles hese models face is by deriving an empirical lower bound on he volailiy of he SDF. Subracing he ] Euler equaions for bond reurns from he one for sock reurns yields: E M,+1 R+1 s R f = 0. Noing ha he lef hand sides of his condiion can be rewrien as Cov M,+1 R+1 s R f +E M,+1 E R+1 s R f, some simple manipulaions yield he following key decomposiion: E R+1 s R f σ R+1 s R f = σ M,+1 E M,+1 corr M,+1, R s +1 R f, 4 where σ denoes he sandard deviaion. Observing ha he correlaion erm is bounded from above in absolue value by 1, we ge E R+1 s R f σ M,+1 σ R+1 s R f E M,+1. 5 The lef hand side of his inequaliy is he Sharpe raio, which measures he expeced excess reurn demanded by invesors per uni sandard deviaion of risk hey bear, which averages abou 0.40 in annual U.S. daa. The righ hand side is called he marke price of risk or he maximum Sharpe raio. This implies ha a consumpion-based model mus be able o generae an SDF wih a coefficien of variaion sandard deviaion normalized by mean of a leas 40 percen o be consisen wih he Sharpe raio observed in he daa. Hansen and Jagannahan 1991 developed and generalized his observaion o provide a volailiy bounds es for poenial candidae models. As discussed in he enry Consumpion-based asse pricing models: empirical performance he majoriy of plausibly calibraed asse pricing models fail his es. When he invesor faces a binding borrowing consrain, she canno increase her consumpion oday by reducing he holdings of asse j. This implies ha her marginal uiliy oday will remain higher han he value implied by he equaliy condiion in 1, and he Euler condiion for ha asse will be an inequaliy: E M,+1 R,+1] j < 1. This relaxes he lower bound on he SDF volailiy derived in equaion 5 cf., Lumer To develop furher implicaions of consumpion based models i is necessary o impose furher srucure on M,+1, which requires being more specific abou i he funcional form and he argumens of he uiliy funcion; ii he sochasic properies of variables affecing marginal uiliy i.e, consumpion, leisure, ec.; and iii he marke srucure. The laer deermines wheher an appropriae aggregaion heorem holds which happens for example when markes are complee in which case C i can be replaced wih aggregae consumpion. Therefore, consumpion based models can be broadly caegorized based on he assumpions hey make on hese hree dimensions. These 3

4 differen models are discussed in he nex enry. 1.1 Predicabiliy The sandard deviaion of he log price/dividend P/D raio of socks is exremely high abou 40 percen per annum in he U.S. daa. In a world wih a consan SDF as would be he case wih risk-neural invesors, i is impossible o raionalize his high volailiy wih he relaively low variabiliy of he underlying dividend sream LeRoy and Porer 1981 and Shiller Le p denoe he log price, d denoe he log dividend, and r denoe he log reurn. Using a firs-order approximaion, Campbell and Shiller 1988 show ha he log P/D raio can be decomposed as follows: p d = consan + E j=1 ρ j 1 d +j r +j ] wih ρ = exppd/1 + exppd and pd denoes he average log P/D raio. The firs erm in he square brackes is referred o as he cash flow componen, and he second par is referred o as he discoun rae componen. This decomposiion implies ha he variance of he log P/D raio can be saed as : varp d = cov p d, ρ j 1 d +j cov p d, ρ j 1 r +j. j=1 j=1 The P/D raio on socks only moves because i predics fuure reurns on socks or because i predics fuure dividend growh. In he daa, mos of he volailiy in P/D raios is due o news abou fuure expeced reurns discoun raes, no due o fuure dividend growh cash flows Campbell 1991 and Cochrane There is a large lieraure ha documens he predicabiliy of sock reurns over longer holding periods, saring wih work by Campbell and Shiller 1988 and Campbell and Shiller 1998, Poerba and Summers 1986 and Fama and French 1988 and Fama and French Oher variables ha predic reurns include he spread beween long and shor bonds Fama and French 1989 and he T-bill rae Lamon More recenly, more aenion has been paid o macro variables ha predic reurns, mos noably in he work by Leau and Ludvigson 2001a who documen ha he consumpion/wealh raio is a powerful predicor of sock reurns. So, he volailiy of P/D raio implies ha excess reurns on socks are highly predicable. In oher words, expeced excess reurns change a lo over ime, even per uni of risk. We use he 4

5 condiional version of he expression in 4 o undersand he implicaions of his finding: E R s +1 R f σ R s +1 R f = σ M,+1 E M,+1 corr M,+1, R+1 s R f, 6 where σ denoes he condiional sandard deviaion. Good models need o produce a lo of ime variaion in he righ hand side of 6 and his happens mosly hrough variaion in he condiional marke price of risk firs erm. This is an upper bound on he condiional Sharpe raio. See also Leau and Ludvigson 2001b on how o measure variaion in he condiional Sharpe raio. Anoher es of consumpion-based asse pricing models is wheher hey are able o generae as much predicabiliy as found in he daa. Examples of early models ha mach he variaion in he condiional marke price of risk include Kandel and Sambaugh 1990, Campbell and Cochrane 2000 and Barberis, Huang, and Sanos More recen work includes he work by Sanos and Veronesi 2005 and Menzly, Sanos, and Veronesi 2004, Piazzesi, Schneider, and Tuzel 2006, Guvenen 2005, Lusig and Van Nieuwerburgh 2005 and Lusig and Nieuwerburgh 2006, Bansal and Yaron These models are discussed in deail in Consumpion Based Asse Pricing Models: Empirical Performance. 5

6 References Bansal, R., and A. Yaron 2004: Risks for he Long-Run: A Poenial Resoluion of Asse Pricing Puzzles, Journal of Finance, 594, Barberis, N., M. Huang, and T. Sanos 2001: Prospec Theory and Asse Prices, Quarerly Journal of Economics, 1161, Breeden, D. T. 1979: An Ineremporal Asse Pricing Model wih Sochasic Consumpion and Invesmen Opporuniies, Journal of Financial Economics, 7, Campbell, J. Y. 1991: A Variance Decomposiion for Sock Reurns, Economic Journal, 101, Campbell, J. Y., and J. H. Cochrane 2000: Explaining he Poor Performance of Consumpion- Based Asse Pricing Models, Journal of Finance, 556, Campbell, J. Y., and R. J. Shiller 1988: The Dividend-Price Raio and Expecaions of Fuure Dividends and Discoun Facors, Review of Financial Sudies, 1, Campbell, J. Y., and R. J. Shiller 1998: Sock Prices, Earnings and Expeced Dividends, Journal of Finance, 43, Cochrane, J. H. 1991: Explaining he Variance of Price-Dividend Raios, Review of Financial Sudies, 52, Fama, E. F., and K. R. French 1988: Dividend Yields and Expeced Sock Reurns, Journal of Financial Economics, 22, : Business Condiions and Expeced Reurns on Socks and Bonds, Journal of Financial Economics, 25, Guvenen, F. 2005: A Parsimonious Macroeconomic Model for Asse Pricing: Habi Formaion or Cross-secional Heerogeneiy?, Working Paper Universiy of Texas a Ausin. Hansen, L. P., and R. Jagannahan 1991: Implicaions of Securiy Markes Daa for Models of Dynamic Economies, Journal of Policiical Economy, 99, Hansen, L. P., and K. Singleon 1983: Sochasic Consumpion, Risk Aversion, and he Temporal Behavior of Asse Reurns, Journal of Poliical Economy, 91, Kandel, S., and R. F. Sambaugh 1990: Expecaions and Volailiy of Consumpion and Asse Reurns, Review of Financial Sudies, 3, Lamon, O. 1998: Earnings and Expeced Reurns, Journal of Finance, 53,

7 LeRoy, S. F., and R. D. Porer 1981: The Presen-Value Relaion: Tess Based on Implied Variance Bounds, Economerica, 49, Leau, M., and S. C. Ludvigson 2001a: Consumpion, Aggregae Wealh and Expeced Sock Reurns, Journal of Finance, 563, b: Measuring and Modeling Variaion in he Risk-Reurn Tradeoff, in Handbook of Financial Economerics, ed. by Y. Ai-Sahalia, and L. P. Hansen. Elsevier. Lucas, R. 1978: Asse Prices in an Exchange Economy, Economerica, 466, Lusig, H., and S. V. Nieuwerburgh 2006: Can Housing Collaeral Explain Long-Run Swings in Asse Reurns?, Working Paper NYU Sern and UCLA. Lusig, H., and S. Van Nieuwerburgh 2005: Housing Collaeral, Consumpion Insurance and Risk Premia: An Empirical Perspecive, Journal of Finance, 603, Lumer, E. 1996: Asse Pricing in Economies wih Fricions, Economerica, 64, Mehra, and E. Presco 1985: The Equiy Premium: A Puzzle., Journal of Moneary Economics, 152, Menzly, L., T. Sanos, and P. Veronesi 2004: Undersanding Predicabiliy, Journal of Poliical Economy, 1121, Piazzesi, M., M. Schneider, and S. Tuzel 2006: Housing, Consumpion, and Asse Pricing, Forhcoming Journal of Financial Economics. Poerba, J. M., and L. H. Summers 1986: The Persisence of Volailiy and Sock Marke Flucuaions, American Economic Review, 765, Rubinsein, M. 1976: The valuaion of uncerain income sreams and he pricing of opions, Bell Journal of Economics, 7, Sanos, J., and P. Veronesi 2005: Financial Sudies, 191, Labor Income and Predicable Sock Reurns, Review of Shiller, R. J. 1981: The Use of Volailiy Measures in Assessing Marke Efficiency, Journal of Finance, 36,