7/27/2019 Conditional Estimation of Linear Asset Pricing Models Using Alternative Marginal Utility Growth Instruments

http://slidepdf.com/reader/full/conditional-estimation-of-linear-asset-pricing-models-using-alternative-marginal 1/3

FSR Forum | september 2009 | FSR Forum | september 2009 |

1 Introduction

This study proposes a new macroeco-nomic theory-derived and productiv-ity-based marginal utility growth proxy that avoids the theoretical di fculty in utility unction specifcation andempirical problems associated withconsumption data used as the primary utility unction input. The theoreticaland empirical characteristics o thisproxy indicate it may be use ul inlinear asset pricing models as an instru-mental variable.

2 Exchange economy model The exchange economy model (Lucas,1978) yields the Euler condition ora representative agent’s intertemporalutility maximization:

where E t is the time t expectations oper-ator, p t is the price at time t , c t is the con-sumption at time t , β is the rate o timepre erence (subjective discount actor),and d t +1 are dividends paid at timet +1. The equality in (1) reveals the cost inmarginal utility ( p t u’ [c t ]) o purchasingthe asset must be equal to the discounted(β) expected gain o the uture payo (u’ t [c t +1 ] (p t +1 + d t +1 )). By dividing both

sides by p t u’ [c t ] and defning R t +1 = ( p t +1

+ d t +1) / pt equation (1) can be expressedin discount actor orm:

The stochastic discount actor (3) is un-observable there ore explicit utility unc-tion assumptions must be made be oreapplying the model. Two requently usedutility unctions, constant relative risk aversion (CRRA) utility and log utility are now used to illustrate theoretical andempirical di fculties associated withmarginal utility growth proxies.

CRRA utility CRRA utility can beexpressed in the power-utility ormu[c ]= c1-θ / (1-θ) whereθ, a constant, is therelative risk aversion coe fcient.

Taking the frst derivative and insertinginto (3):

This discount actor is at the heart o the“equity premium puzzle” identifed by Mehra and Prescott (1985). The puzzlecan be illustrated by the lower bound on

the discount actor coe fcient o varia-

tion CV [m] = σ[m]=E[m] provided by Hansen and Jagannathan (1991):

whereR e is the return in excess o the Treasury Bill rate. The post-war Sharperatio ( E [R e ] / σ [R e ]) or the value- weighted NYSE port olio is about 0.5 while per-capita consumption growthhas a standard deviation o about 1%per year (Cochrane, 2005). Given thediscount actor o equation (4), and set-ting β = 1, the required coe fcient o r isk aversionθ to satis y (5) is approximately 50. However, Mehra (2003) noted alarge body o literature that suggests thecoe fcient o risk aversionθ is “certainly less than 10.” The extremely high risk aversion indicates the volatility o con-sumption growth is too low to explainthe equity premium.CRRA utility does not ft observed data without implausibly high levels o risk aversion, but what about other utility

unction orms? Other more complexutility unctions have been proposedsuch as habit ormation and time-non-separable orms. Un ortunately, the morecomplex orms have proven unsuccess ulin resolving the equity premium puzzle

Conditional estimation of linear assetpricing models using alternative

marginal utility growth instruments 1

The exchange economy model of Lucas yields the familiar Euler condition thatdiscounts expected returns by marginal utility growth. Unfortunately, proxies formarginal utility growth are subject to theoretical and empirical shortcomings

Join our t eam: www.alloptions.n l/l ife

Success is a team effort!

All Options is a leading market makerproviding liquidity to the derivativesmarkets in Europe and Asia.

In only 2 years we have grown from 60to 300+ employees and are now one ofthe largest market makers in the world. And we don’t stop there – this year weare seeking another 50 young talents fortrading careers.

This success is due to our belief insupport for personal achievement anddiscipline in success. Working in unityis at the core of our culture. That’s whywe reward our traders on both theirindividual and team performance.

If you are interested in a challenging andrewarding career in trading check outyour options at: www.alloptions.nl

Trading on the future.

By David J. Moore

1 ) Earlier versions of this paper were presented at Pepperdine University,The University of Memphis,The University of Tennessee,Lynchburg College,and during the 2009 Ph.D.Project Finance Doctoral

Student Association meetings coincident with the 2009 Western Finance Association annual meeting.

>>>

7/27/2019 Conditional Estimation of Linear Asset Pricing Models Using Alternative Marginal Utility Growth Instruments

http://slidepdf.com/reader/full/conditional-estimation-of-linear-asset-pricing-models-using-alternative-marginal 2/3

FSR Forum | september 2009 | FSR Forum | september 2009 | 9

as well (Mehra, 2003).From log utility to CAPM Log utility u [c t ] = ln [c t ] can be shown to map directly into CAPM. Let R t w represent the returnto the market wealth port olio with theS&P 500 return o ten used as an empiri-cal proxy. Cochrane (2005) ound thatgiven log utility,

Sincemt +1 =βu’ [c t +1] = u’ [c t ], R t w+1 =

1/mt +1 ormt +1 = 1/R t w+1. Taking the

linear approximation we arrive at the

CAPM discount actor:

Given the relatively high volatility o R W ,the stochastic discount actor (7) has aninherent advantage over the low volatil-ity consumption growth derived discount

actor (4). However, the inability o CAPM to explain anomalies such as thesize premium, the value premium, andmomentum profts is indicative o modelmis-specifcation. Cochrane (2005)noted that i the discount actor (7) is to

price multiple assets simultaneously, thecoe fcients can not be constant and thecorrect discount actor representation is

Where a0t and a1t are time-varying

coe cients. Cochrane also noted thesetime-varying coe cients account ortime-varying expected returns, variances,covariances, and risk ree rates. Here, Iprovide an additional perspective: the time varying coe cients o (8) also account

or time-varying risk aversion, speci cally fuctuations about θ = 1 given the connec-tion between log-utility and CAPM.

3 Macroeconomic growth modelSince the time-varying coe fcients in (8)ultimately capture time-varying marginalutility growth, I pursue an alternative ex-

pression or marginal utility growth usedto drive the time variability o a0t and a1tusing a discrete time general equilibriummacroeconomic growth model.

The model; The model developed hereollows the closed economy exogenous

growth model without government o King and Rebelo (1999) and is illus-trated in Figure 1.

To ensure the existence o a steady-state,exogenous growth is introduced via laboraugmentation consistent with Sala-i

Martin (1990). As such, the productionunction can be specifed in Cobb-

Douglas orm as

where Yt is GDP, Kt is capital input, Nt

is labor input, At is the random produc-

tivity shock, and Xt is the deterministiccomponent o productivity which growsat a constant (and exogenous) rateγ > 1:

I am not addressing labor-leisure choicein this study; there ore, labor is assumedto be fxed ( N t = 1∀t ) and the production

unction simplifes to:

The Cobb-Douglas production unction was chosen or several reasons. First,by construction this production unc-tion exhibits constant returns to scale,consistent with the empirical fndings o Jorgenson (1972). Second, the empiri-cal evidence noted by Jorgenson (1972)suggests the estimated elasticity o substitution or the constant elasticity o substitution (CES) production unctionis not signifcantly di erent rom unity and there ore the CES reduces to Cobb-Douglas orm. Third, Arroyo (1996)suggests the Cobb-Douglas orm is“probably more descriptive o aggregatetechnological conditions.”

Maximization problem Following themodel o King and Rebelo (1999), alltrending variables are scaled by Xt andtrans ormed variables are denoted by lower case letters (e.g., yt = Yt/Xt). Theinfnitely-lived central planner maxi-mizes discounted expected utility:

wherebt < 0 is the rate o time pre er-ence, subject to several constraints. First,all output is either consumed or invested:

In addition, capital stock evolves accord-ing to the perpetual inventory method:

Where ϑ is the rate o depreciation.Noting y t = Atk t 1-a and combiningconstraints (13) and (14):

the Lagrangian is:

Solution; The frst order conditions o the maximization problem are:

The frst order conditions (17) and(18) can be combined to reveal the key contribution o this study: an alternativeproxy or marginal utility growth:

This expression provides a read-ily observable proxy or unobservablemarginal utility growth that avoids thetheoretically troublesome utility unctionspeci cation and empirically troublesomeconsumption data. Now I proceed to- wards connectingΓ with the time-vary-ing discount actor coe cients a0t and a1t.

4 Empirical construction andcharacteristics of Γ Table 1 summarizes the sources, re-quency, and availability o the data usedto constructΓ . Aggregate macroeconomicdata are obtained rom the Bureau o Economic Analysis National Income andProduct Accounts tables (BEA NIPAtables). Using the BEA NIPA aggregatemacroeconomic data, data, output (Y ) isde ned as gross domestic product (GDP),capital (K ) is de ned as private non-resi-dential xed assets in place, investment( I ) is de ned as private non-residential

xed investment, and consumption (C ) isde ned as non-durable consumption. Allnominal data are converted to real usingthe CPI defator rom the U.S. Bureau o Labor Statistics (BLS).

Un ortunately quarterly capital data areunavailable rom the Bureau o Econom-ic Analysis there ore the series must beestimated. The quarterly capital series isconstructed rom annual capital data andquarterly investment data ollowing theprocedure o Balvers and Huang (2007).Capital in quarter q is computed as:

with Ky;q equal to capital or quarterq in year y, Iy;i equal to investment

or quarter i in year y, and Ky equalto capital at the end o year y. Follow-ing the parametrization o King andRebelo (1999), the value or the rate o time pre erence b is set to 0.984, thelabor share o output _ is set to 2=3, thequarterly growth rate is set to 1.004, andthe quarterly depreciation rate _ is set to0.025. Given this parametrization and

current dataset, E [Γ ] = 1:0140,σ [Γ ] =0:0030, andσ [Γ ] =E [Γ ] = 0:0029.

5 [Un]conditional estimation using Γ The low volatility and coe fcient o variation o Γ render it unsuitable or useas a stand-alone discount actor since it will not satis y the Hansen- Jagannathanbounds. However, the low volatility may be benefcial in a conditional estima-tion rom two perspectives. First, rom ascaled payoff perspective, a low volatility instrument is representative o a easibletrading strategy due to low transactionalactivity and costs. Second, rom ascaled discount factor perspective, when thecoe fcients o discount actor (8) areallowed to vary withΓ , they capturechanges in risk aversion around un ity.I we allow a0t and a1t to be unctions o Γ t, the conditioning in ormation in theoriginal Euler condition can be statedexplicitly:

O course, whenever one discussesinstruments they are exposed to thecriticism o omitting instruments rel-evant to the estimation. However, giventhe equality between the instrumentand marginal utility growth, I proceed

under the assumption that the relevantin ormation is captured by Γ . Cochrane(2005) noted that an unconditionalEuler condition is implied when usinga discount actor with constant coe -fcients, as in equation (7). Thus, i there were some way to express the discount

actor with time-varying coe fcients,equation (8), in a orm with constantcoe fcients, an unconditional Eulercondition could also be used.

To do so, begin by defning a0t and a1t as linear unctions o Γ t:

Inserting these coe cients into (8) yields:

In other words, a conditional estimation(conditioned onΓ t) with a single

actor (R t w+1) is equivalent to anuncon-

ditional estimation with three actors(Γ t, R t

w+1 andΓ tR t w+1):

E[mt+1 R t+1 |Γ t] = 1 → E [m*t+1 R t+1] = 1

where mt+1 is represented by equation (7)and m*t+1 is represented by equation (24).

6 Theoretical evidence of re ned discountfactor ef cacyHere I show that the coe fcient o varia-tion (CV) o the refned discount actor with time-varying coe fcients (m*t+1 ) islikely to be greater than that o the origi-nal discount actor with fxed coe fcients(mt+1). The larger CV su ggests a greaterlikelihood o satis ying the Hansen-Jag-annathan bounds.For notational simplicity, let X =Γ t and >>>

F

igure 1: Graphical representation of general equilibrium macro growth model Table 1: Data Items, sources, and availability

The rst order conditions (17) and (18)

can be combined to reveal the key contribution of this study: an alternative

proxy for marginal utility growth

7/27/2019 Conditional Estimation of Linear Asset Pricing Models Using Alternative Marginal Utility Growth Instruments

http://slidepdf.com/reader/full/conditional-estimation-of-linear-asset-pricing-models-using-alternative-marginal 3/3

FSR Forum | september 2009 | 10 FSR Forum | september 2009 | 11

Y =R t w+1 . Dropping subscriptsand

rewriting the two discount actors:

It is quite common in fnancial literatureto examine excess returns: R t

e = R t – R t

where R represents the risk ree rate. The Euler condition to excess returns isEt [mt+1 R et+1] = 0 Note that in this ormE [(2m)R e] = 0 is also true. As such,the coe fcients o m are not identifed. There ore some orm o normalizationmust be chosen. Let a0 = 1 and a00 = 1 inequations (25) and (26), respectively:

Although the choice o normalizationseems arbitrary, Cochrane (2005) notedthat the choice is purely one o conve-nience and the pricing errors are inde-pendent o the choice o normalization.Another simplifcation is to constrainthe expected value o the coe fcientso (28) to the values in (27). Beginning with the coe fcient on Y :

Proceeding to the intercept term:

The last equation reveals a01 = 0 since E[X] > 0. The static CAPM and dynamicCAPM discount actors with the unity intercept normalization and constrainedcoe fcients are summarized below:

Now the goal is to demonstrate thecoe fcient o variation or m* is greaterthan that o m:

Computing the expected values o mand m*:

Recall that X is the marginal utility growth proxy and Y is the equity marketreturn. Since marginal utility growth ispositively correlated with equity returns we know cov [X; Y ] > 0 and there ore:E [m*] < E [m] Hal o the battle isfnished. We now know that, all elseconstant, the smaller denominator o

CV [m*] will translate into a larger CV than m. Now lets proceed to the nu-merator. The variance o m and m* canbe computed as:

Given the complexity o expanding var[XY ] and cov [Y,XY ] it is di fcult torigorously prove inequality (33). How-ever, we do know that var [XY ] > 0 andit can be shown that cov [Y,XY ] > 0(Moore, 2009). Thus it is the empiricalresearchers task to arrive at coe fcientsa10 and a11 such that inequality (33)holds. The task does not appear to bemonumental given the unambiguously smaller denominator and quite likely

larger numerator o CV [m*].

7 Conclusion The macroeconomic growth model yields a productivity-based proxy ormarginal utility growth that bypassestwo key drawbacks o traditional mar-ginal utility growth proxies. First, thedi fculty in obtaining explicit utility

unctions, which are unobservable, isbypassed by using an observable pro-ductivitybased proxy. Second, the morereliably measured production data avoidsthe problem measurement error associ-ated with consumption data. A refneddiscount actor was constructed usingthe necessarily timevarying coe fcients

expressed as linear unctions o the new marginal utility growth proxy. The new discount actor was shown to mapdirectly into an unconditional estimation

ramework. In particular, the time-vary-ing coe fcients o the refned discount

actor account or time-varying expectedreturns, variances, covariances, risk reerates, and risk aversion. Theoretical evi-dence was provided to demonstrate thegreater likelihood o the refned discount

actor to satis y the Hansen-Jagannathanbounds. As such, the refned discount

actor with time-varying coe fcientscould price those assets that the standardCAPM discount actor with fxed coe -fcients can not. |||

ReferencesArroyo, Cristino R., 1996, Testing a production-based asset- pricing model, Empirical Inquiry pp. 357–377.Balvers, Ronald J., and Dayong Huang, 2007, Productivity-based asset pricing: Theory and evidence, Journal o FinancialEconomics 86, 405–445. Cochrane, John H., 2005, Asset Pricing(Princeton University Press).Hansen, Lars P., and Ravi Jagannathan, 1991, Implications o security market data or models o dynamic economies, Journalo Political Economy 99, 225–262. Jorgenson, Dale W., 1972, Investment behavior and theproduction unction, Bell Journal o Economics 3, 220–251.King, Robert G., and Sergio T. Rebelo, 1999, Resuscitating real

business cycles, in John B. Taylor, and Michael Wood ord, ed.:Handbook o Macroeconomics (Amsterdam: North Holland).Lucas, Jr., Robert E., 1978, Asset prices in an exchange economy,Econometrica 46, 1426–14.Mehra, Rajnish, 2003, The equity premium: Why is it a puzzle?,Financial Analysts Journal pp. 54–69., and Edward Prescott, 1985, The equity premium: A puzzle, Journal o Monetary Economics15, 145–161.Moore, David J., 2009, Use ul statistics properties, http://djmphd.googlepages.com/statProperties.pd . Sala-i Martin, Xavier, 1990,Lecture notes on economic growth, Working Paper No. 3563.

The macroeconomic growth model yields a productivity-based proxy for marginal utility growth that bypasses two key drawbacks of traditional marginal utility growth proxies

Nu lees je wel eens over staatsbedrijven.Binnenkort mag jij ze beheren.

Financiën zoekt startendebedrijfseconomen

www.werkenbijhetrijk.nl

Wij bieden je van meet af aan veel ruimte vooreigen verantwoordelijkheid. Het kan dan ook zomaar gebeuren dat je direct na je studie staats-bedrijven mag beheren. Dat moet je willen, dat moet je kunnen. Iets voor jou? Toptalent is vanharte welkom. Zeker als je binnenkort als bedrijfs-econoom afstudeert. Kijk voor meer informatieop www.minfin.nl. Je sollicitatie mail je naarrecruitment @minfin.nl of je belt 070-3428532.

Het Rijk is mede-eigenaar van 35 grote onder-nemingen. Het ministerie van Financiën is namensde Staat aandeelhouder en verantwoordelijk voorbijvoorbeeld de verkoop van het vervoersbedrijf Connexxion. Maar ook de oprichting van de onder-neming die de Zuidas moet gaan ontwikkelen of de aankoop van een aandelenbelang in het Rotterdamse Havenbedrijf. Op dit ministerie werk je altijd aan uitda gende proje cten met grote ma at-schappelijke gevolgen. Ook als starter, want je doet direct mee als volwaardig teamlid. Dit betekent wel dat wij veel van jou verwachten.Bij Financiën tel je meteen mee.