Interest Rates and Asset Prices

Transcription

1 Interest Rates and Asset Prices Martin Ellison MPhil Macroeconomics, University of Oxford 1 Introduction The course so far has concentrated in how macroeconomic variables, such as consumption, investment, output and inflation evolve over time. Our efforts have focussed on characterising the dynamics in these variables and trying to understand the underlying forces which produce these fluctuations. Aside from references to interest rates we have made little mention of financial variables. Earlier lectures introduced money into our standard models but the focus in this lecture is on financial variables, e.g. stock prices, bond prices, interest rates, the term structure of interest rates, etc. In particular we will extend our analysis to see what implications our neoclassical model has for these real financial variables. The use of neoclassical models to analyse financial variables has generally been seen as less controversial than the same program aimed at economic variables. Many economists believe that goods and credit markets contain fundamental imperfections that make the neoclassical analysis irrelevant. However, at least until recently it is widely perceived that financial markets come closest to approximating the ideal market structure of neoclassical macroeconomics. However, as we shall see the performance of the neoclassical model in explaining financial variables is little better than in explaining economic variables. 2 Key readings The assigned reading for this lecture is Romer Chapter 7. The seminal paper in this literature is Lucas(1978) Asset Prices in an Exchange Economy Econometrica but this is an extremely demanding read which I would not expect you to even try. The seminal assessment of the theory is Mehra and Prescott(1985) The equity premium: A puzzle Journal of Monetary 1

2 Economics. A useful article which offers a general overall framework is Campbell(1986) Bond and Stock Returns in a Simple Exchange Model Quarterly Journal of Economics which after several careful readings may help to clarify matters. 3 Key concepts Equity Premium, Term Structure, Consumption CAPM, Asset Pricing Models 4 Econometric evidence The crucial link in economic models of asset pricing is between consumption and rates of return. If an individual is to invest in an asset they have to be prepared to defer some consumption now for consumption in the future. Therefore at the core of economic models of assetpricesisalinkbetweenconsumptiongrowthandratesofreturn. Table1showssome basic facts regarding US consumption, the return on US Treasury 90 day bills (a relatively safeshorttermsecurity),thereturnontheusstockmarketandthedifferencebetweenthe equity return and the return on the Treasury Bill(called the equity premium). 2

3 Cons growth Return Tbills Return stocks Equity premium Mean S.D. Mean S.D. Mean S.D. Mean S.D Table 1: Consumption and Rates of Return (from Mehra and Prescott, JME, 1985) The numbers clearly differ between decades and in the case of both consumption growth and treasury bills there is some evidence that volatilities have declined over time. However, the following facts stand out: (i) Consumption growth displays relatively small volatility, ranging between -0.25% and 3%withastandarddeviationof3.57 (ii) Equities are more volatile than short term bonds (with a standard deviation around three times larger) 3

4 (iii)ineverysingledecadeinvestorsearnedahigherreturnonequitiesthantheydidon shorttermbonds. Overthewholeperiodtheequitypremiumwasonaverage6.2%,withthe shortterminterestratebeing0.8%andthereturnonequities7% 5 Consumption Capital Asset Pricing Model In Lecture 1 we examined the consumer s intertemporal first order condition(ec). This was: U (c t )=E t βu (c t+1 )R t+1 (1) whereu ( )denotesthemarginalutilityofconsumption,ristherateofreturnonanasset and β is the discount factor. The intuition behind this expression is very simple. A consumer isdecidingwhethertoforegooneunitofconsumptionnowandinsteadpurchaseoneunitofan assetwhichwillyieldanuncertainreturnrnextperiodandenabletheconsumertohaver extra units of consumption next period. The consumer will choose their consumption/assets so that the marginal utility of today s consumption is equal to what an extra unit of the asset would yield in utility next period. Notice that if rates of return are positive then U (c t+1 )<U (c t )andsoconsumptionisincreasingovertime(u ( )<0). Equation(1)tells us something about the joint determination of rates of return and consumption, alternatively wecouldthinkof(1)asexplainingconsumptiongrowthforagivenrateofreturn(aswedid inlecture1)orasexplainingratesofreturnforagivenpathofconsumption,asweshalldo now. If a consumer has more than one asset that they can invest in, equation (1) must hold foreachandeveryasset. Foreach assettheconsumerhastoequatethelossfromforegone consumption with the gain from additional consumption in the next period which depends uponthe assetrate ofreturn. Consideran economywheretherearetwoassets, itmustbe the case that: U (c t )=E t βu (c t+1 )R 1t+1 U (c t )=E t βu (c t+1 )R 2t+1 (2) Assumethatasset1isa safe assetwhileasset2isrisky. By safe wemeanthatthe returnonasset1isknownoneperiodinadvance. From(2)wehave: 0=E t U (c t+1 )(R 2t+1 R 1t+1 ) 0=E t U (c t+1 )E t (R 2t+1 R 1t+1 )+cov(u (c t+1 ),(R 2t+1 R 1t+1 )) (3) 4

5 wherewehaveusedtheresultthate(xy)=e(x)e(y)+cov(x,y). Becauseasset1isa safeassetweknowthate t R 1t+1 =R 1t+1 alsobecauseitissafeitinvolvesnorisk/uncertainty andsodisplaysnocovariancewithu (c t+1 ). Thereforewecanre-write(3)as: E t R 2t+1 =R 1t+1 cov(u (c t+1 ),R 2t+1 ) E t U (c t+1 ) (4) Equation(4)isattheheartoftheConsumptionCAPMmodelanditsaysthatiftherate of return on a risky asset is positively correlated with the marginal utility of consumption thenthatassetwillearnalowerrateofreturnthanthesafeasset. Incontrast,assetswhich have a negative correlation with marginal utility will earn a greater return than the safe asset. remembering that marginal utility is decreasing in consumption, this implies that assets which tendtopayahighreturnwhenconsumptionislowearnaloweraveragerateofreturn,whereas assets which pay a high return when consumption is high earn above average rates of return. Why?Effectivelyanassetwhichpaysahighrateofreturnwhenconsumptionislowactsasan insurance policy. When consumption is low, marginal utility is high and agents would like to havemoreincometospend-inotherwordswhenabadeventhappensyouwouldideallylike an insurance policy that pays out. Therefore assets which have a positive covariance between their rate of return and marginal utility offer insurance benefits. As a result agents will be preparedtoholdthemeveniftheyearnalowerrateofreturnthanotherassets. Bycontrast, assets with a negative covariance have their highest rates of return when consumption is high, thereforetobepersuadedtoholdtheseassetstheirrateofreturnhastobehigherthanother assets. 5.1 Equity prices Wecanalsouse(1)todrawsomeimplicationsaboutequityprices. Inparticularwewillshow howundercertainassumptionswecanexpressanequitypriceasadiscountedsumoffuture dividends. Thereturnonequityisgivenby(p t+1 +d t+1 )/p t wherepisthepriceof ashare and d is the dividend payment received by the share. Therefore re-writing(1) and using our expressionfore(xy)wehave: 1=βE t ( pt+1 +d t+1 p t ) ( ) ( ) U (c t+1 ) U (c t+1 ) E t +βcov U (c t ) U (c t ),p t+1+d t+1 p t To derive a simple equation for equity prices we need to make very strong assumptions. Firstly,weneedtoassumethatthecovariancetermin(5)iszeroandsecondlywealsoneed (5) 5

6 toassume thatthe expectation of the ratio of marginal utilities is constant. For this tobe truewerequirethattheutilityfunctionislinear(e.g. U(C)=α+δC). InthiscaseU (c)=δ andsoe t U (c t+j )=δ j. Ifwemaketheseassumptionsthen(5)canbewrittenas: E t (p t+1 +d t+1 )= p t β (6) which says that when allowance is made for discounting and dividends then the share price is unpredictable- a variant of the famous random walk result. Moreover, we can use recursive forward substitution to re-write this equation as: p t = β j E t d t+j (7) j=1 sothatthecurrentsharepriceisequaltothepresentdiscountedvalueofallfutureexpected dividends. Equation(7) is a much-tested implication of various asset pricing models. To see howwemovefrom(6)to(7),re-write(6)as: Butthesameexpressionfort+1gives: p t =βe t p t+1 +βe t d t+1 so by substitution we have: p t+1 =βe t+1 p t+2 +βe t+1 d t+2 p t = βe t (βe t+1 p t+2 +βe t+1 d t+2 )+βe t d t+1 = β 2 E t p t+2 +βe t d t+1 +β 2 E t d t+2 IfwecontinuallysubstituteforE t p t+j wearriveat(6)-theequitypricereflectsthediscounted stream of future dividends. Notice however the extreme assumptions we have made toarriveat(7)-thecovariancetermbeingsettozeroandtheutilityfunctionbeinglinear. Neither of these assumptions can be empirically justified and without them it is not the case that the equity price equals the present discounted value of future dividends. 5.2 Term structure Equation(1) can also be generalised to assets which involve more than one period of investment (i.e. twoyearbonds). Consideraconsumerwhoisdecidingwhethertoinvestinaj period 6

7 bond which will earn return R jt over the next j periods. The same logic as we used for equation (1) tells us that the consumer will equate the lost utility from lower consumption this period with higher consumption gained in j periods time so that the Euler equation is: [ ] E t β ju (c t+j ) U (c t ) R jt =1 (8) Atanymomentintimethetermstructureisdefinedas{R 1t,R 2t,...R nt },inotherwords a sequence of interest rates on bonds of different maturities. The term structure is extremely important to policymakers and financial markets as it reveals what the market expects the future interest rate to be, e.g. the difference between R 2t (the return on a two year bond) and R 1t (the return on a one year bond) must contain information on what interest rates willbeinyear2. Thisideaformsthebasisoftheexpectationstheoryofthetermstructure. This theory basically says R 2t = R 1t E t R 1t+1 in other words investors must earn the same returnfrominvestinginatwoyearbondastheyexpecttoearnfrominvestinginaoneyear bondnowandthenreinvestingtheproceedsinaoneyearbondnextyear. Thistheoryhas been thoroughly tested and its strict implications found not to hold. While the term structure does tell us something about future rates the correlation is not perfect. To see this consider equation(8)wherej=2andwehaveslightlyre-writtentheequationsothat: 1 =E t β U (c t+1 ) (c t+2 ) R 2t U (c t ) βu U (c t+1 ) (9) An important statistical result is the Law of Iterated Expectations or LIE. This says that E t (E t+1 X t+1+j )=E t X t+1+j. WhileLIElooksforebodingitisactuallyaverysimpleresult. Itsaysthatifwearetoforecasttodaywhatwethinkourforecastwillbeofavariableinthe future, then our best forecast of tomorrow s forecasts is simply our current forecast. We can therefore use this result to re-write(9) as: 1 R 2t =E t β U (c t+1 ) (c t+2 ) U (c t) βu U (c t+1 ) 1 R 2t =E t β U (c t+1 ) U (c E t) t+1β U (c t+2 ) U (c t+1 ) 1 R 2t =E t β U (c t+1 ) U (c R 1 t) 1t+1 (10) where we have again used (8) for j = 1 in the last line. Equation (10) is a generalised version of the expectations theory of the term structure. If the covariance term is zero(which would happen if U( ) were linear) then (10) is exactly the standard expectations model of the term structure - the return on a two year bond equals the return on a one year bond times the expected return on a one year bond next period. However, more generally there 7

8 is a covariance term reflecting the fact that the consumer dislikes uncertainty. The intuition behind the covariance term is not surprisingly similar to that in the consumption CAPM. If U t+1 /U t and R 1 1t+1 are negatively correlated then the one period bond tends to pay a high rateofreturnwhenthemarginalutilityofconsumptionishigh. Ifthisisthecasethenthe returnonatwoyearbondmustbegreaterthanfromsimplyinvestingintwoconsecutiveone year bonds, to compensate the consumer from losing this insurance effect. This is exactly what(10)says,ifthecovariancetermisnegativethenr 1 2t <R 1 1tE t R 1 1t+1 implyingthatr 2t is greater. While (1) tells us something about the term structure we need to make more specific assumptionsifweareabletosayanythingmorepreciseabouttheslopeofthetermstructure -thatisdointerestratesonbondsincreaseordecreasewithmaturity? Assume that the utility function is given by constant relative risk aversion(crra) so that U(c)=c 1 σ /(1 σ). Inthiscasewecanwrite(8)as: [ ( ) σ E t β j ct+j R jt] =1 (11) c t Beforewecansayanythingpreciseaboutthetermstructureweneedtomakeonemore assumption, and that is that consumption growth and interest rates are distributed jointly log normal. This is a standard trick in modern macro and often leads to very tractable analyticalexpressions.whatdoesitmean? IfX isdistributedlognormalthenlog(e t X)= E t logx +σ 2 /2, where σ is the standard deviation of logx. It should be stressed that this assumption of joint log-normality does have economic implications. We are essentially assuming something about the tastes and technology of the economy and the type of economic fluctuations they produce. If we apply this formula to(11) and re-arranging we have that: lnr jt = jlnβ+σe t ln(c t+j /c t ) 1 2 var( σln(c t+j/c t )) (12) Assuming that on average consumption grows by η per period we can then use (12) to calculate an average one year interest rate associated with a j period bond. lnr jt j = lnβ+ση 1 2j var( σln(c t+j/c t )) (13) Equation(13)saysthattheaverageyieldonajperiodbonddependsonthreeterms: the discount rate, mean consumption growth and a variance term. The first reflects the discount rate. Becauseβ<1,lnβ<0andsotheyieldonajperiodbondisincreasinginthediscount factor. The intuition behind this is simple- consumers discount the future and so place less 8

9 weight on the future marginal utility of consumption. Therefore in order to persuade the consumertoholdabondtherateofreturnneedstoatleastmatchtherateoftimepreference. However,thiseffectisthesameforbondsofallmaturitiesandsodoesnotaffecttheslopeof thetermstructure(thetermstructurebeingaplotofessentiallylnr jt /j againstj. The second term is the expected average consumption growth over the next j periods. If consumption is expected to grow strongly over the next j periods then the ratio of marginal utilityinj periodstimeandnowwillbelessthan1. Thereforethegreaterthisconsumption growth,thehigherthej periodinterestrateneedstobetopersuadeagentstogiveupeven more consumption today in return for higher t+j consumption. We have assumed that consumption growthis expected tobe thesame1,2orj years out. However, this need not be the case. For instance, at the bottom of a recession consumption growth over the next fewyearscanbeexpectedtobehigherthanoverthenext20years. Thereforefrom(13)the average yield on short term bonds should exceed that on 20 year bonds in the depth of a recession. The final term reflects the variance of consumption growth. If consumers are not characterised by certainty equivalence then increases in uncertainty affect their behaviour. The CRRA utility function does not display certainty equivalence and so uncertainty has an important role to play. Here an increase in uncertainty causes the bond rate to fall, because the greater the uncertainty the more that consumers value the certain payoff provided by the bond. Whether the term structure is upward or downward sloping depends on whether the numerator or denominator of the third term increases the most with the maturity j. Since consumption growth in US data is positively autocorrelated, we would expect the numerator to rise faster than the denominator. In other words, this model delivers a downward sloping term structure. This is contrary to the upward-sloping term structure usually observed in data. Intuitively, the term structure slopes downwards in the model for insurance reasons. Suppose that consumption growth is subject to a negative shock sometime between periods tandt+j. Theworseningoutlookforconsumptiongrowthwillcauseinterestratestofall, which implies that the long bond will increase in price. Consumers will therefore have a capital gain with which to offset their reduced consumption. Therefore bonds offer a hedge against consumptionrisk. From(13)wecanseethatthegreatertheuncertaintythereisaboutj periodaheadconsumptionthelowerthereturnonaj periodbond. Onceagainthisisbecause consumers are willing to earn a lower return on bonds because of the hedging characteristics they offer. Those of you who are interested in how well stochastic dynamic general equilibrium 9

10 models succeed in reproducing the observed behaviour of the term structure would do well to read Wouter den Haan The term structure of interest rates in real and monetary economies, Journal of Economic Dynamics and Control 1995, pp The final term reflects the variance of consumption growth, rates of return and the covariance between them. If consumers are not characterised by certainty equivalence then increases in uncertainty affect their behaviour. The CRRA utility function does not display certainty equivalence and so uncertainty has an important role to play. Once again the reason why uncertainty influences the rate of return is for insurance reasons. To see this consider the case where consumption growth overthe next j periods is expected to be low. Fromourearlier analysisweknowthisimpliesthatratesofreturnoverthenextj periodswillbelow. Fora bond this implies that bond prices will be relatively high. Conversely if consumption growth isexpectedtobehighbondpriceswillberelativelylow. Thereforebondsofferahedgeagainst consumptionrisk. From(13)wecanseethatthegreatertheuncertaintythereisaboutj periodaheadconsumptionthelowerthereturnonaj periodbond. Onceagainthisisbecause consumers are willing to earn a lower return on bonds because of the hedging characteristics they offer. Those of you who are interested in how well stochastic dynamic general equilibrium models succeed in reproducing the observed behaviour of the term structure would do well to read Wouter den Haan The term structure of interest rates in real and monetary economies, Journal of Economic Dynamics and Control 1995, pp Lucas Asset Pricing Model While the last section gave a number of insights regarding the relationship between consumption, rates of return and uncertainty it was not a general equilibrium analysis. We tended to move between treating consumption as a given and then seeing what determined rates of return or fixing rates of return and analysing what happens to consumption. However, what we really want to analyse is the joint determination of both. Unfortunately this requires numerical simulations for plausible economic models. However, Lucas(1978) offers a very abstract model whichavoidstheproblem. IntheLucasmodeltheonlyformofcapitalaretreeswhichbear fruit. Unfortunately the fruit produced by these trees can only be used for consumption and not investment purposes. Therefore in this economy output must equal consumption(output is simply the crop of fruit). From period to period the crop varies randomly (presumably becauseofweather). Theideahereistointerpretthetreeasanassetwhichyieldadividend 10

11 streamforallfutureperiods(thedividendsbeingthecrop)andthequestioniswhatpriceto attach to the asset. The questions Lucas tries to answer are actually more ambitious than this explanation might suggest. What Lucas was trying to arrive at were asset pricing formulae. That is given certain information about the economy, i.e. value of productivity shocks, capital stock, etc. how could you convert these into a formula for determining asset prices. Further, Lucas was interested in asset pricing rules which formed a Rational Expectations Equilibrium. That is if everyone used these asset pricing rules then everyone would choose appropriate capital stocks and consumption such that the prices predicted by these pricing rules actually materialised. However, we shall consider only very simple examples of the Lucas paper. Thereturntoholdingatreeis(p t+1 +d t+1 )/p t where disdividends(crop)andpisthe priceofthetree. OurusualEulerequation(1)holdssothat: ( )( ) pt+1 +d t+1 U (c t+1 ) E t β =1 U (c t ) p t Becausethefruitisperishableitmustbethecasethateachperiodthecropisconsumed (d t =c t )sothatwecanre-writethisequationas: ( ) U (d t+1 ) p t =E t β (p U t+1 +d t+1 ) (14) (d t ) Ifweusethisequationtokeepsubstitutingoutp t+j ontherighthandsideweeventually arrive at the results that: p t =E t β ju (d t+j ) U (d t ) d t+j (15) j=1 Thisisasimplegeneralisationof(6)sothattheassetpriceisstillequaltothediscounted sumoffuturedividendsbutnowtheconsumerusesadiscountratewhichdependsuponthe marginal utility of consumption. In this model when dividends are high they are give a lower weight(u (d)islow)becauseconsumptionisalreadyhighandthehighoutputisnotvaluedso highlycomparedtoalowoutputsituation. IfwemakethestrongassumptionthatU(c)=lnc then(15) becomes: p t = E t β j d t p t = j=1 or (16) β 1 β d t In other words, the share price simply depends upon today s dividend. This is an extreme case but (16) gives an example of an asset pricing function (i.e. feed in today s dividend 11

12 and out comes equity price) and also illustrates how this function crucially depends upon the utilityfunction. The reason why (16) depends only on current dividends is due tothe fact that future dividends are discounted completely. Announcements of future dividends have two effects: firstly, they increase the price of the share, secondly they increase future discount rates. In this simple logarithmic model these two facts example cancel out leaving the share price to depend only on current dividends. Notice that even though the equity price depends only on the current dividend the model is completely forward looking and characterised by Rational Expectations. Therefore even though most of the underlying model is the same we arrive at a very different result from(6). This in part justifies Lucas focus on asset pricing rules- clearly asset prices will differ strongly under different assumptions about the underlying economy. 7 Empirical evidence In Lecture 1 we briefly discussed some empirical rejections of (1) (consumption being predictable by income, the fact that the serial correlation properties of rates of return and consumption are very different). However, here we shall discuss less the failures of the model with respect to consumption but to rates of return uncertainty. The Mehra and Prescott (1985) paper takes a very simple Lucas asset pricing model and calculates plausible values of uncertainty, risk aversion and consumption and asks what the model predicts for the riskless rateofreturnandtheequitypremium. Inthedatatheequitypremiumisover6%butthey find that for a variety of models and assumptions the consumption CAPM cannot generate an equity premium higher than 0.4%. In other words, the model fails miserably to generate enoughexcess returninequity. Togive theintuitionbehindthisresultletutilitybe of the CRRA form, i.e. U(c) = c 1 σ /(1 σ) where σ is the coefficient of relative risk aversion. Given the lack of variability in consumption growth shown in Table 1, to explain large equity premiums itis necessary to assume verylarge amounts of riskaversion so thatσ is a large number(around60-80whereasinthedataitappearstobebetween1and5). Inotherwords, toexplainwhyriskyassetsearnsuchahighrateofreturnwhenthereisnotmuchriskinthe economyyouhavetoassumeagentsareextremelyriskaverse andwould prefernottohold risky assets. However, from(1) and using our CRRA assumption we can write: lnc t =α+ 1 σ R t+u t (17) InterpretingR t asthereturnonthesafeassetwecanseetheproblem. Toexplainthehigh equitypremiumσneedstobeverylarge,butthen(17)saysthattounderstandconsumption 12

13 growth of 1.88% per annum we require R/σ to be large. But if R is small (which Table 1 shows it is) and σ is large then we cannot explain the observed magnitude of consumption growth(theαin(17)isalsosmall). Inotherwords,ifweexplainthehighequitypremiumby highriskaversionwehaveanotherproblem-whyistheriskfreeratesolow? Whereasifwe explainthelowriskfreeratebylowervaluesofσtheequitypremiumpuzzleresurfaces,why doconsumersneedtoberewardedsomuchtoholdequities? A number of proposal have been examined for explaining the equity premium. Here I briefly discuss three of them. 7.1 Preferences The first is to change the specification of the utility functions. Two popular strands here are to introduce habits and also to move away from expected utility theory Non-expected utility theory Weexplaintheequitypremiumvia(17). However,theterm1/σin(17)istheintertemporal elasticity of substitution. In other words it tells us how much consumption a consumer is willing to reallocate between time periods in response to the interest rate, When we assume CRRA the intertemporal elasticity of substitution is the inverse of the coefficient of relative risk aversion. More generally whenever the utility function satisfies expected utility theory there is an inverse relationship between the intertemporal elasticity and risk aversion. This is an unfortunate restriction as risk aversion and the intertemporal elasticity measure to different things. Risk aversion is about how agents compare consumption in different states of the world whereas intertemporal substitution is about how agents compare consumption at different pointsintime. Inresponsetothisanumberofpeople(i.e. EpsteinandZin(1989)Journalof Political Economy) have investigated non-expected utility functions which do not impose this inverse relationship between risk aversion and intertemporal substitution. While non-expected utilityhasgonesomewaytosolvingtheequitypremiumandriskfreepuzzleitssuccesshas been limited. Firstly, while estimates of risk aversion from this approach are higher than with standard expected utility models they are still not high enough to explain the extent of the equity premium. Explaining the equity premium simply requires counterfactually high risk aversion. Secondly, estimates of the intertemporal elasticity of substitution are approximately the same regardless of whether you use expected or non-expected utility. Therefore the low risk free rate puzzle remains. 13

14 7.1.2 Habits Constantinides(1990) Journal of Political Economy shows that the equity premium and risk free rate puzzles can be explained by assuming habits in the utility function and without recourse to very high levels of risk aversion. The effect of introducing habits is that utility depends not just upon current consumption but also recent consumption. For instance, a CRRA utility function with habits would be: 1 1 σ (c t φc t 1 ) 1 σ (18) where φ reflects the importance of habits. In the presence of habits consumers have even more reason to want to smooth. Equation (1) reveals that the key variable for rates of return is the Marginal Rate of Substitution(MRS), that is the ratio of marginal utilities. In the absence of habits, marginal utility just depends n current consumption. However, in the case of habits marginal utility depends on consumption in several periods and so becomes more volatile. This makes the MRSmorevolatileaswellasreturnsanddependingonthedegreeofriskaversioncanexplain alargeequitypremium. Thetrickhereissimple-Table1tellsusthatconsumptionisn tvery volatile. Therefore to explain the equity premium via a volatile MRS we must make sure that marginal utility does not just depend on current consumption. As this explanation makes clear this habit-based explanation seems an excellent candidate for explaining the two asset market anomalies. However, as shown in Boldrin, Christiano and Fisher ( Asset Pricing Lessons for Modelling Business Cycles American Economic Review 2001)thisisonlypartlythecase. Inthecaseofanendowmenteconomy(withoutanycapital) habit-based utility functions can explain the equity premium and the risk free puzzle. However, onceproductionandalaboursupplychoiceisintroducedthisisnolongerthecase. Thereason whythisisthecaseisquitestraightforward. Intheproductionmodelwithcapitalandalabour supply decision agents have additional ways of smoothing their marginal utility. For instance, when output is high they can choose to invest more rather than raise consumption and similarly ifconsumptionishightheycanworkharderbytakinglessleisure. Alloftheseactionsserve toreducethevolatilityofthemrsandsogoagainstexplainingtheequitypremium. 7.2 Market structure The Mehra-Prescott (1985) paper examines a general equilibrium model where all markets are open. Therefore one reason why the model predictions might fail is that some markets 14

15 are not open. For instance, some consumers may be unable to borrow. If this is the case then a consumer s consumption will be correlated with their income in every period, and as a result there will be some individual specific income risks which will influence an individual s consumption. If there existed perfect borrowing opportunities or insurance possibilities then these idiosyncratic income risks would not influence consumption. The introduction of borrowing constraints can explain both the low risk free rate and the high equity premium. The risk free rate is the interest rate which ensures equilibrium in the deposit/loan market, that is where savings equals loans. However, if an economy is characterised by borrowing constraints then loans made are very small and so to ensure equilibriuminthedepositmarketitmustbethecasethatsavingsarealsosmall. Theonly waythiscanbeachievedisbyhavingverylowinterestrates. Thereforeinaneconomywith borrowing constraints the risk free rate is very low(see Huggett(1993) Journal of Economic Dynamics and Control). Borrowing constraints can also explain high values of the equity premium. Because of borrowing constraints individual consumption is more volatile than it otherwise would have been. This is because individual specific income risks cannot be diversified away through borrowing. Therefore consumers are already bearing more risk than theywouldliketoiftherewerecompletemarkets. Thereforeinordertotakeonevenmore riskbyholdingequitytheyneedtoberewardedwithveryhighratesofreturn. Borrowing constraints/incomplete markets have therefore always been seen as the most likely explanation for the equity premium puzzle. However, this claim has been questioned. Telmer(1993) Journal of Finance and D. Lucas(1994) Journal of Monetary Economics both examine the effect that various incomplete market assumptions have on the risk free rate and the equity premium. They find that only if borrowing opportunities are completely absent is it possible to explain the equity premium puzzle. Basically these papers find that agents only needaccesstooneassetwhichtheycansellshort(borrow)oversomerange(i.e. thereisstill a borrowing constraint) for them to be able to avoid large amounts of diversifiable risk. In other words, markets need to be seriously incomplete to explain the equity premium puzzle. Ifonlyafewassetmarketsareopenthisstillenablesassetpricestoapproximateveryclosely thosepredictedbyacompletemarketsrepresentativeagentmodel 1. 1 However, Heaton and Lucas (Journal of Political Economy 1996 Evaluating the effects of incomplete markets on risk sharing and asset prices ) suggests that if transaction costs are large in asset markets and also agents face persistent idiosyncratic shocks then a substantial proportion of the equity premium can be explained. 15

16 7.3 Immobile factors of production Boldrin, Christiano and Fisher(2001 American Economic Review) argue that habits combined with immobile factors of production can explain asset market puzzles. To understand why this isthecaseitisusefultoreturntothemodelwithhabitsintheendowmenteconomy. Boldrin et al identify two features which any model must possess in order to explain asset market puzzles. The first is that consumers must have frequent motivation to buy and sell assets in order to smooth consumption. The second is that for some reason consumers desire to trade assets is restricted. In the endowment economy with habits both these features are present. Because of habit formation marginal utility is very volatile and so for a given consumption variability the stronger are habits the more consumers wish to trade in assets. However, in an endowment economy there is a fixed supply of capital. As a consequence variations in thedemandforassetsleadtolargechangesinassetprices. Asaconsequencethismodelcan explain asset price puzzles. However as soon as we introduce production into the model the supply of capital becomes perfectly elastic and asset prices hardly change at all in response to demand variations. Hence the production model with habits cannot explain the asset market puzzles. Armed with this intuition Boldrin et al argue that the way to explain the asset market puzzles in the context of a production economy with habits is to introduce some rigidities whichfrustratethedesireofconsumerstotradeinassets. Inordertodothistheyintroduce a two sector economy: one sector produces capital goods and the other consumer goods. To introduce rigidities they assume that the capital employed in each sector needs to be chosen in advance. As a consequence capital cannot move between sectors immediately in the aftermath of a shock. To introduce additional frictions they also assume that the labour employed in each sector has to be fixed in advance. Their simulations suggest that these modifications go a significant way to explaining asset price puzzles. 8 Conclusion We have shown how the neoclassical model links consumption(not output) and rates of return on different assets and how particular importance is place on risk and covariance. These are extremely elegant theories which have been widely used in the finance literature. However, as was the case for the neoclassical model s ability to explain non-financial variables the model fails on a number of important empirical dimensions. Understanding these failures is the 16

17 subject of much research but as yet no clear consensus regarding how to proceed has been achieved. 17