TWO TREES: ASSET PRICE DYNAMICS INDUCED BY MARKET CLEARING. John H. Cochrane Francis A. Longstaff Pedro Santa-Clara

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1 TWO TREES: ASSET PRICE DYNAMICS INDUCED BY MARKET CLEARING John H. Cochrane Franci A. Longtaff Pedro Santa-Clara Abtract If tock go up, invetor may want to rebalance their portfolio. But invetor cannot all rebalance. Expected return may need to change o that the average invetor i till happy to hold the market portfolio depite it changed compoition. In thi way, imple market clearing can give rie to complex aet market dynamic. We tudy thi phenomenon in a very imple model. Our model ha two Luca tree. Each tree ha i.i.d. dividend growth, and the repreentative invetor ha log utility. We are able to give analytical olution to the model. Depite thi imple etup, price-dividend ratio, expected return, and return variance vary through time. A dividend hock lead to underreaction in ome tate, a expected return rie and price lowly adjut, and overreaction in other. Expected return and exce return are predictable by price-dividend ratio in the time erie and in the cro ection, roughly matching value effect and return forecating regreion. Return generally diplay poitive erial correlation and negative cro-erial correlation, leading to momentum, but the oppoite ign are poible a well. A hock to one aet dividend affect the price and expected return of the other aet, leading to ubtantial correlation of return even when there i no correlation of cah flow and giving the appearance of contagion. Market clearing allow the invere portfolio problem to be olved, in which the weight of the aet in the market portfolio are inverted to olve for the parameter of the aet return generating proce. Current Verion: October 3. Graduate School of Buine, Univerity of Chicago. The Anderon School at UCLA. John Cochrane gratefully acknowledge reearch upport from an NSF grant adminitered by the NBER and from the CRSP. We are grateful for the comment and uggetion of Michael Brandt, George Contantinide, Vito Gala, Lar Peter Hanen, John Heaton, Jun Liu, Monika Piazzei, Rene Stulz, Pietro Veronei, and eminar participant at the Univerity of Chicago. We are alo grateful to Bruno Miranda for reearch aitance. All error are our reponibility.

2 . INTRODUCTION When tock do better than bond, growth tock do better than value tock, or the Nadaq doe better than the S&P 5, there i much talk about rebalancing portfolio. But everyone cannot rebalance. Expected return mut adjut, or other moment mut change, o that the average invetor i till happy to hold the market portfolio, depite it larger weight in the aet that ha gone up. In thi way, the market-clearing condition can induce dynamic in tock price and return, even when the underlying preference are table and the underlying cah flow are i.i.d. To characterize dynamic induced by market clearing, we examine a imple model with two Luca tree. Each tree give a dividend tream that follow a geometric Brownian motion. The repreentative invetor ha log utility and conume the um of the tree dividend. We obtain explicit cloed-form olution for price and expected return. With one tree, of coure, thi model lead to a contant price-dividend ratio, a contant interet rate, and i.i.d. return. With two tree, price and return dynamic emerge. Aggregate conumption diverifie acro the two tree, o it i more volatile when dividend hare are near zero or one than when the hare are more equal. Thi time-varying volatility of conumption growth, and therefore of the dicount factor, generate time-varying expected return and volatility for the market, the individual aet, and a time-varying rikle rate. Whyithinew? Intraditionalfinance model uch a the CAPM or ICAPM, the rate of return i exogenou and independent of the cale of invetment. Thee model implicitly aume linear technologie output i a linear function of capital with no adjutment cot or irreveribilitie. In thee model, we can all rebalance, by intantly and cotlely tranferring capital to other production technologie or to conumption; we can all leave portfolio weight contant in the face of return hock. In reality, however, market portfolio weight do change over time (e.g. tock v. bond). Thu, any realitic model, with at leat hort-run adjutment cot, irreveribilitie, and other impediment to aggregate rebalancing, will contain ome market-clearing induced dynamic of the ort we iolate and tudy in a pure exchange economy. The aet price and return dynamic of the two-tree model are imilar to thoe found in the empirical aet pricing literature. Firt, we find that a poitive dividend hock, which increae current price and return, alo typically raie ubequent expected return and exce return, a the intuition of the firt paragraph ugget. Thu, return typically diplay poitive autocorrelation or momentum, price typically eem to underreact or not to fully adjut to dividend new, and to drift upward for ome time after that new. Interetingly, however, there are alo parameter, horizon, and region of the tate pace that give the oppoite ign, leading to mean-reverion, price overreaction, and downward drift, uch a that found after IPO, and exce volatility of price and return. Second, when one aet ha a poitive dividend hock, the expected return of the other aet typically decline. We ee a negative cro-erial correlation, which Lo and MacKinlay (99) and Lewellen () argue i an important part of the momentum effect. We ee movement in an aet price even with no new about that aet dividend, another ource and form of apparent exce volatility. Finally, we ee that aet return can be highly correlated with each other even when their

3 underlying dividend are independent. A common factor or contagion emerge in return even though there i no common factor in cah flow. Third, ince price-dividend ratio vary depite i.i.d. dividend growth, price-dividend ratio forecat return in the time erie and in the cro ection, i.e. we ee value and growth effect. Thu, imple market-clearing mechanic generate imultaneouly hort-run momentum (poitive autocorrelation) and long-run mean reverion (valuation ratio forecat return) in individual tock and in the tock market a a whole. While thee aet price and return dynamic are qualitatively imilar to the tylized fact of the empirical literature, our two-tree log utility model doe not typically offer a quantitatively compelling match. In our informal calibration, typical tatitic uch a autocorrelation are between a factor of two and an order of magnitude maller than their counterpart in the empirical literature. Any log utility model will of coure not match the large mean and volatility of return with the low volatility of aggregate conumption and dividend growth. Alo, none of our dynamic effect are pricing puzzle, a all expected return are explained by the conumption-baed model or the CAPM in a log utility model. (The CAPM in our model doe require properly meaured conditional beta, a time-varying market premium, and ue of the entire conumption claim a a reference return rather than jut the tock portfolio. Puzzle would till be generated by the failure to incorporate any of thee ide condition.) The point of thi paper, however, i not to provide a quantitatively convincing match to a wide range of aet pricing puzzle. Rather, it i to how how market-clearing logic, a heretofore ignored ource of aet pricing dynamic, can produce intereting dynamic even in a very tylized model two geometric Brownian tree and log utility. Thi mechanim will be an important part of more complex and realitic (but alo le tranparent) model. Put another way, the one-tree model with log utility i a workhore, depite it obviou unrealim, ince it i imple to olve and it illutrate o many of the economic principle that underlie more quantitatively realitic model. Our two-tree model fill a imilar niche. Many of it prediction, while matching the ign reported in the empirical literature, do not match the magnitude. Yet it i a imple, tractable model with cloed-form olution, which capture and clearly diplay intereting economic effect which will alo be preent in more complex model. Market clearing ha not played a central role in traditional dynamic model of aet pricing. Model with exchange economie typically aume a ingle aet in nonzero upply (Luca (978)). Production economie idetep the iue by allowing the upply of aet to change elatically with invetor demand (Cox, Ingeroll, and Ro (985)). However, a recent literature examine and price multiple long-lived dividend tream in nonzero net upply, generalizing tandard ingle-tree model. Santo and Veronei (), Banal, Dittmar, and Lundblat (), and Menzly, Santo, and Veronei (3) offer Luca-type model with multiple aet. Their model have intereting dynamic for dividend hare, intereting preference (e.g. habit), and they are deigned to addre quantitatively a variety of tylized fact. The paper cloet to our i Santo and Veronei (). They pecify log utility and two tree, which they interpret a aet and labor income. Like our, their model predict that expected return, price-dividend ratio, beta, and o forth hould vary a function of the dividend hare. They reproduce and extend a number of empirical fact involving aet price dynamic. In particular, they how how the hare of labor income in total conumption perform well in forecating tock return. The main difference i that they pecify the hare proce to be mean reverting, which exogenouly build dynamic into the dividend procee. Thi ingredient i potentially an additional ource of aet price dynamic. We focu on a model with a impler ingredient, i.i.d. dividend growth, in order to focu on the quetion: what dynamic come from the imple logic of market clearing alone? To provide additional perpective on the role of the market-clearing condition, we how that it

4 enable the olution of the invere portfolio-choice problem. In the uual portfolio-choice problem, we are given the return proce and then olve for the optimal portfolio. In the invere portfolio-choice problem, we are given the optimal portfolio and then olve for the moment of return that upport thoe holding. We verify that the olution to the invere portfolio-choice problem give the ame reult a our endowment economy calculation. The conundrum that we cannot all rebalance, o return cannot in general be i.i.d. how up everal place in finance. Roenberg and Ohlon (976) and Cheng and Grauer (98) note that relative price and return cannot vary ex-pot if one pair the CAPM with contant hare upply.. THE SINGLE-ASSET BENCHMARK To provide a benchmark for comparion, we begin by reviewing the traditional ingle-aet model. The aet pay a dividend tream dd D = µdt + σ dz, () with contant coefficient µ and σ, and where Z i a tandard Brownian motion. Unle neceary for clarity, we uppre time indice, e.g. dd dd t, etc. The repreentative invetor ha log utility, U t = E t e δτ ln (C t+τ ) dτ. () Thi i an endowment economy, o price adjut until conumption equal the dividend, C = D. The invetor firt-order condition implie that marginal utility i a dicount factor that price aet, M t = e δt D t. (3) The price P t of the aet, which i the market portfolio claim to aggregate conumption, i given by P t = M t+τ E t C t D t M t D t+τ dτ = E t e δτ D t+τ D t+τ dτ = δ. (4) Since price are proportional to conumption (which equal the dividend), price appreciation i the ame a dividend growth, dp P = dc C = dd D, (5) Black and Litterman (99) ue the market-clearing condition to back out expected return from knowledge of the covariance matrix and the market capitalization weight of the different aet. He and Leland (993) impoe a market-clearing condition to derive dynamic of aet price that are conitent with equilibrium. 3

5 o that the total intantaneou return R t i R t = dp t P t + D t P t dt = (µ + δ) dt + σ dz. (6) The expected return and return variance are contant, E t [ R t ] = (µ + δ) dt, (7) Var t [ R t ] = σ dt. (8) Thu, dividend growth rate and tock return are i.i.d. through time in thi ingle-aet model. The intantaneou interet rate i given by dmt rdt = E t M t ddt = δ dt + E t D t ddt Var t D t = (δ + µ σ ) dt. (9) The rikle aet i in zero net upply. We ee the tandard dicount rate (δ), conumption growth (µ), and precautionary aving (σ )effect. Since the rikle rate i contant, the entire term tructure i contant and flat. 3. THE TWO-ASSET MODEL Now conider the ame economy with two aet but no other modification. A before, dividend follow imple geometric Brownian motion, dd i D i = µ i dt + σ i dz i, () where i =,, and the correlation between dz and dz i ρ dt. Aggregate conumption i the um of the two dividend C = D + D. The invetor ha log utility a in Equation (). The dividend hare, = D D + D, () i a natural tate variable for the two-tree model. We firt derive the dynamic of the dividend hare. We then find the interet rate and price the market portfolio from the aggregate conumption proce. Finally, we price the individual aet and find return, expected return, and variance of return. 4

6 3. Dividend Share Dynamic. An application of Itô Lemma to Equation () and () give the dynamic of the dividend hare, d = ( ) µ µ σ +( )σ +( )ρσ σ dt + ( )(σ dz σ dz ). () The drift of the dividend hare proce in () i zero when =, κ, or, where κ = µ µ + σ ρσ σ σ + σ ρσ σ. (3) When κ lie between zero and one, the drift i poitive from zero to κ, bringing the hare up toward κ, and negative from κ to one, bringing the hare down toward κ. The top left panel of Figure illutrate the drift proce for a cae in which dividend dynamic are ymmetric; the bottom left panel of Figure illutrate the drift proce for a cae in which the dividend for the firt aet are more volatile than for the econd aet. Thu, the drift term can induce pattern of mean reverion in the dividend hare that are not preent in the underlying dividend procee. 3 The diffuion coefficient in Equation () i quadratic, implying that change in the dividend hare are mot volatile when = /. Initially, it may eem urpriing that imple geometric Brownian dividend give rie to a complex hare proce with a cubic drift and a quadratic diffuion. However, thee propertie of hare dynamic reult directly from the nonlinearity of the hare a a function of the dividend. The denity f( τ ) of the dividend hare τ period ahead conditional on it current value i given by 4 where f( τ ) = τ ( τ ) πη τ exp [ln (/( )) ln ( τ /( τ )) ντ] η, (4) τ Thetochaticproceforthehare i a member of the important cla of Wright-Fiher diffuion. In an intereting parallel to our two-aet model, thee type of diffuion are often applied in genetic theory to characterize the evolution of gene in a population of two genetic type. For example, Karlin and Taylor (98) Ch. 5, p preent an example of the Wright-Fiher gene frequency diffuion model in which the fraction of gene in a population follow a proce with drift and diffuion term that are repectively third- and econd-order polynomial in the fraction, jut a in Equation (). Alo ee Crow and Kimura (97) for other example and a dicuion of the aymptotic propertie of thee model. 3 The cubic drift of our hare proce i alo cloely related to that of the tochatic Ginzburg-Landau diffuion ued in uperconductivity phyic to model phae tranition. See Kloeden and Platen (99) and Katoulaki and Kho (). 4 Rather than olve the Kolmogorov or Fokker-Planck equation aociated with Equation () directly, we expre the hare a an invertible function of the lognormally ditributed ratio D /D,i.e. = ( + D /D ). Since the ratio of lognormal i lognormal, we then olve for the denity uing a tandard change of variable. 5

7 ν = µ µ σ/+σ /, η = σ + σ ρσ σ. Note that ν dt = E[d ln(d /D )] and η dt =Var[dln(D /D )]. The mean and variance of log dividend growth drive the ditribution. Although the underlying dividend are lognormal, the dividend hare itelf i not lognormal. From Equation (4), one can verify analytically that the ditribution of τ can be either unimodal or bimodal, and can diplay either poitive or negative kewne. A one might expect of the hare formed from two geometric Brownian motion, the dividend hare i trictly between zero and one for all finite horizon, provided current dividend are nonzero. An increae in the current dividend hare hift the ditribution of τ toward larger value for all finite τ. The right hand panel of Figure plot the conditional denity of the dividend hare given in Equation (4) for everal horizon. In the top right panel, the dividend dynamic are ymmetric and the initial dividend hare equal 5 percent. In thi cae, the ditribution of the dividend hare i initially unimodal, but eventually pread out and begin to look more uniformly ditributed. After 5 year, the ditribution become bimodal a the probability of being in the neighborhood of zero and one lowly increae. The bottom right panel of Figure how denitie for an aymmetric cae in which the firt aet ha a higher volatility than the econd and the initial dividend hare i percent. Here, the ditribution of the dividend hare become kewed toward the right, and large value for the dividend hare become more likely. The larger volatility of the maller dividend proce give it ome chance of overtaking the initially larger dividend proce. A the horizon increae further, however, the ditribution tend to hift back toward maller value. The dividend hare i both peritent and volatile for the parameter value we examine. The conditional mean hare doe not move quickly over time, a een both in the mall value of the drift (top panel of Figure ) and in the mean of the denitie (middle and bottom panel of Figure ). Thi i peritence. However, the denitie in Figure pread out rapidly, o change in the hare are very volatile. Thi model doe not poe a tationary hare ditribution. The hare of one of the aet will alway gradually decline to the point that the other aet become dominant in the market. 5 A degenerate long-run hare ditribution may eem counterintuitive, but it i not necearily an unrealitic feature of the model. Firt, it may be hard to tell. Given realitic parameter value, the mean time until the hare of one firm i le than, ay, five or ten percent of it initial value may be on the order of centurie. Thu, thi apect of the cah flow tream may have little effect on it preent value. Second, firm do in fact diappear over long period of time. A one recent example, Fama and French (3) find that more than 5 percent of eaoned firm are delited from the tock market for poor performance during a typical decade. The right generalization may be to allow the birth of new tree, not to preclude the death of old tree. 3. Conumption Dynamic. Aggregate conumption C = D + D follow 5 Thi feature parallel the aymptotic propertie of Wright-Fiher gene frequency model in which one of the two gene type ultimately become fixed in the population. 6

8 dc = dd + dd, = µ D dt + µ D dt + σ D dz + σ D dz, (5) o that dc C = µ + µ ( ) dt + σ dz + σ ( ) dz. (6) Since the dynamic of conumption depend on the tate variable, conumption growth i no longer i.i.d. through time. Mean conumption growth, E t dc C =[µ + µ ( )] dt, (7) i the hare-weighted mean of the dividend growth rate. Conumption volatility, Var t dc C =[σ + σ( ) +ρσ σ ( )] dt, (8) i lower for intermediate value of the dividend hare, a conumption i then diverified between the two dividend. 3.3 The Rikle Rate. We find the intantaneou (zero net upply) interet rate a before rdt = δ dt + E t dc C Var t dc C. (9) Subtituting the moment of the conumption dynamic into Equation (9) give r = δ + µ + µ ( ) σ σ ( ) ρσ σ ( ). () Thu, the rikle rate varie over time, a a quadratic function of the tate variable. The rikle rate i lower for intermediate value of the dividend hare becaue dividend diverification lower conumption volatility, which lower the precautionary aving motive. Since the interet rate i not contant, the term tructure i not flat. 3.4 Market Price and Return. A i uual in log utility model, the price P m of the aggregate conumption tream C = D + D (the market portfolio) i given by the imple expreion P mt C t = E t e δt C t+τ C t+τ dτ = δ. () 7

9 Thi calculation, which i the ame a in the ingle-aet model, i valid for all conumption dynamic. Since aggregate conumption equal aggregate dividend, Equation () alo implie that the market price-dividend ratio i contant. A before, the price of the market i proportional to aggregate conumption which implie that the price appreciation of the market i dp m P m = dc C. () Since the total intantaneou return R M on the market equal price appreciation plu the dividend yield (R m = dpm P m + C P m dt = dc C + δdt), Equation (5) implie R m = [δ + µ + µ ( )] dt + σ dz + σ ( ) dz. (3) The expected market return and variance are no longer contant, E t [ R m ] = [δ + µ + µ ( )] dt, (4) Var t [ R m ] = σ + σ ( ) + ρσ σ ( ) dt. (5) The expected return equal the ubjective dicount rate δ plu expected conumption growth, which i the hare-weighted average of the dividend growth rate µ and µ. The variance of the market return equal the variance of conumption growth, and reflect diverification between the two aet cah flow. Finally, ubtracting the expreion for the rikle rate in Equation () from the expected return on the market in Equation (4) indicate that the equity premium equal the variance of the market, E t [ R m ] rdt =Var t [ R m ], (6) a uual for log utility model. From Equation (5), the variance of the market i a convex quadratic function of the dividend hare. Thi mean that the equity premium i alo time varying, and generally increae a the market become more polarized. 3.5 Aet Price. We focu on the firt aet. The econd aet i ymmetric. From the uual Euler condition, the price P of the firt aet i given by, P t = E t e δτ C t C t+τ D t+τ dτ. (7) Recalling the definition of the dividend hare from Equation (), thi reult can be expreed a P t C t = E t e δτ t+τ dτ 8. (8)

10 Formally, valuing the individual aet i identical to the rik-neutral pricing of an aet that pay a cah flow equal to the dividend hare, and with a dicount rate δ. The price-conumption ratio i thu an exponentially-weighted average of the expected dividend hare. The dividend hare play a imilar role in many tractable model of long-lived cah flow, including Santo and Veronei (), Banal, Dittmar, and Lundblad (), Menzly, Santo, and Veronei (3), and Longtaff and Piazei (3). Thee paper, however, exogenouly pecify a proce for that facilitate the computation of the expectation in Equation (8). We olve for the aet price from Equation (8) in three way. Firt, we evaluate the double integral (expectation and time) directly, after changing the order of integration. Second, we derive the differential equation for the price-conumption ratio that reult from the tandard intantaneou pricing condition, and olve it. Third, we write the conventional log utility portfolio-choice problem, but rather than olving for portfolio weight given aet price and return ditribution, we olve for the aet price that determine given portfolio weight: the invere portfolio-choice problem. All three olution, of coure, give the ame anwer. The firt two approache are preented in Section and of the Appendix repectively. The third approach i dicued in Section 5. The price of the firt aet a a function of the dividend hare i P t C t = ψ( γ) F, γ; γ; + ψθ F, θ; +θ;, (9) where ψ = ν +δη γ = ν ψ η θ = ν + ψ η and where ν and η are a defined in Equation (4). F (α, β; γ; z) i the tandard hypergeometric function (ee Abramowitz and Stegum (97) Chapter 5). The hypergeometric function i defined by the power erie F (α, β; γ; z) =+ α β γ α(α +) β(β +) z + z + γ(γ +) α(α +)(α +) β(β +)(β +) z (3) γ(γ +)(γ +) 3 The hypergeometric function ha an integral repreentation, which can be ued for numerical evaluation and a an analytic continuation beyond z <, F (α, β; γ; z) = Γ(γ) Γ(β)Γ(γ β) w β ( w) γ β ( wz) α dw; Re(γ) >Re(β) >. (3) The derivative of the hypergeometric function, needed for Itô lemma calculation, ha the imple form 9

11 d αβ F (α, β; γ; z) = dz γ F (α +, β +;γ +;z). (3) Thi formula can be derived by differentiating the term of the power erie in Equation (3) (ee alo Gradhteyn and Ryzhik (), 9., 9.). The price P of the econd aet i ymmetric to that of the firt, P t = C t ψ( + θ) F, +θ; +θ; ψγ F, γ; γ;. (33) 3.6 Aet Return. Let R denote the intantaneou return on the firt aet. Given the explicit price function in Equation (9), the functional form of it derivative from Equation (3), and the hare proce in Equation (), R i given by a direct application of Itô Lemma, R = δ + µ + µ ( )+(ρσ σ σ + η ) Φ() dt +σ [ + Φ()] dz σ [ +Φ()] dz, (34) where Φ() = A() B(), A() = γ γ + +θ F, γ; γ; F, γ;3 γ; F, +θ; +θ;, B() = γ F, γ; γ, +, θ F θ; +θ;. From thi equation, it follow that both the mean return and return volatility vary with the tate variable, but in a more complex way than i the cae for the market: E t [ R ] = δ + µ + µ ( )+(ρσ σ σ + η ) Φ() dt, (35) Var t [ R ] = σ [ + Φ()] + σ [ +Φ()] ρσ σ [ + Φ()][ +Φ()] dt. (36)

12 Section 3 of the Appendix how that the limit of Φ() a ieitherorθ, depending on whether θ i greater than or le than one. Uing thi reult, it follow that the expected exce return of the firt aet need not converge to zero a. Similarly, the volatility of the firt aet return need not converge to the volatility of it cah flow a. A, however, the firt aet become the market and it expected return and variance converge to the value given in Equation (4) and (5). 4. ASSET-PRICING IMPLICATIONS In thi ection we characterize the model olution by plotting price-dividend ratio, expected return, return volatility, etc. a a function of the ingle tate variable: the dividend hare of the firt aet. To illutrate and quantitatively evaluate the model, we preent pecific numerical example baed on three cae. Throughout thee example, we fix the ubjective dicount factor δ to., and we et the correlation between the dividend procee to zero. The three cae are: The Symmetric Cae. In thi cenario, dividend for the two aet follow identical geometric Brownian motion, µ = µ =., and σ = σ =.. Thi i the natural implet cae to tart with. We can alo view thi cae a a market in which there are two large primary ector, uch a financial and indutrial. TheAymmetricCae. In thi cenario, dividend volatility for the firt aet i higher than for the econd aet, µ = µ =., σ =.4, and σ =.. We can view the firt aet a an individual firm or a mall ector and the econd aet a the ret of the market. For thi interpretation, the region of the tate pace with a low dividend hare i the mot intereting. We can alo think of the econd aet a a firm with unuually low cah flow volatility, uch a a regulated utility. For thi interpretation, high value of the firt aet hare (low value of the econd aet hare) are the mot intereting. Finally, we can think of the firt aet a relatively volatile traded ecuritie, and the econd aet a relatively afe but le liquid ecuritie uch a human capital, real etate, etc. The Stock-Bond Cae. In thi cenario, we pecify µ =.3, µ =., σ =., and σ =.. The econd tree i a level perpetuity (in poitive net upply) with no dividend rik. We allow the firt tree higher mean dividend growth, a tock dividend typically grow over time while bond (perpetuity) coupon do not. Dividend growth of three percent rather than two percent a in the other cae produce clearer plot. Thi parameterization allow u to addre what i perhap the mot important rebalancing and portfolio iue of all, that of tock v. bond. It alo allow u to addre tock market dynamic that come from market clearing in the overall market for capital, while the return on the overall wealth portfolio remain i.i.d. In each plot, the top panel preent the ymmetric cae; the middle panel, the aymmetric cae; and the bottom panel, the tock-bond cae. 4. Expected Return and Exce Return. Figure plot expected return and the rikle rate. Figure 3 plot expected exce return. The often trong dependence of expected return on the dividend hare hown in thee figure implie that both expected return and exce return can diplay a great deal of time variation even though expected dividend growth rate are contant. A we expect from Equation (), the rikle rate in Figure i a quadratic function of the dividend hare. The ymmetric cae in the top panel how that the rikle rate i higher for intermediate hare,

13 where dividend are better diverified, conumption volatility i lower, and thu the precautionary aving motive i lower. In the aymmetric cae in the middle panel, the quadratic rikle rate i hifted to the left, a the firt aet i more volatile. There i till ome diverification effect, however, a the maximum rikle rate i interior. In the tock-bond cae of the bottom panel, conumption growth i alo zero and rik free when the riky aet hare i zero, o the interet rate equal the dicount rate, ten percent. The interet rate rie to an interior maximum, firt following the greater mean conumption growth due to the greater mean dividend growth of the firt aet, but then falling a that aet greater volatility induce precautionary aving. The market expected return in the top two panel of Figure i a contant, a the market pricedividend ratio and mean conumption growth rate are contant. In the bottom panel, the market expected return rie with mean conumption growth a a linear function of the hare. The market expected exce return in Figure 3 i then the mirror image of the quadratic rikle rate. Alo, the market expected exce return i proportional to conumption volatility in any log utility model, and conumption volatility i lower for intermediate hare. The expected exce return for the two individual aet in the ymmetric cae (Figure 3, top) are monotonic and approximately linear in the dividend hare. To undertand thi behavior, recall that expected exce return repreent rik premia, reflecting the covariance of return with the dicount factor. With log utility, E t [ R ] rdt=cov t R, dc. (37) C Now, return hock come from dividend growth hock and hock to the valuation of dividend, 6 R = dp P + D P dt = D P dt + dd D + d(p /D ) P /D + dd D d(p /D ) P /D. (38) Thu, we can expre the covariance of return with conumption growth a dd E t [ R ] rdt=cov t, dc D C d(p /D ) +Cov t, dc. (39) P /D C Since conumption growth i the hare-weighted um of dividend growth rate, the firt term i linear in the dividend hare, dd Cov t, dc = σ D C. (4) 6 To derive Equation (38), expre P a D (P /D ) and apply Itô Lemma. Since dividend growth i i.i.d., the firt two term on the right hand ide of Equation (38) decribe the effect on return of current and expected future change in cah flow. The remaining term, and epecially the third, capture dicount rate effect, the effect on return of change in the dicount rate applied to future cah flow. The firt two term alo decribe return with no change in the tate variable, and the P /D term decribe the effect of the changing tate variable on return.

14 When the firt aet ha a hare of zero, the covariance of it dividend growth with that of aggregate conumption compoed entirely of the uncorrelated dividend of the other aet i zero. A the hare increae, the covariance of the firt aet dividend with aggregate conumption increae linearly. In thi way, the approximate linearity of expected exce return in Figure 3 i natural, and it how that cah-flow beta linear in dominate the covariance of return with conumption growth in thee cae. The deviation from linearity een in Figure 3 repreent the uually maller effect of valuation beta, expected return correponding to covariance of the change in the price-dividend ratio with aggregate conumption. Thee deviation from linearity are perhap the mot intereting part of the model. In the aymmetric cae, the valuation effect are larger. A hown in the middle panel of Figure 3, the expected exce return of the firt aet i no longer monotonic, a it decline lightly near a hare of one. The left-hand cale of the middle panel i larger, reflecting much larger variation in expected exce return. Thi come from the larger variance of dividend and hence conumption growth. More dramatically, the expected exce return of the econd aet decline with it hare through much of the range. The expected exce return no longer decline to zero a it hare decline to zero, on the right hand ide of the lower panel. Here, the entire expected exce return i driven by valuation rik, the covariance of the price-dividend ratio with aggregate conumption, even though the dividend rik, covariance of dividend growth with aggregate conumption, vanihe. In the tock-bond cae of the lower panel, the expected exce return of the tock how the uual near-linearity. The long-term bond now alo how a varying expected exce return, depite a rikle cah flow. Thi reult i driven entirely by dicount rate effect of coure. The expected exce long-term bond return the term premium can be both poitive and negative, a found in bond data by Fama and Bli (987). A in reality, term premia are much maller than expected exce tock return, ince there i no premium for cah-flow rik. The individual-aet expected return in Figure how a roughly quadratic pattern. We can now mot eaily undertand thi pattern a the nearly linear expected exce return of Figure 3 plu the quadratic rikle rate. The behavior of expected return drive many of the reult that follow. A poitive hock to the firt aet dividend ha an immediate effect on the aet price. However, thi hock alo change the dividend hare, and hence, affect the aet expected return. Where expected exce return rie in their dividend hare, we expect to ee poitive autocorrelation and momentum of return. Where expected return decline in the dividend hare, we expect to ee negative autocorrelation and meanreverion. The plot are not monotonic, o both ign are poible. We calculate autocorrelation below, and find thi intuition i roughly correct. (It not exactly correct ince hock to the econd aet dividend affect the price and return of the firt aet.) Changing expected return take the form of further expected change in price, ince dividend growth i i.i.d. Thu, dividend hock may have long-lating price and return effect. Where expected return increae in the hare, price will eem initially to underreact and lowly to incorporate dividend new. Where expected return decline, price will eem to overreact. 4. Price-Dividend Ratio. Figure 4 plot price-dividend ratio a a function of the dividend hare of the firt aet. From Equation (), the price-dividend ratio for the market i contant, and equal ten in all three cae. Price-dividend ratio for the individual aet vary widely, however, and need not be monotonic in the hare. 3

15 Since dividend growth i i.i.d., price-dividend ratio are driven entirely by expected return. Since the hare i highly autocorrelated, today expected return capture a great deal of the future expected return that drive price-dividend ratio. Hence, the price-dividend ratio in Figure 4 are eentially the invere of the expected return of Figure, and can be undertood a uch. In the upper panel of Figure 4, the limit of the price-dividend ratio for the firt aet i 6.67 at zero, and then decline a the hare increae. Pat a hare of /, it become even le than the market price-dividend ratio of ten, and then rie lightly to finih at ten when the firt aet become the entire market. In the aymmetric cae of the middle panel, the hape are the ame, but the magnitude are quite different. Interetingly, the price-dividend ratio for the econd aet increae without bound a the hare approache one. The tock-bond cae how variation in the bond price-dividend ratio (the invere of the coupon yield), depite no cah flow uncertainty. In each cenario, aet have lower expected return and higher price-dividend ratio when their hare of the total dividend approache zero than otherwie. Aet are more highly valued when they repreent a mall hare of total dividend, and hence, are more valuable from a diverification perpective. Thu, mall firm are growth firm, in the ene of having high valuation and low expected return. Thi phenomenon i particularly trong for a mall firm with lower dividend volatility, uch a the econd aet in the aymmetric cae in Figure, whoe expected return i mall and whoe price-dividend ratio goe to infinity a it hare goe to zero. Our calibration of the model do not diplay a eparate mall firm effect. We do not have eparate ize and value dimenion to the cro ection. In our calibration, mall firm generate low average return and exce return, whether mall refer to the dividend hare (a een in Figure and 3), or to market value (not hown). Of coure, it i likely in reality that mall firm have cah flow that are enitive to aggregate condition, rather than the uncorrelated cah flow we have pecified. Thi additional ingredient can eaily produce high average return. The decline of the price-dividend ratio with ize i not monotonic however, and we ee interior minima in the price-dividend ratio plot. The price-dividend ratio of the firt aet mut be below the price-dividend ratio of the market when the ratio of the econd aet i above the ratio of the market, ince the hare-weighted average price-dividend ratio equal that of the market. A it hare approache one, however, the price-dividend ratio of an aet mut converge to the market price-dividend ratio. The price-dividend ratio of the econd aet in the middle panel of Figure 4 give ome inight into the trange behavior of that aet expected exce return in Figure 3. In the far right region, where the low-volatility econd aet i a mall fraction of the market, it price-dividend ratio i a trongly loped and nonlinear function of the hare. Thu, mall change in the hare reult in large change in the aet valuation, which i why the valuation covariance term in Equation (39) i o important. By taking limit of the hypergeometric function, we how in Section 3 of the Appendix that lim P D = δ + ν η /. (4) Thi expreion hold for θ >, which implie that the denominator i poitive. If θ, the limit i. For the cae with ρ =andµ = µ, thi expreion implifie to lim P = D δ σ. (4) 4

16 Thu, the price-dividend ratio of the firt aet diverge to if the variance of the econd aet i greater than the dicount factor, and vice vera. Given δ =., the cutoff i σ =. =.36. Our aymmetric cae with σ =.4 i well above that cutoff, o the econd price-dividend ratio doe go to a Figure ugget. The price-dividend ratio increae at a rate le than or equal to /, however, ince the hare-weighted average of the price-dividend ratio mut equal the market price-dividend ratio of ten, and the market value mut decline to zero a. To provide ome intuition for why the price-dividend ratio can become infinite in thi model, recall from Equation (8) that the price-conumption ratio i an exponentially-weighted average of the expected dividend hare. Since D = C, the price-dividend ratio i imply / time the priceconumption ratio, and we can write P = E t e δτ D t+τ t dτ. (43) Thu, the price-dividend ratio can be expreed a an exponentially-weighted average of expected dividend hare growth rate. When the dividend hare of an aet i expected to grow at a rate fater than e δτ, the integral in Equation (43), and hence the price-dividend ratio, can diverge. Thi feature parallel the reult from the claical Gordon growth model in which the price-dividend ratio for a tream of dividend can be infinite if the dividend growth rate exceed the dicount rate. Now, ince, cannot growth fater than e δτ forever, for any finite initial. Hence, the price-dividend ratio i finite for any finite. Inthelimita, however, we alo have t+τ, but the latter occur at a lower rate, o that the hare doe grow ufficiently fat to give an infinite price-dividend ratio. 4.3 Price-Dividend Ratio and Expected Return. Comparing expected return and exce return in Figure and 3 with price-dividend ratio in Figure 4, one upect that price-dividend ratio forecat expected return and exce return, ince they vary in invere way with the dividend hare. Following up on thi intuition, Figure 5 plot expected return and exce return veru the dividend-price ratio. In all three cae, expected return are not far from a linear function of the dividend-price ratio. The quadratic hape of expected return and the quadratic hape of price-dividend ratio a function of the hare about offet. Expected exce return how intereting nonlinearitie at high dividend-price ratio, correponding to low price-dividend ratio at high dividend hare. The price-dividend ratio i not a monotonic function of the hare (Figure 4). Hence, even when the expected exce return i a monotonic function of the hare, wewilleethe intereting nonlinear relation hown in Figure 5. The aymmetric calibration in the middle panel i ueful for thinking about individual firm v. the market, and hence in conidering cro-ectional relation from the empirical literature. We ee here that value firm with high dividend-price ratio alo have high average return and exce return. The tock-bond calibration in the lower panel allow u to compare the model to regreion of tock market return on dividend yield, following Fama and French (988). Both calibration give a reaonable quantitative a well a qualitative fit. In each cae, the lope i about one a one percentage point increae in the dividend-price ratio correpond to a one percentage point increae in expected return and exce return. Thi i about the magnitude uggeted by both invetigation of the cro ection of tock and time erie regreion of return on dividend-price ratio for the tock market a a whole. For a cro-ectional example, Cohen, Polk, and Vuolteenaho (3, Table, p. 69) report a regreion coefficient of log one-year return on log book-market ratio of.7, where the regreion i taken acro book-market orted portfolio. The coefficient of about one for return on the dividendprice ratio in Figure 5 correpond to a coefficient of about.4 for log return on the log dividend-price 5

17 ratio, linearizing around a typical four percent dividend-price ratio. Since price rather than book or dividend i the important right-hand-ide variable in thee regreion, thee coefficient are comparable. The model thu produce if anything a lightly tronger regreion than found by Cohen, Polk, and Vuolteenaho. Fama and French (99, Table IV, p. 44) ort firm by book-market and report average return of the book-market orted portfolio. A regreion of Fama and French average monthly percent return on their log book-market ratio acro their 3 portfolio give a coefficient of.5, and a plot how a reaonably linear relation. One can alo ee thi reult in the imple pread: Fama and French average return vary from.3 percent to.83 percent acro portfolio, and log book-market varie from. to., giving a lope of about.5. Converting to annual net (not percent) return, Fama and French evidence implie a coefficient of about.5 =.6, a little higher thi time than the.4 uggeted by Figure 5. In time-erie regreion for overall tock market indice, Fama and French (988, Table III, p. ) report coefficient between.35 and 5.37 in the full ample. However, the high return of the 99 depite ever lower dividend-price ratio ha brought down the etimate omewhat. Cochrane (, Ch. ) urvey the evidence and argue for a coefficient of about two. Other expre even lower view. A in the empirical literature, the return regreion and the exce return plot of Figure 5 produce quite imilar coefficient. Variation in price-dividend ratio i driven by variation in rik premia more than by variation in the rikle rate. The tock-bond calibration contrat tarkly to reult we would obtain with the total market, i.e. wealth portfolio or conumption claim. In thi model, the total market price-dividend ratio i contant. It cannot forecat the equally contant market expected return, nor the time-varying market expected exce return. However, Fama and French (988) regreion apply to the tock market only. We ee in the bottom panel of Figure 5 that the tock portion of the market may be forecatable from it price-dividend ratio, even when the total market i not. In thi cae, the tock market i ubject to market-clearing induced dynamic: a it hare get larger, it expected return rie (Figure and 3) to induce invetor to keep holding the larger hare. The price-dividend ratio fall, reflecting and forecating that higher expected return. In um, imple market-clearing mechanic generate both hort-run continuation and momentum in tock, and imultaneouly long-run mean reverion a evidenced by dividend-yield predictability in both individual tock and in the tock market a a whole. 4.4 Return Volatility. Figure 6 plot the tandard deviation of market and individual-aet return a a function of the dividend hare. In the one-tree model, return volatility i contant and equal dividend volatility. Here, market return volatility mirror conumption growth volatility. Market volatility i generally lower for intermediate value of the dividend hare where the market diverifie between the two ecuritie. In the ymmetric cae (top panel of Figure 6), return volatility tart at the. volatility of the dividend proce when the hare i zero, but then varie a an S haped function of the dividend hare. For dividend hare greater than about.8, the volatility of aet return exceed the volatility of the underlying cah flow or dividend. Thi reult implie exce volatility in the ene that return volatility i higher than the volatility of the fundamental cah flow. There are two force at work here. Firt, thi i the region in which expected return in Figure are a declining function of the hare, o price overreact to dividend. A poitive hock to the firt aet dividend thu raie the price of the firt aet by more than the dividend hock. Second, hock to the econd aet dividend affect the price of the firt aet even with no new about the firt aet dividend. 6

18 For hare below.8, return volatility i le than dividend growth volatility. Here, expected return are a poitive function of the hare in Figure, o price underreact to dividend new. In thi region, thi effect i large enough to overwhelm volatility in the firt aet return induced by hock to the econd aet dividend. In the aymmetric cae (the middle panel of Figure 6), the firt aet return volatility follow a imilar S haped function of the hare. The econd aet diplay about the ame volatility a that of the aet in the top panel, though it harder to ee in the necearily larger cale of the middle panel. Interetingly, the econd aet return volatility i le than it dividend volatility even in the limit a it hare goe to zero, on the right hand ide of the bottom panel. Thi reult again reflect the exploive behavior of it price-dividend ratio in thi region. In thi region, a mall increae in dividend bring a much lower price-dividend ratio, o the return i le than the dividend increae. The volatility of the firt aet in the tock-bond cae i imilar to the above cae. However, there i only a very mall and barely viible region of exce volatility, where return volatility i larger than dividend growth volatility. Thi occur for two reaon. Firt, there i only a mall region in Figure 3 where expected exce return decline in the hare, where a hock to the firt aet dividend caue price to rie more than dividend. Second, there are no hock to the econd aet dividend to move the price of the firt aet in the abence of new about it dividend. In the tock-bond cae, the econd aet how an intereting volatility pattern with two lobe. The price-dividend ratio in Figure 4 i a non-monotonic function of the hare, and there are no hock to the bond dividend. Thu, where the bond price-dividend ratio i declining, bond return are negatively correlated with tock dividend and aggregate conumption growth. Where the bond price-dividend ratio i riing, bond return are poitively correlated with aggregate conumption growth. Where the bond price-dividend ratio i flat, bond return are rikle, a neither the price nor the dividend can change. For thi reaon there are two hump and a zero in the plot of the tandard deviation of bond return. The volatility of the tock i larger than the volatility of the total market, which here equal conumption growth. Thi i an important feature of the data. If we think of the total market a including bond, o a typical tock hare i.6, thi effect i not a quantitative match, however, a the model conumption growth volatility i till percent. If we think of the total market a including all wealth, uch a real etate, human capital, etc., then a reaonable hare i. or le, and the model doe begin to capture the fact that tock return with σ = 8 percent are much more volatile than conumption growth with σ < 4 percent. However, thi reult i fairly mechanical in thi calibration of the model, a the price effect are mall. Stock return volatility i driven here primarily by the volatility of it dividend tream, where actual aggregate tock market return volatility of about 6 percent i ubtantially higher than (and le correlated with) it roughly percent dividend growth volatility. Comparing the expected exce return in Figure 3 with the return volatility in Figure 6, we ee that Sharpe ratio vary coniderably over time a the dividend hare varie. 4.5 Market Beta. With log utility, expected return follow a conditional CAPM and conumption CAPM. Figure 7 plot the intantaneou beta. Of coure, beta are implied by the expected exce return plot of Figure 3. However, it till worthwhile to conider beta directly. The trong variation of beta with hare in Figure 7 how that beta will vary over time for individual tock and portfolio. For the ymmetric cae in the top panel, the beta for the firt aet i zero when the hare i zero. A we have een, the return covariance i equal to the dividend covariance here, and the firt aet dividend are uncorrelated with thoe of the econd aet, which i now the entire market. A the hare 7

19 increae, however, the firt aet contribute more to the total market return and it beta increae correpondingly. The beta eventually become greater than one for value of the hare above 5 percent. A the hare approache one, the beta begin to decreae and eventually converge to one. When the hare i one, the firt aet i the entire market and it beta with itelf ha to equal one. The nearly linear expected exce return plot of Figure 3 i thu compoed of thi intereting nonlinear beta and the quadratic market expected exce return of Figure 4. It initially puzzling that the beta can be greater than one. However, the hare-weighted average beta mut be one. Thu, if the beta of the econd aet i le than one, the beta of the firt aet mut be greater than one. More generally, a regreion of x on x +( )y with x, y independent and identically ditributed give a coefficient +( ), which exceed one for >/,oabetagreater than one i not unexpected. The aymmetric cae in the middle panel of Figure 7 i imilar, but a uual hifted to the left. The firt aet beta increae more rapidly and reache a maximum of about.5 near a 5 percent hare. Interetingly, the beta for the econd aet remain well above zero even in the right-hand limit a the econd aet hare approache zero. Thi mirror the behavior found in the previou graph: the expected exce return approache a nonzero limit, and the price-dividend ratio explode. Although the dividend tream become rikle and uncorrelated with aggregate conumption, change in the valuation of that tream are not rikle, and remain correlated with aggregate conumption. There i no contradiction here with the identity that hare averaged beta mut equal one, a the hare of the econd aet approache zero. The tock-bond cae in the bottom panel how ubtantially different behavior. A the hare approache one, the tock beta approache one and the bond beta approache zero, ince the tock become the entire wealth portfolio. For lower hare, the tock beta increae dramatically while the bond beta pae through zero and become negative. In thi calibration, the only hock are hock to the tock (the firt aet ) dividend. Figure 4 how that the bond price-dividend ratio i a declining function of the hare to about =.35 and a riing function thereafter. Hence, we expect a negative beta for the econd aet on the firt aet dividend below =.35 and a poitive beta thereafter, a we ee in Figure 7. The large magnitude of the firt aet beta alo follow. When the hare i mall, a one percent market return require a change in the firt aet dividend of about one percent of total market wealth. Such a change i a large proportional change, implying a huge return for the firt aet. Beta i the change in the aet return correponding to a one percentage point change in the market return, and it will hence be a large number in thi region. Interetingly, the beta of the econd aet doe not have to converge to one a the econd aet become the whole market, ince the whole market become rikle in that limit. 4.6 Serial Correlation of Return. Figure 8 graph the conditional erial correlation of one-year aet return a a function of the initial dividend hare, Corr t [R t,t+,r t+,t+ t = ], where R t,t+ denote the dicrete-time return from time t to time t +. To calculate thi correlation, we imulate,, path uing the ame random number generator eed for each initial value of the dividend hare. We ue the initial value of the dividend throughout the year. A before, ince the hare doe not have a well-defined unconditional denity, we cannot preent the unconditional correlation and other moment, o we preent correlation conditional on the current tate t. Aet return are erially correlated in all cae. In the ymmetric cae hown in the top panel, the erial correlation are generally poitive, indicating poitive momentum. The erial correlation are generally larget in the region where expected return rie with the dividend hare, a one would expect. However, nonlinearitie in the hare proce and the fact that the firt aet return are alo driven 8