Two Trees. John H. Cochrane University of Chicago. Francis A. Longstaff The UCLA Anderson School and NBER

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1 Two Tree John H. Cochrane Univerity of Chicago Franci A. Longtaff The UCLA Anderon School and NBER Pedro Santa-Clara The UCLA Anderon School and NBER We olve a model with two i.i.d. Luca tree. Although the correponding one-tree model produce a contant price-dividend ratio and i.i.d. return, the two-tree model produce intereting aet-pricing dynamic. Invetor want to rebalance their portfolio after any change in value. Becaue the ize of the tree i fixed, price mut adjut to offet thi deire. A a reult, expected return, exce return, and return volatility all vary through time. Return diplay erial correlation and are predictable from price-dividend ratio. Return volatility differ from cah-flow volatility, and return hock can occur without new about cah flow. Return that are independent over time are the tandard benchmark for theory and empirical work in aet pricing. Yet, on reflection, i.i.d. return eem impoible with multiple poitive net upply aet. If a tock or a ector rie in value, invetor will try to rebalance away from it. But we cannot all rebalance, a the average invetor mut hold the market portfolio. It eem that the ucceful aet expected return mut rie, or ome other return moment mut change, in order to induce invetor to hold more of the ucceful ecuritie. We characterize the aet price and return dynamic that reult from thi market-clearing mechanim in a imple context. We olve an aet-pricing model with two Luca 1978 tree. Each tree dividend tream follow a geometric Brownian motion. The repreentative invetor ha log utility and conume the um of the two tree dividend. Price adjut o that invetor are happy to conume the dividend. We obtain cloed-form olution for price, expected return, volatilitie, correlation, and o forth. Depite it Graduate School of Buine, Univerity of Chicago, and NBER. The UCLA Anderon School and NBER. John Cochrane gratefully acknowledge reearch upport from an NSF grant adminitered by the NBER and from the CRSP. The author are grateful for the comment and uggetion of Chri Adcock, Yacine Aït-Sahalia, Andrew Ang, Ravi Banal, Geert Bekaert, Peter Boaert, Michael Brandt, George Contantinide, Vito Gala, Mark Grinblatt, Lar Peter Hanen, John Heaton, Jun Liu, Ian Martin, Anna Pavlova, Monika Piazzei, Rene Stulz, Raman Uppal, Pietro Veronei, anonymou referee, and many eminar and conference participant. We are alo grateful to Bruno Miranda for reearch aitance. Addre correpondence to Franci Longtaff, UCLA/Anderon School, 110 Wetwood Plaza, Lo Angele, CA ; United State or C The Author Publihed by Oxford Univerity Pre on behalf of the Society for Financial Studie. All right reerved. For Permiion, Pleae doi: /rf/hhm059 Advance Acce publication November 20, 2007

2 The Review of Financial Studie/ v 21 n imple ingredient, and although the correponding one-tree model produce a contant price-dividend ratio and i.i.d. return, the two-tree model diplay intereting dynamic. The two tree can repreent indutrie or other characteritic-baed grouping. The two tree can repreent broad aet clae uch a tock veru bond, or tock and bond veru human capital and real etate. The two tree can repreent two countrie aet market, providing a natural benchmark for aet market dynamic in international finance. 1 Valuation ratio and market value of the two tree vary over time, o portfolio trategie that hold aet baed on value/growth, mall/large, momentum, or related characteritic give different average return. Many intereting reult continue to hold a one tree become vanihingly mall relative to the other, o the model ha implication for expected return and dynamic of one aet relative to a much larger market. Underlying the dynamic, we find that expected return typically rie with a tree hare of dividend, to attract invetor to hold that larger hare. A a reult, a poitive dividend hock, which increae current price and return, alo typically raie ubequent expected return. Thu, return tend to 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. However, there are alo parameter and region of the tate pace in which expected return decline a function of the dividend hare, leading to mean reverion, price overreaction, and downward drift, with correponding exce volatility of price and return. When one aet ha a poitive dividend hock, thi hock lower the hare of the other aet, o the expected return of the other aet typically decline. We ee negative cro-erial correlation. We ee movement in the other aet price even with no new about that aet dividend, a dicount rate effect, and another ource and form of apparent exce volatility. Finally, we ee that aet return can be poitively contemporaneouly correlated with each other even when their underlying dividend are independent. The lower expected return raie the price of the other aet. A common factor or contagion emerge in aet return even though there i no common factor in cah flow. Becaue price-dividend ratio vary depite i.i.d. dividend growth, pricedividend ratio forecat return in the time erie and in the cro ection. Thu, we ee value and growth effect: high price-dividend ratio aet have low expected return and vice vera. We ee that time in which a given aet ha a high price-dividend ratio are time when that aet ha a low expected return. Thinking of the two tree a tock veru all other aet bond, real etate, human capital, price-dividend ratio forecat tock index return. Thee effect coexit with the poitive hort-run autocorrelation of return decribed above. 1 Hau and Rey 2004, Guibaud and Coeurdacier 2006, and Pavlova and Rigobon 2007 are ome recent article that ue multiple-tree framework to model countrie. 348

3 Two Tree However, although thee dynamic are uperficially reminicent of thoe in the empirical aet pricing literature, and although we have ued the terminology of that literature to help decribe the model dynamic, we do not claim that our model quantitatively matche the empirical literature. Our ingredient the log utility function, the pure-endowment production tructure, and epecially the geometric Brownian motion dividend procee are imple but empirically unrealitic. Our model i imple becaue our goal i purely theoretical: to undertand the dynamic induced by the market-clearing mechanim. Other paper in the emerging literature that price multiple cah-flow procee, including Banal et al. 2002, Menzly et al. 2004, and Santo and Veronei 2006, include non-i.i.d. dividend procee and temporally noneparable preference in order to better fit ome apect of the data. Other paper in the emerging generalequilibrium literature uch a Gome et al and Gala 2006 add an intereting invetment and production ide to endogenize cah-flow dynamic. For our aim, le i more. If we were to add thee kind of ingredient, we would no longer ee what dynamic reult from market clearing alone veru what dynamic come from the temporal tructure of preference, cah flow, or technology. We alo would be forced to more complex and le tranparent olution method. A the tandard one-tree model, though unrealitic, deliver ueful inight into key aet-pricing iue, thi imple two-tree model can iolate market-clearing effect that will be part of the tory in more complex model, and allow u a clearer economic intuition for thoe effect. Higher rik averion and greater number of tree are deirable extenion that would not by themelve introduce dynamic. The firt might raie the magnitude of market-clearing dynamic, providing a better fit to the data, and the econd i clearly important in it own right. However, our olution method work only for log utility and two tree. Wa it a mitake to believe in the logical poibility if not the reality of i.i.d. return for all thee year, given that our world doe contain multiple nonzero net upply aet? The anwer i no; it i poible to contruct multiple nonzero net upply aet model with i.i.d. return. For example, thi can occur if the upply of aet change intantly to accommodate change in demand. If a price rie i intantly matched by a hare repurchae, and a price decline i intantly matched by a hare iuance, then the market value of each ecurity can remain contant depite any variation in price. No change in return moment i neceary, becaue no change in the market portfolio occur. In thi ituation, invetor can and do collectively rebalance. In economic language, thi ituation i equivalent to the aumption of linear technologie. Output i a linear function of capital with no adjutment cot, diminihing return, or irreveribilitie, in contrat to our aumption of fixed endowment tree, hare upply. Invetor can then intantly and cotlely tranfer phyical capital from ucceful project to unucceful one or to conumption, keeping all market weight contant. Such a linear technology 349

4 The Review of Financial Studie/ v 21 n aumption i explicit, for example, in Cox et al Mot imply put, we can jut write the i.i.d. rate of return procee or technologie, and ak invetor how much they want to hold, allowing them collectively to hold a much or a little a they want at any time. The reulting model will of coure have i.i.d. return. Although uch model are poible, thee clearly unrealitic upply or technological underpinning of i.i.d. return are perhap not often appreciated. It i a mitake to think that i.i.d. return emerge from a multiperiod verion of the uual market-equilibrium derivation of the CAPM, in which demand adjut to a fixed upply of hare. And a the lat example make clear, we hould really think of model that deliver i.i.d. return in thi way a aet quantity model, not aet pricing model. They are model of the compoition of the market portfolio, becaue the aet price and expected return are given exogenouly. Which i the right aumption? In reality, market portfolio weight do change over time. Thu, a realitic model hould have at leat ome hort-run adjutment cot, irreveribilitie, and other impediment to aggregate rebalancing. It will therefore contain ome market-clearing dynamic of the ort we iolate and tudy in our imple two-tree exchange economy. On the other hand, new invetment i made, new hare are iued, and ome old capital i allowed to depreciate or i reallocated to new ue. Thu, a realitic model cannot pecify a pure endowment tructure. It need ome mechanim that allow aggregate rebalancing in the long run. The dynamic of the ort we tudy will apply le and le at longer horizon. In any cae, thi dicuion and the example of thi paper make clear that the technological underpinning of aet pricing model are more important to aet-price dynamic than i commonly recognized. Dynamic do not depend on preference alone. 1. Model and Reult 1.1 Model etup The repreentative invetor ha log utility, [ ] U t = E t e δτ ln C t+τ dτ. 1 0 There are two tree. Each tree generate a dividend tream D i dt. The dividend follow geometric Brownian motion with identical parameter, dd i D i = µ dt + σ dz i, 2 where i = 1, 2, and dz i are tandard Brownian motion, uncorrelated with each other. To make the notation more tranparent, we uppre time indice, for example, dd i /D i dd it /D it, etc., unle needed for clarity. Alo, we focu on the firt aet and uppre it index, for example, dd/d = dd 1 /D 1. Finally, 350

5 Two Tree ince intantaneou moment are of order dt, we will typically omit the dt term in expreion for moment. Thi i an endowment economy, o price adjut until conumption equal the um of the dividend, C = D 1 + D 2. Thi economy i a traightforward generalization of the well-known one-tree model, and the one-tree model i the limit of our two-tree model a either tree become dominant. In the one-tree model, the price-dividend ratio i a contant, P/D = 1/δ, and return are i.i.d. 1.2 Dividend hare dynamic The relative ize of the two tree give a ingle tate variable for thi economy. We find it convenient to capture that tate by the dividend hare, = D 1 D 1 + D 2. 3 Expected return and other variable are function of tate, o we can undertand their dynamic by undertanding thoe function of tate and undertanding how the tate variable evolve. Applying Itô Lemma to the definition in Equation 3, we obtain the dynamic for the dividend hare proce, d = 2σ 2 1 1/2 dt + σ1 dz 1 dz 2. 4 The drift of the dividend hare proce i an S-haped function. The drift i zero when equal 0, 1/2, or 1. The drift i poitive for between 0 and 1/2 and i negative for between 1/2 and 1. Thu, there i a tendency for the dividend hare to mean revert toward a value of 1/2. The two dividend-growth rate are independent, and there i no force raiing an aet dividend-growth rate if it hare become mall. Mean reverion in the hare reult from the nonlinear nature of the hare definition in Equation 3 through econd-order Itô Lemma effect the drift i zero when σ 2 = 0. In general, however, the drift i mall o the dividend hare i a highly peritent tate variable the path of it conditional mean tend only lowly back to 1/2. For example, at = 1/4, the drift i 3/32 σ 2, and with σ = 0.20 that implie a drift of only percentage point per year. Share volatility i a quadratic function of the hare, larget when the tree are of equal ize, = 1/2, and declining to zero at = 0or = 1. Share volatility i ubtantial. For example, at = 1/2, and with σ = 0.20, hare volatility i five percentage point per year. In turn, thi mean that if expected return are function of the hare, then they have the potential to vary ignificantly over time. The dipering effect of volatility overwhelm the mean-reverting effect of the drift, o thi hare proce doe not have a tationary ditribution. A hare proce with a tationary ditribution might be more appealing, but that would require putting dynamic in the dividend procee, o that mall aet catch up. We want to make it clear that all dynamic in thi model come from 351

6 The Review of Financial Studie/ v 21 n market-clearing, not from dynamic of the input. We return to the iue of the nontationary hare and it effect on aet pricing below, after we ee how aet price behave in the model. 1.3 Conumption dynamic In the one-tree model, conumption equal the dividend, o conumption growth i i.i.d. with contant mean E[dC/C] = µ and variance Var[dC/C] = σ 2. In the two-tree model, aggregate conumption C = D 1 + D 2 follow dc C = µ dt + σ dz 1 + σ 1 dz 2. 5 Conumption growth i no longer i.i.d. Mean conumption growth i till contant, but conumption volatility i a convex quadratic function of the hare, [ ] dc Var t = σ 2 [ ]. 6 C Conumption-growth volatility i till σ 2 at the limit = 0 and = 1, but decline to one-half that value at = 1/2. Volatility i lower for intermediate value of the dividend hare a conumption i then diverified between the two dividend. 1.4 The rikle rate The invetor firt-order condition imply that marginal utility i a dicount factor that price aet, i.e., M t = e δt C t. 7 The intantaneou interet rate i given from the dicount factor by [ ] [ ] [ ] dmt dc dc rdt= E t = δ dt + E t Var t. 8 M t C C In the one-tree model, conumption equal the dividend o we have r = δ + µ σ 2. 9 We ee the tandard dicount rate δ, conumption growth µ, and precautionary aving σ 2 effect. Becaue the rikle rate i contant, the entire term tructure i contant and flat. We can compute a rikle rate, even though the rikle aet i in zero net upply. Subtituting the moment of the conumption dynamic, Equation 5, into Equation 8, the interet rate in the two-tree model i r = δ + µ σ 2 [ ]

7 Two Tree Thu, the rikle rate varie over time a a quadratic function of the dividend hare. The rikle rate i higher for intermediate value of the dividend hare becaue dividend diverification lower conumption volatility, which lower the precautionary aving motive. Becaue the interet rate i not contant, the term tructure i not flat. 1.5 Market portfolio price and return A i uual in log utility model, the price-dividend ratio V M for the market portfolio or claim to conumption tream i a contant [ ] V Mt P Mt C t = 1 C t E t = E t [ 0 0 M t+τ M t e δτ C t+τ C t+τ dτ C t+τ dτ ] = 1 δ. 11 Thi calculation i the ame for the one-tree and two-tree model, and it i valid for all conumption dynamic. The total intantaneou return R M on the market equal price appreciation plu the dividend yield, R M = dp M P M + C P M dt = dc C + δ dt. 12 [In the econd equality, we ue the fact from Equation 11 that P M = C/δ.] Subtituting in conumption dynamic, we find for the one-tree model that the market return i i.i.d. with contant expected return and variance: R M = µ + δ dt + σ dz, 13 E t [R M ] = µ + δ, 14 Var t [R M ] = σ For the two-tree model, the conumption dynamic in Equation 5 imply R M = µ + δ dt + σ dz 1 + σ 1 dz The expected market return and variance are now E t [R M ] = µ + δ, 17 Var t [R M ] = σ 2 [ ]. 18 The expected return on the market i the ame a in the one-tree cae, but the variance of the market return equal the variance of conumption growth, which i now a quadratic function of the dividend hare, declining for intermediate hare. 353

8 The Review of Financial Studie/ v 21 n Subtracting the rikle rate in Equation 10 from the expected market return in Equation 17 how that the equity premium equal the variance of the market return: E t [R M ] r = Var t [R M ], 19 a i uual in log utility model. From Equation 18, the variance of the market i a convex quadratic function of the dividend hare. Thi fact mean that the equity premium and market Sharpe ratio are alo time varying, and increae a the market become more polarized. 1.6 The price-dividend ratio The price P of the firt aet i given by [ ] P t = E t e δτ C t D t+τ dτ. 20 C t+τ 0 Again, we uppre aet and time ubcript unle neceary for clarity, and we focu on the firt aet becaue the econd follow by ymmetry. For the remainder of thi ection, we impoe the parameter retriction δ = σ 2. Thi retriction, in conjunction with the ymmetry of the aet, give much impler formula and more tranparent intuition than the general cae. Thi retriction i not unreaonable: with σ = 0.20, δ = In the following ection, we treat the general cae that break thi retriction, allow the tree to have different value of µ and σ, and allow correlated hock. We preent formula uing whichever of δ or σ 2 give a more intuitive appearance. Uing the definition of the dividend hare in Equation 3, Equation 20 can be rewritten to give the price-conumption ratio for the firt aet: [ P ] C = E t e δτ t+τ dτ Valuing the aet i formally identical to rik-neutral pricing uing the dicount rate δ of an aet that pay a cah flow equal to the dividend hare. The dividend hare play a imilar role in many tractable model of long-lived cah flow, including Banal et al. 2002, Menzly et al. 2004, Longtaff and Piazzei 2004, and Santo and Veronei Solving Equation 21 with hare dynamic from Equation 4, the pricedividend ratio of the firt aet i V P D = 1 P C = 1 2δ [ ] ln1 ln The Appendix give a hort proof, along the following line. Firt, becaue follow a Markov proce, Equation 21 implie that P/C i a function of. Uing Itô Lemma and the dynamic for from Equation 4, we obtain an 354

9 Two Tree expreion for E[dP/C] in term of the firt and econd derivative of P/C. Second, from Equation 21, the price-conumption ratio atifie E [ d ] P = δ P. 23 C C Equating thee two expreion for E[dP/C], we obtain a differential equation for P/C. We then verify that Equation 22 olve thi differential equation. The formula in Equation 22 and ubequent one are all given in term of elementary function, and thu they can be characterized traightforwardly. However, it i eaier and make for better reading imply to plot the function, o we follow that route in our analyi. For comparability, we ue the parameter value δ = 0.04, µ = 0.02, and σ = 0.20 throughout all the plot preented in thi ection. Figure 1 plot the price-conumption and price-dividend ratio for δ = The price-conumption ratio lie cloe to a linear function of the dividend hare. Thi behavior i largely a cale effect; larger dividend command higher price. The lightly S-haped deviation from linearity are thu the mot intereting feature of thi graph. The price-conumption ratio initially i higher than the hare and then fall below the hare. We anticipate that the firt aet will have a high price at low hare and a low price at high hare. The price-dividend ratio varie greatly from the contant value P/D = 1/δ = 25 of the one-tree model. Thi variation of the price-dividend ratio a a function of the dividend hare drive the return dynamic that follow. The price-dividend ratio i equal to the price-conumption ratio divided by hare, o it behavior i driven by the mall deviation from linearity in the price-conumption ratio function. The price-dividend ratio in Figure 1 i larget for mall hare, decline until it value i le than the price-dividend ratio of the market portfolio tarting at = 0.5, and then increae to equal the market price-dividend ratio of 25 when = 1. Thi initially urpriing nonmonotonic behavior i neceary, becaue the contant price-dividend ratio of the market portfolio mut equal the hareweighted mean of the price-dividend ratio of the individual aet. If the price-dividend ratio of the firt aet i greater than that of the market at a hare of, ay, 0.25, then by ymmetry, it mut be le than that of the market at a hare of It mut then recover to equal the market price-dividend ratio when it i the market at = 1. The S-hape of the price-conumption ratio about the 45-degree line i exactly ymmetric in thi way, reflecting the ymmetry of Equation 22. The price-dividend ratio increae rapidly a the dividend hare decreae toward zero. The baic mechanim for thi behavior i a decline in rik premium. A the firt aet hare decline, it dividend become le correlated with aggregate conumption. Thi fact lower their rik premium and dicount rate, 355

10 The Review of Financial Studie/ v 21 n Figure 1 Price-conumption ratio and price-dividend ratio The top panel preent the price-conumption ratio of the firt aet, in the imple cae, uing parameter µ = 0.02, δ = σ 2 = The bottom panel preent the price-dividend ratio. The dahed line in the top panel give the 45-degree line; the dahed line in the bottom panel give the price-dividend ratio of the market portfolio and of the one-tree model, 1/δ = 25. raiing their valuation. An aet with a mall hare i more valuable from a diverification perpective. A 0, the price-dividend ratio rie to infinity. A 0, the firt tree dividend become completely uncorrelated with conumption, becaue conumption conit entirely of the econd tree dividend. A a reult, the firt tree i valued a a rik-free ecurity. In thi parameterization, the = 0 limit of the interet rate equal the dividend-growth rate r = µ, o the price-dividend ratio of the growing dividend tream explode. Although the rie in the pricedividend ratio a 0 i generic, becaue the aet become le riky, the 356

11 Two Tree left limit i finite for other parameterization in which the interet rate exceed the mean dividend-growth rate. 1.7 Aet return and moment Return follow now that we know price. Again uppreing aet and time ubcript, the intantaneou return R of the firt aet i R D P dt + dp P. 24 For both calculation and intuition, it i convenient to expre thi return in term of the price-dividend or valuation ratio, R = 1 V dt + dd D + dv V + dd dv D V. 25 The return equal the dividend yield, the dividend growth, the change in valuation or growth in the price-dividend ratio, and an Itôterm. Although the price-dividend ratio i contant in the one-tree model, the lat two term in Equation 25 are zero and the aet return i jut the dividend yield plu dividend conumption growth. With two aet, a we have een, the price-dividend ratio i no longer contant and varie through time a the relative weight of the aet evolve, o the latter two term matter. Both the mean and variance of return vary over time a function of the tate variable. Now we can find return moment. Taking the expectation of Equation 25, the expected return of the firt aet i E t [R] = 1 [ ] [ ] [ dd dv dd V dt + E t + E t + Cov t D V D, dv V The intantaneou variance of the firt aet return i [ dd Var t [R] = Var t D + dv ] V [ ] [ ] [ dd dv dd = Var t + Var t + 2Cov t D V D, dv V ]. 26 ]. 27 In the ingle-aet model with a contant valuation ratio, the variance of return i equal to the variance of dividend growth, the firt term in Equation 27. In the two-aet model, variation in the price-dividend ratio can provide additional return volatility through the econd term in Equation 27. However, if the price-dividend ratio i trongly negatively correlated with dividend growth if an increae in dividend and thu hare trongly reduce the price-dividend ratio then return volatility can be le than dividend-growth volatility through the third term in Equation 27. Applying Itô Lemma to the price-dividend ratio V in Equation 22 and uing the hare proce in Equation 4 give the following expreion for 357

12 The Review of Financial Studie/ v 21 n moment of dv/v in Equation 26 and 27: [ ] dv E t = δ ln, 28 V V [ ] dv Var t = 2 [ δ ] ln 2, 29 V δ 1 V [ dd Cov t D, dv ] [ = δ ] ln. 30 V 1 V Subtituting the expected dividend-growth rate E t [dd/d] = µ and Equation 28 and 30 into Equation 26 and rearranging give u the expected return a a function of tate, E t [R] = µ + 2δ1 + 1 ln 1 V. 31 Subtracting the rikle rate in Equation 10 from the expected return in Equation 31 give the expected exce return E t [R] r = 2δ ln 1 V. 32 Similarly, ubtituting from Equation 29 and 30 into Equation 27 give the return variance Var t [R] = δ [ δ ] 1 ln δ 1 V Figure 2 plot the expected return given in Equation 31 a a function of the dividend hare. Expected return rie with the hare, reach an interior maximum, and then decline lightly. Thi behavior of expected return in Figure 2 mirror the behavior of the price-dividend ratio in Figure 1. With contant expected dividend growth, expected return are the only reaon price-dividend ratio vary at all, o low expected return mut correpond to high price-dividend ratio. One way to undertand the nonmonotonic behavior of expected return, then, i a the mirror image of the nonmonotonic behavior of the price-dividend ratio tudied above. Expected return mut be higher than the market expected return for high hare near = 0.8 o that expected return can be lower than the market expected return for mall hare. The behavior of expected return a a function of tate in Figure 2 drive aet return dynamic. A poitive hock to the firt aet dividend increae the dividend hare. For hare value below about 0.80, thi event increae expected return. In thi range, then, a poitive dividend hock lead to a tring of expected price increae. Price will eem to underreact and lowly 358

13 Two Tree Figure 2 Expected return and component E[R] give the expected return of the firt aet a a function of the hare of the firt aet. The remaining line give component of thi expected return. 1/V give the dividend-price ratio. E[dD/D] give the expected dividendgrowth rate. E[dV/V] give the expected change in the price-dividend ratio. Cov[D,V] give Cov[dV/V,dD/D], the covariance of dividend growth with price-dividend ratio hock. incorporate dividend new. To the extent that own-dividend hock dominate other-dividend hock a a ource of price movement, we expect to ee here poitive autocorrelation and momentum of return. For hare value above about 0.80, however, expected return decline in the dividend hare. Here, a poitive own-dividend hock lead to lower ubequent expected return; we ee overreaction to or mean reverion after the dividend hock, and we expect to ee negative autocorrelation, mean reverion, and exce volatility of return. Equation 26 expree the expected return a a um of four component. The individual component are alo plotted in Figure 2. A illutrated, the dividend yield 1/ V i generally the larget component of expected return, followed cloely by the expected growth rate of dividend E[dD/D]. For mall dividend hare, the negative covariance between dividend and the valuation ratio reduce the expected return ubtantially. The negative covariance appear becaue a poitive hock to dividend increae the hare, and thi ha a trong negative effect on the valuation ratio a per Figure 1. The expected change in the valuation ratio i mall, generally poitive, and ha it larget effect on the expected return for mall to intermediate value of the dividend ratio. All three of thee component change ign or lope at the point where the price-dividend ratio tart riing a a function of hare, for high hare value. 359

14 The Review of Financial Studie/ v 21 n Expected exce return repreent rik premia, reflecting the covariance of return with conumption growth, [ E t [R] r = Cov t R, dc ]. 34 C We can expre thi rik premium a the um of a cah-flow beta, correponding to the covariance of dividend growth with conumption growth, and a valuation beta, correponding to the covariance of valuation hock with conumption growth. Subtituting Equation 25 into Equation 34, we have E t [R] r = Cov t [ dd D, dc C ] + Cov t [ dv V, dc C ]. 35 We can evaluate the term on the right-hand ide uing the dividend-growth and conumption procee in Equation 2 and 5, [ dd Cov t D, dc ] = σ 2 = δ, 36 C [ dv Cov t V, dc ] [ dd = Cov t C D, dv ] 2 1 V [ = 1 2 δ ] ln V In the lat equality, we have ubtituted from Equation 30. The cah-flow beta expree what would happen if price-dividend ratio were contant. Then, the covariance of return with conumption growth would be exactly proportional to the covariance of dividend growth with conumption growth. Thi covariance would of coure be larger preciely a the firt tree dividend provide a larger hare of conumption. Thu, the cah-flow beta i linear in the hare. The valuation beta i more intereting, a it capture the fact that pricedividend ratio change a well, and change in valuation that covary with the conumption growth generate a rik premium. Valuation beta capture the fact that return dynamic change in expected return, which change valuation pill over to the level of the expected return, a in Merton Figure 3 plot expected exce return from Equation 35, the rik-free rate from Equation 10, and the cah-flow and valuation beta from Equation 36 and 37, repectively. A hown, the expected exce return tart at zero, but then increae rapidly a the dividend hare increae. Expected exce return rie uniformly following a poitive dividend hock, o we expect to ee the poitive autocorrelation dynamic throughout the hare range uing thi meaure of coure, expected exce return remain poitive a 0 if one allow correlated cah flow. 360

15 Two Tree Figure 3 Expected exce return and component E[R] r give the expected exce return of the firt aet a a function of it hare. Cov[D,C] give the covariance of dividend and conumption-growth hock, Cov[dD/D,dC/C]. Cov[V,C] give the covariance of price-dividend ratio and conumption-growth hock, Cov[dV/V,dC/C]. The latter two component add up to the expected exce return. The rikle rate i given by r. The valuation beta can have either ign. It i lightly negative for dividend hare value between about 0.50 and For mall hare, the valuation beta i dominant. For mall hare, the rik premium i due primarily to change in valuation correlated with the market dicount rate effect rather than change in dividend or cah flow correlated with the market. An oberver might be puzzled why there i o much return correlation, beta, and expected return in the face of o little correlation of cah flow. In the limit a 0, the expected exce return collape to zero. One can how thi fact analytically by taking limit of the above expreion. In that limit, the firt tree dividend are completely uncorrelated with conumption. However, a Figure 3 demontrate, the expected exce return rie very quickly from zero and i ubtantial even for very mall hare. For example, the expected exce return i already 1/2% at = 0.1%. Formally, the derivative of expected exce return with repect to hare rie to infinity a 0. Therefore, the market-clearing-induced rik premium and return dynamic remain important for mall aet. Figure 4 plot the return volatility given in Equation 33 along with the component of that volatility from Equation 27. Dividend-growth Var[dD/D] give a contant contribution of 20% volatility. Change in valuation Var[dV/V ] add a mall amount of volatility at mall hare value, where the valuation in Figure 1 i a trong function of the hare. The negative 361

16 The Review of Financial Studie/ v 21 n Figure 4 Return volatility and component The olid line with ymbol labeled Var[R] give the variance of the firt aet return a a function of the firt aet hare. The vertical axi label give percent tandard deviation. However, the vertical axi plot variance, o that the variance component add up. The horizontal olid line labeled Var[dD/D] give the dividend-growth component of return variance. The dahed line labeled Var[dD/D]+Var[dV/V] add the dividend-growth variance and the variance of the price-dividend ratio. Adding the third, uually negative, component Cov[dD/D,dV/V] give the overall return variance. value of the covariance term in Equation 33 i a larger effect and puhe overall variance below dividend-growth variance for < The price-dividend ratio i a declining function of hare here, o a poitive dividend hock lower the price-dividend ratio Figure 1. A a reult, return price plu dividend move le than dividend themelve. A the hare increae, however, the covariance term eventually become poitive, where the price-dividend ratio rie with the hare, and add to the total return variance. Thu, there i a mall region of exce volatility in which the volatility of the aet return exceed the volatility of the underlying cah flow or dividend. Again, the derivative of volatility with repect to hare become infinite a 0, o market-clearing effect apply to very mall aet. Becaue the price-dividend ratio, expected return, and expected exce return are all function of the hare, we can ubtitute out the hare and plot expected return and exce return a function of the dividend yield. Figure 5 preent the reult. Both expected return and expected exce return are increaing function of the dividend yield. Therefore, dividend yield or price-dividend ratio forecat return in the time erie and in the cro ection. Expected return how a nicely linear relation to dividend yield through mot of the relevant range. Expected exce return how an intriguing nonlinear relation. The lope 362

17 Two Tree Figure 5 Dividend yield and expected return The olid line plot the firt aet expected return veru it dividend yield. The dahed line plot the firt aet expected exce return veru it dividend yield. Symbol mark the point = 0, = 0.1, = 0.2, etc., tarting from the left. of the return line i about 1.6 through the linear portion. A lope of one mean that higher dividend yield tranlate to higher expected return one-for-one. Higher lope mean that a high dividend yield forecat valuation increae a well. 1.8 Market beta With log utility, expected return follow a conditional CAPM and conumption CAPM. Thu, we can alo undertand expected exce return by reference to the aet beta and the market expected exce return. In the ingle-tree model, the aet i the market, and it beta equal one. In the two-tree model, the beta of each aet varie over time with the dividend hare. Uing the fact from Equation 12 that the market return equal conumption growth plu the dicount rate, we have β = Cov t [R, R M ] Var t [R M ] = Cov t [R, dc/c]. 38 Var t [R M ] Subtituting the covariance from Equation 36 and 37, and uing the market return variance in Equation 18, we obtain ln 1 V β = 2σ σ [ ]

18 The Review of Financial Studie/ v 21 n Figure 6 Market beta and market rik premium The olid line labeled β give the beta on the market portfolio of the firt aet return. It value are plotted on the left vertical axi. The dahed line labeled E[R M ] r give the expected exce return of the market portfolio. It value are plotted on the right vertical axi. We preent Equation 39 in term of σ 2, which i more intuitive for a econd moment, but thi formula i only valid under the retriction σ 2 = δ of thi ection. Figure 6 plot beta a a function of the dividend hare, revealing intereting dynamic. A hown, the beta i zero when the hare i zero. A the hare increae, the beta rie quickly, in fact infinitely quickly a 0. A we can ee in Figure 3, thi rie i due to the large valuation beta for mall aet. A the hare rie, the beta continue to rie almot linearly. Here, the nearly linear cah-flow beta of Figure 3 i at work: the firt aet contribute more to the total market return and it beta begin to increae correpondingly. At a hare = 0.5, the beta become greater than one, and then decline until it become one again when the firt aet i the entire market. A before, we can tart to undertand thi nonmonotonic behavior by aggregation: the hareweighted average beta mut be one, o if the mall aet ha a beta le than one, the larger aet mut have a beta larger than one. A the hare approache one, however, the beta begin to decreae and converge to one becaue the firt aet become the market a 1. The expected exce return of Figure 3 i equal to the beta of Figure 6 time the market expected exce return; E t [R M ] r = σ 2 [ ] from Equation 18 and 19 and i alo included in Figure 6. The decline in the market expected exce return from = 0to = 1/2 account for the 364

19 Two Tree lightly lower rie in expected exce return in Figure 3 compared to the rie in market beta in Figure 6 in thi region. The rie in market expected exce return from = 1/2 to = 1 offet the decline in beta hown in Figure 6, allowing the nearly linear rie in expected exce return hown in Figure 3. In um, although a conditional CAPM and conumption CAPM hold in thi model, one mut make reference to time-varying expected exce return, expected exce market return, and market beta in order to ee the relation predicted by the CAPM or conumption CAPM. 1.9 Return correlation The return of the two aet can be correlated even though their dividend are independent. To ee thi fact, we can write from Equation 25, Cov t [R 1, R 2 ] = Cov t [ dd1 D 1, dd 2 D 2 + Cov t [ dd2 D 2, dv 1 V 1 ] [ dd1 + Cov t, dv ] 2 D 1 V 2 ] [ dv1 + Cov t, dv 2 V 1 V 2 ]. 40 Becaue the dividend are independent, the firt term on the right-hand ide i zero. If the price-dividend ratio V i for the two aet were contant, the remaining three term on the right-hand ide would alo be zero and the return of the two aet would be uncorrelated. However, the price-dividend ratio vary over time and are correlated with each other and with the dividend. Thu, the correlation of the aet return i generally not equal to zero. Figure 7 plot the correlation between the aet return a a function of the dividend hare. A hown, the return have a correlation above 25% for mot of the range of dividend hare, even though the underlying cah flow are not correlated. The mechanim are traightforward. If tree two enjoy a poitive dividend hock, that event raie aet two return. However, it alo lower aet one hare. Lowering the hare typically raie the price-dividend ratio i.e., give rie to a poitive return for aet one. Thi tory underlie the Cov t [dd 2 /D 2, dv 1 /V 1 ] component of Equation 40, graphed a the [V 1, D 2 ] line of Figure 7. A expected it give a poitive contribution to correlation for hare below = 0.8, in which the price-dividend ratio in Figure 1 i riing in the hare. Adding the ymmetrical contribution to correlation from the effect of aet one dividend on aet two valuation give the dahed line marked [V 1, D 2 ] + [V 2, D 1 ] in Figure 7, and we ee that thee two effect are mot of the tory for the overall correlation, marked [R 1, R 2 ]in Figure 7. Again, the correlation are zero in the limit a 0or 1, but the derivative become infinite at thee limit o market-clearing-induced correlation are large for vanihingly mall aet. 365

20 The Review of Financial Studie/ v 21 n Figure 7 Return correlation The olid line with triangle labeled [R 1, R 2 ] give the conditional correlation between the two aet return, given the dividend hare of the firt aet. The remaining line give component of that correlation, uing the decompoition [ of Equation 40. For example, [V 1, D 2 ] give the component of correlation correponding to dv1 Cov V, dd 2 1 D ] Autocorrelation How important are the return dynamic in the two-tree model? How much do expected return vary over time? Do market-clearing return dynamic vanih for mall aet? A one way to anwer thee quetion, Figure 8 preent the intantaneou autocorrelation of return. In dicrete time, the autocorrelation i the regreion coefficient of future return on current return, Cov[R t+1, R t ]/Var[ R t ] = Cov[E t R t+1, R t ]/Var[ R t ]. We compute a continuou-time conditional counterpart to the econd expreion, Cov t [der t, R t ]/Var t [R t ]. R t denote the intantaneou expected return. ER t denote the expected return that i a function of the tate a plotted in Figure 2. Thu, we can apply Itô Lemma to find der t and the covariance follow. The reult expree how much a return hock raie ubequent expected return. The pattern of autocorrelation hown in Figure 8 i conitent with the pattern of expected return and exce return illutrated in Figure 2 and 3. The autocorrelation of return i poitive where the expected return in Figure 2 rie with the hare, and the autocorrelation i negative where expected return in Figure 2 decline with hare. Expected exce return in Figure 3 rie with hare throughout, and we ee a poitive autocorrelation throughout. Thi reult i not automatic, a a poitive return can alo be caued by an increae in the other aet dividend, which typically lower the expected return. Figure 8 366

21 Two Tree Figure 8 Return autocorrelation The return line i calculated a Cov[dER t, R t ]/Var[ R t ] where R t i the intantaneou return, and ER t = E t [R t ] i the expected value of the intantaneou return. It meaure how much a unit intantaneou return raie the ubequent expected return. The exce line doe the ame calculation for expected exce return. how that thi effect i dominated by the own-dividend hock, allowing a poitive autocorrelation to emerge. The magnitude of the autocorrelation i mall, reaching only one percentage point. 2 In the limit 0, both autocorrelation vanih. However, the lope of the autocorrelation tend to infinity a 0, o again market-clearing effect remain important for vanihingly mall aet. 2. The General Model Thi ection preent the general cae of the two-aet model. Dividend till follow geometric Brownian motion, but we allow different parameter, dd i D i = µ i dt + σ i dz i. 41 The correlation between dz 1 and dz 2 i ρ, not necearily zero. We alo allow the dicount rate and the volatility of dividend growth to differ, σ 2 δ. 2 Autocorrelation i a common and intuitive meaure of return predictability. However, lagged return do not contain all information that i available at time t. It i poible for return to be predictable for example, by dividend yield while not autocorrelated. The variance of expected return, which i the variance of the numerator of R 2 in a multivariate return-forecating regreion, i a more comprehenive meaure. However, our computation of the continuou-time counterpart to thi quantity Var[dER t ] doe not differ enough from Figure 8 to warrant preentation. 367

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