CAPITAL ASSET PRICING MODEL 871 repurchase, then from(1)and (2), [P(t +h)-P(o)]P(c)is the rate of return on the asset over the period, in units of the consumption good. I Since movements from equilibrium to equilibrium through time involve both price and quantity adjustment, a complete analysis would require a description of both the rate of return and change in asset value dynamics. To do so would require a specification of firm behavior in determining the supply of shares, which in turn would require knowledge of the real asset structure (i. e, technology whether capital is"putty"or"clay"; etc. In particular, the current returns on firms with large amounts(relative to current cash flow) of non-shiftable capital with low rates of depreciation will tend to be strongly affected by shifts in capital ization rates because, in the short run, most of the adjustment to the new equi- librium will be done by prices 6. Since the present paper examines only investor behavior to derive the demands assets and the relative yield requirements in equilibrium, 2 only the rate of return dynamics will be examined explicitly. Hence, certain variables, taken as exogeneous in the model, would be endogeneous to a full-equilibrium system From the assumption of continuous trading(Assumption 7), it is assumed that the returns and the changes in the opportunity set can be described by continuous time stochastic processes. However, it will clarify the analysis to describe the processes for discrete trading intervals of length h, and then, to consider the limit as h tends to zero We assume the following ASSUMPTION 8: The vector set of stochastic processes describing the opportunity set and its changes, is a time-homogeneous Markov process ASSUMPTION 9: Only local changes in the state variables of the process are ASSUMPTION 10: For each asset in the opportunity set at each point in time t, the expected rate of return per unit time, defined by EE[(P(t +h)-P(t)/P(t]h to define two quantities, such as number of shares and price per share, to distinguish between the two ways in which a firms value can change. The eturn part, (I), reflects new additions to wealth, while( 2)reflects a reallocation of capital alternative assets. The former is important to the investor in selecting his portfolio while the latter ning equilibrium through time. The definition of price per share sed here(except for cash dividends) corresponds to the way open-ended, mutual funds determine sset value per share, and seems to reflect accurately the way the term is normally used in a portfolio 12 While the analysis is not an equilibrium one in the strict sense because we do not develop the upply side, the derived model is as muc xchange"model of Mossin 9]. Because his is a one-period model, he could take supplies as fixed. To assume this over time is i3 While it is not necessary to assume that the processes are independent of calendar time, nothing of content is lost by it. However, when a state variable is declared ant in the tex mean non-stochastic. Thus, the term"constant"is used to describe va has content downl ued stube to sT oR ems ae ondtp23013024426AM
CAPITAL ASSET PRICING MODEL 871 repurchase, then from (1) and (2), [P(t + h) - P(t)]/P(t) is the rate of return on the asset over the period, in units of the consumption good." Since movements from equilibrium to equilibrium through time involve both price and quantity adjustment, a complete analysis would require a description of both the rate of return and change in asset value dynamics. To do so would require a specification of firm behavior in determining the supply of shares, which in turn would require knowledge of the real asset structure (i.e., technology; whether capital is "putty" or "clay"; etc.). In particular, the current returns on firms with large amounts (relative to current cash flow) of non-shiftable capital with low rates of depreciation will tend to be strongly affected by shifts in capitalization rates because, in the short run, most of the adjustment to the new equilibrium will be done by prices. Since the present paper examines only investor behavior to derive the demands for assets and the relative yield requirements in equilibrium,'2 only the rate of return dynamics will be examined explicitly. Hence, certain variables, taken as exogeneous in the model, would be endogeneous to a full-equilibrium system. From the assumption of continuous trading (Assumption 7), it is assumed that the returns and the changes in the opportunity set can be described by continuoustime stochastic processes. However, it will clarify the analysis to describe the processes for discrete trading intervals of length h, and then, to consider the limit as h tends to zero. We assume the following: ASSUMPTION 8: The vector set of stochastic processes describing the opportunity set and its changes, is a time-homogeneousl 3 Markov process. ASSUMPTION 9: Only local changes in the state variables of the process are allowed. ASSUMPTION 10: For each asset in the opportunity set at each point in time t, the expected rate of return per unit time, defined by oc-Et[(P(t + h) -P(t))/P(t)]/h " In an intertemporal model, it is necessary to define two quantities, such as number of shares and price per share, to distinguish between the two ways in which a firm's value can change. The return part, (1), reflects new additions to wealth, while (2) reflects a reallocation of capital among alternative assets. The former is important to the investor in selecting his portfolio while the latter is important in (determining) maintaining equilibrium through time. The definition of price per share used here (except for cash dividends) corresponds to the way open-ended, mutual funds determine asset value per share, and seems to reflect accurately the way the term is normally used in a portfolio context. 12 While the analysis is not an equilibrium one in the strict sense because we do not develop the supply side, the derived model is as much an equilibrium model as the "exchange" model of Mossin [29]. Because his is a one-period model, he could take supplies as fixed. To assume this over time is nonsense. 13 While it is not necessary to assume that the processes are independent of calendar time, nothing of content is lost by it. However, when a state variable is declared as constant in the text, we really mean non-stochastic. Thus, the term "constant" is used to describe variables which are deterministic functions of time. This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:44:26 AM All use subject to JSTOR Terms and Conditions
872 ROBERT C. MERTON and the variance of the return per unit time, defined by 02= E[([P(t h)-P(t)]/P(t)-ah)2)/h exist, are finite with 0>0, and are(right) continuous functions of h, where"Er "is the conditional expectation operator, conditional on the levels of the state variable at time t. In the limit as h tends to zero, a is called the instantaneous expected return nd o the instantaneous variance of the return Assumption 8 is not very restrictive since it is not required that the stochastic processes describing returns be Markov by themselves, but only that by the expansion of the state"(supplementary variables)technique [7, P. 262]to include (a finite number of) other variables describing the changes in the transition probabilities, the entire (expanded) set be Markov. This generalized use of the Markov assumption for the returns is important because one would expect that e required returns will depend on other variables besides the price per share (e.g, the relative supplies of assets) Assumption 9 is the discrete-time analog to the continuous-time assumption of continuity in the state variables (i.e, if X(t+ h)is the random state variable, then with probability one, limh-o[X(t+ h)-X(t)]=O). In words, it says that over small time intervals, price changes(returns)and changes in the opportunity set are small. This restriction is non-trivial since the implied"smoothness"rules out Pareto- Levy or Poisson-type jump processes. 14 Assumption 10 ensures that, for small time intervals, the uncertainty neither rashes out(i.e,a-=0) nor dominates the analysis (i.e, 02= oo). Actually Assumption 10 follows from Assumptions 8 and 9(see[13, p. 321) If we let X(o stand for the vector stochastic process, then Assumptions 8-10 imply that, in the limit as h tends to zero, X(t) is a diffusion process with continuous state-space changes and that the transition probabilities will satisfy a(multi dimensional)Fokker-Planck or Kolmogorov partial differential equation Although these partial differential equations are sufficient for study of the transition probabilities, it is useful to write down the explicit return dynamics in stochastic difference equation form and then, by taking limits, in sto differential equation form. From the previous analysis, we can write the ynamIcs as P(t+ h)-P(t) P(t) =mh+oy(t)√h, where, by construction, E,()=0 and E, (y2)=1, and y(t)is a purely random process;that is, y(t) and y(t s), for s>0, are identically distributed and mutually 14 While a similar analysis can be performed for l-type processes(see Kushner [18] and Merton[25])and for the subordinated processes of Press[30] and Clark[6], most of the results derived under the continuity assumption will not obtain in these cases has content downl ued stube to sT oR ems ae ondtp23013024426AM
872 ROBERT C. MERTON and the variance of the return per unit time, defined by a2 _ Et[([P(t + h) - P(t)]/P(t) -och)2]/h exist, are finite with a2 > 0, and are (right) continuous functions of h, where "Et" is the conditional expectation operator, conditional on the levels of the state variables at time t. In the limit as h tends to zero, ox is called the instantaneous expected return and U2 the instantaneous variance of the return. Assumption 8 is not very restrictive since it is not required that the stochastic processes describing returns be Markov by themselves, but only that by the "expansion of the state" (supplementary variables) technique [7, p. 262] to include (a finite number of) other variables describing the changes in the transition probabilities, the entire (expanded) set be Markov. This generalized use of the Markov assumption for the returns is important because one would expect that the required returns will depend on other variables besides the price per share (e.g., the relative supplies of assets). Assumption 9 is the discrete-time analog to the continuous-time assumption of continuity in the state variables (i.e., if X(t + h) is the random state variable, then, with probability one, limh,O [X(t + h) - X(t)] = 0). In words, it says that over small time intervals, price changes (returns) and changes in the opportunity set are small. This restriction is non-trivial since the implied "smoothness" rules out Pareto-Levy or Poisson-type jump processes.14 Assumption 10 ensures that, for small time intervals, the uncertainty neither "washes out" (i.e., a 2 = 0) nor dominates the analysis (i.e., c2 = oo). Actually, Assumption 10 follows from Assumptions 8 and 9 (see [13, p. 321]). If we let {X(t)} stand for the vector stochastic process, then Assumptions 8-10 imply that, in the limit as h tends to zero, X(t) is a diffusion process with continuous state-space changes and that the transition probabilities will satisfy a (multidimensional) Fokker-Planck or Kolmogorov partial differential equation. Although these partial differential equations are sufficient for study of the transition probabilities, it is useful to write down the explicit return dynamics in stochastic difference equation form and then, by taking limits, in stochastic differential equation form. From the previous analysis, we can write the returns dynamics as (3) P(t + h)-P(t) = cih + oy(t),/h, where, by construction, Et(y) = 0 and E,(y2) = 1, and y(t) is a purely random process; that is, y(t) and y(t + s), for s > 0, are identically distributed and mutually 14 While a similar analysis can be performed for Poisson-type processes (see Kushner [18] and Merton [25]) and for the subordinated processes of Press [30] and Clark [6], most of the results derived under the continuity assumption will not obtain in these cases. This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:44:26 AM All use subject to JSTOR Terms and Conditions
CAPITAL ASSET PRICING MODE 873 independent. 5 If we define the stochastic process, z(t), b z(+)=()+y(/h, then z(t)is a stochastic process with independent increments. If it is further assumed that y(t)is Gaussian distributed, 6 then the limit as h tends to zero of z(t h) z(o) describes a Wiener process or Brownian motion. In the formalism of tochastic differential equations dz= y(t)vdr In a similar fashion, we can take the limit of (3)to derive the stochastic differential equation for the instantaneous return on the ith asset as dP P a: dt +o dzi Processes such as(6)are called Ito processes and while they are continuous, they are not differentiable. 17 From(6), a sufficient set of statistics for the opportunity set at a given poi time is(ai, 0i, Pul where Pi is the instantaneous correlation coefficient between the Wiener processes dzi and dz. the vector of return dynamics as described in (6)will be Markov only if a Oi, and Pi were, at most, functions of the P's. In general, one would not expect this to be the case since, at each point in time, equilibrium clearing conditions will define a set of implicit functions between equilibrium market values, v(t)= Ni(t)P(o), and the ai, Ui, and pii. Hence, one would expect the changes in required expected returns to be stochastically related to changes in market values, and dependence on P solely would obtain only if changes in N(changes in supplies)were non-stochastic. Therefore, to close the system, we append the dynamics for the changes in the opportunity set over time namely da= ai dt +bi dq do=fi dt gi dx where we do assume that (6)and(7, together, form a Markov system, with dq and dx; standard wiener processes It is sufficient to assume that the y(r are uncorrelated and that the higher ord thesis of Samuel son[33] and Fama [10]. See Merton and Samuelson [27] for further discussion 16 While the Gaussian as nalt aking the assumption is more apparent than real, since it can be shown the ocesses can be described as functions of Brownian motion (see Feller [13, p. 326] and Ito and 17 See Merton [25] for a discussion of Ito processes in a f stochastic differential equations of the Ito type, see Ito and McKean [16]. McKean [22], and Kushner [18] ply of shares as well ther factors such as new technical developments. The particular derivation of the dzi in the text implies that the pu are constants. However, the analysis could be generalized by appending an ad ditional set of dynamics to include changes in the pi has content downl ued stube to sT oR ems ae ondtp23013024426AM
CAPITAL ASSET PRICING MODEL 873 independent." If we define the stochastic process, z(t), by (4) z(t + h) = z(t) + y(t) Ih, then z(t) is a stochastic process with independent increments. If it is further assumed that y(t) is Gaussian distributed,'6 then the limit as h tends to zero of z(t + h) - z(t) describes a Wiener process or Brownian motion. In the formalism of stochastic differential equations, (5) dz _ y(t)/dt. In a similar fashion, we can take the limit of (3) to derive the stochastic differential equation for the instantaneous return on the ith asset as dP. (6) dp = ai dt + vi dzi Pi Processes such as (6) are called Ito processes and while they are continuous, they are not differentiable. 17 From (6), a sufficient set of statistics for the opportunity set at a given point in time is {ai, vi, pij} where pij is the instantaneous correlation coefficient between the Wiener processes dzi and dzj. The vector of return dynamics as described in (6) will be Markov only if ai. vi, and pij were, at most, functions of the P's. In general, one would not expect this to be the case since, at each point in time, equilibrium clearing conditions will define a set of implicit functions between equilibrium market values, Vi(t) = N#(t)Pf(t), and the oci, vi, and pij. Hence, one would expect the changes in required expected returns to be stochastically related to changes in market values, and dependence on P solely would obtain only if changes in N (changes in supplies) were non-stochastic. Therefore, to close the system, we append the dynamics for the changes in the opportunity set over time: namely, (7) dai = ai dt + bi dqi, doi = f dt + gi dxi, where we do assume that (6) and (7), together, form a Markov system,18 with dqi and dxi standard Wiener processes. 15 It is sufficient to assume that the y(t) are uncorrelated and that the higher order moments are o(1/hA). This assumption is consistent with a weak form of the efficient markets hypothesis of Samuelson [33] and Fama [10]. See Merton and Samuelson [27] for further discussion. 16 While the Gaussian assumption is not necessary for the analysis, the generality gained by not making the assumption is more apparent than real, since it can be shown that all continuous diffusion processes can be described as functions of Brownian motion (see Feller [13, p. 326] and It6 and McKean [16]). 17 See Merton [25] for a discussion of It6 processes in a portfolio context. For a general discussion of stochastic differential equations of the It6 type, see It6 and McKean [16], McKean [22], and Kushner [18]. 18 It is assumed that the dynamics of a and a reflect the changes in the supply of shares as well as other factors such as new technical developments. The particular derivation of the dzi in the text implies that the Pij are constants. However, the analysis could be generalized by appending an additional set of dynamics to include changes in the pij. This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:44:26 AM All use subject to JSTOR Terms and Conditions