《投资学原理 Investments》课程教学资源（阅读资料）an intertemporal capital asset pricing model

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Econometrica, VoL 41, No 5.(September, 1973) AN INTERTEMPORAL CAPITAL ASSET PRICING MODELI An intertemporal model for the capital market behavior by an arbitrary functions for assets are derived, and it is shown that, unlike the one-period model, current unities. After aggregating demands and requiring market clearing, the equilibrium re lationships among expected returns are derived, and contrary to the classical capital asset hey have no systematic or market risk 1. INTRODUCTION ONE OF THE MORE important developments in modern capital market theory is the Sharpe-Lintner-Mossin mean-variance equilibrium model of exchange,com- monly called the capital asset pricing model. 2 Although the model has been the basis for more than one hundred academic papers and has had significant impact on the non-academic financial community, it is still subject to theoretical and empirical criticism. Because the model assumes that investors choose their port folios according to the Markowitz [21] mean-variance criterion, it is subject to all the theoretical objections to this criterion, of which there are many. 4 It has also been criticized for the additional assumptions required, especially homogeneous expectations and the single-period nature of the model. The proponents of the model who agree with the theoretical objections, but who argue that the capital market operates"as if"these assumptions were satisfied are themselves not beyond criticism. While the model predicts that the expected excess return from holding an asset is proportional to the covariance of its return with the market This paper is a substantial revision of parts of [24] presented in various forms at the NBER Con ference on Decision Rules and Uncertainty, d at the Wells Fargo Conference on Capital Market Theory, San Francisco, July, 1971. I am grate useful discussions, and Robert K. Merton for editorial assistance. Aid from the National Science Foundation is gratefully acknowledge 2 See Sharpe[38 and 39], Lintner [19 and 20], and Mossin [29]. while more general and elegant than the capital asset pricing model in many ways, the general equilibrium model of Arrow [1] and 41]) The"growth optimum"model of Hakansson [15] can be formulated as an although it is consistent with expecte that the model fits the data about as well as the capi et pricing For academic references, see Sharpe [39] and the Jensen [17] survey article. For a summary of or a See Sharpe [39, pp. 77 a list of the assumptions require 867 content donal use fibre to S R ems we Cond stp2301302 44:26AM

Econometrica, Vol. 41, No. 5, (September, 1973) AN INTERTEMPORAL CAPITAL ASSET PRICING MODEL' BY ROBERT C. MERTON An intertemporal model for the capital market is deduced from the portfolio selection behavior by an arbitrary number of investors who aot so as to maximize the expected utility of lifetime consumption and who can trade continuously in time. Explicit demand functions for assets are derived, and it is shown that, unlike the one-period model, current demands are affected by the possibility of uncertain changes in future investment opportunities. After aggregating demands and requiring market clearing, the equilibrium relationships among expected returns are derived, and contrary to the classical capital asset pricing model, expected returns on risky assets may differ from the riskless rate even when they have no systematic or market risk. 1. INTRODUCTION ONE OF THE MORE important developments in modern capital market theory is the Sharpe-Lintner-Mossin mean-variance equilibrium model of exchange, commonly called the capital asset pricing model.2 Although the model has been the basis for more than one hundred academic papers and has had significant impact on the non-academic financial community,' it is still subject to theoretical and empirical criticism. Because the model assumes that investors choose their portfolios according to the Markowitz [21] mean-variance criterion, it is subject to all the theoretical objections to this criterion, of which there are many.4 It has also been criticized for the additional assumptions required,5 especially homogeneous expectations and the single-period nature of the model. The proponents of the model who agree with the theoretical objections, but who argue that the capital market operates "as if" these assumptions were satisfied, are themselves not beyond criticism. While the model predicts that the expected excess return from holding an asset is proportional to the covariance of its return with the market 1 This paper is a substantial revision of parts of [24] presented in various forms at the NBER Conference on Decision Rules and Uncertainty, Massachusetts Institute of Technology, February, 1971, and at the Wells Fargo Conference on Capital Market Theory, San Francisco, July, 1971. I am grateful to the participants for helpful comments. I thank Myron Scholes and Fischer Black for many useful discussions, and Robert K. Merton for editorial assistance. Aid from the National Science Foundation is gratefully acknowledged. 2 See Sharpe [38 and 39], Lintner [19 and 20], and Mossin [29]. While more general and elegant than the capital asset pricing model in many ways, the general equilibrium model of Arrow [1] and Debreu [8, Ch. 7] has not had the same impact, principally because of its empirical intractability and the rather restrictive assumption that there exist as many securities as states of nature (see Stiglitz [41]). The "growth optimum" model of Hakansson [15] can be formulated as an equilibrium model although it is consistent with expected utility maximization only if all investors have logarithmic utility functions (see Samuelson [36] and Merton and Samuelson [27]). However, Roll [32] has shown that the model fits the data about as well as the capital asset pricing model. 3 For academic references, see Sharpe [39] and the Jensen [17] survey article. For a summary of the model's impact on the financial community, see [42]. 4 See Borch [4], Feldstein [12], and Hakansson [15]. For a list of the conditions necessary for the validity of mean-variance, see Samuelson [34 and 35]. See Sharpe [39, pp. 77-78] for a list of the assumptions required. 867 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

868 ROBERT C. MERTON ortfolio (its"beta"), the careful empirical work of Black, Jensen, and Scholes [3 has demonstrated that this is not the case. In particular, they found that"low beta"assets earn a higher return on average and"high beta "assets earn a lower return on average than is forecast by the model.6 Nonetheless, the model is still used because it is an equilibrium model which provides a strong specification of the relationship among asset yields that is easily interpreted, and the empirical evidence suggests that it does explain a significant fraction of the variation in asset returns This paper develops an equilibrium model of the capital market which( has the simplicity and empirical tractability of the capital asset pricing model (ii)is consistent with expected utility maximization and the limited liability of assets and (ii) provides a specification of the relationship among yields that is more onsistent with empirical evidence. Such a model cannot be constructed without costs. The assumptions, principally homogeneous expectations, which it holds in common with the classical model, make the new model subject to some of the me criticisms o The capital asset pricing model is a static(single-period )model although it is cnerally treated as if it holds intertemporally. Fama [9] has provided some justification for this assumption by showing that, if preferences and future invest- ment opportunity sets are not state-dependent, then intertemporal portfolio maximization can be treated as if the investor had a single-period utility function However, these assumptions are rather restrictive as will be seen in later analysis Merton [25] has shown in a number of examples that portfolio behavior for an intertemporal maximizer will be significantly different when he faces a changing investment opportunity set instead of a constant one The model presented here is based on consumer-investor behavior as described in [25], and for the assumptions to be reasonable ones, it must be intertemporal Far from a liability, the int tertemp poral nature of the model allows it to capture effects which would never appear in a static model, and it is precisely these effects which cause the significant differences in specification of the equilibrium relation among asset yields that obtain in the new model and the classical model 2. CAPITAL MARKET STRUCTURE It is assumed that the capital market is structured as follows assumption 1. All assets have limited liabilit ASSUMPTION 2: There are no transactions costs, taxes, or problems with in- diuisibilities of assets b Friend and Blume [14]also found that the empirical capital market line was"too fat. "Their planation was that the borrowing-lending assumption of the model is violated. Black [2]provides an alternative explanation based on the assumption of no riskless asset. Other less important, stylized facts in conflict with the model are that investors do not hold the same relative proportions of risky assets, and short sales occur in spite of unfavorable institutional requirement 7 Fama recognizes the restrictive nature of the assumptions as evidenced by discussion in Fama and Miller [11]. has content downl ued stube to sT oR ems ae ondtp23013024426AM

868 ROBERT C. MERTON portfolio (its "beta"), the careful empirical work of Black, Jensen, and Scholes [3] has demonstrated that this is not the case. In particular, they found that "low beta" assets earn a higher return on average and "high beta" assets earn a lower return on average than is forecast by the model.6 Nonetheless, the model is still used because it is an equilibrium model which provides a strong specification of the relationship among asset yields that is easily interpreted, and the empirical evidence suggests that it does explain a significant fraction of the variation in asset returns. This paper develops an equilibrium model of the capital market which (i) has the simplicity and empirical tractability of the capital asset pricing model; (ii) is consistent with expected utility maximization and the limited liability of assets; and (iii) provides a specification of the relationship among yields that is more consistent with empirical evidence. Such a model cannot be constructed without costs. The assumptions, principally homogeneous expectations, which it holds in common with the classical model, make the new model subject to some of the same criticisms. The capital asset pricing model is a static (single-period) model although it is generally treated as if it holds intertemporally. Fama [9] has provided some justification for this assumption by showing that, if preferences and future investment opportunity sets are not state-dependent, then intertemporal portfolio maximization can be treated as if the investor had a single-period utility function. However, these assumptions are rather restrictive as will be seen in later analysis.7 Merton [25] has shown in a number of examples that portfolio behavior for an intertemporal maximizer will be significantly different when he faces a changing investment opportunity set instead of a constant one. The model presented here is based on consumer-investor behavior as described in [25], and for the assumptions to be reasonable ones, it must be intertemporal. Far from a liability, the intertemporal nature of the model allows it to capture effects which would never appear in a static model, and it is precisely these effects which cause the significant differences in specification of the equilibrium relationship among asset yields that obtain in the new model and the classical model. 2. CAPITAL MARKET STRUCTURE It is assumed that the capital market is structured as follows. ASSUMPTION 1: All assets have limited liability. ASSUMPTION 2: There are no transactions costs, taxes, or problems with indivisibilities of assets. 6 Friend and Blume [14] also found that the empirical capital market line was "too flat." Their explanation was that the borrowing-lending assumption of the model is violated. Black [2] provides an alternative explanation based on the assumption of no riskless asset. Other less important, stylized facts in conflict with the model are that investors do not hold the same relative proportions of risky assets, and short sales occur in spite of unfavorable institutional requirements. X Fama recognizes the restrictive nature of the assumptions as evidenced by discussion in Fama and Miller [11]. 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 MODEL 869 ASSUMPTION 3: There are a sufficient number of investors with comparable wealth levels so that each investor believes that he can buy and sell as much of an asset as ASSUMPTION 4: The capital market is always in equilibrium (i.e, there is no ading at non-equilibrium prices ASSUMPTION 5: There exists an exchange market for borrowing and lending at ASSUMPTION 6: Short-sales of all assets, with full use of the proceeds, is allowed ASSUMPTION 7: Trading in assets takes place continually in time ASSUMPTIONS 1-6 are the standard assumptions of a perfect market, and their merits have been discussed extensively in the literature. Although Assumption 7 is not standard, it almost follows directly from Assumption 2. If there are no costs to transacting and assets can be exchanged on any scale, then investors would prefer to be able to revise their portfolios at any time(whether they actually do so or not). In reality, transactions costs and indivisibilities do exist, and on iven for finite trading- interval(discrete-time) models is to give implici if not explicit, recognition to these costs. However, this method of avoiding the problem of transactions costs is not satisfactory since a proper solution would almost certainly show that the trading intervals are stochastic and of non-constant length. Further, the portfolio demands and the resulting equilibrium relationships will be a function of the specific trading interval that is chosen.An investor making a portfolio decision which is irrevocable ("frozen")for ten years, will choose quite differently than the one who has the option(even at a cost) to revise his portfolio daily. The essential issue is the market structure and not investors' tastes, and for well-developed capital markets, the time interval between successive market openings is sufficiently small to make the continuous-time assumption a good approximation 3. ASSET VALUE AND RATE OF RETURN DYNAMICS Having described the structure of the capital market, we now develop the dy- namics of the returns on assets traded in the market It is sufficient for his decision ple example from the expectations theory of the term structure will is well known(see, e.g, Stiglitz [40 )that bonds cannot be priced lect a"fundamental "period(usuall h)to equate expected ret learly, the prices which satisfy this relationship will be a function of h Similarly, the demand functions of investors will depend on h. We have chosen for our interval the mallest h possible. For processes which are well defined for every h, it can be shown that the limit of every discrete-time solution as h tends to zero, will be the continuous solutions derived here(see depends on the particula odered. F ude ks small is for what h does the become sul compact in the son [35] sense? content donal use fibre to S R ems we Cond stp2301302 44:26AM

CAPITAL ASSET PRICING MODEL 869 ASSUMPTION 3: There are a sufficient number of investors with comparable wealth levels so that each investor believes that he can buy and sell as much of an asset as he wants at the market price. ASSUMPTION 4: The capital market is always in equilibrium (i.e., there is no trading at non-equilibrium prices). ASSUMPTION 5: There exists an exchange market for borrowing and lending at the same rate of interest. ASSUMPTION 6: Short-sales of all assets, with full use of the proceeds, is allowed. ASSUMPTION 7: Trading in assets takes place continually in time. ASSUMPTIONS 1-6 are the standard assumptions of a perfect market, and their merits have been discussed extensively in the literature. Although Assumption 7 is not standard, it almost follows directly from Assumption 2. If there are no costs to transacting and assets can be exchanged on any scale, then investors would prefer to be able to revise their portfolios at any time (whether they actually do so or not). In reality, transactions costs and indivisibilities do exist, and one reason given for finite trading-interval (discrete-time) models is to give implicit, if not explicit, recognition to these costs. However, this method of avoiding the problem of transactions costs is not satisfactory since a proper solution would almost certainly show that the trading intervals are stochastic and of non-constant length. Further, the portfolio demands and the resulting equilibrium relationships will be a function of the specific trading interval that is chosen.8 An investor making a portfolio decision which is irrevocable ("frozen") for ten years, will choose quite differently than the one who has the option (even at a cost) to revise his portfolio daily. The essential issue is the market structure and not investors' tastes, and for well-developed capital markets, the time interval between successive market openings is sufficiently small to make the continuous-time assumption a good approximation.9 3. ASSET VALUE AND RATE OF RETURN DYNAMICS Having described the structure of the capital market, we now develop the dynamics of the returns on assets traded in the market. It is sufficient for his decision 8 A simple example from the expectations theory of the term structure will illustrate the point. It is well known (see, e.g., Stiglitz [40]) that bonds cannot be priced to equate expected returns over all holding periods. Hence, one must select a "fundamental" period (usually one "trading" period, our h) to equate expected returns. Clearly, the prices which satisfy this relationship will be a function of h. Similarly, the demand functions of investors will depend on h. We have chosen for our interval the smallest h possible. For processes which are well defined for every h, it can be shown that the limit of every discrete-time solution as h tends to zero, will be the continuous solutions derived here (see Samuelson [35]). 9 What is "small" depends on the particular process being modeled. For the orders of magnitude typically found for the moments (mean, variance, skewness, etc.) of annual returns on common stocks, daily intervals (h = 1/270) are small. The essential test is: for what h does the distribution of returns become sufficiently "compact" in the Samuelson [35] sense? 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

870 ROBERT C. MERTON making that the consumer- investor know at each point in time: (i)the transition probabilities for returns on each asset over the next trading interval (the investment opportunity set); and (i) the transition probabilities for returns on assets in future periods (i.e, knowledge of the stochastic processes of the changes in the inves ment opportunity set). Unlike a single-period maximizer who, by definition, does not consider events beyond the present period, the intertemporal maximizer in selecting his portfolio takes into account the relationship between current period returns and returns that will be available in the future. For example, suppose that the current return on a particular asset is negatively correlated with changes in n, higher return on the asset if, ex post, yield opportunities next period are lowe tha A brief description of the supply side of the asset market will be helpful in understanding the relationship between current returns on assets and changes in the investment opportunity set. An asset is defined as a production technology which is a probability dis- tribution for cash flow (valued in consumption units )and physical depreciation, as a function of the amount of capital, k(t)(measured in physical units, e.g., number of machines), employed at time t. The price per unit capital in terms of the consumption good is P(), and the value of an asset at time t, v(o), equals Pk(o)K(o) The return on the asset over a period of length h will be the cash flow, x, plus the value of undepreciated capital, (1-a)p(t + h)K(t)(where a is the rate of physical depreciation of capital), minus the initial value of the asset, v(t). The total change in the value of the asset outstanding, v(t +h)-v(t), is equal to the sum of the return on the asset plus the value of gross new investment in excess of cash flow, R(+h)K(t+h)-(1-A)K()]-X. c Each firm in the model is assumed to invest in a single asset and to issue.one class of securities, called equity. o Hence, the terms"firm"and"asset"can be used interchangeably. Let N(t) be the number of shares of the firm outstanding and let P(t)be the price per share, where N(t)and P(t)are defined by the difference quations, (1)P(t+h)≡[X+(1-APk(t+h)K()]/N(t) (2)N(t+h)≡N(t)+[P(t+h)[K(t+h)-(1-A)k(t)-Ⅺ]/P(t+h), subject to the initial conditions P(O)=P, N(O)= N, and v(O)= N(O)P(O). If we ssume that all dividend payments to shareholders are accomplished by share io It is assumed that there are no economies or diseconomies to the"packaging of assets (ie ng mor held a portfolio of thefirms"in the text. Similarly, it is assumed that all financial leveraging and other capital structure differences are carried out by investors(possibly through financial in has content downl ued stube to sT oR ems ae ondtp23013024426AM

870 ROBERT C. MERTON making that the consumer-investor know at each point in time: (i) the transition probabilities for returns on each asset over the next trading interval (the investment opportunity set); and (ii) the transition probabilities for returns on assets in future periods (i.e., knowledge of the stochastic processes of the changes in the investment opportunity set). Unlike a single-period maximizer who, by definition, does not consider events beyond the present period, the intertemporal maximizer in selecting his portfolio takes into account the relationship between current period returns and returns that will be available in the future. For example, suppose that the current return on a particular asset is negatively correlated with changes in yields ("capitalization" rates). Then, by holding this asset, the investor expects a higher return on the asset if, ex post, yield opportunities next period are lower than were expected. A brief description of the supply side of the asset market will be helpful in understanding the relationship between current returns on assets and changes in the investment opportunity set. An asset is defined as a production technology which is a probability distribution for cash flow (valued in consumption units) and physical depreciation, as a function of the amount of capital, K(t) (measured in physical units, e.g., number of machines), employed at time t. The price per unit capital in terms of the consumption good is Pk(t), and the value of an asset at time t, V(t), equals Pk(t)K(t). The return on the asset over a period of length h will be the cash flow, X, plus the value of undepreciated capital, (1 - t)Pk(t + h)K(t) (where A is the rate of physical depreciation of capital), minus the initial value of the asset, V(t). The total change in the value of the asset outstanding, V(t + h) - V(t), is equal to the sum of the return on the asset plus the value of gross new investment in excess of cash flow, Pk(t + h)[K(t + h) - (1 - A)K(t)] - X. Each firm in the model is assumed to invest in a single asset and to issue.one class of securities, called equity.'0 Hence, the terms "firm" and "asset" can be used interchangeably. Let N(t) be the number of shares of the firm outstanding and let P(t) be the price per share, where N(t) and P(t) are defined by the difference equations, (1) P(t + h) [X + (1 - %)Pk(t +h)K(t)]/N(t) and (2) N(t + h) N(t) + [Pk(t + h)[K(t + h) - (1 - A)K(t)] - X]/P(t + h), subject to the initial conditions P(O) = P, N(O) = N, and V(O) = N(O)P(O). If we assume that all dividend payments to shareholders are accomplished by share 10 It is assumed that there are no economies or diseconomies to the "packaging" of assets (i.e., no "synergism"). Hence, any "real" firm holding more than one type of asset will be priced as if it held a portfolio of the "firms" in the text. Similarly, it is assumed that all financial leveraging and other capital structure differences are carried out by investors (possibly through financial intermediaries). 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 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

874 ROBERT C. MERTON Under the assumptions of continuous trading and the continuous structure of the stochastic processes, it has been shown that the instantaneous, first two moments of the distributions are sufficient statistics. 9 Further, by the existence and boundedness of a and o, p equal to zero is a natural absorbing barrier ensuring limited liability of all assets For the rest of the paper, it is assumed that there are n distinct20 risky tantaneously risk-less"asset. Instantaneously risk -les that, at each instant of time, each investor knows with certainty that he can earn rate of return r(t)over the next instant by holding the asset (i., 0n+1=0 and n+1=r(o). However, the future values of r(t)are not known with certainty (i.e bn+1*0 in (7). We interpret this asset as the exchange asset and r(t)as the instantaneous private sector borrowing(and lending) rate. Alternatively, the asset could represent (very) short government bonds 4. PREFERENCE STRUCTURE AND BUDGET EQUATION DYNAMICS We assume that there are K consumer-investors with preference structures as described in [25]: namely, the kth consumer acts so as to max EUc(s,s」ds+Bwk(T where"Eo " is the conditional expectation operator, conditional on the current value of his wealth, W(0)=w are the state variables of the investment oppor tunity set, and T is the distribution for his age of death(which is assumed to be independent of investment outcomes). His instantaneous consumption flow at age t is ck(t). I Uk is a strictly concave von Neumann-Morgenstern utility function for consumption and B is a strictly concave"beq or utility-of-terminal wealth function Dropping the superscripts(except where required for clarity), we can write the accumulation equation for the kth investor as dW=∑ w, w dP /P+(y-c)dr, where W,= N P/W is the fraction of his wealth invested in the ith asset, N, is the number of shares of the ith asset he owns, and y is his wa me. Substituting 9 Since these are sufficient statistics, if there are n I assets and n is finite, then our assumption of a finite vector for X is satisfied Distinct"means that none of the assets'returns can be written as an (instantaneous) linear ombination of the other assets returns Hence, the instantaneous var Because the paper is primarily interested in finding equilibrium conditions for the model assumes a single consumption good. The model could be generalized by c a vector ucing as state variables the relative prices, while the unfavorable shifts in relative consumption goods prices (i. e, in the consumption opportunity sei2 See Merton[25] for a derivation of (9). has content downl ued stube to sT oR ems ae ondtp23013024426AM

874 ROBERT C. MERTON Under the assumptions of continuous trading and the continuous Markov structure of the stochastic processes, it has been shown that the instantaneous, first two moments of the distributions are sufficient statistics.'9 Further, by the existence and boundedness of a and a, P equal to zero is a natural absorbing barrier ensuring limited liability of all assets. For the rest of the paper, it is assumed that there are n distinct20 risky assets and one "instantaneously risk-less" asset. "Instantaneously risk-less" means that, at each instant of time, each investor knows with certainty that he can earn rate of return r(t) over the next instant by holding the asset (i.e., an,+1 = 0 and ?Cn + =_r(t)). However, the future values of r(t) are not known with certainty (i.e., bn+ #= 0 in (7)). We interpret this asset as the exchange asset and r(t) as the instantaneous private sector borrowing (and lending) rate. Alternatively, the asset could represent (very) short government bonds. 4. PREFERENCE STRUCTURE AND BUDGET EQUATION DYNAMICS We assume that there are K consumer-investors with preference structures as described in [25]: namely, the kth consumer acts so as to (8) max Eo J' u[ck(s), s] ds + Bk[Wk(Tk), Tki], where "Eo" is the conditional expectation operator, conditional on the current value of his wealth, Wk(0) = Wk are the state variables of the investment opportunity set, and Tk is the distribution for his age of death (which is assumed to be independent of investment outcomes). His instantaneous consumption flow at age t is ck(t).21 Uk is a strictly concave von Neumann-Morgenstern utility function for consumption and Bk is a strictly concave "bequest" or utility-of-terminal wealth function. Dropping the superscripts (except where required for clarity), we can write the accumulation equation for the kth investor aS22 n+ 1 (9) dW= wi wWdP/Pi + (y-c)dt, where wi NiPi/W is the fraction of his wealth invested in the ith asset, Ni is the number of shares of the ith asset he owns, and y is his wage income. Substituting 19 Since these are sufficient statistics, if there are n + 1 assets and n is finite, then our assumption of a finite vector for X is satisfied. 20 "Distinct" means that none of the assets' returns can be written as an (instantaneous) linear combination of the other assets' returns. Hence, the instantaneous variance-covariance matrix of returns, Q = [ij], is non-singular. 21 Because the paper is primarily interested in finding equilibrium conditions for the asset markets, the model assumes a single consumption good. The model could be generalized by making c" a vector and introducing as state variables the relative prices. While the analysis would be similar to the onegood case, there would be systematic effects on the portfolio demands reflecting hedging behavior against unfavorable shifts in relative consumption goods prices (i.e., in the consumption opportunity set). 22 See Merton [25] for a derivation of (9). 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 MODEL 875 for dPJlPi from (6), we can re-write (9) as n _ n (10) dW= Zwi(i - r) + r Wdt + Z wiWaidzi + (y - c)dt, where his choice for wl, w2,... , w, is unconstrained because w1 + can always be chosen to satisfy the budget constraint . = 1. From the budget constraint, W = 1 +'NiPi, and the accumulation equation (9), we have that n+1 (11) (Y -C c )dt = dNi(Pi + dPi), i.e., the net value of new shares purchased must equal the value of savings from wage income. 5. THE EQUATIONS OF OPTIMALITY: THE DEMAND FUNCTIONS FOR ASSETS For computational simplicity, we will assume that investors derive all their income from capital gains sources (i.e., y = 0), 23 and for notational simplicity, we introduce the state-variable vector, X, whose m elements, xi, denote the current levels of P, a, and a. The dynamics for X are written as the vector Ito process, (12) dX = F(X)dt + G(X)dQ, where F is the vector [ft , f2,... fn], G is a diagonal matrix with diagonal elements [g1, g2..., gm], dQ is the vector Wiener process [dql, dq2,... , dqm], ij is the instantaneous correlation coefficient between dqi and dzj, and vij is the instantaneous correlation coefficient between dqi and dqj. I have shown elsewhere24 that the necessary optimality conditions for an investor who acts according to (8) in choosing his consumption-investment program are that, at each point in time, (13) 0 = max [U(c,t) + J, + Jw[( wi(oci - r) + r) W - c m n n + Jifi + -WEE WiWjO(ijW2 1 1 1 m n 'm m + 1 Eiwwjwgiujqij + 2 E E 1 2 Jijgigjvijj, 1 1 23 The analysis would be the same with wage income, provided that investors can issue shares against future income, since we can always redefine wealth as including capitalized future wage income. However, since institutionally this cannot be done, the introduction of wage income will cause systematic effects on the portfolio and consumption decisions. 24 See Merton [23 and 25]. J(W, t, X) _ max E,{f[ U(c, s) ds + B[W(T), T]} and is called the "derived" utility of wealth function. Substituting from (14) and (15) to eliminate wi and c in (13) makes (13) a partial differential equation for J, subject to the boundary condition J(W, T, X) = B(W, T). Having solved for J, we then substitute for J and its derivatives in (14) and (15) to find the optimal rules (wi, c). 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

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