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《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

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