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《投资学原理 Investments》课程教学资源（阅读资料）equilibrium in a capital asset market

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Econometrica, vol. 34, No. 4(October, 1966) EQUILIBRIUM IN A CAPITAL ASSET MARKETI BY JAN MOSSIN 2 paper investigates the properties of a market for risky assets on the basis of a model of general equilibrium of exchange, where individual investors seek to maximize preference functions over expected yield and variance of yield on their port folios. A theory of market risk premiums is outlined and it is shown that general equilibrium implies theexistence of a so-called"market line, "relating per dollarexpected yield and standard deviation of yield. The concept of price of risk is discussed in terms of the slope of this line IN RECENT YEARS several studies have been made of the problem of selecting optimal (6, 81, and others). In these models the investor is assumed to possess a preference ordering over all possible portfolios and to maximize the value of this preference ordering subject to a budget restraint taking the prices and probability distributions of yield for the various available assets as given data From the point of view of positive economics, such decision rules can, of course be postulated as implicitly describing the individual's demand schedules for the different assets at varying prices. It would then be a natural next step to enquire into the characteristics of the whole market for such assets when the individual demands are interacting to determine the prices and the allocation of the existing supply of assets among individuals These problems have been discussed, among others, by Allais [1], Arrow [2] Borch [3], Sharpe [7], and also to some extent by brownlee and Scott [5] Allais model represents in certain respects a generalization relative to the model to be discussed here. In particular, Allais does not assume general risk aversion This generalization requires, on the other hand certain other assumptions that we shall not need in order to lead to definite results Arrow's brief but important paper is also on a very general and even abstract level. He uses a much more general preference structure than we do here and also allows differences in individual perceptions of probability distributions. He then proves that under certain assumptions there exists a competitive equilibrium which is also Pareto optimal Borch has investigated the problem with special reference to a reinsurance 1 Revised manuscript received December, 1965. 2 The author is indebted to Karl borch, Jacques Dreze and Sten Thore for their valuable comments and suggestions has content downl ued stube to sT oR ems aecondtp23013020-0 AM

Econometrica, Vol. 34, No. 4 (October, 1966) EQUILIBRIUM IN A CAPITAL ASSET MARKET' BY JAN MOSSIN2 This paper investigates the properties of a market for risky assets on the basis of a simple model of general equilibrium of exchange, where individual investors seek to maximize preference functions over expected yield and variance of yield on their portfolios. A theory of market risk premiums is outlined, and it is shown that general equilibrium implies the existence of a so-called "market line," relating per dollar expected yield and standard deviation of yield. The concept of price of risk is discussed in terms of the slope of this line. 1. INTRODUCTION IN RECENT YEARS several studies have been made of the problem of selecting optimal portfolios of risky assets ([6, 8], and others). In these models the investor is assumed to possess a preference ordering over all possible portfolios and to maximize the value of this preference ordering subject to a budget restraint, taking the prices and probability distributions of yield for the various available assets as given data. From the point of view of positive economics, such decision rules can, of course, be postulated as implicitly describing the individual's demand schedules for the different assets at varying prices. It would then be a natural next step to enquire into the characteristics of the whole market for such assets when the individual demands are interacting to determine the prices and the allocation of the existing supply of assets among individuals. These problems have been discussed, among others, by Allais [1], Arrow [2], Borch [3], Sharpe [7], and also to some extent by Brownlee and Scott [5]. Allais' model represents in certain respects a generalization relative to the model to be discussed here. In particular, Allais does not assume general risk aversion. This generalization requires, on the other hand, certain other assumptions that we shall not need in order to lead to definite results. Arrow's brief but important paper is also on a very general and even abstract level. He uses a much more general preference structure than we do here and also allows differences in individual perceptions of probability distributions. He then proves that under certain assumptions there exists a competitive equilibrium which is also Pareto optimal. Borch has investigated the problem with special reference to a reinsurance 1 Revised manuscript received December, 1965. 2 The author is indebted to Karl Borch, Jacques Dreze, and Sten Thore for their valuable comments and suggestions. 768 This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:20:50 AM All use subject to JSTOR Terms and Conditions

CAPITAL ASSET MARKET market. He suggests, however, that his analysis can be reversed and extended to a more general market for risky assets. The present paper may be seen as an attempt in that direction. The general approach is different in important respects, however, particularly as concerns the price concept used. Borch's price implies in our terms that the price of a security should depend only on the stochastic nature of the yield, not on the number of securities outstanding. This may be accounted for by he particular characteristics of a reinsurance market, where such a price concept eems more reasonable than is the case for a security market. A rational person ill not buy securities on their own merits without considering alternative invest- ments. The failure of Borchs model to possess a Pareto optimal solution appears to be due to this price concept Generality has its virtues, but it also means that there will be many questions to which definite answers cannot be given. To obtain definite answers, we must be willing to impose certain restrictive assumptions. This is precisely what our paper ttempts to do, and it is believed that this makes it possible to come a long way towards providing a theory of the market risk premium and filling the gap between demand functions and equilibrium properties Brownlee and Scott specify equilibrium conditions for a security market very similar to those given here, but are otherwise concerned with entirely different problems. The paper by Sharpe gives a verbal-diagrammatical discussion of the determination of asset prices in quasi-dynamic terms. His general description of the character of the market is similar to the one presented here, however, and h main conclusions are certainly consistent with ours. But his lack of precision in the specification of equilibrium conditions leaves parts of his arguments somewhat indefinite. The present paper may be seen as an attempt to clarify and make precise ome o 2. THE EQUILIBRIUM MODEL Our general approach is one of determining conditions for equilibrium ofexchange of the assets. Each individual brings to the market his present holdings of the various assets, and an exchange takes place. We want to know what the prices must be in order to satisfy demand schedules and also fulfill the condition that pply and demand be equal for all assets. To answer this question we must first derive relations describing individual demand. Second, we must incorporate these relations in a system describe I equilibrium. Finally, we want to discuss properties of this equilibrium We shall assume that there is a large number m of individuals labeled i, (i=I 2, .., m). Let us consider the behavior of one individual. He has to select a portfolio of assets, and there are n different assets to choose from, labeled j, (j=1, 2,. n The yield on any asset is assumed to be a random variable whose distribution is known to the individual. moreover, all individuals are assumed to have identic has content downl ued stube to sT oR ems aecondtp23013020-0 AM

CAPITAL ASSET MARKET 769 market. He suggests, however, that his analysis can be reversed and extended to a more general market for risky assets. The present paper may be seen as an attempt in that direction. The general approach is different in important respects, however, particularly as concerns the price concept used. Borch's price implies in our terms that the price of a security should depend only on the stochastic nature of the yield, not on the number of securities outstanding. This may be accounted for by the particular characteristics of a reinsurance market, where such a price concept seems more reasonable than is the case for a security market. A rational person will not buy securities on their own nmerits without considering alternative investments. The failure of Borch's model to possess a Pareto optimal solution appears to be due to this price concept. Generality has its virtues, but it also means that there will be many questions to which definite answers cannot be given. To obtain definite answers, we must be willing to impose certain restrictive assumptions. This is precisely what our paper attempts to do, and it is believed that this makes it possible to come a long way towards providing a theory of the market risk premium and filling the gap between demand functions and equilibrium properties. Brownlee and Scott specify equilibrium conditions for a security market very simnilar to those given here, but are otherwise concerned with entirely different problems. The paper by Sharpe gives a verbal-diagrammatical discussion of the determination of asset prices in quasi-dynamic terms. His general description of the character of the market is similar to the one presented here, however, and his main conclusions are certainly consistent with ours. But his lack of precision in the specification of equilibrium conditions leaves parts of his argulments somewhat indefinite. The present paper may be seen as an attempt to clarify and make precise some of these points. 2. THE EQUILIBRIUM MODEL Our general approach is one of determining conditions for equilibrium of exchange of the assets. Each individual brings to the market his present holdings of the various assets, and an exchange takes place. We want to know wllat the prices must be in order to satisfy demand schedules and also fulfill the condition that supply and demand be equal for all assets. To answer this question we must first derive relations describing individual demand. Second, we must incorporate these relations in a system describing general equilibrium. Finally, we want to discuss properties of this equilibrium. We shall assume that there is a large number m of individuals labeled i, (i= 1, 2, ..., im). Let us consider the behavior of one individual. He has to select a portfolio of assets, and there are n different assets to choose from, labeled j, (j= 1, 2, ..., n). Tne yield on any asset is assumed to be a random variable whose distribution is known to the individual. Moreover, all individuals are assumed to have identical This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:20:50 AM All use subject to JSTOR Terms and Conditions

770 perceptions of these probability distributions. The yield on a whole portfolio is, of course,also a random variable. The portfolio analyses mentioned earlier assume that, in his choice among all the possible portfolios, the individual is satisfied to be guided by its expected yield and its variance only. This assumption will also be made in the present paper. It is important to make precise the description of a portfolio in these terms. It is obvious(although the point is rarely made explicit)that the holdings of the various assets must be measured in some kind of units. The Markowitz analysis, for exam ple, starts by picturing the investment alternatives open to the individual as a point set in a mean-variance plane, each point representing a specific investment opportunity. The question is: to what do this expected yield and variance of yield refer? For such a diagram to make sense, they must necessarily refer to some unit common to all assets. An example of such a unit would be one dollar's worth of investment in each asset. Such a choice of units would evidently be of little use for our purposes, since we shall consider the prices of assets as variables to be deter- mined in the market. Consequently, we must select some arbitrary"physical"unit of measurement and define expected yield and variance of yield relative to this unit.If, for example, we select one share as our unit for measuring holdings of Standard Oil stock and say that the expected yield is u and the variance o2,this means expected yield and variance of yield per share; if instead we had chosen a hundred shares as our unit, the relevant expected yield and variance of yield would have been 100 4, and 10,000 a, respectively. We shall find it convenient to give an interpretation of the concept of"yield by assuming discrete market dates with intervals of one time unit. The yield to be considered on any asset on a given market date may then be thought of as the value per unit that the asset will have at the next market date(including possible accrued dividends, interest, or other emoluments). The terms "yield""and"future value"may then be used more or less interchangeably ye shall, in general, admit stochas assets. But the specification of the stochastic properties poses the problem of identification of"different"assets. It will be necessary to make the convention that two units of assets are of the same kind only if their yields will be identical. 3 This assumption is not crucial for the analysis, but simplifies it a good deal. It also seems doubtful whether the introduction of subjective probabilities would really be useful for deriving propositions about market behavior. In any case, it may be argued as borch [3, p. 439] does Whether two rational persons on the basis of the same information can arrive at different evalua- tions of the probability of a specific event, is a question of semantics. That they may act differently on the same information is well known, but this can usually be explained assuming that the two 4 Acceptance of the von Neumann-Morgenstern axioms leading to their theorem on measur- able utility), together with this assumption, implies a quadratic utility function for yield(see (4D) But such a specification is not strictly necessary for the analysis to follow, and so by the principle of Occams razor has not been introduced has content downl ued stube to sT oR ems aecondtp23013020-0 AM

770 JAN MOSSIN perceptions of these probability distributions.3 The yield on a whole portfolio is, of course, also a random variable. The portfolio analyses mentioned earlier assume that, in his choice among all the possible portfolios, the individual is satisfied to be guided by its expected yield and its variance only. This assumption will also be made in the present paper.4 It is important to make precise the description of a portfolio in these terms. It is obvious (although the point is rarely made explicit) that the holdings of the various assets must be measured in some kind of units. The Markowitz analysis, for example, starts by picturing the investment alternatives open to the individual as a point set in a mean-variance plane, each point representing a specific investment opportunity. The question is: to what do this expected yield and variance of yield refer? For such a diagram to make sense, they must necessarily refer to some unit common to all assets. An example of such a unit would be one dollar's worth of investment in each asset. Such a choice of units would evidently be of little use for our purposes, since we shall consider the prices of assets as variables to be determined in the market. Consequently, we must select some arbitrary "physical" unit of measurement and define expected yield and variance of yield relative to this unit. If, for example, we select one share as our unit for measuring holdings of Standard Oil stock and say that the expected yield is ,u and the variance a2, this means expected yield and variance of yield per share; if instead we had chosen a hundred shares as our unit, the relevant expected yield and variance of yield would have been 100 4e, and 10,000 a2. respectively. We shall find it convenient to give an interpretation of the concept of "yield" by assuming discrete market dates with intervals of one time unit. The yield to be considered on any asset on a given market date may then be thought of as the value per unit that the asset will have at the next market date (including possible accrued dividends, interest, or other emoluments). The terms "yield" and "future value" may then be used more or less interchangeably. We shall, in general, admit stochastic dependence among yields of different assets. But the specification of the stochastic properties poses the problem of identification of "different" assets. It will be necessary to make the convention that two units of assets are of the same kind only if their yields will be identical. 3 This assumption is not crucial for the analysis, but simplifies it a good deal. It also seems doubtful whether the introduction of subjective probabilities would really be useful for deriving propositions about market behavior. In any case, it may be argued, as Borch [3, p. 439] does: "Whether two rational persons on the basis of the same information can arrive at different evaluations of the probability of a specific event, is a question of semantics. That they may act differently on the same information is well known, but this can usually be explained assuming that the two persons attach different utilities to the event." 4 Acceptance of the von Neumann-Morgenstern axioms (leading to their theorem on measurable utility), together with this assumption, implies a quadratic utility function for yield (see [4]). But such a specification is not strictly necessary for the analysis to follow, and so, by the principle of Occam's razor, has not been introduced. This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:20:50 AM All use subject to JSTOR Terms and Conditions

CAPITAL ASSET MARKET The reason for this convention can be clarified by an example. In many lotteries (in particular national lotteries), several tickets wear the same number. When a number is drawn, all tickets with that number receive identical prizes. Suppose all tickets have mean u and variance o of prizes. Then the expected yield on two tickets is clearly 2u, regardless of their numbers. But while the variance on two tickets is 2o when they have different numbers, it is 4o when they have identical numbers. If such lottery tickets are part of the available assets, we must therefore identify as many"different""assets as there are different numbers (regardless of the fact that they have identical means and variances). For ordinary assets such as corporate stock, it is of course known that although the yield is random it will be same on all units of each stock We shall denote the expected yield per unit of asset j by u; and the covariance between unit yield of assets j and k by ak. We shall also need the rather trivial gula ption that the covariance matrix for the yield of the risky assets is nonsin- An individual's portfolio can now be described as an n-dimensional vector with elements equal to his holdings of each of the n assets. We shall use x to denote individual i's holdings of assets j (after the exchange), and so his portfolio may be written(x1,x2,…,x) One of the purposes of the analysis is to compare the relations between the prices and yields of different assets. To facilitate such comparisons, it will prove useful to have a riskless asset as a yardstick. We shall take the riskless asset to be the nth. That it is riskless of course means that onk=0 for all k. But it may also be suggestive to identify this asset with money, and with this in mind we shall write specifically un=l, i.e., a dollar will(with certainty) be worth a dollar a year from now We denote the price per unit of asset by pi. Now, general equilibrium conditions are capable of determining relative prices only: we can arbitrarily fix one of the prices and express all others in terms of it. We may therefore proceed by fixing the price of the nth asset as g, 1. e,Pn=g. This means that we select the nth asset as numeraire. We shall return to the implications of this seemingly innocent con vention below ons and conventions, the expected yield on individual uitm and the variance (2)y2=∑∑oxx As mentioned earlier, we postulate for each individual a preference ordering has content downl ued stube to sT oR ems aecondtp23013020-0 AM

CAPITAL ASSET MARKET 771 The reason for this convention can be clarified by an example. In many lotteries (in particular national lotteries), several tickets wear the same number. When a number is drawn, all tickets with that number receive identical prizes. Suppose all tickets have mean M and variance a2 of prizes. Then the expected yield on two tickets is clearly 2ji, regardless of their numbers. But while the variance on two tickets is 2a2 when they have different numbers, it is 4a2 when they have identical numbers. If such lottery tickets are part of the available assets, we must therefore identify as many "different" assets as there are different numbers (regardless of the fact that they have identical means and variances). For ordinary assets such as corporate stock, it is of course known that although the yield is random it will be the same on all units of each stock. We shall denote the expected yield per unit of assetj by jt3 and the covariance between unit yield of assets j and k by ai k- We shall also need the rather trivial assumption that the covariance matrix for the yield of the risky assets is nonsingular. An individual's portfolio can now be described as an n-dimensional vector with elements equal to his holdings of each of the n assets. We shall use xJ to denote individual i's holdings of assets j (after the exchange), and so his portfolio may be written (xl, xi, ..., xi). One of the purposes of the analysis is to compare the relations between the prices and yields of different assets. To facilitate such comparisons, it will prove useful to have a riskless asset as a yardstick. We shall take the riskless asset to be the nth. That it is riskless of course means that ank = 0 for all k. But it may also be suggestive to identify this asset with money, and with this in mind we shall write specifically Pun=1, i.e., a dollar will (with certainty) be worth a dollar a year from now. We denote the price per unit of assetj byp,. Now, general equilibrium conditions are capable of determining relative prices only: we can arbitrarily fix one of the prices and express all others in terms of it. We may therefore proceed by fixing the price of the nth asset as q, i.e., P n = q. This means that we select the nth asset as numeraire. We shall return to the implications of this seemingly innocent convention below. With the above assumptions and conventions, the expected yield on individual i's portfolio can be written: n-I (1) Y1 L tjxi+Xn j=i and the variance: n-I n-i (2) Y2 =i x jaX Xaa j=1 a=1 As mentioned earlier, we postulate for each individual a preference ordering This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:20:50 AM All use subject to JSTOR Terms and Conditions

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