ECONOMETRICA IOURV4I DF TIr TCNOMTERIr 54CItTY Equilibrium in a Capital Asset Market Author(s): Jan Mossin Source: Econometrica, Vol 34, No. 4(Oct, 1966), pp. 768-783 Published by: The Econometric Society StableUrl:http://www.jstor.org/stable/1910098 Accessed:11/09/201302:20 Your use of the JSTOR archive indicates your acceptance of the Terms Conditions of Use, available at JStOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support(@jstor. org Rl The Econometric Socie i collaborating with JSTOR to digitize, preserve and extend acess to Ecomometrica 的d http://www.jstororg This content downloaded from 202. 115.118.13 on Wed, I I Sep 2013 02: 20: 50 AM All use subject to STOR Terms and Conditions
Equilibrium in a Capital Asset Market Author(s): Jan Mossin Source: Econometrica, Vol. 34, No. 4 (Oct., 1966), pp. 768-783 Published by: The Econometric Society Stable URL: http://www.jstor.org/stable/1910098 . Accessed: 11/09/2013 02:20 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. . The Econometric Society is collaborating with JSTOR to digitize, preserve and extend access to Econometrica. http://www.jstor.org 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
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
772 JAN MOSSIN (utility function) of the form: over all possible portfolios, 1. e,, we postulate that an individual will behave as if he were attempting to maximize U. with respect to the form of Ui, we shall assume that it is concave, with the first derivative positive and the second negative. This latter assumption of general risk aversion seems to be generally accepted in the literature on portfolio selection. The investor is constrained, however, to the points that satisfy his budget equation ∑p/(x-动)+q(x-x)=0, old "portfolio should equal total outlays on the"new"portfolio Formally, then, we postulate that each individual i behaves as if attempting to maximize(3), subject to(4),(1), and (2). Forming the Lagrangean v=f(y,y)+0∑1(x-动)+8(x we can then write the first-order conditions for the maxima for all i as ax=+2f1x+时p=0 〔=1,…,n-1) f+0 n-1 ∑p(对-对)+(x一x where fi and fi denote partial derivatives with respect to yi and y2, respectively Eliminating 8 this can be written as 两-p/ 1), 点2 In(5), the-filIfi is the marginal rate of substitution dy2/dyi between the variance has content downl ued stube to sT oR ems aecondtp23013020-0 AM
772 JAN MOSSIN (utility function) of the form: (3) U =PA(Y, YD) over all possible portfolios, i.e., we postulate that an individual will behave as if he were attempting to maximize Ui. With respect to the form of Ui, we shall assume that it is concave, with the first derivative positive and the second negative. This latter assumption of general risk aversion seems to be generally accepted in the literature on portfolio selection. The investor is constrained, however, to the points that satisfy his budget equation: n-i (4) EPij(XJ-XJ) +q(xn-5n) = ? s where XJ are the quantities of asset j that he brings to the market; these are given data. The budget equation simply states that his total receipts from the sale of the "old" portfolio should equal total outlays on the "new" portfolio. Formally, then, we postulate that each individual i behaves as if attempting to maximize (3), subject to (4), (1), and (2). Forming the Lagrangean: VZ=fJ(4Y, y2)+0 ' Pi i)+g(Xn- K) we can then write the first-order conditions for the maxima for all i as: avi n-1 av f4 +2fP Eoje4+O pj=o (j=1, ..., n-1), = f'+Olq=O, avi n-1 -=E pj (Xj-Xj)~ + q(x - 5in) = O wherefl' and f2i denote partial derivatives with respect to y' and y', respectively. Eliminating 0', this can be written as: i2Eafja Xx (5) _=a, n-1), f2 1ju-pj/q n-i (6) E Pj (XJ-Xj) + q(xn-Xn')= j=t In (5), the -f;i/f2' is the marginal rate of substitution dyi ldy' between the variance 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 773 and mean of yield. Equations(5)and(6)constitute, for each individual, n equations describing his demand for the n assets. To determine general equilibrium, we must also specify equality between demand and supply for each asset. These market clearing conditions can be writter (7) As we would suspect, one of these conditions is superfluous. This can be seen by first summing the budget equations over all individuals ∑P(x一x)+q∑(x-x=0 ∑p∑(x-x+q∑(xn-x)=0 Suppose that (7) were satisfied for all j except n. This would mean that the first term on the left of (8) vanishes, so that Hence also the nth equation of (7 )must hold. We may therefore instead write ∑x=习 (j=1,,n-1) where x denotes the given total supply of asset j: x =2i=1X This essentially completes the equations describing general equilibrium. The system consists of the m equations(4), the m(n-1)equations(5)and(6), and the (n-1)equations(7); altogether (mn+n-1)equations. The unknowns are the mn x and the(n-1)prices pi We have counted our equations and our unknowns and found them to be equal in number. But we cannot rest with this; our main task has hardly begun. We shall bypass such problems as the existence and uniqueness of a solution to the system and rather concentrate on investigating properties of the equilibrium values of the variables, assuming that they exist. We may observe, first of all, that the equilibrium allocation of assets represents allocation to increas individuals utility without at the same time reducing the utility of one or more other individuals. This should not need any explicit proof, since it is a well known general property of a competitive equilibrium where preferences are concave, We should also mention the problem of nonnegativity of the solution to which we shall return at a later stage has content downl ued stube to sT oR ems aecondtp23013020-0 AM
CAPITAL ASSET MARKET 773 and mean of yield. Equations (5) and (6) constitute, for each individual, n equations describing his demand for the n assets. To determine general equilibrium, we must also specify equality between demand and supply for each asset. These market clearing conditions can be written: m (7') (x (XJ-Cj) = O Oj= 1,.I. n). As we would suspect, one of these conditions is superfluous. This can be seen by first summing the budget equations over all individuals: m n-1 m E pj (x.- j+ q E (x'- 5n)=O 0 1j= or n-1 m m (8) Epj E (x' J + q E (x' 0n j=i i=i i=i Suppose that (7') were satisfied for all j except n. This would mean that the first term on the left of (8) vanishes, so that m X i= 1 Hence also the nth equation of (7') must hold. We may therefore instead write: m (7) Z xj =xj (j=1,..., n-1), i=1 where Xj denotes the given total supply of asset j: &j = mim= lj. This essentially completes the equations describing general equilibrium. The system consists of the m equations (4), the ni (n - 1) equations (5) and (6), and the (n -1) equations (7); altogether (mn + n -1) equations. The unknowns are the mn quantities xJ and the (n-1) prices pj. We have counted our equations and our unknowns and found them to be equal in number. But we cannot rest with this; our main task has hardly begun. We shall bypass such problems as the existence and uniqueness of a solution to the system and rather concentrate on investigating properties of the equilibrium values of the variables, assuming that they exist. We may observe, first of all, that the equilibrium allocation of assets represents a Pareto optimum, i.e., it will be impossible by some reallocation to increase one individual's utility without at the same time reducing the utility of one or more other individuals. This should not need any explicit proof, since it is a well known general property of a competitive equilibrium where preferences are concave. We should also mention the problem of nonnegativity of the solution to which we shall return at a later stage. 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
JAN MOSSIN 3. RISK MARGINS The expected rate of return r, on a unit of a risky asset can be def u/(1+ri)=Pi, i.e., r;=(/pi)-1,(=l,,n-1). Similarly, the rate of return of a unit of the riskless asset rn is defined by 1/ (1+rm=g, i.e, rn=1 g-1. with our earlier interpretation of the riskless asset in mind, rn may be regarded as the pure rate of interest. The natural definition of the pure rate of interest is the rate of return on a riskless asset. In general, we may think of the rate of return of any asset as separated into two parts: the pure rate of interest representing the "price for waiting, and a remainder, a risk margin, representing of risk”Whe yield of the riskless asset at I and decided to fix its current price at g, we thereby implicitly fixed the pure rate of interest. And to say that the market determines only lative asset prices is seen to be equivalent to saying that the pure rate of interest is not determined in the market for risky assets. Alternatively, we may say that the asset market determines only the risk margins. The risk margin on asset j, mi, is defined by =-1n=-D/q To compare the risk margins of two assets j and k, we write m=此一PqD nk We now make use of the equilibrium conditions. From (5)we have ∑nx21∑xx2 Summing over i and using(7), we then get 4j-Pjiq Fk-pk/q These equations define relationships between the prices of the risky assets in terms of given parameters only. We can then write Pkk k∑如又Px Now, ijEajaia is the variance of yield on the total outstanding stock of asset j <-, is similarly the total value, at market prices, of all of asset j. Let us denote n.se magnitudes by V, and R;, respectively. In equilibrium, therefore, the risk reins satisfy has content downl ued stube to sT oR ems aecondtp23013020-0 AM
774 JAN MOSSIN 3. RISK MARGINS The expected rate of return rj on a unit of a risky asset can be defined by ,uj/(l +rj)=pj, i.e., rj=( pjlpj)- 1, (j= l, ..., n- 1). Similarly, the rate of return of a unit of the riskless asset rn is defined by 1/(1 +rn)=q, i.e., rn= l/q- 1. With our earlier interpretation of the riskless asset in mind, rn may be regarded as the pure rate of interest. The natural definition of the pure rate of interest is the rate of return on a riskless asset. In general, we may think of the rate of return of any asset as separated into two parts: the pure rate of interest representing the "price for waiting," and a remainder, a risk margin, representing the "price of risk." When we set the future yield of the riskless asset at 1 and decided to fix its current price at q, we thereby implicitly fixed the pure rate of interest. And to say that the market determines only relative asset prices is seen to be equivalent to saying that the pure rate of interest is not determined in the market for risky assets. Alternatively, we may say that the asset market determines only the risk margins. The risk margin on asset], in, is defined by mj = rj- r =t Hi pjlq Pj To compare the risk margins of two assets j and k, we write: mj - __ _ lj- __ pjlq _P Pk Ink Pk Pklq P j We now make use of the equilibrium conditions. From (5) we have: >3 Ef YCX~ aa Zik,X E Uko, Xa (9) q (j,k=1,... ,n-1). pj - pjlq 1-k Pklq Summing over i and using (7), we then get: A, ufjxa Xa ,U ka XaT (10) a a yUj p;lq Ilk Pklq These equations define relationships between the prices of the risky assets in terms of given parameters only. We can then write: m i _ _ _ _ _ _ _ mj Xj ,> . fja 5Xa __ _ _ x cx Pk Xk4 Mk Xk54 Eka Xa Pj Xj Now, x; jc is the variance of yield on the total outstanding stock of asset j; pjxj is slmilarly the total value, at market prices, of all of asset j. Let us denote these magnitudes by Vj and Rj, respectively. In equilibrium, therefore, the risk margins satisfy: 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 (11) i R mx Rk (,k=1, 1), i.e., the risk margins are such that the ratio between the total risk compensation paid for an asset and the variance of the total stock of the asset is the same for all assets 4. COMPOSITION OF EQUILIBRIUM PORTFOLIOS We can now derive an important property of an individuals equilibrium port When(10)is substituted back in(9), the result is x2∑k2x (12)∑0y Now define for each individual z=xi/,(=l,,n-1),i.e, zj is the proportion of the outstanding stock of asset j held by individual i. Further, let that 2abja=l. Then (12)can be written (13)∑b2=∑bk2z (,k=1,…,n-1) It is easily proved that these equations imply that the z are the same for all j 4 What this means is that in equilibrium, prices must be such that each individual will hold the same percentage of the total outstanding stock of all risky assets. This percentage will of course be different for different individuals, but it means that if an individual holds, say, 2 per cent of all the units outstanding of one risky asset, he also holds 2 per cent of the units outstanding of all the other risky assets. note that we cannot conclude that he also holds the same percentage of the riskless asset; this proportion will depend upon his attitude towards risk, as expressed by his utility function. But the relation nevertheless permits us to summarize the description of an individuals portfolio by stating (a) his holding of the riskless asset, and(b) the percentage z held of the outstanding stock of the risky assets. We 5 Let the common value of the n-1 terms 2a=i bja zd be d, and let ca be the elements of the inverse of the matrix of the bia(assuming nonsingularity). It is well known that when sabia=1 then also 2acja-1. The solutions for the ff are then: zf= 2acaa'=a'2acja=a, which proves our proposition has content downl ued stube to sT oR ems aecondtp23013020-0 AM
CAPITAL ASSET MARKET 775 (1) mjRi _ Mk Rk (j, k=1, ...,I n-1), Vi Vk i.e., the risk margins are such that the ratio between the total risk compensation paid for an asset and the variance of the total stock of the asset is the same for all assets. 4. COMPOSITION OF EQUILIBRIUM PORTFOLIOS We can now derive an important property of an individual's equilibrium portfolio. When (10) is substituted back in (9), the result is: Z 7jaX~a UkaXXa (12) a - a Z 5fja x Z?ka Xa a a Now define for each individual zJ =X/X (j=1, ..., n- 1), i.e., zJ is the proportion of the outstanding stock of asset j held by individual i. Further, let bia = jaXa Z ijaXa a so that Zabja= 1. Then (12) can be written (13) Zbjaz =Z bka (j, k-1, ... 1) . a a It is easily proved5 that these equations imply that the zJ are the same for all j (equal to, say, zi), i.e., (14) zJ-Zk=4z (j, k=1, ..., n-1). What this means is that in equilibrium, prices must be such that each inidividual will hold the same percentage of the total outstanding stock of all risky assets. This percentage will of course be different for different individuals, but it means that if an individual holds, say, 2 per cent of all the units outstanding of one risky asset, he also holds 2 per cent of the units outstandlng of all the other risky assets. Note that we cannot conclude that he also holds the same percentage of the riskless asset; this proportion will depend upon his attitude towards risk, as expressed by his utility function. But the relation nevertheless permits us to summarize the description of an individual's portfolio by stating (a) his holding of the riskless asset, and (b) the percentage zi held of the outstanding stock of the risky assets. We 5 Let the common value of the n-I terms La= I bja z be a', and let cj, be the elements of the inverse of the matrix of the bc,, (assuming nonsingularity). It is well known that when Efbj= 1, then also 4.cja = 1. The solutions for the z, are then: z= Eaciaa'= aiaC;a = ai, which proves our proposition. 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
776 JAN MOSSIN lso observe that if an individual holds any risky assets at all (i. e, if he is not so averse to risk as to place everything in the riskless asset), then he holds some of every asset. (The analysis assumes, of course, that all assets are perfectly divisible. Looked at from another angle, 13)states that for any two individuals r and s, and any two risky assets j and k, we have xilxk=x/xk, i.e., the ratio between the holdings of two risky assets is the same for all individuals With these properties of equilibrium portfolios, we can return to the problem of onnegativity of the solution. With risk aversion it follows from (5) that P The sum of such positive terms must also be positive, i.e. ∑σn又2/ >0 But then also, 2, 2/Eajaia>0, so that the a' of footnote 4 is positive, which then implies z>0. Hence, negative asset holdings are ruled out Our results are not at all unreasonable. at any set of prices, it will be rational for investors to diversify. Suppose that before the exchange takes place investors generally come to the conclusion that the holdings they would prefer to have of some asset are small relative to the supply of that asset. This must mean that the price of this asset has been too high in the past. It is then only natural to expect the exchange to result in a fall in this price, and hence in an increase in desired holdings What the relations of (14)do is simply to give a precise characterization of the ultimate outcome of the equilibrating effects of the market process 5. THE MARKET LINE The somewhat diffuse concept of a"price of risk"can be made more precise and meaningful through an analysis of the rate of substitution between expected yield and risk(in equilibrium). Specification of such a rate of substitution would imply line in a mean-standard deviation plane and characterizes it by saying: In equi librium, capital asset prices have adjusted so that the investor, if he follows rational procedures(primarily diversification), is able to attain any desired point along a capital market line"(p 425). He adds that". some discussions are also consistent with a nonlinear(but monotonic)curve"(p. 425, footnote) We shall attempt to formulate these ideas in terms of our general equilibrium ystem. As we have said earlier, a relation among points in a mean-variance diagram makes sense only when the means and variances refer to some unit common to all assets,for example, a dollar's worth of investment. We therefore had to reject such has content downl ued stube to sT oR ems aecondtp23013020-0 AM
776 JAN MOSSIN also observe that if an individual holds any risky assets at all (i.e., if he is not so averse to risk as to place everything in the riskless asset), then he holds some of every asset. (The analysis assumes, of course, that all assets are perfectly divisible.) Looked at from another angle, (13) states that for any two individuals r and s, and any two risky assets j and k, we have xJ/x =xJ/x', i.e., the ratio between the holdings of two risky assets is the same for all individuals. With these properties of equilibrium portfolios, we can return to the problem of nonnegativity of the solution. Withl risk aversion it follows from (5) that The sum of such positive terms must also be positive, i.e., E fa X ( q)>?- But then also, IXx/ Z >aj0xc> 0, so that the a' of footnote 4 is positive, which then implies z >0. Hence, negative asset holdings are ruled out. Our results are not at all unreasonable. At any set of prices, it will be rational for investors to diversify. Suppose that before the exchange takes place investors generally come to the conclusion that the holdings they would prefer to have of some asset are small relative to the supply of that asset. This must mean that the price of this asset has been too high in the past. It is then only natural to expect the exchange to result in a fall in this price, and hence in an increase in desired holdings. What the relations of (14) do is simply to give a precise characterization of the ultimate outcome of the equilibrating effects of the market process. 5. THE MARKET LINE The somewhat diffuse concept of a "price of risk" can be made more precise and meaningful through an analysis of the rate of substitution between expected yield and risk (in equilibrium). Specification of such a rate of substitution would imply the existence of a so-called "market curve." Sharpe illustrates a market curve as a line in a mean-standard deviation plane and characterizes it by saying: "In equilibrium, capital asset prices have adjusted so that the investor, if he follows rational procedures (primarily diversification), is able to attain any desired point along a capital market line" (p. 425). He adds that "... some discussions are also consistent with a nonlinear (but monotonic) curve" (p. 425, footnote). We shall attempt to formulate these ideas in terms of our general equilibrium system. As we have said earlier, a relation among points in a mean-variance diagram makes sense only when the means and variances refer to some unit common to all assets, for example, a dollar's worth of investment. We therefore had to reject such 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
