The pricing of capital assets in a multiperiod world

Journal of Banking and Finance 7 (1983) 31--44. North-Holland Publishing Company

THE PRICING OF CAPITAL ASSETS IN A MULTIPERIOD WORLD Marshall E. BLUME* The University of Pennsylvania, Philadelphia, PA 19104, USA Received March 1981, final version received September 1982 This paper proposes a general framework for the pricing of capital assets in a multiperiod world. Under quite general conditions, the analysis shows that the equilibrium expected nominal return on any asset can always be expressed as the sum of the risk-free rate and various risk premiums. The first risk premium is identical to the usual risk premium in the Sharpe-LintnerMossin capital asset pricing model. The mathematical forms of all the remaining risk premiums are identical even though each individual risk premium may be present for a different reason.

I. Introduction

Recent empirical tests of the traditional capital asset pricing model of Sharpe, Lintner and Mossin have found that this model is not fully consistent with observed returns on common stock? Partly in resPonse, various authors have developed alternative pricing models. For instance, in a multiperiod world, Merton (1973) postulates a changing investment opportunity set as a function of a stochastic interest rate. In another early extension, Mayers (1972) allows for human capital. Many other extensions have been proposed, but any attempt to review everyone would require an article in itself. The purpose of this paper is to propose a general framework in which to interpret these various extensions and by implication to demonstrate their conceptual similarities. Specifically, the expected nominal return on any asset in equilibrium can be viewed as the sum of the nominal risk-free asset, and four basic kinds of premiums. The first premium is the one common to the traditional capital asset pricing model and is associated with the usual beta coefficient of that model. The remaining three types of premiums are associated respectively with: (a) state variables which condition the evaluation of end-of-period nominal wealth, (b) departures of the joint probability distribution of end-of-period random variables from normality, *The author would like to thank Professors Eugene Fama, Andre Farber, Irwin Friend, E. Han Kim, Stuart M. Turnbull, and Randolph Westerfield for their helpful comments. The financial support of the Rodney L. White Center for Financial Research is gratefully acknowledged. 1For example, see Blume and Friend (1973) or Fama and MacBeth (1973). 0378-4266/83/$03.00

32

M.E. Blume, The pricing of capital assets in a multiperiod world

and (c) institutionally imposed constraints, such as the non-marketability of certain types of assets. Although these last three types of premium stem from different sources, they all lead ultimately to premiums of the same mathematical form. It should be noted that the analysis of this paper makes no contribution as to which of these premiums need to be included. To make statements about the explicit risk premiums would require detailed assumptions as to investors' preference functions and the investment and consumption opportunity sets. Nonetheless, whatever model is assumed, the resulting equilibrium relationship for the expected nominal return can be expressed in the mathematical forms developed in this paper. Following a brief review of the use of dynamic programming to solve a multiperiod consumption and investment problem will be the main results of the paper. First, it will be shown that any seemingly one-period problem derived from a multiperiod problem can be transformed into an equivalent problem of minimizing the variance of return subject to various constraints. This transformed problem might be viewed as a generalized version of the usual mean-variance portfolio problem associated with the name of Markowitz. Then, the first order conditions of this equivalent problem will be aggregated over all investors to yield the desired equilibrium relationship. This relationship shows that the expected rate of return on an individual asset can be expressed as a sum of the nominal risk-free rate and various risk premiums. The paper will conclude with several examples illustrating the interpretation of this equilibrium relationship.

2. The efficient set and equilibrium

In 1952, Markowitz observed that any feasible portfolio would be one of two types: a portfolio which no risk averse investor would want to hold or a portfolio which would be suitable for some risk averse investor. He termed the latter set of portfolios 'the efficient set'. Introducing a risk-free asset, Sharpe (1964) developed an equilibrium pricing relationship for individual assets. Black (1972) developed a similar kind of relationship in the absence of a risk-free asset. Others have developed equilibrium relationships in a multiperiod setting. In addition to Merton's (1973) early piece, examples include Long (1974), Kraus and Litzenberger (1975), Stapleton and Subrahmanyam (1978a) in conjunction with the comment by Kreps and Wilson (1980), Breeden (1979), and Bhattacharya (1981). Following a brief review of the reduction of a multiperiod problem to a one-period problem, Markowitz's concept of the efficient set will be generalized so as to capture the essence of a multiperiod problem and then equilibrium pricing relationships for individual assets will be developed.

M.E. Blume, The pricing of capital assets in a multiperiod world

33

2.1. The seemingly one-period problem In an uncertain world with non-continuous trading, it is well known that an investor's multiperiod consumption-investment problem can be reduced to a seemingly one-period problem. Early examples include Mossin (1968), Hakansson (1970), and Fama (1970). Specifically, the backwards optimization process of dynamic programming induces for investor k with current wealth Wo, a seemingly one-period utility function fo, defined over current consumption Co, end-of-period wealth Wx, and an appropriate set of state variables S~, whose end-of-period values are unknown as of the current time but resolved at the end of the period. Exactly what is an appropriate set of st,ate variables has been the subject of much of the past literature on multiperiod models, and this paper will add nothing to this literature. Under some conditions, there is no need to include state variables. Fama (1970) has given one set of conditions for the exclusion of state variables: namely, that (a) future consumption and investment opportunities are known at the current time, and (b) the multiperiod utility function itself is not state dependent. 2 State variables may need to be included in fo if one or both of these assumption are violated. 2.2. The generalized efficient set Without being explicit as to the exact reason, let it be assumed that at least one state variable must be included in the one-period recursive function fo- In this case, the investor would select Co and his portfolio of assets so as to

max E[fo(C o, wl, S0],

(1)

subject to the pursuit of a feasible policy. Of importance for the following analysis, Fama (1970) has proven, except in one special but uninteresting case, 3 that if the multiperiod utility function is strictly concave in the consumption vector, fo will be strictly concave in Co and wl, and such concavity will be so assumed. Moreover, to simplify the analysis, it will be assumed that the state vector $1 consists of only one element; namely sl. Generalizing the results to a state vector with more than one element is straightforward. If the random variables in fo are jointly normal, 4 E(fo) can be rewritten as 2Fama and MacBeth (1974) provide a similar but somewhat less restricted set of assumptions. 3Fama (1976). 4Without being more explicit as to the utility function and investment opportunities, it is not possible to assess the reasonableness of assuming that w 1 and sl are jointly normal. However, the reader is referred to Stapleton and Subrahmanyam (1978a) for an analysis of the potential difficulties in assuming in a multiperiod setting that even wl is normal. For example, in a multiperiod world, they show that 'if the cash flows are normally distributed and not independent over time, the prices will not, in general, be normal'.

34

M.E. Blume, The pricing of capital assets in a muhiperiod world

a deterministic function, say g, of the expected values and the variancecovariance matrix of wl and s~.5 Thus, under the assumption of normality, (1) can be rewritten as max g[Co,E(wi), E(si), var(wl), cov(wl, si), var(si)],

(2)

subject to the pursuit of a feasible policy. Since (2) is derived from (1), it will give the same solution as (1). Moreover, the arguments E(sl) and var(sx) are outside the control of the individual investor and known as of the current time. As a consequence, these two variables need not be included as explicit arguments in g. The strict concavity offo in Co and wl implies that the partial derivative of g with respect to var(wi) is negative. 6 Thus, the very definition of a partial derivative implies that the investor of this paper faced with the choice between two feasible investment consumption strategies promising the same current consumption, the same expected end-of-period wealth, and the same covariance between wl and sx would select that strategy with the .smaller variance of end-of-period wealth. The same kind of proposition would hold in the one-period world of Markowitz except that there would be no reference to the covariance between w~ and sl since in this world the only arguments of g subject to control would be Co, E(wi), and var(wi). Thus, of all feasible consumption-investment strategies with the same values of Co, E(Wl), and cov(wi, sl), the investor of this paper would select that strategy with the smallest possible variance of end-of-period wealth. Likewise, of all feasible consumption-investment strategies with the same values of Co, var(w0, and cov(wl, s0, the investor would select that strategy with the largest possible expected value of end-of-period wealth. 7 The set of 5Cf. Tobin (1959), Long (1974). 6The following establishes the proposition: define h as

where x and y are jointly normally distributed with means ~x and ~y, standard deviations ax and at, and correlation coefficient p and where zx and zy are standardized normal variates with correlation coefficient P. The function f is assumed strictly concave in its first argument for fixed values of the second. Take the partial derivative of h with respect to a~ to obtain

The term in brackets may be recognized as the expectation of z~df/O(# + axZx) conditional on zr Because of the concavity o f f in its first argument and the symmetry of the normal distribution, this conditional expectation is negative. Integrating with respect to zy preserves the negativity. The proposition in the text follows from the fact that ah/Oa and Oh/c3a2 are of the same sign. 7Since this property will not be used in the derivation of the macrorelationships, a formal proof is omitted.

M.E. Blunw, The pricing of capital as,sets in a multiperiod world

35

all feasible portfolios satisfying both of these properties would correspond to a generalized version of Markowitz's efficient set; any other feasible portfolio would be inefficient. Before formulating the mathematical form of this generalized version of Markowitz's efficient set, let us consider the effect of relaxing the normality assumption. If relaxed, g would in general have to be defined over all moments, assuming that they exist, to assure that the maximization of g would give the same solution as (1). To obtain a more manageable problem, E(fo) might be approximated with a function, g, defined over a limited number of moments, s Again, for given values of all the variables in g other than var(wl), g would be maximized by minimizing var(wl). 9 The examples in the next section will explicitly consider the case of including skewness in the optimization process. To develop a program for the generalized version of Markowitz's efficient set, define the following variables: N = the number of assets available at time 0, a k = the dollar amount of asset i held by investor k at time 0 after any revision of the portfolio, ri = the total return relative on asset i from time 0 to time 1, defined as 1.0 plus the rate of return, #~ = the expected value of ri, ~ij = the covariance between ri and rj. To distinguish among investors, Wo, wl, and Co will be superscripted by k. Although /~i and aij could be superscripted by k to allow for heterogeneous expectations and equilibrium relationships derived, the resulting relationships have little intuitive economic appeal. Thus, homogeneous expectations will be assumed. As argued, among all feasible portfolios which have the same values for the arguments of g other than var(w]), an investor would choose that portfolio with the minimum value of var(w]). In the usual Markowitz oneperiod model, the investor would want to minimize var(w]) subject to a constraint on E(w k) and a feasibility constraint to insure that the available funds after fixed consumption are fully invested. By varying E(w~) over all 8There is vast literature on this approximation. For example, cf. Samuelson (1970) for a continuous-time treatment and Tsiang (1972) for a discrete-time treatment. 9The proof in footnote (6) which showed that 8g/Ovar(wl) is negative relied on the assumption of normality and specifically the symmetry of the distribution and thus in general is not applicable to the case in which E(fo) is approximated by a function g. To establish the negativity of ~g/Ovar(wl), set c o to its optimal value and expand fo about E(w0 and E(sO, take expected values, and finally drop all terms involving moments not in g. The partial derivative of this expansion with respect to var(wl) is negative. A more complete discussion is contained in Arditti (1967).

M.E. Blume, The pricing of capital assets in a multiperiod world

36

feasible values, the investor would obtain the efficient set. In the more general case discussed in this paper, there would be additional arguments in g which would have to be held constant as well as possibly additional feasibility constraints, such as non-marketability of specific assets. Although more than one additional constraint may be required to hold constant the additional variables in g and to maintain feasibility, the following analysis will assume for simplicity that only one additional constraint is required. The results are easily generalizable to more than one additional constraint. Similarly to the Markowitz case, varying the values of the arguments in g, other than var(u/[~), over all feasible values and identifying the minimum variance portfolio would trace out a generalized version of an efficient set. In the notation just introduced, the mathematical programming problem of minimizing the variance of end-of-period wealth, holding all the other arguments in g constant and maintaining feasibility, takes the form k k min~.~, aiajtTij , l

subject to

j

Ej a s # s -

E4 . s

(3)

d ( a ~ , . . . , a ~ ) = d k.

The first two constraints are the usual ones in the standard one-period model of Markowitz. The third constraint represents any additional constraint required to hold constant additional arguments of g or to maintain feasibility. In some cases, the third constraint could be non-linear, such as a constraint on the value of the third moment of end-of-period wealth. In other cases, it would be linear. The notation in (3) permits the third constraint to be non-linear. To summarize so far, there are only three possible sources from which this third type of constraint could arise. The three sources are: (1) state variables which condition the evalution of end-of-period wealth, (2) departures from normality of the probability distributions of end-ofperiod wealth and the state variables which would force an investor to consider moments other than the first and second, (3) feasibility constraints such as the prohibition of certain types of transactions like marketability restrictions. If 2~ is the Lagrange multiplier of investor k for constraint q, (3) can be rewritten as

M.E. Blume, T h e pricing o f capital assets in a multiperiod world

minh = ~ ~

ai

a~iaij- 22

37

aj#j-

J

- 2 2 k [ ~ ak--(~0--ck) 1 -22kEd(ak~,..., ak)--dk].

(4)

Now, problem (4) can be used to separate all potential investmentconsumption strategies into two types: those strategies which do not minimize var(w~l) for given values of E(wk), ~o--ck, and dk, and those that do. No investor would ever pursue a strategy of the first type. The investor's optimal strategy will be of the second t y p e - - n a m e l y , an efficient strategy. The solution to (4) is given by setting the (N + 3) partial derivatives of h to zero. These resulting (N + 3) equations consist of the three constraints given in (3) and N equations of the form k

k

k

^k k

1 -- 1, • .., N.

tTija j - - )'ukti - - ~'w - - £ddi = 0,

(5)

J

The symbol d k represents the partial derivative of d with respect to a/.k U n l e s s d is linear and of the same form for all investors, the value of this partial derivative would in general differ from investor to investor, necessitating the k superscript. Moreover, if d is non-linear, d k will generally be a function of the amounts invested in each of the assets. If there is no risk-free asset in nominal terms and the variance-covariance matrix has full rank, the optimal values of a k for the efficient portfolio which maximizes the investor's expected utility will be given by k "" k ai=2k~aupj+2,,

J

~'~ij.jv

J

k 2d2aiJd k,

i = 1,..,N,

(6)

J

where trij are the elements in the inverse of the variance-covariance matrix and where the Lagrange multipliers and the dk's are evaluated at the optimal values of the ak's. Eq. (6) states that the optimal amount invested in each asset can be viewed as a weighted sum of three vectors or f u n d s - - a threefund result. If, however, there is a risk-free asset, it can still be shown that the amount invested in each asset is a weighted sum of three vectors or f u n d s - - a similar three-fund results, x° These funds are not unique, and more generally, it can be shown that there will be as many funds as constraints in order to span the efficient set of strategies. k N+ t°Let the Nth asset be the risk-free asset. Then the Nth equation of (5) takes the form: 2,# k k 2 w + 2 d d s = O . Using this to eliminate 2~ from the first ( N - I ) equations of (5), one can show that the amounts invested in the first ( N - 1) assets can be expressed as the weighted sum of two portfolios. The amount placed in the Nth asset would be the difference between what is available for investment and what is invested in the first ( N - 1 ) assets• k

38

M.E. Blume, The pricing of capital assets in a multiperiod world

If the third constraint is linear and the coefficients are the same for all investors, the portfolio of any investor can be decomposed into a weighted sum of three portfolios, which in turn will be the same for every investor. If the function d is such that the k superscript is required, the third fund would differ from one investor to another. Unless some additional assumptions are made as to how the dk's vary from one investor to another and hence the third mutual fund varies from one investor to another, a three-fund decomposition, though formally correct, may have little economic meaning.

2.3. A n e q u i l i b r i u m

relationship

Finally, assuming that each investor has reached equations in (5) can be manipulated and aggregated to relationship for the expected return of each asset. On there is a risk-free rate and the additional constraint is coefficients for each investor, the appendix contains equilibrium relationship I~i = r y + (#,,, -- r y) fli + ~[( di - d :) - fli( d,. - d : )],

his equilibrium, the yield an equilibrium the assumption that linear with the same a derivation of the

(7)

where kt,, is the expected return on the market portfolio of all assets including the risk-free asset, 11 fli is the usual beta coefficient of the capital asset pricing model defined as the ratio of cov(ri, r,.) to aZ(r,.), V is the ratio of ~k 2k to ~k 2k, and d,, is the average of the individual d,'s weighted in proportion to their market values. Without the last term generated by the third constraint, (7) would be the same as the traditional capital asset pricing model. If there were four instead of three constraints, there would be a further additional term of the form 6 [ ( e i - e : ) - f l ~ ( e r , - e : ) ] , the same form as the term associated with the third constraint. As in the first additional term, the ei's represent the coefficients of a linear constraint and 6 the ratio of appropriate sums of Lagrange multipliers. In sum, the first two constraints of the minimization problem generate the traditional capital asset pricing model; each additional linear constraint generates a term in the form of the third term in (7). If, however, the third constraint is non-linear or linear with different coefficients from one investor to another, the equilibrium relationship for the pg's would have the same first two terms as (7) but a different third term, 11Alternatively, #,~ is sometimes taken as the expected return on the market portfolio of risky assets. The effect is to multiply the expected risk premium on the market by the reciprocal of the ratio of the market value of risky assets to the market value of all assets and to multiply the beta coefficient by this ratio, leaving the product unchanged.

39

M.E. Blume, The pricing of capital assets in a muhiperiod world

namely 1

Ak

Akdk--Z'~ddi--fli k

Wj

,~kdk

Z 2kd k fk k

,

(8)

where wfs are weights proportional to the market value of each asset j and summing to 1.0 over all assets including the risk-free asset. If the d~'s had the same value for all investors, the k superscripts could be dropped and ~k 2k factored out from the braces. Since y is defined as the ratio of Zk/~k to ~-'~k/~k, the resulting expression would be the same as in the linear case examined above. The trick in applying these equilibrium relationships is the specification of which additional constraints are empirically important. The next section gives examples of several possible additional constraints, but does not evaluate them empirically.

3. Examples The equilibrium relationships given by (7) and (8) are compatible with many extensions of the traditional capital asset pricing model. The following examples illustrate several possible extensions. 3.1. Merton's model

Following the suggestion of Merton (1973), it might be postulated that the information needed to describe the state of theworld in the one-period problem as derived from the multiperiod consumption problem can be adequately summarized by the changes in the risk-free nominal rate from that applicable at time 0 to that at time 1. If the change in the risk-free rate, dr f, were jointly normally distributed with the other random variables in the one-period problem, the third constraint in (4) could be expressed as

cov(r , Arl) = cove,

(9)

is the optimal value for investor k. Since cov(rf,Arf)=O by the definition of a risk-free rate and the constraint is linear, the equilibrium relationship can be written directly from (7) as where

cov k

'~d E(ri) = r l + fli[E(r,.)- r f ] + -7--[cov(ri, dr f ) - fli cov(r,., dry)].

(10)

Eq. (10) can be rewritten in perhaps a more intuitive way by expressing Ar I

40

M.E. Blume, The pricing of capital assets in a multiperiod world

as a function of r,. as

(11)

Ar f = ~ o + 41rrn + e,

where 41 is defined as cov(Ar:,r,,,)/tr2(r,.), 4o as E(Ar:)--41E(r,.), and e is a mean-zero disturbance independent of r,.. Recognizing that ~ is the same as Ar:-E(Ar:lr,.), using the definition of fli, and after some rearrangement of terms, one can rewrite (11) as ~d

E(ri) = r: + fli[E(r,,,)- r :] + .---cov[ri, Ar f - E(Ar I r,,)]. 2u

(12)

In words, (12) states that the expected return on asset i is the sum of the risk-free rate, a risk premium proportional to fli, and an adjustment for the relationship of the return on the asset to changes in the risk-free rate which cannot be explained by the actual market return. Note that any change in the risk-free rate which can be explained by the actual market return has already been incorporated in the market risk premium as adjusted by fli.

3.2. Long's model Long (1974) has proposed a multiperiod model to incorporate the effects of inflation and the term structure of interest rates upon stock prices. In his model, the appropriate vector of state variables consists of the end-of-period prices of each of K consumption goods and the end-of-period prices of T - 1 zero-coupon discount bonds of differing maturities but with more than one period to go. He does not include the end-of-period prices of common stocks in the state variable on the assumption that these prices convey no additional information about future investment and consumption opportunities to that already contained in the end-of-period prices of the consumption goods and the discount bonds. At the start, Long's analysis parallels that of this paper. He takes the first order conditions of a maximization problem equivalent to (2) in this paper but with additional state variables and then aggregates these conditions over all investors. By so doing, he obtains an equation similar to (A.1) in the appendix of this paper. From this point on, his derivation and resulting equations differ from those in this paper. Appealing to properties of multivariate normality, Long derives the following equilibrium condition for the expected returns of common stock i: K

T

Z 4,k k+ n = 2 6,.'7.,

k=l

(13)

M.E. Blume, The pricing of capital assets in a multiperiod world

41

where /~s is the expected return on the stock market and not the expected return on all assets as it would be in the usual capital asset pricing model. The coefficients hi, {~k, k = 1,..., K}, and {6~,,n = 2,..., T} are respectively the multiple regression coefficients of the end-of-period prices of stock i on the end-of-period values of the stock market, the end-of-period prices of the K consumption goods, and the end-of-period prices of the T--1 zero coupon discount bonds. Thus, the bi in (13) is not the fli of the capital asset pricing model. The pricing equation given by (13) highlights the expected returns from bearing specific types of risks through the parameters (~s-rl), {Yk, k= I,...,K}, and {r/,, n = 2 , . . . , T}. In contrast, the pricing eq. (7) in this paper contains the usual capital asset pricing model explicitly and highlights the deviations of the expected returns of individual risky assets from these terms either for the reasons implicit in Long's model or for reasons of feasibility or non-normality. The next two examples illustrate these last two reasons. 3.3. Rubinstein' s model Rubinstein (1973) has examined the effect of skewness in the pricing of capital assets. In the framework of this paper, Rubinstein's model would imply that the third constraint of (3) would be of the form E

;J/

k - ak#~ a~r~

(Wk)2 = d k,

(14)

Defining rpk as the rate of return for the portfolio of investor k, (14) can be rewritten as

Z

cos%,

(15)

g) = d,

J

where cos(r~,rkp,r k) is the coskewness the left-hand side of (15) shows that the partial derivative differ from equilibrium equation is obtained by

of rj, ~, and rp. k Taking the derivative of d k equals cos(ri, rp, k ~). Since the values of investor to investor, the appropriate substituting into (8).

3.4. StapIeton and Subrahmanyam's model Non-marketable assets are readily incorporated into the framework developed here. As an example, Stapleton and Subrahmanyam (1978b) have explored the effect of a government investment, a predetermined portion of which must be held by each investor, upon the equilibrium pricing relationships. If the Nth asset is riskfree and the N - l s t asset is the

42

M.E. Blume, T h e pricing of capital assets in a multiperiod worm

government investment of which investor k must hold h k dollars, this nonmarketability restriction can be incorporated into (3) by rewriting the third constraint as a~ _ 1 = hk.

(16)

In the notation of (3), di is 1.0 for i= N - 1 and 0.0 otherwise. From (7), the equilibrium relationship can be immediately written as

E(Fi) =

rf

+ ~i[E(rm)- r f ]

~d ~U

+ -~--(d i - ~iX~l-

1)"

(17)

Parenthetically, eq. (17) is not equivalent to that developed by Mayers (1972) for a non-marketable asset like human capital. Mayers assumes that the expected return per dollar of human capital as well as the variance of these returns and the covariances with other returns may differ from one investor to another. Formally, Mayers' assumption is equivalent to heterogeneous expectations for the returns on human capital, so that aij and /~i in (3) would be superscripted by k whenever i or j is N - 1 . In addition, Mayers defines the market portfolio r,, over only marketable assets and thus the aggregation of the first order conditions in the appendix would have to be modified to obtain Mayers' exact formula.

4. Conclusion

Beginning with a multiperiod utility function defined over consumption and perhaps state variables over which an investor has no control, the paper showed how the problem of maximizing the expected value of this function could be reduced to a problem of minimizing the variance of returns subject to a set of constraints--a generalized version of the usual mean-variance portfolio problem. Aggregating the solutions from this problem yielded a general equilibrium relationship for the pricing of capital assets. As long as one maintains the objective of maximizing the expected value of a general multiperiod utility function as used here, any attempt to make the traditional capital asset pricing model a more realistic description of the world must be equivalent to making some assumption about one or more of the following three areas: (a) state variables which condition the one-period utility function derived from the multiperiod problem, (b) departures from normality, or (c) feasibility constraints. The results of this paper show how any type of assumption about these areas can be incorporated directly into a pricing equation for the expected returns of individual assets.

M.E. Blume, The pricing of capital assets in a multiperiod world

43

Appendix: Derivation of the equilibrium relationship (11) Eq. (7) is obtained as follows: If d~ is the same for all, sum (5) over k and express the resulting equation as i=l,...,n,

4# j : l

(A.1)

where 2~, is defined as 2k 2kx and Vj is the total market value of asset j given by ~k a~. Letting V,, be the market value of all assets, the above equation can be multiplied by the ratio V~/V,, and aggregated over i to give an expression involving the expected return on the market portfolio, #m: 1

n Vii ~,

(TijVj=/Am.~t_,~mql_'~d~-~Vii

(A.2)

The summation on the left-hand side of (A.1) may be recognized as V~,.cov(r~, r,,), where r,. is the return on the market portfolio; and the double summation on the left-hand side of (A.2) as Vm var(rm). Dividing the second equation into the first yields after some simplification (A.3) where fli is defined as the ratio of cov(r i, rm) t o var(r,.) and d m as the average of the individual di's weighted by their market values. Up to this point, the derivation has not explicitly used the assumption that asset N is riskfree. Thus, (A.3) holds in the absence of a risk-free rate. After aggregation, (A.3) implies that the return on the risk-free asset FzN or r f, satisfies -- 2w/2 u = r y + (2a/2u) . dr,

(A.4)

where dy denotes the coefficient on the Nth asset in the linear constraint. By substituting (A.4) in (A.3) and rearranging terms, one obtains eq. (7) of the text. If dk differs from one investor to the next, replace d~ in (A.1) by dk. Repeating virtually the same mathematical manipulations, one would obtain the desired results. The generalization of the equilibrium to four or more linear or non-linear constraints is straightforward.

References Arditti, F.D, 1967, Risk and the required return on equity, Journal of Political Economy 22. 1~7 36.

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M.E. Blume, The pricing of capital assets in a multiperiod world

Bhattacharya, S., 1981, Notes on multiperiod valuation and the pricing of options, Journal of Finance 26, 163-180. Black, F., 1972, Capital market equilibrium with restricted borrowing, Journal of Business 45 444-454. Blume, M.E., and I. Friend, 1973, A new look at the capital asset pricing model, Journal of Finance 28, 19-33. Breeden, D.T., 1979, An intertemporal asset pricing model with stochastic consumption and investment opportunities, Journal of Financial Economics 7, 265-296. Fama, E.F., 1970, Multiperiod consumption-investment decisions, American Economic Review 60, 163-174. Fama, E.F., 1976, Multiperiod consumption-investment decisions: A correction, The American Economic Review 66, 723-724. Fama, E.F. and J.D. MacBeth, 1973, Risk return and equilbrium: Empirical tests, Journal of Political Economy 81, 607--636. Fama, E.F. and J.D. Macbeth, 1974, Tests of the multiperiod two parameter model, Journal of Financial Economics 1, 43-66. Hakansson, N., 1970, Optimal investment and consumption strategies under risk for a class of utility functions, Econometrica 38, 587-607. Kraus, A. and R.H. Litzenberger, 1975, Market equilbrium in a multiperiod state preference model with logarithmic utility, Journal of Finance 30, 1213-1227. Kreps, D.M. and R. Wilson, 1980, Temporal resolution of uncertainty in Stapleton and Subrahmanyam's multiperiod equilibrium asset pricing model, Econometrica 48, 1565-1566. Long, J.B., 1974, Stock prices, inflation, and intrest rates, Journal of Financial Economics 1, 131-170. Markowitz, H.M., 1952, Portfolio selection, Journal of Finance 17, 77-91. Markowitz, H.M., 1959, Portfolio selection: Efficient diversification of investments (Wiley, New York). Mayers, D., 1972, Non-marketable assets and capital market equilibrium under uncertainty, in M.C. Jensen, ed., Studies in the theory of capital markets (Praeger, New York). Merton, R.C., 1973, An intertemporal capital asset pricing model, Econometrica 41, no. 5, 867 887. Mossin, J., 1968, Optimal multiperiod portfolio policies, Journal of Business 41, no. 2, 215-229. Rubinstein, M.E., 1973, The fundamental theorem of parameter-preference-security valuation, Journal of Finance and Quantitative Analysis 8, 61-70. Samuelson, P., 1970, The fundamental approximation theorem of portfolio analysis in terms of means, variances, and higher moments, Review of Economic Studies 37, 537-542. Sharpe, W.F., 1964, Capital asset prices: A theory of market equilibrium under conditions of risk, Journal of Finance 19, 425-442. Stapleton, R.C. and M.G. Subrahmanyam, 1978a, A multiperiod equilbrium asset pricing model, Econometrica 46, 1077-1096. Stapleton, R.C. and M.G. Subrahmanyam, 1978b, Capital market equilibrium in a mixed economy, optimal public sector investment decisions rules, and the social risk of discount, Quarterly Journal of Economics, no. 368, 399-411. Tobin, J., 1959, Liquidity preference as behavior towards risk, Review of Economic Studies 2526, 65-86. Tsiang, S., 1972, The rationale of the mean-standard deviation analysis, skewness preference and the demand for money, American Economic Review 62, 354-371.