Necessary Conditions for the CAPM

File: 642J 221801 . By:CV . Date:19:03:97 . Time:10:09 LOP8M. V8.0. Page 01:01Codes: 4089 Signs: 2436 . Length: 50 pic 3 pts, 212 mm

Journal of Economic Theory � ET2218

journal of economic theory 73, 245�257 (1997)

Necessary Conditions for the CAPM*

Jonathan B. Berk-

School of Business Administration, University of Washington,Box 353200, Seattle, Washington 98195

Received December 23, 1994; revised March 8, 1996

The general restrictions on all economic primitives (i.e., (a) endowments,(b) preferences, and (c) asset return distributions) that yield the CAPM under theexpected utility paradigm are provided. These results are then used to derive theclass of restrictions on preferences and the distribution of asset returns alone thatprovides the CAPM. We also show that the conditions that provide the CAPM andderived preferences over mean and variance are equivalent. Consequently, thispaper also resolves the question of when mean�variance maximization is consistentwith expected utility maximization. Journal of Economic Literature ClassificationNumbers: G12, G11, D50. � 1997 Academic Press

1. INTRODUCTION

The single-period Capital Asset Pricing Model (CAPM) was originallyderived under the assumption that investors have utility functions overonly the mean and variance of their end of period consumption [23,14, 17]. At the time, the only conditions that were known to generatethese preferences under the expected utility paradigm were investors withquadratic utility functions or assets with normally distributed returns. Sincethen, Ross [21] showed that the CAPM can be derived from any two-fundseparating distribution and that the normal distribution is a special case ofthese distributions. However, the complete set of conditions under theexpected utility paradigm that are necessary and sufficient for the CAPM

article no. ET962218

2450022-0531�97 �25.00

Copyright � 1997 by Academic PressAll rights of reproduction in any form reserved.

* This research was supported by the SSHRC (Grant 410-94-0725) as well as by a grantfrom the Bureau of Asset Management, UBC. The author would like to thank Kerry Back,Avi Bick, Jim Brander, Murray Carlson, Kent Daniel, Larry Epstein, Joel Feldman, RickGreen, Burton Hollifield, Vasant Naik, Bryan Routledge, Raman Uppal, an anonymousreferee, and the associate editor for their insights, comments, and suggestions. A copy of thispaper, in addition to any other working paper by the author, is available on the author'sWWW home page: ``http:��weber.u.washington.edu�tberk.''

- E-mail: berk�u.washington.edu.


are not known. In this paper we completely characterize the set of alleconomic primitives, that is, endowments, preferences, and asset returndistributions, that yield the CAPM in equilibrium under the expectedutility paradigm.

Both Allingham [1] and Nielsen [18] provide sufficient conditions forexistence of CAPM equilibria. However, these conditions are difficult tointerpret within the expected utility paradigm because both authors assumethat all agents have preferences over mean and variance alone. We showthat if, for any wealth level, agents are assumed to be risk averse andalways prefer more to less then, under the expected utility paradigm, theseconditions are no more general than Ross' conditions.

A related literature, which has as its goal the answer to the question ofwhen expected utility maximization is consistent with mean�variancemaximization, has its foundation in Tobin's [24] original conjecture thatany two-parameter distribution will provide indirect preferences over meanand variance and Feldstein's [10] lognormal counter-example. It has sincegrown into a research topic unto itself, separate from the literature on theCAPM.1 We show that the restrictions on primitives that provide theCAPM in equilibrium are equivalent to the conditions that provide indirectutility functions over mean and variance alone in equilibrium. Conse-quently, this paper also provides the conditions under which expectedutility maximization and mean�variance analysis are consistent.

The paper proceeds as follows: Section 2 describes the model and Section 3derives the main result��the necessary and sufficient conditions that providethe CAPM. All non-trivial proofs are collected in the Appendix.

2. THE MODEL

Traditionally the CAPM has been derived in a one-period, single con-sumption good, exchange economy. Each agent } # K has a von Neuman�Morgernstern utility function, E[U}(x})], over final consumption x} # X} ,where X}/L2, the vector space of real-valued random variables whosevariances exist. U} : R � R is assumed to be continuous, concave, and dif-ferentiable with E[ |U $}(z)|]<� for any z # L2. It is traditional to assume,in addition, that U} is strictly increasing everywhere. However, in theCAPM literature this assumption is not imposed (e.g., quadratic utilityfunctions). Therefore, at this point we will make no further assumptionson U} .2

246 JONATHAN B. BERK

1 See [24, 10, 25, 7, 2, 4, 12, 13, 16]. Recently, Lo� ffler [15] has extended this literaturebeyond the expected utility paradigm.

2 Strictly speaking, the traditional literature on the CAPM does not even assume concavitysince an agent need not be risk averse in order to be variance averse (see [8, pp. 90, 95; 15]).


At the beginning of the period, agents trade in I<� securities that payoff at the end of the period in the consumption good. Each security, i # I,is uniquely defined by its return Ri # R, where the returns set R/L2. Theriskless asset is always assumed to be an element of I and is labeled 0.3

For convenience we will define r#R0 .A portfolio is a vector : # RI with �I

i=0 :i=1. The return of a portfolio,:, is just the sum of the return of its constituent securities, R:=�I

i=0 :i Ri .Initially, each agent has wealth W} which is invested in the endowmentportfolio e}. Since each agent's endowment is a portfolio, the total endow-ment is also a portfolio. This portfolio, :m#�} # K W}e}��} # K W} , hasreturn Rm (with R� m#E[Rm] and _2

m#var(Rm)) and is known as themarket portfolio. We assume that the value of the total endowment is non-zero4 and adopt the price normalization �} # K W}=1, that is, the totalendowment's price is 1. Any portfolio : is said to be a levered position inthe market portfolio if there exists an a # R such that, :=a:m+(1&a) e0 ,where e0 is an I-vector with its first component equal to 1 and the rest 0.A portfolio : is said to be mean�variance efficient in a return set R if it hasthe minimum variance of all portfolios that have its expected return.

We are now ready to define the equilibrium in this economy. Theportfolio, :}, that generates agent }'s consumption, x}=W} �I

i=0 :}i Ri , is

strictly supported by a return set R, if for any portfolio #z # RI withz=Wz �I

i=0 #zi Ri ,

E[U}(z)]>E[U}(x})] O Wz>W} . (1)

The set of all consumption portfolios, [:}]} # K , is said to be market-clearing if

:} # K

W}:}= :} # K

W}e}. (2)

A set of consumption portfolios and a return set ([:}]} # K , R) is an equi-librium, if for every } # K, :} is strictly supported by the return set R andmarkets clear.

Agent }'s consumption portfolio x} , is defined to be admissible if thefollowing two properties are satisfied: (i) U$}(x})>0 (i.e., marginal utility

247NECESSARY CONDITIONS FOR THE CAPM

3 The existence of a riskless asset is assumed in the traditional version of the CAPM.However, like the traditional results in the CAPM literature, the results in this paper do notrely on this assumption. Similar results can be derived for Black's [3] version of the CAPM.

4 This assumption is standard in the CAPM literature in which it is not generally assumedthat the total endowment is positive in every state (e.g., normally distributed returns).


is strictly positive in every state), and (ii) x} is an interior point in X} .5 Ifan agent's consumption, x} , is admissible, then the portfolio, :}, thatgenerates this consumption is also admissible. An equilibrium ([:} ]} # K , R)is an admissible equilibrium if the equilibrium set of consumption portfolios[:}]} # K is admissible.

The CAPM is said to hold in an equilibrium, ([:}]} # K , R), if, and onlyif, (i) the equilibrium is admissible, (ii) the market portfolio, :m, is mean�variance efficient for R, and (iii) every agent }'s consumption portfolio, :},is a levered position in the market portfolio.

Although this definition of a CAPM equilibrium is consistent with otherdefinitions in the literature, past researchers have not defined the equi-librium in precisely this way. For instance, [18, 19, 1] provide existenceconditions for CAPM equilibria by deriving conditions for the existence ofgeneral equilibria when agents have preferences over mean and variancealone. They therefore implicitly define the model to be any equilibrium thatresults in an economy in which agents have preferences over mean andvariance alone. Since the object of the paper is to derive the set of condi-tions on primitives that provide the CAPM, we cannot define the model interms of restrictions on primitives.

We define the CAPM in terms of the distinct characteristics of the equi-librium in which agents have preferences over mean and variance: (i) eachasset's expected return is a linear function of the expected return of themarket portfolio, and (ii) every agent's portfolio consists of a combinationof only the market portfolio and the riskless asset. Admissibility is addedbecause it is traditional in the CAPM literature to make two additionalassumptions. First, it is well known that restricting agents to preferencesthat are increasing in mean and decreasing in variance alone, does notnecessarily imply that all agents prefer more to less. Consequently, thetraditional literature on the CAPM does not assume that agents' utilityfunctions are strictly increasing everywhere. However, most derivations ofthe CAPM implicitly assume that, in equilibrium, agents prefer more to less(see, for example, [11, p. 96]). Second, since the CAPM pricing result relieson every agent's ability to make marginal trade-offs in equilibrium, no agentcan be constrained because she is on the boundary of her choice set.

3. THE RESTRICTIONS THAT SUPPORT THE CAPM

Before the complete set of primitives that provide the CAPM can bederived, a parsimonious characterization of the asset span is needed.


5 Alternatively, one could assume that :} is interior to its feasible subset. The latter assump-tion would be preferred if one has non-negative consumption in mind, because the interior ofthe set of non-negative random variables with finite variance is empty.


Without loss of generality, assume that a set of [[$j ]Jj=1 , [=i ]i # I]/L2

exists such that, for any Ri # R,

Ri=R� i+ :J

j=1

bji $j+=i (3)

with

R� i=E[Ri] (4)

bji # R, j=1, ..., J (5)

E[=i | Rm]=0 (6)

E[$j]=0, j=1, ..., J (7)

E[$k $j]={1, k= j,0, k{ j,

k, j=1, ..., J (8)

and, in addition, for i # I

:I

i=1

:mi =i=0. (9)

$j is referred to as the j th factor and =i as the residual of the i th security.The reason that (3)�(9) can always be satisfied is that no restriction isplaced on J, the number of factors. By setting the number of factors equalto the number of risky securities (which allows =i#0, \i # I), (3)�(9) aretrivially satisfied. We will henceforth refer to any [[$j ]J

j=1 , [=i ] i # I ] thatsatisfies (3)�(9) as a factor structure. Any factor structure in which thenumber of factors, J, is the minimum number of factors required to satisfy(3)�(9) will be referred to as a minimal factor structure.

Now, for a minimal factor structure 2=[[$j ]Jj=1 , [=i ] i # I] and con-

sumption y # L2, define the class of functions of y that are orthogonal tothe factors:

12[ y]=[# | #: R � R, #( y) # L2, E[#( y) $j]=0 \j=1, ..., J ]. (10)

This class of functions is clearly non-empty.6 Furthermore, for any givenconsumption y, the set 12[ y] is determined by the span of the factors. Ingeneral, since even a minimal factor structure is not uniquely determined,the set 12[ y] depends on the choice of factor structure. However, an


6 #( y)#k is a member of this class.


exception occurs when y is generated by a portfolio that is a levered positionin the market portfolio:

Lemma 3.1. For any a, b # R and any two minimal factor structures2=[[$j ]J

j=1 , [=i ]i # I ] and 4=[[*j ]Jj=1 , [!i ]i # I],

12[aRm+b]=14[aRm+b].

The proof of this lemma is straighforward and is left to the reader. In viewof this lemma we will henceforth drop the 2 subscript on the set 12[ y]whenever y is generated by a portfolio that is a levered position in themarket.

Given a risk free rate r, define 5(r) to be the set of all market-clearingconsumption allocations in which every agent }'s consumption, x} , is alevered position in the market portfolio and is admissible, i.e.,

5(r)#{x | x}=C}Rm+(W}&C}) r \} # K, :} # K

C}=1, x is admissible= .

The following proposition and its corollaries are the main contributionsof the paper:

Proposition 3.1 (Necessary and Sufficient Conditions for the CAPM).The CAPM holds in an equilibrium if and only if every agent }'s utilityfunction satisfies

U $}(C} Rm+(W}&C}) \)

=#}(C}Rm+(W}&C}) \)&E[#}(C}Rm+(W}&C}) \)]

[_2m�(R� m&\)]+R� m

Rm , (11)

where [C}Rm + (W} & C}) \]} # K # 5( \), \ # R, and #} # 1[C}Rm +(W}&C}) \] for each agent } # K. Furthermore, the value of \ that solvesthe above equations is r, the risk free rate in the economy.

Corollary 3.1. Assume that Rm has a continuous, strictly increasing,distribution function with support (D, U )�R. Then Eq. (11) in the aboveproposition can be replaced with

U}(z)=A}(z)&E[#}(C} Rm+(W}&C}) \)]

[_2m�(R� m&\)]+R� m _z2&2$W}&C})z

2C} & (12)

for z # (C}D+(W}&C}) \, C}U+(W}&C}) $, where A$}(z)=#}(z).



Proof. Letting z=C}Rm+(W}&C}) \ in (11), integrating, and recal-ling that expected utility functions are unique up to an affine transformationprovides (12). K

Remark. This proposition provides the conditions in terms of what istraditionally taken, in the CAPM literature, as one of the primitives of theeconomy, namely, the set of asset returns. In most general equilibriummodels the asset return set would not be regarded as a primitive since itrequires knowledge of equilibrium prices to compute it. Note, however,that asset returns enter in only three places: Rm , 1[ } ], and W} . Underthe price normalization (i.e., the total endowment has price 1) Rm is not afunction of prices. Since the factors are also not a function of prices(they are a function of the asset span which does not depend on prices),1[C}Rm+(W}&C}) \] also does not depend on prices. Finally, W} canbe defined without any knowledge of equilibrium prices by valuing eachagent's asset endowment in the return set for which the market portfolio ismean�variance efficient. Consequently, no knowledge of equilibrium pricesis required to check the conditions in the proposition.

It follows immediately from Proposition 3.1 that indirect preferences overmean and variance alone are not only sufficient, but also necessary for theCAPM. To see why, note that (11) implies that in a CAPM equilibrium, thenon-linear part of each agent's marginal utility function prices the assets atzero. Consequently, the non-linear part of every agent's marginal utility func-tion can be deleted without affecting the equilibrium. That is, all agents'utility functions can be replaced with quadratic functions, so the conditionsthat provide the CAPM and mean�variance maximization are identicalin this economy. The proposition therefore answers the question of whenexpected utility maximization is consistent with mean�variance maximization.

It has long been recognized (see [9]) that the presence of financial (i.e.,zero net supply) assets with non-linear payoffs such as options pose achallenge to the CAPM since the distribution of these assets returns areclearly not normal (or more generally two-fund separating). The questionthen arises: Under what conditions can a zero net supply asset with anarbitrary return distribution be added to an economy without upsetting anexisting CAPM equilibrium? Proposition 3.1 provides the answer to thequestion. Since the asset span only enters through 1, the conditions in theproposition remain unchanged whenever the part of the return of the newasset not spanned by the factors is a residual; i.e., if the return of thenew asset is denoted Ri , then the CAPM will continue to hold so long asE[Ri&R� i&�J

j=1 bji $j | Rm]=0.When the CAPM holds, both the market and agents' marginal utility

functions price assets. Consequently, their projections onto the factors mustbe the same. The following corollary proves this:



Corollary 3.2. In an equilibrium of the economy, the CAPM holds if,and only if,

;}(x})=c} bm , c} # R, \} # K, x # 5(r), (13)

where the jth element of ;}(x}) # RJ is ;}j(x})=Cov(U $}(x}), $j) andbm=(bm1 , bm2 , ..., bmJ ) is the market portfolio's sensitivity to the factors(the market betas).

From Proposition 3.1 and its corollaries it is apparent why the CAPMholds under the two well-known conditions. Equation (12) implies that theCAPM can only hold if all agents' utility functions differ from a quadraticfunction by a term whose derivative is orthogonal to every factor in theeconomy. The quadratic function itself trivially satisfies this condition.Similarly, when J=1, ;} and bm are scalars and so (13) does not restrict;} . Thus, any two-fund separating distribution (where one factor is theriskless asset) satisfies the corollary.

Finally, we sketch7 how the results in this paper can be used to answera question posed by Ross [22, p. 889]: What joint restrictions on preferencesand distributions provide the CAPM? This question was motivated by theobservation that quadratic utility and two-fund separating distributions,by themselves, are necessary and sufficient for the CAPM. That is, Ross[21] showed that in the absence of any further assumptions on primitives,the CAPM can only hold under the two-fund separating distributionalassumption, and it follows immediately from [5] that in the absence ofany further assumptions on primitives, the CAPM can only hold under thequadratic preference assumption. Since assumptions on endowments areruled out on economic grounds, the only potential set of undiscovered condi-tions were joint restrictions on preferences and distributions.

Equation (13) implies, for any two factors k and l, E[U $}(x})(bmk $l&bml $k)]=0. Assume all agents have polynomial utility functions of orderN. This equation then becomes,

:N&1

n=0

E _a}n \%}+;} :

J

j=1

bmj $j+n

(bmk $l&bml $k)&=0, (14)

where %} and ;} are defined implicitly. By expanding each term on the left-hand side of the above series, it becomes clear that the only way it will bezero for any endowment allocation (i.e., for every bmj) and any two factorsk and l is if for every positive integer m�N and every non-negative integer[nj ]J

j=1 such that �Jj=1 nj=m,


7 A complete derivation is available on the author's WWW home page.


E _ `J

j=1

$njj &={

0 if any nj is odd,

(15)

(nk&1) E _$2l $nk&2

k `j{k, l

$njj &

if nl=0 and every nj is not odd,

\nl&1nk+1+ E _$nl&2

l $nk+2k `

j{k, l

$njj & o.w.

These restrictions (i.e., polynomial utility functions of order N and (15))therefore define a class of joint restrictions on preferences and distributionsthat provide the CAPM. Since the only analytic functions with finite powerseries expansions are the polynomials, every other analytic function musthave an infinite power series expansion. It therefore immediately followsthat the only way the CAPM can hold when utility functions are assumedto be analytic but not polynomial, is if either J=1 or (15) holds withN=�. As was pointed out above, the case for N=� (i.e., any analyticfunction) was solved by Ross [21]. Thus no other joint restrictions onpreferences and distributions exist that provide the CAPM.

Intuitively, if agents' utility functions are polynomials of order N or less,agents will not care about any moment above the N th. Therefore, to getthe CAPM to hold, all central moments from the third to the Nth need tobe restricted so that agents are indifferent to them. The above momentrestrictions on the distribution of the factors (i.e., (15)) effects this. The casewhen J=1 corresponds to Ross' two-fund separating distributions. WhenJ{1, and the utility functions are not polynomials, then the CAPM canonly hold if all factor moments satisfy (15). An example of such aneconomy is asset returns that are elliptically distributed.8 As the reader canverify, every moment of the elliptical distributions satisfies (15). These jointrestrictions therefore provide an intuitive link between quadratic utilities onthe one hand, and elliptical distributions on the other hand. To get theCAPM, the power series coefficients of agents' utility functions and thecentral moments of the factor distributions must be restricted. Quadraticutility and elliptical distributions represent the extreme position of usingonly one of these strategies at a time.

Unfortunately, the complete set of restrictions that provide the CAPMare no more realistic than the previously known set. Even if consumptionis restricted to be positive, no finite order polynomial is both strictlyincreasing and strictly concave in the positive domain. When the class ofutility functions is expanded to include any analytic function that is


8 See [6] or [20]. Note that although these authors assume that all assets are elliptically dis-tributed, all that is actually required is that the factors be elliptically distributed.


everywhere strictly concave and strictly increasing, Ross [21] has shown thatthe CAPM will only hold if return distributions are two-fund separating.Ross' conditions therefore provide the only realistic way, within the expectedutility paradigm, to get the CAPM (or mean�variance preferences).

APPENDIX

A. Proof of Proposition 3.1

Proof. Dybvig and Ingersoll [9] have shown that :m is mean�varianceefficient if and only if

r=E _Ri {1&R� m&r

_2m

(Rm&R� m)=& \i # I. (16)

For any } # K in an admissible equilibrium ([11, p. 66]),

1E[U$}(x})]

E[U$}(x}) } Ri]=r \i # I. (17)

Therefore, when the market portfolio is mean�variance efficient in equi-librium then both (16) and (17) hold so:

E _{U$}(x})&E[U$}(x})] _1&(R� m&r)

_2m

(Rm&R� m)&= Ri&=0 \i # I.

(18)

Using (3), (6), (9), and that J is the minimum number of factors, it can beshown that the only way (18) can be satisfied9 is if

E _{U$}(x})+E[U$}(x})] Rm(R� m&r)

_2m = $j&=0 \j. (19)

The above derivation shows that the CAPM holds in an equilibrium if andonly if (19) holds for every } when x # 5(r).

We next show that when x # 5(r), (11) implies (19) and so the aboveargument allows us to conclude that the CAPM holds. Setting \=r in (11)provides:

U$}(x})=#}(x})&E[#}(C} Rm+(W}&C}) r)]

[_2m�(R� m&r)]+R� m _x}&r(W}&C})

C} & . (20)


9 A proof of this fact is available on the author's WWW home page.


Taking expectations (for every }),

E[U$}(x})]=E[#}(x})]&E[#}(C}Rm+(W}&C}) r)]

[_2m�(R� m&r)]+R� m

__E[x}]&r(W}&C})C} & . (21)

Using (20) and (21) it is straightforward to verify that (19) is satisfied sothe CAPM holds.

Last, we need to show that if the CAPM holds in at least one equi-librium of the economy, then the conditions in the proposition will besatisfied. If the CAPM holds in equilibrium, then (19) holds for every }when x # 5(r). This implies that

U$}(x})+E[U$}(x})] _&2m Rm(R� m&r)=#}(x}), (22)

where #} # 1[x}]. Taking expectations of both sides of (22) and solving forE[U$}(x})] provides:

E[U$}(x})]=E[#}(x})]

1+R� m(R� m&r)�_2m

=E[#}(C}Rm+(W}&C}) r)]

1+R� m(R� m&r)�_2m

. (23)

Substituting (23) into (22) and writing x} out explicitly, we have that forevery },

U$}(C}Rm+(W}&C}) r)=#}(C}Rm+(W}&C}) r)

&E[#}(C}Rm+(W}&C}) r)]

[_2m�(R� m&r)]+R� m

Rm , (24)

which is (11). K

B. Proof of Corollary 3.2

Proof. (1) Necessity: By Proposition 3.1, if the CAPM holds inequilibrium then (11) holds. Thus, c}=&E[#}(C}Rm+(W}&C}) r)]�([_2

m�(R� m&r)]+R� m). (2) Sufficiency: Take the allocation in 5(r) forwhich the projection of every agent's marginal utility function onto thefactors is a constant multiple of the market betas and project agent }'smarginal utility function at this allocation onto the market

U$}(x})=:}+c}Rm+=}(x}). (25)



where,

x}=C} Rm+(W}&C}) r (26)

E[=}(x})]=E[=}(x}) $j]=0, \j=1, ..., J (27)

c}=E[U$}(x})(Rm&R� m)]

_2m

(28)

:}=E[U$}(x})]&c}R� m . (29)

Note that (27) follows from the fact that the projection of every agent'smarginal utility function onto the factors is a constant multiple of themarket projection onto the factors. Let

#}(x})#=}(x})+:}. (30)

so that #} # 1[C}Rm+(W}&C}) r]. Using the first-order condition (i.e.,(17)) gives E[U$}(x})(Rm&R� m)]=E[U$}(x})(r&R� m)], and so (28)becomes

c}=E[U$}(x})(r&R� m)]

_2m

=&E[U$}(x})](R� m&r)

_2m

. (31)

Substituting (30) and (31) into (25), taking expectations, and solving forE[U$}(x})] provides,

E[U$}(x})]=E[#}(x})]

(1+[(R� m&r) R� m�_2m])

. (32)

Substituting (32) into (31) provides another expression for c} . Using thisexpression (as well as (30)) in (25) provides,

U$}(x})=#}(x})&E[#}(x})]

[_2m�(R� m&r)]+R� m

Rm , (33)

which, together with (26), provides (11). By Proposition 3.1 the CAPMholds. K

REFERENCES

1. M. Allingham, Existence theorems in the capital asset pricing model, Econometrica 59(1991), 1169�1174.

2. G. O. Bierwag, The rationale of the mean�standard deviation analysis: Comment, Amer.Econ. Rev. 64 (1974), 431�33.



3. F. Black, Capital market equilibrium with restricted borrowing, J. Bus. 45 (1972),444�454.

4. K. Borch, The rationale of the mean�standard deviation analysis: Comment, Amer. Econ.Rev. 64 (1974), 428�30.

5. D. Cass and J. E. Stiglitz, The structure of investor preferences and asset returns, andseparability in portfolio allocation: A contribution to the pure theory of mutual funds,J. Econ. Theory 2 (1970), 102�160.

6. G. Chamberlain, A characterization of the distributions that imply mean�variance utilityfunctions, J. Econ. Theory 29 (1983), 185�201.

7. J. S. Chipman, The ordering of portfolios in terms of mean and variance, Rev. Econ. Stud.40 (1973), 167�90.

8. D. Duffie, ``Security Markets Stochastic Models,'' Academic Press, San Diego, CA, 1988.9. P. H. Dybvig and J. E. Ingersoll, Mean�variance theory in complete markets, J. Bus. 55

(1982), 233�252.10. M. S. Feldstein, Mean�variance analysis in the theory of liquidity preference and portfolio

selection, Rev. Econ. Stud. 36 (1969), 5�12.11. J. E. Ingersoll, ``Theory of Financial Decision Making,'' Rowman 6 Littlefield, Totowa,

NJ, 1987.12. H. Levy, The rationale of the mean�standard deviation analysis: Comment, Amer. Econ.

Rev. 64 (1974), 434�41.13. H. Levy and H. M. Markowitz, Approximating expected utility by a function of mean and

variance, Amer. Econ. Rev. 69 (1979), 308�17.14. J. Lintner, The valuation of risky assets and the selection of risky investments in stock

portfolios and capital budgets, Rev. Econ. Statist. 47 (1965), 346�382.15. A. Lo� ffler, Variance aversion implies +-_2-criterion, working paper, Graduiertenkolleg des

FB Wirtschaftswissenschaften, Humboldt Universita� t, 1994.16. J. Meyer, Two-moment decision models and expected utility maximization, Amer. Econ.

Rev. 77 (1987), 421�30.17. J. Mossin, Equilibrium in a capital asset market, Econometrica 35 (1966), 768�783.18. L. T. Nielsen, Existence of equilibrium in CAPM, J. Econ. Theory 52 (1990), 223�231.19. L. T. Nielsen, Equilibrium in CAPM without a riskless asset, Rev. Econ. Stud. 57 (1990),

315�324.20. R. Owen and R. Rabinovitch, On the class of elliptical distributions and their applications

to the theory of portfolio choice, J. Finance 38 (1983), 745�752.21. S. A. Ross, Mutual fund separation in financial theory��The separating distributions,

J. Econ. Theory 17 (1978), 254�286.22. S. A. Ross, The current status of the capital asset pricing model (CAPM), J. Finance 33

(1978), 885�901.23. W. Sharpe, Capital asset prices: A theory of capital market equilibrium under conditions

of risk, J. Finance 19 (1964), 425�442.24. J. Tobin, Liquidity preference as behavior towards risk, Rev. Econ. Stud. 25 (1958), 65�86.25. S. C. Tsiang, The rationale of the mean�standard deviation analysis, skewness preference,

and the demand for money, Amer. Econ. Rev. 62 (1972), 354�71.


Date post:	14-Jan-2017
Category:	Documents
Upload:	lequynh
View:	218 times
Download:	1 times

Necessary Conditions for the CAPM

Documents