Theoretical Economics Letters, 2013, 3, 306316 Published Online December 2013 (http://www.scirp.org/journal/tel) http://dx.doi.org/10.4236/tel.2013.36052 Open Access TEL wMPS Risk Aversion and the CAPM* Phelim P. Boyle1, Chenghu Ma2# 1School of Business and Economics, Wilfrid Laurier University, Waterloo, Canada 2School of Management & Fudan Finance Center, Fudan University, Shanghai, China Email: #machenghu@fudan.edu.cn Received September 29, 2013; revised October 29, 2013; accepted November 6, 2013 Copyright © 2013 Phelim P. Boyle, Chenghu Ma. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In accordance of the Creative Commons Attribution License all Copyrights © 2013 are reserved for SCIRP and the owner of the intellectual property Phelim P. Boyle, Chenghu Ma. All Copyright © 2013 are guarded by law and by SCIRP as a guardian. ABSTRACT This paper establishes general conditions for the validity of mutual fund separation and the equilibrium CAPM. We use partial preference orders that display weak form mean preserving spread (wMPS) risk aversion in the sense of Ma (2011). We derive this result without imposing any distributional assumptions on asset returns. The results hold even when the market contains an infinite number of securities and a continuum number of traders, and when each investor is permitted to hold some (arbitrary) finite portfolios. A proof of existence of equilibrium CAPM is provided for finite economies by assuming that when preferences are constrained on the market subspace spanned by the risk free bond, the market portfolios admit continuous utility representations. Keywords: CAPM; wMPS; Risk Aversion; Infinite & Incomplete Market; NonGaussian Returns 1. Introduction This paper provides general conditions for the classical CAPM as an equilibrium model in economies with a fric tionless market. First, we show that, if equilibrium exists, then the asset returns must satisfy the CAPM if all in vestors are risk averse in a sense of meanpreserving spread1 (Theorem 1). Second, we prove the existence of equilibrium satisfying the CAPM for economies with wMPS risk averse investors, without assuming complete markets (Theorem 2). Our results advance the literature and lead to deeper understanding of the extent to which the CAPM holds in equilibrium: 1) preferences satis fy wMPS risk aversion, and such preferences can be incomplete and nontransitive; 2) the population space, or investors’ type space, can be finite or continuum infi nite; and 3) the market may contain a finite, or a count able infinite number of securities, while the market port folio is composed of a (fixed) finite number of risky as sets2. It has been known for a long time that if all investors have meanvariance preferences, then the CAPM holds ([57]). It is also known that meanvariance preferences persist for general probabilistic sophisticated expected and nonexpected utility functions when asset returns are elliptically distributed ([8,9]). It is, therefore, of particu lar interest to explore if the CAPM holds when prefer ences do not fall into the meanvariance class, particu larly when asset returns may follow arbitrary distribu tions3. Since the existence of equilibrium CAPM docu *This paper has benefitted from discussions with Jonathan Berk, Ber nard Cornet, Darrell Duffie, Editya Goenka, Chiaki Hara, Atsushi Kajii Juan Pedro Gómez, Patrick Leoni, Ning Sun, Dong Chul Won, and Zaifu Yang. Ma acknowledges financial supports from the Economic and Social Science Research Council, UK (R000223337), the Nature Science Foundation of China (70871100, 71271058), and the Fudan Finance Center (EZH4301102/020). #Corresponding author. 1An investor exhibits wMPS risk aversion if, for all random payoffs and XYX , the investor would prefer to whenever X Y 0Cov and . The notion of meanpreservingspead used in this paper is weaker than the notion of MPS in [1,2], and is thus denoted wMPS. Our wMPS risk averse preferences correspond to the strict variance averse preferences used by [3]. , 0X 2The validity of CAPM, along with existence of equilibrium CAPM, for the case of a finite number of securities and a finite number of investors was documented in [4], Chapter 5. This paper can be thus regarded as a generalization of [4] to infinite economies. 3In Duffie’s book [3] it contains a derivation of CAPM. Duffie’s deri vation rests on several explicit and implicit technical assumptions on asset returns: first, the existence of a continuous linear pricing rule in the market span; second, the market subspace is assumed to be a closed subset in 2. Whilst the linearity of the pricing rule is necessarily im lied by the noarbitrage condition as part of the equilibrium conditions, the closedness assumption on the market space is somewhat arbitrary.
P. P. BOYLE, C. H. MA 307 mented in literature can be largely built up on the finite specification with respect to the number of tradable secu rities and the number of investors, it is thus also desirable to consider infinite economies. In comparison with the above cited work, this paper makes no distributional assumption on asset returns, and purely focuses on investor’s risk preferences over the random payoffs. The assumption of wMPS risk aversion is appealing because 1) it captures investor’s psycho logical aversion towards “increase in risk” in a natural way, 2) wMPS risk averse preferences are not restricted to certain classes of expected or nonexpected utility functions. Therefore, they are not subject to criticisms such as the wellknown Allais Paradox and other defi ciencies associated with expected utility functions. It is noted that expected utility functions may violate wMPS risk aversion—and it could even be argued that this con stitutes another drawback of expected utility functions. In fact, we shall show that CAPM holds even when inves tors’ preferences are not complete, nor transitive. The relationship between meanvariance utility func tions and the wMPS risk averse preferences has been analyzed. It is shown in [10] that, when the market con tains a finite number of risky assets (but no less than three) and when the market span forms a convex cone of 2 if the wMPS risk averse preference is represented by continuously Frechétdifferentiable (in the 2norm) utility functions, then the preference must admit a mean variance utility representation. Nevertheless, one may still want to treat the meanvariance preferences as a sub set of the wMPS risk averse preferences. In fact, a bi nary relationship satisfying the wMPS risk aversion property constitutes a partial order, which may not admit an utility representation. As an illustrative example, con sider the following Lexicographic binary relation defined on : For a given positive integer , 2n Y if dominates by the first order stochas Y tic dominance (i.e., dist XY , where 0 and 0 ); or, if there exists such that (a) 2mn ii Y 1 1 m for , and (b) 0im 1 m m 1 m Y . In this binary relation, in order to rank random payoffs and Y, which may not dominate each other by first order stochastic domi nance, one may need to compare their moments up to the th order. The binary relation displays wMPS risk aversion property, yet it does not admit a meanvariance representation! Indeed, most of the analyses to be carried out in this paper do not rely on the assumption on the existence of a meanvariance utility representation. n Equilibrium with continuum of traders was thoroughly studied in literature ([1113]). [13] also contains an ab stract treatment of equilibrium with incomplete orders. Existence of equilibrium with incomplete and nontransi tive preferences (with finite number of traders) traces to [14,15]. The wMPS risk averse preferences assumed in this paper is neither complete, nor transitive, thus falls into the category studied in [14]. Also, in this paper we adopt general population type space with arbitrary po pulation mass distribution functions which may accom modate the case of finite, countable infinite, as well as a continuum range of investors. It is noted that the existing proofs of equilibrium pro vided in these earlier studies cited above are all with re spect to the standard Warasian equilibrium in a context of certainty. In contrast, the existing proof provided in this paper is for a particular stochastic finance economy. Specifically, we assume that the financial market can be incomplete, and that the economy consists of an arbitrary (finite, countable infinite or continuum) number of inves tors and each investor has (incomplete and nontransitive) a preference order displaying wMPS risk aversion. The key assumption underlying the existing proof in our pa per is that when investor’s preference is restricted to the market subspace of efficient portfolios, it admits a con tinuous utility representation, even though an utility rep resentation over the entire market span may not exist. From technical point of view, our existing proofs are very much in line with those in the literature ([16,17]). [16] assumes a finite number of meanvariance investors and a complete market, while [17] assumes a finite num ber of investors and that each investor has an wellde fined utility function defined on the market subspace spanned by the risk free bond and a common risky asset —the market portfolio. The literature contains several other existing proofs for the equilibrium CAPM, all re lying crucially on the assumption of meanvariance pref erences (See, for instance, [18,19]). The derivation of CAPM provided in this paper fol lows the heritage of the traditional approach ([5,6]). It is based on the relevance of meanvariance efficient fron tier to investor’s optimal portfolio holdings. Following the standard treatment in literature ([18,20]) we restrict investors to hold portfolios involving in only a finite number of securities even though the market may contain an infinite number of tradable securities. The finiteness assumption on the market portfolio is not restrictive since, in practice, the market portfolio is an index of a finite number of stocks. Derivative securities written on the stocks and indeces of stocks, which represent a large or even an infinite number of traded financial securities, are not in the composite of the market portfolio. Our model thus provides a useful mechanism to price, which pro vides not only those primitive securities in the composite of the market portfolio, but also securities out of the composite of the market portfolio. The remainder of the paper is organized as follows: Section 2 describes the model and summarizes the main Open Access TEL
P. P. BOYLE, C. H. MA 308 results. Section 3 introduces the notion of generalized efficient frontiers. Section 3 also discusses wMPS risk averse investor’s optimal portfolio choice problem and its relevance to the generalized efficient frontier. Section 4 includes a formal derivation of the CAPM, along with the twofund separation theorem. The proofs of the exis tence of equilibrium are outlined in Section 5. Section 6 concludes the paper. 2. Outline of the Model and Main Results This section describes the basic framework and summa rizes the main results of the paper. We start with several useful notations. Let and f be two realvalued functions (vectors). We write if fg tgt for all ; if tfg tgt for all t and gf ; and g if tgt for all t. We consider a twoperiod exchange economy with a frictionless capital market and heterogenous agents. The uncertainty is summarized by a probability state space with probability measure . The topological properties of the state space are otherwise not specified. There exists a countable set of nonredundant risky secu rities available for trade4. Let , 1,, ,Jj #J j be the number of tradable (risky) securities. Security is associated with a statecontingent random payoff . Security 0 is a risk free bond with unit payoff in all future states (i.e., ). Let , 2 j 0 1 p be the price for security . The (total) return on security is j j thus given by 2, j j j Rp (if ). The 0 j p return for the risk free bond is denoted R. Let # be a set of admissible portfolios on 5. We restrict to consist of portfolios that involve only a finite number of risky assets; that is, #:sup: 0. J j jJ (1) Here, represents the number of shares in risky security . The market span consists of all possible portfolio payoffs obtainable by trading: j 2, . 2,: s.t.dd (2) Following the standard literature, we may refer the composite of a portfolio by its weight allocated to each of the risky assets, as well as the number of shares in vested in the assets. Precisely, let be a copy of . Let be the initial wealth. Let 0 W be a portfolio with representing the proportion of initial wealth invested in risky security . The number of shares in j security held is thus given by j0j W p , and the amount invested in the risk free bond is 011W , where # 1 is the vector of unit elements. For each , the portfolio return is j j jJ RR R f R, with expected portfolio return and standard deviation respec tively denoted by and . There is a bunch of investors, indexed by tT . can be taken as a closed interval, say T 0,1 , ohe Eucli dean space . Theopulation distribution is summa rized by a positive measure on such t for each Borel set BT, population mass belonging to B. Typt investr is endowed with an vector of shares n p t hat, T e B is th o e et . Investors express preferences over . The typet investor’spreference is summa rized by a partial binary relation t on . With these, the exchange economy is summarized by ,,;,,, t tT Tet . (3) Economy is finite if the state space contains only finite elements; otherwise, it is infinite. For finite economies, it is sensible to assume the number of non redundant securities is finite and no more than the num ber of states. Throughout this paper, we make the fol lowing assumptions for : A1 0 0 and the payoffs for the risky securities have a positive definite (infinitedimensional) covariance matrix ; that is, for all , , and x 00xx 00xx x . A2 satisfies Feller’s property: Let be an open set in , and O O be the closure of . For any realval ued integrable function on , O fTO tyy ,f to be continuous on , O a.s., implies ,d T yfyt t to be continuous on ; particularly, for all O0 yO , d 00 lim,lim , yy yy TT d ytf yt t t whenever the limit on the right hand side converges6. A3 is :eT measurable with 0et . Moreover, Tet 0dt m and 0 mm . A4 is :T measurable. And, for each , is strictly monotonic and homothetic, and displays wMPS risk aversion in the sense that t t t Y when ever dist YX , where 0 and Cov ,0X , and that the preference strictly holds if 0 . 4By nonredundancy, we mean that for each security , its payoff jj cannot be duplicated by holding a finite portfolio of other tradable securities. 5Much of the analysis below applies for a general admissible set such as defined in Section 3, which may contain portfolios with an infinite number of securities. 2 When trading takes place at time 0, after observing security prices # 0,,, j pp p investor t 6The Feller’s condition is purely technical. It obviously holds true when the economy consists of a finite number of investors. In this case, corresponds to a measure of finite supports. Open Access TEL
P. P. BOYLE, C. H. MA 309 may assess his initial wealth and revise his portfolio holding accordingly. 0 Wt pet is feasible (for ) if . A feasible allocation t 0 pWt is optimal (for ) if t t for all that is feasible for . By strict monotonicity of the preference relation, at the optimum, the budget constraint must hold with equality; that is, . t 0 0 Wtp If security prices are all nonzero as must be the case when , the optimality condition can be expressed in terms of portfolio weights 00Wt , allocated to each risky securities. For instance, for , we have is optimal if, for all t RR m :T d Ttt . The market equilibrium for consists of a price vector and an asset allocation that is p measurable such that (i) p :T ; and (ii) for all , is optimal w.r.t. for type t inves tors. t t Again, for the case when equilibrium security prices are all nonzero, and when investors’ initial wealth at are all positive, the equilibrium condition can be equiva lently stated in terms of portfolio allocation p with condition (i) replaced by d 0jm Tj tW p tt for all , in addition to the optimality condition (in term of j ) for all type investors. Let be the market portfolio with m , m jj m pjJ p m j . Let be the return of the market m R portfolio, and let mm R . We state the first main result of the paper: Theorem 1 Suppose conditions A1A4 hold. If the economy achieves its equilibrium at (with p0 j p for all ) with equilibrium portfolio allocation j , then (1) there exists :T d1t , that is measurable and satisfies , such that Tt tm t for all ; tT (2) the capital asset pricing model (CAPM) holds , f R fm R (4) where Cov Var ,m m RR R. Theorem 1(1) is about mutual fund separation of equilibrium allocation—investors of all type optimally hold a proportion of the market portfolio, in addition to a position in the risk free bond. It extends the well known Black's twofund separation theorem [21] to this infinite economy. The proof of the statement is based on insight on the mathematical properties of the efficient frontier covered in Section 3. The notion of efficient frontier, direct extension to that of Markowitz ([22,23]) by con sidering the possibility of an infinite number of tradable securities. Proof of the equilibrium CAPM outlined in Theorem 1(2) is summarized in Section 4. To find an equilibrium price vector, we which is formally introduced below in Definition 3.1, is a work with a set of “normalized prices” satisfying the CAPM, and the term “normalized prices” will be made precise in Section 5. Let # be the set of normalized price vector satisfyingAPM. We shall show that the set the C forms a straight line on the vector space # (Re: Lem mas 4 and 5), and that, for each p, iestor’s opti nv expre the market subspace spanned by th mal portfolio, if it exists, must bessed as a combi nation of the risk free bond and the market portfolio (Re: Lemma 8). Denote by0,m nd and the risk free boe market portfolio. We write :, . m xy xy 0,m(5) Recall that, by assumption A3, m is finite and , m is bounded. To prove e existence of assume further that investor’s preference restricted on 0,m admits an utility representation7. Put this formally A5 For eact equilibrium, we th h, the restriction of on ad m t 0,m nd,its a utility representation: 0, : tm U. A the utility function ,t xyU x m y tinuous and strictly quasico Assumption A5 is logi is con nve. ca cally consistent with the wMPS ris ite and satisfies A1A5. Th heorem 2 is contained in Section 5. For th e restriction ofon the market subspace sp we see that A5’ implies A5 when se 3. MV Analysis and Portfolio Choice variance k averse behavioral assumption (A4) since there exists no wMPS dominating relations for each bundles in 0,m . The second main result of this paper concerns the ence of equilibrium CAPM. Theorem 2 Suppose is fin exist en, the equilibrium exists and, at equilibrium, the CAPM holds. The proof of T e validity of Theorem 2, we may replace A5 with A5’ below: A5’ Th t anned by those efficient portfolios (if exists) admits a utility representation. In fact, by Lemma 8, curity prices are governed by the CAPM. This section starts with a discussion of mean efficiency and generalized meanvariance efficiency as an infinite dimensional extension to Markowitz mean variance efficiency. It follows with an indepth discus sion on the relevance of the meanvariance efficiency to 7Since payoffs within the market subspace do not display mean pre serving spead to each other, assumption A5 is thus logically consistent with assumption A4. Open Access TEL
P. P. BOYLE, C. H. MA 310 optimal portfolio choice by wMPS risk averse investors. 3.1. Efficient Frontier price vector Taking as given an arbitrary# p with 0 j p for all j so that the asset returns are wellde r all tradle securities. Let # fined foab be an ad missible portfolio space. For each , the portfolio return has its mean return and standaviation respec tively given by rd de 1 2 1, , ff RR where and represent the vector of expected re arturns a the viancecovariance matrix of the risky returns. Similarly, for two arbitrary risky portfolios nd and , the covariance of the two portfolio returns is , . For pure risky portfolios we have 11. The fowing definition of efficient portfolio and effi ci For oll ent frontier for applies whether or not there is a risk free asset. Definition 10 , 0 is said to be effi cient at 0 if 0 : 0 arg min . The curve ,:is efficient (6) is referred to as the meanvariance efficient fronti roy of ef t er, or simply the “efficient frontier”, with respect to . The next proposition describes a general ppert ficient portfolios. Proposition 1 Le 0 be an efficient portfolio with mean 0 . For all with 0, we have: RR 0 with 0 and Cov 0 0, .R Proof. Consider the set of portfolios nvex combina 1: formed by co 0 tions of 0 and . The rn se portfolios all have the same mean retu given by 0 Since 0 is efficient at 0 with standard deviation0 , 0 1 m achieve its minimum at 0; us t that is, 22 0argmin 22 0 21, 1 0 The first order condition leads to 2 00 , . Let 0 RR . We have: 0 and Co 0 v ,0R . ■ Notice thf does nuire assumption th ite portfolios, w e that is relevant to choices made by at this prooot req on e finiteness of the number of securities, nor requires the covariance matrix to be nonsingular. Since investors are restricted to hold fin e shall naturally pay special attention to the efficient frontier with respect to . In contrast to the finitensional case orig considered by Markowitz in [22,23], we do encounter one technical dif ficulty; that is, when the number of tradable securities is infinite the efficient frontier could be empty and thus not well defined. There is another cas e diminally wMPS risk averse investors, and is of particular in terest. This refers to the efficient frontier for 2 . Here, # 2: J is an Hilbert space with inner product , and norm , . The portfolio space forms t not closed su a dense bubset of 2. Since elements in 2 are with possibly infimber of securi tiesficient frontier w.r.t. 2, denoted nite nu , the ef , is called the generalized efficient frg.e.f.). Unlike ontier ( , we shall show that is well de fined under general conditions (see Proposition 3). Similar to the finite dimensional case, an analytic ex pression for the g.e.f. can be obtained. As it turns out, fairly forms a hyperbola on the μσ plane in absence of free asset. And, in presence of risk free asset, risk is well defined when the mean return for the minimuisk portfolio differs from the risk free interest rate. For this latter case, m r is composed of the generalized tangent ray and the reflection of the generalized tangent ray. Ac cordingly, the classical BlackTobin mutual fund separa tion theorems (see [21] and [24]) extend to this infinite dimensional setting for 2. 3.2. Portfolio Choice mber of a risky asset, it is well When there is a finite nu known that the optimal portfolio for meanvariance in vestors, if it exists, must be located on the Markowitz meanvariance efficient frontier. The existing literature tells us little about the relevance of efficient frontier for portfolio choices made by investors whose preferences are not in the meanvariance class8. The difficulties in establishing such relevancy are well known: First, an in vestor’s optimal portfolio may not exist even though the efficient frontier is well defined. This occurs, for exam ple, when the security prices violate the noarbitrage con ditions which are necessary for the existence of an opti mal portfolio for all investors with increasing and con tinuous utility functions. Second, even if optimal portfo lio exists, the efficient portfolio with mean return corre sponding to that of the optimal portfolio may not exist. This occurs, for instance, when the mean return of the minimum variance portfolio is equal to the risk free rate. Finally, when the optimal portfolio and efficient portfolio both exist, it is still not obvious if the investor would choose to optimally hold the efficient portfolio because 8See [25,26] for results on twofund separation for investors with risk averse expected utility functions. They prove that, the separating port folios, if they exist, must be on the efficient frontier. See also [27] fo conditions on asset returns and the expected utility functions that are sufficient for both twofund separation and the CAPM. Open Access TEL
P. P. BOYLE, C. H. MA 311 the investor may care about higher moments beyond the first two. In this section, we study the optimal choice behavior for risk averse investors and explore the rele vance of the efficient portfolios in their optimal choices. Here, we restrict our attention to investors who are risk averse in the sense of wMPS risk aversion. Recalling first the definition of wMPS risk aversion. An investor is said to display wMPS risk aversion if Y whenever dist YX , where 0 and Cov X , 0 . The rence hold0strict prefes if . In this case, we say that Y is identical in distribo a wMPS of ution t . For wMPS risk averse investors, our next result fol lows as a corollary to Proposition 1. Proposition 2 Let be an optimal portfolio holding for a wMPS risk ave investor. Let rse be the portfo lio mean return for the optimal portfolio . Then, if the efficient portfolio at exists, the optimal portfolio must be efficient. It is well known that, with a finite number of securities, efficient frontiers are welldefined and all efficient port folios can be expressed as a convex combination of two efficient portfolios. The optimal portfolio holdings for wMPS risk averse investors can be easily characterized because they would have to be located on the efficient frontier. When investors face an infinite number of in vestment opportunities #J , it is not clear if mu tual fund separation holdt, as to be illustrated below, in presence of infinite number of risky assets, the efficient frontier for is generically not defined (non existence), and the timal portfolio correspondence (valued in ) for the wMPS risk averse investors is generically epty. We first conside s. In fac po m r the case when the market contains pu d rely risky portfolios. We impose the following two conditions on the coefficients9: C1 is positive definite, an 11 2 ,1 ; C2 Nndegeneracy: o is not proportional to 1. nd e following im Suppose conditions C1 and C2 hold. Fo Under conditions C1 a C2, the g.e.f is well defined, and th 2g portant risk decomposition result holds for the general ized portfolios: Proposition 3 r all 2 , there exists a unique 0 such that R be expressed as a wMPS of R0 . Moreover, for a0 ll , the generalized efficient portfolio at 0 is given by 111 00 ,,1eA 2 (7) where 1 ,, ee is a positive definite matrix. 22 Proof. The first statement follows the same argument as in Proposition 1. Condition C2 implies is not proportional to 1e 1 , which in turn implies 1111 ,ee , and matrix has a welldefined inverse 1 . The quadratic optimization problem 2 211 0 min:,1and,e can be readily solved with the Lagrangean method, and the optimal solution is given by Equation (7). ■ Notice that, the riskdecomposition theorem holds for all generalized risky portfolios in , thus in par ticular for risky portfolios in . That is, re turn for each risky portfolio 2 2 in must admit as a wMPS to that of a generalized efficient portfolio 02 on . This implies that wMPS risk averse investors would tend to hold generalized efficient portfolios if they were allowed to hold an infinite number of securities. In other words, investors would, in general, not be satiated with holding any arbitrary finite number of securities10. This results in (generic) nonexistence of optimal portfolio holdings for wMPS risk averse inves tors who are restricted to hold finite portfolios in . Secondly, we consider the case when the market con tains a risk free bond, and maintain conditions C1 and C2 for the risky assets. Let 11 2 1 1, 1 11. If R , then the generalized tangent portfolio would be well defined and be given by 11 2 11 1 1. 1, 1 f m f R R (8) When m contains an infinite number of nonzero elements (that is, m ), all generalized efficient port folios (except the risk free bond) would not be admissible. This causes the generic nonexistence of optimal portfo lios for wMPS risk averse investors who are restricted to hold finite portfolios. For the extreme case when the generalized tangent portfolio is finite (that is, m ), 10However, there are two exceptions to this statement: (a) when #J , the g.e.f. g reduces to the Markowitz efficient frontier, and all generalized efficient portfolios become efficient; (b) when #J and the (normalized) risky assets contain just a finite number distinct expected returns. 11Here, is the mean return for the generalized minimum variance ortfolio 1 2 1 1 1 . 9We consider normalized securities that forms an orthonormal basis o the market span . Each normalized security is a finite portfolio in . The set of normalized securities are uncorrelated to each other, and they generate the same market span from the original set of securi ties . For normalized securities, is diagonal with positive di agonal elements, its inverse is also well defined and diagonal. 1 Open Access TEL
P. P. BOYLE, C. H. MA 312 the tangent ray and its reflection constitute the efficient frontier w.r.t. in presence of an risk free bond. Based on this discussion, the optimal portfolio hold ings for wMPS risk averse investors can be character ized. Proposition 4 Suppose conditions C1 and C2 hold. Consider an investor with monotonic and wMPS risk averse preference. a) The investor’s optimal portfolio, if it exists, must have an expected return no less than the risk free interest rate, R . b) When R , the optimal portfolio, if it exists, must be on the generalized efficient rays and be ex pressed as a combination of the risk free bond and the generalized tangent portfolio m . In this case, the gen eralized tangent portfolio must be finite; that is, m . c) When R , the optimal portfolio, if not risk free, does not exist. If the investor were allowed to hold an infinite number of securities, say , the optimal portfolio would exist and be expressed as a combination of the risk free bond and the generalized tangent portfolio. Since the generalized tangent portfolio may not be finite, hence does not belong to , the optimal demand correspon dence in for an wMPS risk averse investor can be empty. 2 4. Proof of Theorem 1 This section builds upon our earlier results to derive the CAPM with wMPS risk averse investors. Recall that investors have homogeneous beliefs and so they will perceive the same generalized efficient frontier as de scribed in the previous section. Further to the twofund separation property (Proposition 4(b)), to prove the va lidity of the CAPM we shall first show that, in equilib rium, (a) the generalized tangent portfolio exists and be longs to , and (b) the generalized tangent portfolio coincides with the market portfolio. Lemma 1 Existence of equilibrium implies R . Proof. Suppose to the contrary that the equilibrium exists with R . By Proposition 4(c), the optimal portfolio would be either given by the risk free bond, or not exist. Since all investors investing in the risk free bond will necessarily violate the market clearing condi tions for the risky assets, we thus conclude that, the opti mal portfolios do not exist for at least one investor. The latter contradicts the assumption on the existence of equili brium. Therefore, in equilibrium, we must have R . ■ Since R , in equilibrium the generalized tangent portfolio m is well defined. We can further identify the generalized tangent portfolio to coincide with the market portfolio (in equilibrium) following the standard separat ing portfolio arguments. So, we may state without proof the following claim. Lemma 2 Suppose economy has an equilibrium that is supported by nonzero equilibrium security prices ( 0 j p for all jJ ). The equilibrium generalized tangent portfolio must be finite ; and, in par ticular, it must be given by the market portfolio; that is, m m m 12. The following risk decomposition theorem is an infi nitedimensional generalization to that of Huang and Litzenberger [26], Chapter 3.18 & 3.19. Lemma 3 Suppose R , the generalized tangent portfolio m is well defined; in particular, for all 2 it holds true that13 2 ,m m ff m RRR R (9) where has a zero mean and is uncorrelated with ; in particular, m R 2 ,. m fm m RR f (10) Lemmas 1, 2 and 3 together lead to the equilibrium CAPM that is valid for all generalized portfolios, par ticularly for those admissible finite portfolios in as a subset of 2 . This concludes the validity to Theo rem 1(2). Since, in equilibrium, the generalized tangent portfolio is given by the market portfolio which, by assumption A3, is finite, the equilibrium efficient frontier for with #J (and with a risk less asset) is welldefined and is given by the tangent ray and its reflection ray. This, in turn, implies (by Proposition 4(b)) the validity of two fund separation; that is, in equilibrium, each investors optimally holds a portfolio that involves a combination of the risk free bond and the market portfolio. This con firms the validity of Theorem 1(1). Notice further that the CAPM remains valid as an equilibrium model if wMPS risk averse investors are allowed to hold an infinite number of securities, say port folios belonging to 2 . This is because, under con ditions C1 and C2, the efficient frontier w.r.t. 2 is welldefined, and because wMPS risk averse investors would all invest in the riskfree asset and the efficient tangent portfolio. Following the same argument as in Lemma 3, the tangent portfolio must coincide with the 12We can further show that, if equilibrium exists with for all 00 i W i, then we must have mf and the market portfolio m must be on the efficient ray. 13Notice that, when 0 j p for all , the risky returns have a non singular variancecovariance matrix as long as the covariance matrix for the risky payoffs j is nonsingular. This last assumption is to ensure that none of the tradable securities are redundant and that 0 m . Open Access TEL
P. P. BOYLE, C. H. MA 313 market portfolio14. This would let us conclude the valid ity of the CAPM along with the twofund separation for . 2 #J 0 π0 5. Existence of Equilibrium We now proceed with the proof of the existence of equi librium (Theorem 2). We assume finite economies with a finite number of states and a finite number of assets . To find the equilibrium, we shall restrict security prices to satisfy the CAPM. Lemma 4 If equilibrium exists, then there exists and such that, for all 1 π0x 01 ˆ πwith πππ, xc (11) where ˆ c mm m , 0 π10 and 1 π 10 mm m . The CAPM pricing rule in Lemma 4 is well known (see, for instance, [3]). Here, is a discount factor. Therefore, given the random payoff for the market port folio, and given the expressions for 0 and 1, the equilibrium pricing rule is fully determined by π π π 1 and m , which are the prices of the risk free bond and that of the market portfolio. Lemma 5 If is an equilibrium discount factor, then for all positive constant , is also an equilibrium discount factor. π 0kπk By Lemmas 4 and 5, we can normalize the equilibrium discount factor in (11) to be such that 01 and write for some ππ1 ππr 0,1r , , where 1r ˆ rrc . Denote by the pricing rule resulting from . Let rx rπ ,r be the positive price vector for the risky assets. To prove the existence of equilibrium, we need to show that there exists an such that, given prices de termined by (11) with r rππ, the optimal portfolio exists for each investor and satisfies the market clearing conditions. To ensure existence of an optimal portfolio, we shall restrict so that the pricing rule (11) satisfies the no arbitrage condition. By the fundamental theorem of fi nance (see [28]), the noarbitrage condition is equivalent to the existence of 2,vr vr such that π0r rv and is orthogonal to the market span 15. For the case when , we can choose π0r 0vr . Let be the subset of for which noarbitrage condition holds. We have O 0,1 Lemma 6 The set 0O is an open convex subset of 0,1 ; in particular, we can write 0,Or for some 01rO . Proof. The proof proceeds in five steps. First, 0O with 01π and 0v . Second, 1. This is be cause, with O ˆ 1cππ , the price for the market portfolio is negative: 1, 0. m m 0,1 r This violates the noarbitrage condition since, by assumption A3, . Third, there exists s.t. 0 m rO . Since is a finite, m takes finite possible values, and has a finite norm, denoted m . Since m is risky, we have mm . Let 0,1 m m r mm m 0, m rr . For all we have 11 11 0; mm m mm m rr r π that is, 0rπ with 0vr rO . Fourth, we show that is connected: Let O with and π0rvr vr . For any 0,1a we have av r and πa ar 0vrπar ar O 1avr . So, . Finally, 0O is open. For any 0 0rO with 0r 00 and . For sufficiently small, set πrv 0 vr 0 0 vrvr for all ,r 00 rr . We have 000 00 πrvr ˆˆ ππ.rvrrrcrvrc Since 0 πrvr 0 is strictly positive and takes fi nite values, when is sufficiently small, the right hand side, namely 00 ˆ πrvrc , must be strictly posi tive. This yields π 0,Or 0rvr. Consequently, we can write for some 00 ,1r. ■ We understand that, the noarbitrage condition is vio lated at r . For all rO , let and mr mr be respectively the generalized tangent portfolio and the market portfolio. For all # x, let diag be the diagonal matrix with jth diagonal element given by . Lemma 7 For all 0rO , the generalized tangent portfolio mr is welldefined and is given by 11 0 11 0 diag,,1,. ,,1 m rr rrr 14Here, of course, we must assume 2 m . , r r (12) 15We see that is orthogonal to 2,v 2,i 0vd for all . We write whenever is ortho gonal to . The set is the orthogonal complement of . d v v Open Access TEL
P. P. BOYLE, C. H. MA 314 Proof. Since all risky securities are with nonnegative payoffs, the noarbitrage condition implies the price vec tor ,r to be strictly positive. By assumption A1, 0 is positive definite, so the covariance matrix for asset returns is well defined, and is positive definite. We can further verify that, for all 0,rO 1,1 rrR . This in turn implies, by Proposi tion 4, the existence of the efficient rays with a well de fined generalized tangent portfolio given by 1 2 11 1 1 1, 1 f m f R r R in which 2 11 1 1,10 f R for 0. The de rO sired expression for the generalized tangent portfolio is valid because 11 0 diag ,diag , ,diag ,1 ,1 ,1diag,. f rr rrR r rr We can further show that Lemma 8 For all , the generalized tangent portfolio is given by the market portfolio: 0rO m mrr . Proof. The market portfolio is diag , , m m m r rr . Since both the generalized tangent portfolio and the market portfolio have unit length, and since diag ,r is nonsingular, it suf fices to show that m is proportional to 1 0 , ,1 r r . In fact, with 0 ,1 ˆ ,1 11 m m rrr c rr r we ob tain 1 0 , 1 ,1 mm r r rr as desired. ■ In the light of this observation, by the twofund sepa ration theorem, all wMPS risk averse investors will choose to hold a combination of the market portfolio and the risk free bond. By assumption A3, the market portfo lio is finite, hence the efficient frontier exists. Let be the efficient frontier that is composed of combinations of the risk free bond and the market portfolio. Therefore, for all , the portfolio pay offs generated by efficient portfolios must coincide with the market subspace 0,m that is spanned by the risk free bond and the market portfolio. : m rara r By assumption A5, there exists a utility representation on 0,m that is given by ,m t xyU xy . With initial wealth ,ret 0 Wr,t , the shares in vested in the bond and the market portfolio can be ex pressed as 0,1 ,1 Wrta xr and 0, ,m Wrta yr , where is the proportion (of the wealth) invested in the market portfolio. Since, by assumption, a 0> te ,0t , the positivity of the pricing rule implies 0 for all Wr rO . By the homotheticity of the utility function (Re: assumption A4), the optimal choice problem re duces to max , at Var, where 1 ,,1 , m tt m aa Var Urr . For any , assumptions A4 and A5 together imply that r t taV ar,, is measurable in and con tinuous in t a . By the strictquasiconcavity of r, t aVa O r , the optimal solution (if it exists) must be unique. Since no arbitrage condition is satisfied for (Re: Lemma 6), by the Fundamental Theorem of Finance, the optimal solution exists (for rO ). Let ,r 0at be the optimal solution for , where ,amaxrt V O rg r a t . Here, we restrict the number of shares invested in the market portfolio to be nonnegative. By the measurable maximum theorem ([29], Theorem 14.91), we have ,tr t to be measurable. Lemma 9 below shows that is continuous on . rt ,r O Lemma 9 For all rO and , we have: tT 1. 0, 0.t 2. If converges to , then 1 nn r OrO lim,. nn rt 3. ,:tO is continuous. Proof. At 0r the price of the market portfolio is given by m . The expected return of the market portfolio is thus given by the risk free interest rate 10,1 1 . This implies that, for all , the portfolio return a 1 10,1 0, m m aa is a wMPS of the risk free return 1 0,1. Therefore, all risk averse investors would optimally invest in the risk free bond with zero position in the market portfolio. That is, 0, 0t for all .t To prove the second statement, let con verge to 1 nn rO rO . Consider 1 , nn rt . To show lim , nn rt , suppose, to the contrary, that 1 , nn rt has a finite limit point given by . 0 Let 1 kk n be a convergent subsequence that con Open Access TEL
P. P. BOYLE, C. H. MA 315 verges to ,V . We have: for all ,a , tntn n Varrt r, for all n, particularly holds true for the subsequence k n1 k k. Let , by con tinuity of , we have: t V , t r V r , t r Va , which holds true for all . Therefore, a arg , at Var Gr max . This, however, contradicts the emptiness of at . Therefore, we must have . lim , nn rt ,:tO To show that is continuous, it is suf ficient to show that, for all 1 nn r , nn rt O converging to , the resulting sequence converges to rO 1 ,rt . Firstly, we show to constitute a 1 , nn rt 1 nn a , bounded sequence. Suppose, without loss of generality, that . Let be an ar lim , nn rt 0 bitrary sequence that converges to . For any ar rt bitrary let a , n n a xrt for all . For suf n n ficiently large, we have 0,1 n x . The sequence 1 n nn xa xrt 1 nn , converges to ,rta . By the quasiconcavity of , and by the optimality of we have t V , n rt , tn Va , n 1, tnnnnn axrr r Vx t . Let , n it yields ,, , t Vrtar r , t Vrt ,rt . This contra dicts the unique optimality of for the given . rO Now, let 0 be any finite limit point of , n rt 1 n , and let be the convergent subse 1 kk n quence. We have, for all a and for all ,k ,, k tntnn VarVrtr , k k k . Let , by continuity of , we have t V r,VraV tt , rOsup , which holds for all a. Also, for , achieves its maxi ,Var , n rt t a mum uniquely at . Therefore, we conclude that ,rt ,rt ,rt . This implies that has a uni 1 n que limit point ; or, equivalently, ,rt ,rt limnn . This ends the proof of the third statement. ■ As a necessary condition for the existence of a market equilibrium, the market clearing condition for the risky assets implies dt rO ,, , m Tret r ; rt (13) that is, the aggregate investment in all risky assets equals to the value of the market portfolio. We have, Lemma 10 There exists 0 that solves Equa tion (13). Proof. Let ,,d , , mT m rt rett rr r O . By assumption A2 (the Feller’s property) and by Lemma 9, we have : mO to be continuous and to satisfy (i) 0 m 0 and (ii) for all m nn r r lim O r 1 nn converging to . By the Intermediate Va lue Theorem, there exists an such that 0,rr 1 mr ; or, equivalently, ,,d Tret t , m rrt . ■ The validity of the main existence theorem, namely Theorem 2, can be concluded with the proof of Lemma 11: Lemma 11 There exists an such that 0,1r πr constitutes an equilibrium discount factor. Proof. Let 0rO be a solution to Equation (13). Lemma 8, together with Proposition 4, implies that ’s optimal portfolio t t , for the given , is proportional to the market portfolio. We write r ,m trt r. With 0,Wr t,t re we have 0 1, ,d ,,,d T m T rt W rtt rrtrett 0; that is, the net borrowing in the risk free bond is zero. Moreover, for each risky asset , we have j 0,d , ,, d; , j T j mm T j j tW rtt r rt rettr r or, the total number of shares invested in equals to the number of shares outstanding for the security. There fore, the pricing rule resulting from the discount factor j πr constitutes a market equilibrium. This concludes the proof. ■ 6. Concluding Remarks In this paper, we prove that the CAPM holds for econo mies with wMPS risk averse investors. The CAPM model is shown to be valid without imposing any distri butional restriction on asset returns and the number of tradable securities, and to be valid for economies with a continuum type of investors. This approach is compared to multifactor models in the literature based on assump tions on the existence of some exogenous factors’ struc ture in modeling asset returns. Our results suggest that, so long as investors exhibit wMPS risk aversion, the relevance of all those factors that affect asset returns Open Access TEL
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