Mathematics of the Portfolio Frontier

8/6/2019 Mathematics of the Portfolio Frontier

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The Quarterly Review of Economics and Finance44 (2004) 337361

The mathematics of the portfolio frontier:a geometry-based approach

Avi Bick

Faculty of Business Administration, Simon Fraser University, Burnaby, BC, Canada V5A 1S6

Received 24 April 2003; accepted 28 April 2003

Abstract

The mathematics of the portfolio frontier and the Capital Asset Pricing Model (CAPM) are derived

by using Analytical Geometry as the point of departure. The paper provides the Analytical Geometry

theorem which is the analog of the CAPM relationship between frontier portfolios.

2003 Board of Trustees of the University of Illinois. All rights reserved.

JEL classification: G11, G12

Keywords: Mean-variance; Portfolio frontier; Capital Asset Pricing Model

1. Introduction

Markowitzs (1952) and Tobins (1958) theory of portfolio selection is one of the mostimportant pillars of Financial Economics. Built on this theory is the celebrated Capital Asset

Pricing Model (CAPM), developed by Sharpe (1964), Lintner (1985), and Mossin (1966).1Roughly speaking, the model says that investors are compensated on the average for taking risk.The CAPM equation relates the expected rate of return of a security to its risk, as measuredby its beta, which is the (normalized) covariance of two random variables: the return of thegiven security and the return of the market portfolio.

This work is a self-contained expository paper on the basics of Portfolio Theory. At thesame time, this is also a research paper because it contributes at the foundational level by

Further details of the mathematics can be found at. http://www.bus.sfu.ca/homes/avi b/Bick papers.htm.

Tel.: +1-604-291-3748; fax: +1-604-291-4920.E-mail address: [email protected] (A. Bick).

1062-9769/$ see front matter 2003 Board of Trustees of the University of Illinois. All rights reserved.

doi:10.1016/j.qref.2003.04.001
http://www.bus.sfu.ca/homes/avib/Bickpapers.htm


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338 A. Bick / The Quarterly Review of Economics and Finance 44 (2004) 337361

demonstrating that the celebrated CAPM equation is based on a result of Analytical Geometry.Thus the Fundamental Theorem of Portfolio Theory (in our terminology), which entails the

CAPM equation, is just a translation of an Analytical Geometry theorem into the setting ofportfolio selection.

The geometric aspects of portfolio selection, developed in Merton (1972), Gonzales-Gaverra(1973), Roll (1977) and in numerous other works, are well-known.2 However, in this literaturethe geometry is always interpreted as a representation of the portfolio selection problem. Incontrast, our exposition will be based on compartmentalization of the geometry and therequired matrix algebra. The paper is aimed at readers who appreciate an approach where theunderlying pure-mathematics structure is made transparent before any applications.

The paper is organized as follows: Section 2 presents Analytical Geometry results. Section 3

contains the matrix algebra tools that we need later. In Section 4 we discuss the mean-varianceportfolio selection problem, where the goal is to identify the portfolio frontier, namely thoseportfolios with minimal variance among all portfolios with the same expected return. Propertiesof the portfolio frontier are discussed in Section 5. This is just a repetition of the results fromSection 3 in the setting of Section 4. In Section 6, we combine these results with the SlopeComparison Theorem from Section 2, and derive the Fundamental Theorem of PortfolioTheory, which is (in relation to the historical development) the CAPM equation without as-suming an equilibrium. Sections 79 discuss important special cases. Section 10 discusses theconnection between the previous results and market equilibrium. Section 11 is a short summary.

2. The geometric results

Consider the conic section in the -plane, given by

2 = k2[( )2 + b2], (1)

where k > 0, b 0 and R are given parameters. Ifb > 0, the equation can be written instandard form as

2

k2b2 ( )2

b2= 1, (2)

and it represents a hyperbola. We also allow b = 0 in (1), in which case the equation 2 =k2()2 represents two lines intersecting at (0, ). Later in the paper and denote standarddeviation of return and expected return, respectively, and therefore we restrict our attention tothe half-plane 0. Hence Eq. (1) defines a function : R [0, ). (We will use the samenotation for the -coordinate and the function () = (; k2, b2, ), but the meaning will beclear from the context.) This function has a a minimum at = , and the minimal value is() = kb. Thus () = 0 only ifb = 0 and = , otherwise () > 0. It actually makes

more sense to represent the graph in the -plane, but in most of the finance literature on thesubject this is done in the -plane, and we will follow this custom here. This is not essential.

With the given parameters, define C : R R R as follows:

C(, ) := C(, ; k2, b2, ) := k2[( )( ) + b2]. (3)


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A. Bick / The Quarterly Review of Economics and Finance 44 (2004) 337361 339

We note that C satisfies, for , , 1, 2, R

C(, ) = 2

(), (4)

C(1 + (1 )2, ) = C(1, ) + (1 )C(2, ), (5)

C(, ) = C(, ), [C(, )]2 2()2(), (6)

C(, ) = k2b2 = 2(). (7)

The inequality in (6) is a matter of simple algebra, using (1) and (3). The rest is straightforward.Property (5), which is satisfied in both variables, means, as a matter of terminology, that C is abiaffine map.

One can then define : R R R as follows: (, ) = 0 ifC(, ) = 0, otherwise

(, ) :=C(, )

()(). (8)

In light of (6), this is well-defined and 1 1. We add parenthetically, although it is notused at the formal development at this stage, that later in the paper C(, ) will represent thecovariance between two frontier portfolios with expected returns and , respectively, and will be the correlation coefficient.

Next, define Z : R \ {} R via

C(, Z()) = 0. (9)

That is, for = ,

Z() := Z(; , b2) := b2

. (10)

Ifb > 0 then clearly

Z(Z()) = . (11)

We note in this case that > if and only if Z() < , if and only if > Z(). We alsoobtain, by simple algebra,

2() = k2( )( Z()), (12)

2(Z()) = k2(Z() )(Z() ), (13)

where (13) is obtained from (12) by replacing by Z(). In the case b = 0 we have thatZ() = for each = , and the two previous equations clearly remain correct.

Later in the paper C(, ) will be given in the form

C(, ) = (, 1)

1

, (14)

where is a given 2 2 symmetric nonnegative definite matrix with 11 > 0. Then

C(, ) = 11 + 12( + ) + 22 = k2[( )( ) + b2], (15)


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where

:=

1211

, k2 := 11 > 0, (16)

b2 :=1122

212

211=

det()

211 0. (17)

Substituting (16) and (17) in Eq. (10) gives, for = ,

Z() = 12 + 22

11 + 12. (18)

The geometric meaning ofZ is as follows: Ifb = 0, Z() = for each , and Z() is theintersection of the lines (1) with the -axis. Ifb = 0, this is true for the tangent line:

Proposition 2.1. Fix k > 0 and R.

(a) Suppose b > 0 and let = . Then the straight line through (0, Z()) and ((), ) istangent to the hyperbola (1) at((), ).

(b) Equivalently, suppose b > 0 and let r = . Then the straight line through (0, r) and((Z(r)), Z(r)) is tangent to the hyperbola (1) at((Z(r)), Z(r)). In this case the tangencypoint is on the upper arc of the hyperbola, namely Z(r) > , if and only ifr < .

Proof (outline). In part (a), apply elementary Calculus. Part (b) follows from the fact that ifr = Z() then = Z(r). The second statement in (b) is clear.

The main result of this section is this:3

Theorm 2.2 (Slope Comparison Theorem). Fix k > 0, b 0 and R. Then

(a) For each R and = ,

Z() =C(, )

2()( Z()). (19)

(b) Suppose b > 0. Then (19) is equivalent to the property that for each R andr = ,

r =C(, Z(r))

2(Z(r))(Z(r) r), (20)

which can also be written as

r

()= (, Z(r))

Z(r) r

(Z(r)). (21)

To state it in geometric terms: Consider the tangent line to the hyperbola (1) through (0, r) forany r = . Then the tangency point is ((Z(r)), Z(r)), andEq. (21) says:

Slope of line connecting (0, r) to an arbitrary hyperbola point((), )

= (, Z(r)) Slope of line through (0, r) tangent to the hyperbola


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(0,r)

0

0

((),)

((Z

(r),Z

(r))

Fig. 1.

Proof. Write = + (1 )Z() where = ( Z())/( Z()). Expand C(, )as in (5) and use (7) and (9). This will give (19). Applying this equation with r = Z() and = Z(r) gives Eqs. (20) and (21). The verbal equation at the end is just a restatement of(21), combined with the geometric interpretation in Proposition 2.1 (see Fig. 1).

The next proposition, which is a slight generalization of Proposition 2.1, is needed only inSection 9 and may be skipped for now:

Proposition 2.3. Fix k > 0, b > 0 and R. Letr R and let2 := k2b2/[( r)2 + b2].

(a) For each R,

2( r)2 k2[( )2 + b2]. (22)

This is a relationship between a hyperbola and a reflected line, which can also be expressed

as

2(; 2, 0, r) 2(; k2, b2, ). (23)

(b) Ifr = , equality holds in (22) only at = Z(r; b2

, ). This is where the reflected line istangent to the hyperbola.(c) Ifr = , i.e. 2 = k2, then strict inequality holds in (22) for each , and the two half-lines

which constitute the graph of2(; 2, 0, r) are asymptotic to the hyperbola correspondingto 2(; k2, b2, ).


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Proof(outline). To prove (22), move everything in this inequality to the right-hand side and findthe minimum of a quadratic function. Ifr = , this minimum is at the point = Z(r; b2, )

where the quadratic function is equal to zero. If r = , the above quadratic function reducesto a positive constant. The formula for the asymptotes of a hyperbola is well known. In ournotation this is 2 = k2( r)2. This concludes the proof.

We should also mention, although this is not needed as a part of the formal argument, that2 was found from

(Z(r) r)

(Z(r))

2=

( r)2 + b2

k2b2, (24)

where the functions and Z are evaluated with parameters (k2, b2, ). (Use Eqs. (10) and

(13) to substitute Z(r) and 2

(Z(r)), respectively.) In light of Proposition 2.1, the left-handside is the squared slope of the tangent line to the hyperbola (1) through (0, r), provided thatr = .

3. Some matrix algebra preparations

In this section we will summarize some results on matrix inversion by partitioning. Materialon this topic can be found in Faddeev and Faddeeva (1963), Section 24 or Zhang (1999) Section2.2.4,5 In these texts it is assumed that the upper-left block (V below) is invertible, but we also

need the case when this is not necessarily satisfied.

Proposotion 3.1. LetQ andA be (n + m) (n + m) symmetric matrices such that Q has a

zero m m bottom-right submatrix. That is, they can be partitioned in the form

Q =

V Y

Y 0

, A =

, (25)

where V, Rnn andY, Rnm. (The bottom-rightm m submatrix of A is denotedfor convenience. Prime denotes transposition.) Then

(a) A = Q1 if and only ifIn 0

0 Im

= QA =

V + Y V Y

Y Y

, (26)

where Ij is the j j identity matrix. In particular:

V = Y, (27)

V = . (28)

(b) Suppose V is invertible, and so is := YV1Y. Then Q is invertible and

Q1 =

V1(In YY

V1) V1Y

YV1

, (29)

where = 1.


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Proof. The second equality in (26) is a result of block multiplication. Eq. (27) is the upper-rightblock equality in (26). To obtain (28), left-multiply (27) by and use the bottom-right equality

in (26).Under the additional assumptions of part (b), Eq. (27) becomes = V1Y. The next

step in specifying Q1 is to use the upper-left block equality in (26) and conclude that =V1(In YY

V1). Now all that is left to do is to find . The equality in the bottom-left blockof(26) translates to

0 = Y = YV1 YV1YYV1 = YV1 YV1. (30)

This is satisfied if = 1.

Corollary 3.2. Suppose Q, V andY are matrices as in (25). IfV is positive definite andY is ofrankm, then := YV1Y is positive definite. Therefore the conditions (and hence the result)of part(b) ofProposition 3.1 are satisfied. That is, Q1 exists and is given by (29).

Proof. If V is positive definite, it also has a positive definite inverse V1. Now it is an easyexercise to show that is also positive definite. (Or see Theorem 4.2.1 in Golub and Van Loan(1996).) Hence is invertible, and the conclusion follows.

Two other results, for the case m = 2, are needed later:

Corollary 3.3. With notation as in Proposition 3.1, assume that A = Q1 and that V is

nonnegative definite. Then is also nonnegative definite. If is 2 2, this entails that either11 > 0 or11 = 12 = 0.

Proof. The fact that is nonnegative definite is immediate from (28). The rest is true in generalfor a 2 2 symmetric nonnegative definite matrix . Indeed, for each R,

0 (, 1)(, 1) = 112 + 212 + 22, (31)

which clearly entails the desired conclusion.

Proposition 3.4. Suppose Q is an (n + 2) (n + 2) symmetric matrix of the form

Q =

V (E, 1)

(E, 1) 0

, (32)

where V Rnn andE, 1 Rn. Assume thatQ is invertible. As a matter of notation, assumethatQ1 = A is partitioned as in (25). Define, for each R,

w() :=

1

Rn, (33)

where is as in (25). Then

(a) w is an affine function. That is, if = 1 + (1 )2 where 1, 2, R, then

w() = w(1) + (1 )w(2). (34)


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(b) For, R

w()

E = , w()

1 = 1, (35)

Vw() = (E, 1)

1

= (11 + 12)E + (12 + 22)1, (36)

w()Vw() = (, 1)

1

. (37)

(c) For R andx Rn such that 1x = 1,

xVw() = (xE, 1)

1

= w(xE)Vw(), (38)

xVw(xE) = w(xE)Vw(xE). (39)

(d) If in addition, V is nonnegative definite, then

w(xE)Vw(xE) xVx. (40)

Proof. Part (a) is straightforward. To prove part (b), we combine the definition (33) and thebottom-right equality in (26) (with Y = (E, 1)), obtaining

w()(E, 1) = (, 1)(E, 1) = (, 1). (41)

Eq. (36) follows from (27) and (33). Eq. (37) follows from (35) and (36). Eq. (38) follows from(36) and (37). Eq. (39) is a special case of (38) which we need. To prove part (d), open thebrackets in

(x w(xE))V(x w(xE)) 0, (42)

and apply (39).

4. The minimum-variance frontier

We now turn to the problem of identifying the minimum-variance frontier. We are givenn 2 securities (investments), which are to be held over a given period, say between time0 and time 1. A portfolio of these n securities is identified with a column vector w =(w1, . . . , wn)

Wn, where prime denotes transposition and Wn :=

w Rn;

i wi = 1

.The wis represent the proportions of wealth invested in securities i = 1, . . . , n, and they are

allowed to be negative (representing a short position) or above 1. If w,x Wn

, R, thenclearly the affine combination w + (1 )x is also in Wn. In words, a portfolio of portfoliosis a portfolio in the original securities.

The n securities are characterized by a column vector of expected returns (or mean returns)over the period, E = (E1, . . . , En)

Rn and by an n n covariance matrix of the returns,


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denoted V, which is symmetric and nonnegative definite. The expected return of a portfoliow Wn is wE. The covariance between portfolios w,x Wn is wVx, and the variance ofw is wVw. For our purpose, all this can be taken as a definition. We say that w and x areuncorrelated (or orthogonal) ifwVx = 0.

A more detailed background, which is mostly not needed in the sequel, is this: Ownershipof securities is represented by shares, which are traded at time 0 (without transaction costs,in our setting). Let Pj,1 be the time-1 price of one share of security j {1, . . . , n}. We mayassume that securities do not pay dividends or other payouts, or alternatively, that Pj,1 alreadyincludes dividends reinvested in security j. The rate of return (return, for short) of securityj is Rj := (Pj,1 Pj,0)/Pj,0, where Pj,0 is the time-0 price. Prices are specified in some unitsof account, say dollars. Let R := (R1, . . . , Rn)

. We regard time-0 prices as given, and we

model uncertainty at time 1 by assigning a joint probability distribution to (P1,1, . . . , P n,1).(We allow the possibility that one of these securities is riskless, namely that its time-1 price isdeterministic.) Thus R becomes an Rn-valued random variable, and by definition,

E := E[R] Rn, V := E[(R E)(R E)] Rnn, (43)

where E denotes the expectation (mean) operator with the appropriate dimensionality. ThusEj := E[Rj], j = 1,. . . ,n, is the expected return of security j. The (i, j) component of thecovariance matrix V is Vi,j := Cov(Ri, Rj) := E[(Ri Ei)(Rj Ej)], assuming that these arefinite numbers. In particular Vi,i := Var(Ri) := Cov(Ri, Ri). For each portfolio w W

n, its

return over the period is the random variable Rw := wR which has mean E[Rw] = wE. Thecovariance between the returns of portfolios w,x Wn (the covariance between w andx, forshort) is

Cov(Rw, Rx) = E[(wR wE)(Rx Ex)] = wVx. (44)

The variance of portfolio w is Cov(Rw, Rw) = E[(wR wE)2] = wVw, and this also proves

that the matrix V is nonnegative definite.Our goal is to identify the minimum-variance portfolios. That is, for each R we wish to

find the portfolio with minimal variance for that level of expected return. We may envision

a financial advisor who has many clients, all of whom have the same beliefs and prefer highermean and lower variance of return. While the tradeoff between expected return and variancemay vary among these clients, the advisor wishes to present to all of them a reduced menu ofonly those portfolios that make sense. This is the mean-variance frontier, or the portfoliofrontier, as this set, or its representation in the mean-standard deviation plane, is often calledin the literature. As we shall see below, only the upper portion of this set contains portfolioswhich are candidates for optimal mean-variance selection.

Portfolio selection can be narrowed down further by solving for a single portfolio which isoptimal for a given individual. Mathematically, an individuals preferences can be represented

by a utility function of the form U(, ) which is maximized over a given set in R2

. Typicallyit assumed that U is suitably smooth and is decreasing in and increasing in . For example,Best and Grauer (1990) take U(, ) = t 2/2, where t is a risk tolerance parameter. Theminimum-variance frontier can then be found as a second step by varying the parameter t. Seealso the Bodie, Kane, and Marcus (1999) textbook, which uses this utility function in some


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simple cases (e.g., two stocks). However, in this paper we will not go into the individualsoptimization problem and the focus is solely on identifying the whole minimum-variance

frontier.Mathematically, following Merton (1972),6 we wish to solve, for any given R,

Problem(,E, V) : minwRn

1

2wVw, (45)

s.t. wE = , w1 = 1, (46)

where 1 = (1, . . . , 1). To solve that, we use the Lagrangian

L = 12

wVw + 1( wE) + 2(1 w

1). (47)

The standard procedure of differentiating L with respect to w1, . . . , wn, 1 and 2 and equatingthese derivatives to zero gives n + 2 equations in n + 2 unknowns. In matrix notation, thefollowing system is obtained.

V E 1

E 0 0

1

0 0

w

1

2

=

0

1

, (48)

where on the right-hand side 0 Rn. Denote the (n + 2) (n + 2) matrix on the left-hand sideby Q.

The assumptions that we need are as follows:

(A.1) The matrix V is symmetric and nonnegative definite.(A.2) The matrix Q from (48) in invertible. As a matter of notation, assume that Q1 = A is

partitioned as in (25).(A.3) 11 from (25) is positive.

Condition (A.1) is our standing assumption. Conditions (A.2)(A.3) will be corollaries ofother conditions later in the paper, but at this point we regard them as assumptions. Note that(A.2) implies thatE is not a multiple of1. IfE is a multiple of1, sayE = 1, then all portfolioshave the same expected return , and Problem (,E, V) can be solved only for = . Thisis the case of a one-point frontier. Also note that assumption (A.3) rules out the possibilitythat (, 1)(, 1), interpreted below as variance, does not depend on . (See Corollary 3.3.)This is the case of a vertical-line frontier. In this paper we will not elaborate on these twocases.

Assuming (A.1)(A.3), we obtain

w

1

2

= Q1

0

1

=

0

1

, (49)


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where thesecond equalityis a matter of notation. Theminimum-varianceportfoliowith expectedreturn is:

w() = w(;E, V) =

1

. (50)

Proposition 3.4 lists properties of this function. In particular, the covariance between twominimum-variance portfolios is

C(, ) := w()Vw() = (, 1)

1

= k2[( )( ) + b2], (51)

where in the last equality we used the fact that C(, ) is of the form (14), hence it can berepresented as in (15)(17). We can then define Z as in (10) or (18) relative to the above C. Wecan also use (18) and (36) to write, provided that = :

Vw() = (11 + 12)(E Z()1). (52)

For the global-minimum portfolio w() we have, based on (36), (16) and (17):

Vw() =1112

212

11 1 = k2b2 1. (53)

The variance of a frontier portfolio is 2

()= C(, ). Ifb = 0, namely det() = 0, then(51) gives 2() = C(, ) = 0. That is, the portfolio w() has zero variance, so that its meanreturn is in fact a riskfree rate. In this case Eq. (10) says that, for each = , Z() = ,and thus Z() is the riskfree rate.

The notation (E, V), k2(E, V) and b2(E, V) will be used later when we wish to emphasizethe dependence on (E, V) (via ), and likewise we will allow ourselves to switch notationand write C(, ;E, V), 2(;E, V) and Z(;E, V) (instead of C(, ; k2, b2, ) etc., as inSection 2).

5. Properties of the minimum-variance frontier

We remain in the setting ofSection 4. In this section we restate known results, mainly fromSzeg (1980) and Huang and Litzenberger (1998), henceforth H-L.7 The difference is that ourtreatment does not require the explicit solution for the weights vector w, and as a result we donot need to distinguish between two cases, with or without the riskfree rate. We only assumethat (A.1)(A.3) are satisfied. In fact, the results were already presented as matrix algebrapropositions, and all that is left to do here is to give the Portfolio Theory interpretation.

The (E, V)-minimum-variance frontier or (E, V)-portfolio frontier is defined as

Fn(E, V) := {w(;E, V); R} Rn. (54)

We will also refer to the (E, V)-minimum-variance frontier in the -plane, which is the set{((;E,V),); R}. We will abbreviate the terminology or notation when no ambiguityarises.


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The point ((), ) represents the global minimum portfolio. (Recall that , k2 and b2 areas in (16) and (17).) We note that for a > 0, 2( + a) = 2( a) while + a > a. Thismeans that w( a) is dominated by w( + a) in the mean-variance sense. In the language ofportfolio theory, it is inefficient. The efficient portfolios are those in {w( + a; a 0)},represented by the upper portion of the conic section (hyperbola or reflected line).

Going back to the matrix algebra work in Section 3, we interpret Proposition 3.4 in theportfolio selection context:

Fn is closed under affine combinations. Furthermore, any two arbitrary different elementsspan the whole set. This is because, for fixed 1,2 R, 2 = 1, any R can be writtenas = 1 + (1 )2 where = (2 )/(2 1), and (34) holds.

Eq. (38) gives the covariance between an arbitrary portfolio and an arbitrary frontier portfolio.It says that if a frontier portfolio w() is fixed, then its covariance with any portfolio x is thesame as its covariance with the frontier portfolio with the same mean return as x. This is akey observation which will enable us (below) to extend the Slope Comparison Theorem tonon-frontier portfolios.

Eq. (39) says that the covariance of any portfolio with the frontier portfolio which has thesame mean return is equal to the variance of the latter.

The inequality in (40) confirms that any frontier portfolio w() has minimal variance com-pared to all portfolios x with mean return xE = .

It also follows by right-multiplying (53) by x that all portfolios x Wn have the same

covariance with the global-minimum portfolio, and this covariance is necessarily equal tothe variance of the latter portfolio:

xVw() = k2b2 = w()Vw(). (55)

6. The Fundamental Theorem of Portfolio Theory

We remain in the setting ofSection 4, assuming that (A.1)(A.3) hold. Recall that Z is as in(10) or (18) relative to C from (51).

Proposition 6.1. For every frontier portfolio w() with = there exists a unique frontier

portfolio w() which is uncorrelated with it, namely such thatw()Vw() = 0. This is givenby = Z().

Proof. This is just a restatement ofSection 2 results in the portfolio selection setting.

Note. This includes the case b = 0 (that is, det() = 0, 2()= 0), where Z() = .

The concept of beta plays a major role in Portfolio Theory and asset pricing models. In

our framework we define, for x R

n

and y W

n

such that y

Vy > 0,

(x,y) =xVy

yVy. (56)

We obtain:


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Theorm 6.28,9 (The Fundamental Theorem of Portfolio Theory).

(a) For any = andx Wn,

xE = Z() + (x, w())( Z()). (57)

Equivalently, ifx Wn andy,z Fn such thatyE = andyVz = 0, then

xE = zE + (x,y)(yE zE). (58)

(b) This is true, in particular, if a riskless portfolio exists (the case b = 0, that is, det() = 0),whereZ() = zE = is the riskfree rate.

Proof. Eq. (57) is obtained from Eq. (19), applied to C from (51), with = xE, and incor-porating (38). In light of the previous proposition, the equivalence between (57) and (58) isstraightforward, via y = w(), z = w(Z()).

One interpretation (or a sketch of an alternative proof) ofEq. (57) is this: Consider the conicsection {((), ); R} (a hyperbola or a reflected line) corresponding to all the portfoliosgenerated byx Wn and w() Fn. It is inside the n-asset frontier {((), ); R} and itis tangent to it at the point = . Since they have the same tangent line at this point, it does notmatter ifZ() is computed with respect to {((), ); R} or {((), ); R}. Eq. (57)is then obtained by applying Eq. (19) (the slope comparison theorem) to {((), ); R}

at the point = . This will be elaborated upon in Appendix A.Another interpretation: Recall that we regarded E and V as given and we used them to com-

pute the portfolio frontier. Eq. (58) says that we can go backward and infer E from frontierportfolios in the following sense: The expected return of each portfolio (equivalently, the ex-pected return Ej of each security j {1, . . . , n}) can be calculated from (i) its beta relativeto a reference frontier portfolio, and (ii) the expected returns of this frontier portfolio and itsuncorrelated twin. This is important later in Section 10 when we regard expected returns asbeing determined by equilibrium.

7. Special case: n risky securities, Vis positive definite

The previous section contains the main results of the paper. The only purpose of this sectionis to show how an important special case fits in the previous setting. For better integration withthe next section, we switch notation and write instead of V. (In both special cases is apositive definite matrix. In this section is equal to V from Section 4, whereas in the nextsection is a submatrix ofV.)

The assumptions used here are as follows:

(B.1) E is not a multiple of1. That is, rank(E, 1) = 2.(B.2) The matrix (denoted V in (45)) is symmetric and positive definite.

This entails that the matrix Q from Section 4 is invertible, as we shall see below, and thusconditions (A.1)(A.3) from our previous analysis will be satisfied.10


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In what follows we will need the notation:

:= (E, 1)1(E, 1) =

E

1

E 1

1

E11E 111

, (59)

:= (E, ) := det() = (E1E)(111) (11E)2, (60)

r R Er := E r1. (61)

Lemma 7.1. For all r R,

(E, ) = (Er, ) := (Er

1Er)(1

11) (11Er)2. (62)

Proof(outline). Apply straightforward algebra. Another way to see it: Use matrix Calculus toshow that (d/dr)(Er, ) = 0. This implies that (Er, ) = (E0, ).

After these preparations, let us now compute Q1 explicitly. In light ofCorollary 3.2, ispositive definite and its determinant is positive. This corollary also says that Q is invertible,and11

Q1 =

1(In (E, 1)(E, 1)

1) 1(E, 1)

(E, 1)1

, (63)

where

= 1 =1

111 11E

11E E1E

. (64)

This means, in particular, that is positive definite. Later we will use the straightforwardrelationships

11 + 12 = 111E, 12 + 22 =

1E1E. (65)

Now that we have , this specifies the minimum-variance frontier in the -plane as thehyperbola (1) with the parameters (see (16) and (17))

(E, ) :=11E

111, k2(E, ) :=

111

(E, ), (66)

b2(E, ) :=(E, )

(111)2. (67)

We also record a relation between these parameters which will be needed later in Section 9. (Itmay be skipped for now.)

Lemma 7.2. For each r R,

( r)2 + b2

k2b2= Er

1Er. (68)

Proof(outline). from (66) satisfies r = 11Er/111. Substitute this in the left-hand

side of(68), and also b2 and k2 as above, where = (Er, ) is now taken from (62).


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We can now rewrite the expressions for C(, ), Z() and w() in this setting. Let us startwith Eq. (51), which gives the covariance between frontier portfolios. Substituting the geometric

parameters from (66) and (67), we can write the last equality in (51) as:12

C(, ) =111

11E

111

11E

111

+

1

111

= 1(111)1{(11E)(11E) + }. (69)

The variance of a frontier portfolio is then 2()= C(, ).Next, we obtain, for = ,13

Z(;E, ) =

(11E) E1E

(111) 11E = +

E

1E

11E , (70)

where the first equality is a straightforward conversion of (18). The second equality requiressome algebraic manipulation. (Start from the right-hand side.) Another connection between and Z() is as follows:

(11E)(11EZ()) + = 0. (71)

This is obtained by combining (69) with C(, Z()) = 0.Finally, we wish to write the expression for the minimum-variance portfolio with expected

return . Left-multiplying Eq. (36) (with V = ) by 1 and combining with (65) gives:14

w() = 11{(11E)E + (E1E)1} (72)

Alternatively, if = , we can use (52) and (65) to express w() as

w() = 1(11E)1

EZ() =1

11EZ()1EZ(), (73)

where in the second equality we applied (71).One interpretation of(73) is as follows: For a given r = , substitute = Z(r) and r = Z()

in the equation, so that the formula is

r = w(Z(r)) = 111Er

1Er. (74)

Recall that ((Z(r)), Z(r)) is the intersection point of the hyperbola and the tangent line through(0, r). The formula identifies w(Z(r)), the tangency portfolio relative to (0, r) and thus itspecifies the minimum-variance frontier as the set of tangency portfolios generated by varying(0, r) on the -axis.15 The global-minimum portfolio w() is obtained from (74) by takingr , Z(r) , which gives w() = 11/(111) = k2b211. It can be shownthat this in agreement with (53) and (72).

8. Special case: a riskfree rate exists

Another special case of interest is when one of the securities is riskless, while the othersecurities have a positive definite covariance matrix. Here it is convenient to switch notation


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and denote the number of securities by n+1, where n 1. We will write the (n+1)-dimensionalvector of expected returns as

E = (E1, . . . , En, En+1) = (ES, RF)

, (75)

whereES includes only the expected returns of the first n securities (stocks), and the return ofsecurity n + 1, which is riskless (to be formalized in assumption (C.2) below), is denoted RF.Likewise, we will write a portfolio w Wn+1 as w = (wS, wn+1)

, where wS is the columnvector of investments in the stocks, and wn+1 = 1 1

wS is the investment in the riskfree rate.The assumptions used in this section are as follows:

(C.1) E is not a multiple of 1. (Here 1 Rn+1.) Equivalently, ES = RF1. (Here and below,1 Rn

.)(C.2) The (n + 1) (n + 1) covariance matrix V from (45) has an n n upper-left submatrix which is symmetric and positive definite. The (n + 1)-th row and (n + 1)-th columnofV are zero.

Again,asweshallseebelow,thisentailsthatthematrix Q from (48) (an (n+3)(n+3) matrixin our case) is invertible and that 11 (from (25)) is positive, and thus conditions (A.1)(A.3)from our previous analysis will be satisfied.16 Note that we can express the covariance betweentwo portfolios x, w Wn+1 as xVw = xSwS.

We denote E := ES RF1. (In the notation of the previous section, this is ERF relative to

ES.) In addition, := E1E> 0, K := 11E, (76)

J := 1(In EK) = 1 11EE1, (77)

(ES, ) :=11ES

111, (78)

and we note that

1K = 0 11E= 0 RF = (ES, ). (79)

In this sections setting the matrix from (48) becomes

Q =

0 E+ RF1 1

0 0 RF 1

E + RF1

RF 0 0

1 1 0 0

. (80)

The inverse is

Q1 =

J J1 K RFK

1J 1J1 1K 1 + RF1K

K K1 1 1RF

RFK 1 + RFK

1 1RF 1(RF)

2

. (81)


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The matrix Q1 was computed by following the method of Section 3, where blocks areidentified recursively, except that here 4 4 partitions were used. We will not give the details,but the assertion that the above matrix is the inverse can be verified by multiplying it by Q.(The reader may need to use the facts that J is symmetric, JE= 0 and KE= 1.) Admittedly,inverting Q is not the most economical way of solving the problem, but the intention here is topresent the solution as a special case of (49).17

Recall that is the bottom-right 2 2 submatrix ofQ1, and thus (16) and (17), togetherwith (81), translate to

(E, V) = RF, b2(E, V) = 0, k2(E, V) = 1 =

1

E1E

. (82)

The (E, V)-minimum-variance frontier in the -plane is composed of the two half-lines orig-inating from (0, RF) (associated with the portfolio (0, . . . , 0, 1)

), as specified in (1).

Proposition 8.1. For each R, the frontier portfolio which has expected return , namelythe portfolio (wS(), wn+1())

Wn+1, is given by18

wS() = ( RF)K, wn+1() = 1 ( RF)1K. (83)

If = RF and x = (xS, xn+1)

Wn+1 is an arbitrary portfolio, then (restating the Funda-mental Theorem of Portfolio Theory)

xSE= xE RF =

xSwS()

wS()wS()( RF). (84)

Proof. The first part follows from (50):

w() =

wS()

wn+1()

=

K RFK

1K 1 + RF1K

1

. (85)

The second part is just a restatement of Theorem 6.2 part (b).

In the Finance literature, Eq. (84) is typically written in the form

E[Rx] RF =Cov(Rx, Rp)

Var(Rp)(E[Rp] RF), (86)

where the notation Rx is as in Section 4, and p := w().We already saw in the general case in Section 5 that any two different frontier portfolios from

the minimum-variance frontier span the frontier. This applies in this sections setting, with the

frontier Fn+1

(E, V). As it was pointed out by Feldman and Reisman (2003), a computationallyconvenient choice is the riskfree rate and the portfolio (in our notation) y = (yS, 1 1

yS),

whereyS = 1E. They noted that this is always an efficient portfolio. This corresponds to the

portfolio specified in (83) with the choice = RF + . Note that the case 1yS = 0 is not ruled

out. (More on this condition below.)


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9. Comparing portfolio frontiers

We remain in the setting of the previous section, where security n + 1 provides a riskless rateof return RF. Ifn 2, it is interesting to relate the minimum-variance frontier F

n+1(E, V) tothe set Fn(ES, ) obtained by investing only in the stocks. This is the topic of this section. ForF

n(ES, ) to be non-degenerate, we need to assume now that ES is not a multiple of1. (Thisassumption is stronger than (C.1).)

IfRF = (ES, ) (or see (79) for equivalent conditions), we denote

T := Z(RF;ES, ) = RF +E

1E

11E= RF +

1

1K, (87)

where the second equality follows from (70). We call w(T;E, V) Fn+1(E, V) Portfolio T.The following result is just a translation ofProposition 2.3 from the geometry section:

Proposition 9.1. The parameters of the (E, V)-frontier and the (ES, )-frontier are related via

k2(E, V) =1

E1E

=k2(ES, )b

2(ES, )

((ES, ) RF)2 + b2(ES, ), (88)

and hence, for each R,

2(;E, V) 2(;ES, ). (89)

In addition,19

(i) IfRF = (ES, ), equality holds only at = T. That is, the (E, V)-frontier(a reflectedline) is tangent to the(ES, )-frontier(a hyperbola) at((T), T).

(ii) If RF = (ES, ), there is strict inequality in (89) for each , and the (E, V)-frontieris asymptotic to the (ES, )- frontier. In this case k

2 := k2(E, V) = k2(ES, ) and the(E, V)-frontier is the reflected line 2 = k2( RF)

2.

Proof. Eq. (88) is a result of(68) and (82), applied to r = RF. The rest is just a translation ofProposition 2.3 to this sections setting. Of course, Eq. (89) can also be explained by comparing

the feasible sets of portfolios of the two optimization problems.

The same result can be rephrased as a statement on portfolio weights:

Proposition 9.2.

(a) IfRF = (ES, ) then wn+1() = 1 for all , hence the investments in the stocks in every(E, V)-frontier portfolio satisfy wS1 = 0.

(b) IfRF = (ES, ) then there is a unique solution to wn+1(;E, V) = 0, and this is Tfrom (87). That is, portfolio T is a pure-stock(E, V)-frontier portfolio. The correspondingstock investments are

wS(T;E, V) = (T RF)K = (11E)11E. (90)

(c) Assuming thatRF = (ES, ),

wS(T;E, V) = w(T;ES, ). (91)


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In words: The n-vector wT := wS(T;E, V) which describes the stock components ofportfolio T is equal to the solution of the n-security portfolio problem (T,ES, ). Thatis, in the -plane, ((wTwT)1

/2, T) belongs both to the (E, V)- frontier and to the(ES, )-frontier.

Proof. Again, note the conditions equivalent to RF = (ES, ), specified in (79). Parts (a)and (b) of the proposition follow from (83). In Part (c), to see (91), write

w(T;ES, ) = w(Z(RF;ES, );ES, )

=1

11E1E= wS(T;E, V), (92)

where in the second equality we applied (74) (recalling that E= ERF) and in the third equalitywe used (90).

10. The connection to equilibrium

The above results were derived from the point of view of one individual who computes theportfolio frontier. They do not require any assumptions on the beliefs or risk preferences ofother market participants. In addition, securities 1, . . . , n (in the notation of Section 4) may

potentially constitute a subset of all securities which are available. We simply need to interpretWn as all portfolios of securities 1, . . . , n instead of all possible portfolios, and likewisefor Fn. This was evident in the previous section where two different portfolio frontiers werecompared.

The distinction between Portfolio Theory and asset pricing theories, like the celebrated Cap-ital Asset Pricing Model, is that the latter are concerned with aggregate demand by all investorsand infer equilibrium relationships (to be elaborated upon below) between expected returnsand risk characteristics. Such theories typically require assumptions on all market participantsor on a representative individual.

Let us start by assuming a general setting as in Section 4, with n securities which may or may

not include a riskfree security. Assume, in addition, that these are all the securities in the market.For each security j, let Mj := Nj Pj,0, where Nj is the number of outstanding shares and Pj,0is the time-0 price. This is the aggregate value of security j in units of account (dollars). LetM :=

j Mj and mj := Mj /M. Then m := (m1, . . . , mn)

is called the market portfolio.This was the supply side. Let us now look at the demand for securities. We assume that

there are I individuals, denoted i = 1, . . . , I , who invest for the same time horizon and havethe same beliefs. (That is, they believe in the same (E, V).) Note that in the portfolio problem(,E,V), formulated in Section 4, the goal is to identify the minimum-variance frontier, butthis does not specify how each individual makes his specific choice. We will not elaborate on

this here. For further details on individual portfolio choice and its connection to mean-varianceportfolio selection, see Levy and Markowitz (1979), Ingersoll (1987), Meyer and Rasche (1992),Constantinides and Malliaris (1995), Berk (1997) and references therein. For our purposes wewill simply regard it as an assumption that the optimal portfolio of each individual i is a frontierportfolio w(i) which is strictly efficient, i.e., i > (E, V).


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Let Wi > 0 be the time-0 value (wealth, measured in dollars) of individual is portfo-lio, and let W := i Wi. Individual i wishes to hold wj(i)Wi dollars worth of security j,and the sum over all individuals (the aggregate demand) is Dj :=

Ii=1 wj(i)Wi dollars.

Clearly

j Dj = W. The aggregate demand portfolio d Wn is defined by dj := Dj/W.

Then

d =

Ii=1

Wi

W

w(i) = w

I

i=1

Wi

W

i

. (93)

(For the second equality, see Proposition 6.1). As a convex combination,

i(Wi/W )i is also

larger than (E, V), and we conclude that dis also a strictly-efficient frontier portfolio.

We now turn to the concept of equilibrium. For our purposes, this market is at equilibrium ifMj = Dj for each j. (For a more detailed definition of equilibrium, which is beyond what weneed here, see Appendix B.) This implies that M = W, and that m = d. Thus an equilibriumassumption boils down to the fact that the market portfolio is equal to the aggregate demandportfolio. The latter is a strictly-efficient frontier portfolio, and thus the Fundamental Theoremof Portfolio Theory is valid when the market portfolio is used as the reference frontier port-folio. Mathematically, this means that we can write the market portfolio as a frontier portfolioof the form m = w(m), where m is its expected return, and then we can restate Theo-rem 6.2 with = m. (At the same time, the theorem remains correct for any other frontier

portfolios !)More specifically, in the setting of Section 7, without a riskfree rate, Eq. (57) implies that,in equilibrium,

xE = Z(m) +xVm

mVm(m Z(m)). (94)

This is the main result of Blacks (1972) zero-beta model.20 It says that for an arbitraryportfolio x,

Expected return of portfoliox

= Expected return of the frontier portfolioz uncorrelated with the market portfoliom

+ (x,m) Difference between expected returns ofz andm

(See (56) for the definition of.)In the setting of Section 8, let us introduce the additional assumption that the riskfree rate

(security n + 1) has zero net supply. That is, Mn+1 = 0, which means that individuals lend andborrow between themselves. Thus equilibrium in this setting boils down to the assertion thatthe market portfolio is an efficient frontier portfolio which is a pure-stock portfolio. If n 2,it means that the tangency portfolio T from Section 9 is equal to the market portfolio, and it

is efficient. The latter property is possible only if RF < (ES, ). Eq. (57) implies that, inequilibrium,

xE = RF +xVm

mVm (m RF). (95)


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This is the main result of the Sharpe-Lintner-Mossin Capital Asset Pricing Model.21 It says thatfor an arbitrary portfolio x,

Expected return ofx

= RF + (x,m) Difference between expected return of the

market portfolio and the riskfree rate.

11. Summary

We have developed a geometry-first approach to derive the mathematics of the portfolio

frontier and the CAPM.

Notes

1. See also Sharpes (1991) summary.2. See also Szeg (1980), Ingersoll (1987), Huang and Litzenberger (1998) and

Constantinides and Malliaris (1995).3. The theorems title is suggested by the author.4. Note a typographical error in the F&F text, middle of p. 162: a negative sign should

come before A

1

BN.5. In the Portfolio Theory literature, Szeg (1980) uses this method, but with a differentmatrix. See his Appendix E.

6. Textbook coverage is available in Szeg (1980), Ingersoll (1987), Huang andLitzenberger (1998) and Cochrane (2001). See also Sections 1.1 and 1.2 in Steinbach(2001). Our matrix representation below is similar (albeit not in the same order ofrows) to the one in Steinbachs paper. For alternative approaches, see Elton, Gruber,and Padberg (1976), Best and Grauer (1990) and Ttnc (2001). Benninga andCzaczkes (2000) provide spreadsheet calculation methods in addition to theory.

7. Our w, V (or ), E, 1, 1, 2, , Z(), E1E, 11E, 111, , RF, E and

correspond to Szegs x, V, r, e, 1, 2, , 0, , , , 2, , d and dV1d,respectively, and to H-Ls w, V, e, 1, , , E[rp], E[rzc(p)], B, A, C, D, rf, e rf1 andH, respectively.

8. The theorems title is suggested by the author and is not a universally-used name.9. The result is parallel to Eqs. (3.16.2) and (3.19.1) in H-L and to Propositions 3 and 4

in the appendix to Chapter 9 in Benninga and Czaczkes (2000). A short proof for part(b) also appears in Feldman and Reisman (2003).

10. See also Szeg (1980), Appendix C, on the connection between assumptions (B.1) and(B.2) and invertibility ofQ.

11. This is consistent with the solution in H-L, Section 3.9 and with Theorem 1.5 inSteinbach (2001).

12. This is parallel to Eq. (4.4) in Szeg (1980) and Eq. (3.11.1) in H-L.13. The first equality corresponds to Eq. (4.24) in Szeg (1980) and Eq. (3.14.2) in H-L.

The second equality is parallel to Eq. (4.30) in Szegs book.


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14. This is parallel to Szegs Eq. (2.9). To compare to H-Ls results, pp. 6465: w(0) istheir g and w(1) is their g +h.

15. This corresponds to Proposition 1 in the appendix to Chapter 9 in Benninga andCzaczkes (2000).

16. See also Szeg (1980), Appendix F, on the connection between assumptions (C.1) and(C.2) and invertibility ofQ.

17. See Szeg (1980), H-Lor Steinbach (2001) for a direct approach of solving the Problem(,E, V) in the presence of a riskfree rate. Our Q1 is consistent with the solution inTheorem 1.8 in Steinbachs paper.

18. Eq. (83) coincides, after some algebra, with Eqs. (6.10) and (6.11) in Szeg (1980).19. This part is parallel to Theorem (6.54) in Szegs book (where his corresponds to

our T) and to discussion in H-L, Section 3.18. For the graphs, see Szegs Figure 6.4or H-L Figures. 3.18.13.18.3.

20. See also Ingersoll (1987) pp. 9295, H-L Section 4.11.21. See also Ingersoll (1987) pp. 9295, H-L Section 4.13.

Appendix A

In this appendix we briefly outline an alternative proof ofEq. (57) (the fundamental theoremof portfolio theory), which provides additional insights regarding the connection to the slopecomparison theorem.

Fix x Wn and w(0) Fn for some 0 = . We will prove (57) for this x and for = 0,

limiting ourselves to the case xE = 0. Define a biaffine map C : R R R via

C(0, 0) := w(0)Vw(0) = C(0, 0), (A.1)

C(xE,xE) := xVx, C(xE, 0) := xVw(0), (A.2)

which is then extended as in (5) in both variables. Define 2() := C(, ) for any R.This is the variance of the portfolio with mean which is spanned by x and w(0). That is, one

can verify that

xE + (1 )0 = {x + (1 )w(0)}V{x + (1 )w(0)} =

2().

Thus the portfolios spanned by affine combinations of x and w(0) generate the conic section() (a hyperbola or a reflected line) in the -plane, and the corresponding biaffine map isC(, ).

Applying Eq. (19) (the slope comparison theorem) to C(, ), evaluated at = xE and = 0, we obtain

xE Z(0) = C(x

E, 0)C(0, 0)

(0 Z(0)). (A.3)

where Z is defined as in Section 2 with respect to C. In light of(A.1) and (A.2), this translatesto Eq. (57), provided that we show that Z(0) = Z(0).


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Indeed, because of the optimality of the n-security frontier, we must have () () for all. Eq. (A.1) also says that (0) = (0). Thus at = 0, the two curves () and () musthave the same tangent line. (This idea appears in the CAPM proof in Jensen (1972).) Recallingthe interpretation of Z as the intercept of a tangent line, we conclude that Z(0) = Z(0).It should be mentioned that this includes the cases that (i) and are hyperbolas, possiblyidentical, (ii) and are reflected lines, necessarily identical, and (iii) is a reflected line and in a hyperbola.

Appendix B

This appendix, which complements Section 10, will elaborate on the definition of equili-brium in a mean-variance setting, following Nielsen (1988) with slight modifications. Supposethere are n securities which are traded at time 0 in a frictionless market by Iindividuals. Assumethat there are Nj 0 outstanding shares of security j. Individual i is endowed with ei,j sharesof security j, and thus

i ei,j = Nj for each j.

The uncertainty is modeled by assigning a joint probability distribution to (P1,1, . . . , P n,1),the time-1 prices of the n securities. (This includes the possibility that one of the securities isriskless.) This probability distribution, regarded below as fixed, represents the common beliefsof all individuals. For every possible vector of time-0 pricesP0 = (P1,0, . . . , P n,0) (announcedby a Walrasian auctioneer), an individual can compute the joint probability distribution ofthe Rn-valued random variable R = R(P0), where Rj = Pj,1/Pj,0 1 is the return of securityj. This distribution defines the mean-variance parameters as in (43), where now we can regardthe expected returns vector E = E(P0) and the covariance matrix V = V(P0) as functionsofP0.

We assume that individual is portfolio selection problem can be represented as

maxxWn

Ui((xVx)1/2,xE; Wi), (B.1)

where Ui(, ; Wi) is a suitably smooth function which is decreasing in and increasing in

, and may depend as a parameter on the individuals time-0 wealth Wi. (This representationcan actually be obtained as a result from more primitive assumptions on risk preferences. SeeSection 10 for references on this issue.) For a givenP0, individual i computes his time-0 wealthWi(P0) =

j ei,jPj,0 and solves for his optimal portfolio, using E = E(P0) and V = V(P0).

Suppose the solution is xi(P0) = (xi,1(P0), . . . , xi,n(P0)) Wn, and let i(P0) = xi(P0)

E.Such a solution is necessarily of the form

xi(P0) = w(i(P0);E(P0), V(P0)), (B.2)

where the right-hand side is the solution of Problem (i(P0);E(P0),V(P0)) as in Section 4. Inother words, this is the minimum variance portfolio among all portfolios with expected returni(P0). The portfolio can also be expressed in terms of the number of shares in each security.For security j, this number (including the individuals initial endowment) is

Ni,j(P0) =xi,j(P0)Wi(P0)

Pj,0. (B.3)


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We can envision each individual i responding to the declared price vector P0 by submittinghis optimal holdings (Ni,1(P0), . . . , N i,n(P0)) to the auctioneer. This procedure can be repeated

for every time-0 price vector P0. Then P0 is an equilibrium price vector if the market clears,that is, if for j = 1, . . . , n

Ii=1

Nij(P0) = Nj. (B.4)

In the notation ofSection 10, this translates to Dj = Mj for each j. (Multiply the two sides of(B.4) by Pj,0 and use (B.2) and (B.3).)

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Mathematics of the Portfolio Frontier

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