a revised geometry of mean-variance e cient portfoliosehrbar/rgep.pdfa revised geometry of...

A Revised Geometry of

Mean-Variance Efficient PortfoliosHans G. Ehrbar, Economics Department, University of Utah

Published in Metroeconomica 44(3):215–238, 1993

This version (with additional proofs) available at www.econ.utah.edu/ehrbar/research/rgep.pdf

Abstract: A revised definition of efficiency compares portfolios withdifferent net worths. One axiom rules out instantaneous riskless gains,the other asserts that it costs money to buy good portfolios. Unlim-ited short sales are allowed. Portfolios which cannot be improved atany price are called absolutely efficient. The geometry of the mean–standard deviation frontiers is governed by three “efficient set con-stants.” A few results regarding efficient portfolios must be modified,and a new version of the security line equation is proposed. Explicitformulas are given; no need to use different formulas for singular ornonsingular covariance matrix.

Hans G. EhrbarEconomics Department, University of Utah1645 E. Central Campus Dr. FrontSalt Lake City, UT 84112-9300

Direct office phone: (801) 581 7797Phone of Department office: (801) 581 7481Fax: (801) 585 5649E-mail: [email protected]

A Revised Geometry of

Mean-Variance Efficient Portfolios

1. Introduction

In the ongoing controversy around the justification of mean-varianceportfolio theory [Markowitz(1987), pp. 52–68], an axiomatic approachmight offer new insights. In the present treatment, unlimited shortsales are allowed, i.e. portfolios form a vector space. Two simpleaxioms are imposed postulating that there are no opportunities forinstantaneous riskless gains, and that in at least one case more moneybuys better portfolios. The definition of mean-variance dominance isrevised in order to meet the criticism of [Ehrbar(1990)]: a portfolio isnot considered efficient if some other portfolio with lower net worthyields identical or better expected return and standard deviation.

This axiomatric approach yields the most general mathematicalconditions under which mean-variance portfolio theory holds. On theother hand, by going back to first principles, the logical flaws in theusual formulation of this theory pointed out in [Ehrbar(1990)] can beremedied.

Section 2 presents the axioms and formulates them as algebraicconditions satisfied by the expected returns, variances, and net worthsof the assets considered.

As a prelude to the central question, how the expected return andstandard deviation of a portfolio is reflected in its price, Section 3 de-lineates the range of available (µ, σ) combinations. Here we encounterportfolios which minimize σ for a given positive µ, regardless of networth. They are shown to be efficient; they provide a natural gener-alization of the riskless asset—which is different from the well-known“global” minimum variance portfolio (which is not efficient under therevised definition).

While Section 3 discusses an important subset of all efficient port-folios, Section 4 investigates its smallest linear superset, namely, theportfolios minimizing σ subject to a given µ and net worth w. Theseportfolios have been characterized by Merton [Merton(1972)] and Roll[Roll(1977)], for the case of nonsingular covariance matrix, with thehelp of three “efficient set constants.” We propose three alternative

2 Enlarged Version with Additional Proofs

parameters, which remain finite even in the presence of a riskless as-set. In this way, Merton and Roll’s elegant method becomes a generalparametrization of all possible efficient sets. In the present treatment,there is no need to use a special set of formulas if the covariance matrixis singular.

Among the minimum variance portfolios defined in Section 4,Section 5 selects those which are efficient. It also derives variousimplications for which the revised definition of efficiency makes a dif-ference: It is no longer true that all portfolios on the rising branch ofthe frontier are efficient, but now it is true that the net worth of everyefficient portfolio is uniquely determined by its expected return andstandard deviation, and now it is true that a portfolio uncorrelatedwith an efficient portfolio always has a nonnegative expected payoff.

For those portfolios which are not efficient, Section 6 proposesand justifies a new extension of the market line equation to the caseof no riskless asset which fits seamlessly into the framework developedhere and, it will be argued, better preserves the properties which themarket line is famous for if a riskless asset exists.

Section 7 establishes the link between the revised concept of ef-ficiency and the portfolio choice problem of an investor with mean–standard deviation utility function, and Section 8 gives once more abrief summary.

The basic non-trivial proofs are given in the Appendix. An en-larged version of the paper with complete proofs is available from theauthor.

If you are reading this small-printed sentence here, you are holding the enlarged

version in your hands. Everything not contained in the original paper is given in small

print.

2. Basic Definitions and Assumptions

We will consider n different types of capital assets or “securities” overone time period. The n × 1 column vector w = [w1, . . . , wn ]′ givesthe market values of “one” security of each type at the beginning ofthe period. Usually, but not necessarily, these securities are measuredin such units that w is the vector [ 1, . . . , 1 ]′, i.e. every security startsout with market value 1 at the beginning of the period. At the end ofthe period, the securities will be liquidated, and their cash flow forms

Enlarged Version with Additional Proofs 3

a random vector with mean µ = [µ1, . . . , µn ]′, the vector of “expectedpayoffs,” and with a n×n covariance matrix Ω (which may be singu-lar). Note that µi is principal plus interest yielded by “one” securityof the ith type. A portfolio p is defined by its investment vectorx(p) =

[x1(p), . . . , xn(p)

]′, indicating that p consists of xi(p) securitiesof type i. The xi(p) may be negative, i.e. short sales are allowed. Theset of all portfolios forms, therefore, a n-dimensional vector space.The net worth of portfolio p will be denoted w(p) = w′x(p), its ex-pected payoff is µ(p) = µ′x(p), and the standard deviation of its payoff

is σ(p) =√x′

(p)Ωx(p). If w(p) 6= 0, the rate of return in its usual def-

inition is r(p) = (µ(p) − w(p))/w(p), and µ(p)/w(p) = 1 + r(p) will becalled the “rate of payoff.” Portfolios with zero expected return, stan-dard deviation, and net worth will be called “zero” portfolios. Thequotation marks around “zero” are retained as a reminder that theirinvestment vectors are not necessarily the null vector.

While usually the constraint w(p) = 1 is imposed, we allow w(p)

to vary. Our definition of dominance, indicating which portfolios areunambiguously better than others, must therefore take account of thenet worth of the securities:

DEFINITION : Portfolio p dominates portfolio q if µ(p) ≥ µ(q),σ(p) ≤ σ(q), and w(p) ≤ w(q), with at least one of these three inequali-ties being strict. p is called efficient if no portfolio dominates it.

The usual definition, of course, leaves out every reference to networth, which is assumed equal to 1. As a consequence it calls certainportfolios efficient, for which other portfolios with lower net worthcan be found that have better (µ, σ) combinations. As pointed out in[Ehrbar(1990)], the portfolios on the arc gt in Figure 2 should not becalled efficient.

Throughout this paper, we will make the following two assump-tions regarding w, µ, andΩ. One should expect the market to satisfythem if investors care about mean and standard deviation:

ASSUMPTION [i]: Any riskless portfolio with nonnegative ex-pected payoff has nonnegative net worth.

If such a portfolio had negative net worth, then instantaneousriskless gains would be possible at no cost: the purchase of such a


portfolio would give a positive cash flow at the beginning of the periodand a nonnegative cash flow at the end.

ASSUMPTION [ii]: There is at least one portfolio with positivenet worth that is not dominated by portfolios with zero or negative networth.

If this assumption were violated, the investor’s budget constraintwould be irrelevant.

These two assumptions can be translated into two simple alge-braic conditions:

PROPOSITION [1]: Assumptions [i] and [ii] hold if and only ifa vector γ exists with

(Ω + µµ′ )γ = w (1)

and

µ′γ > 0. (2)

The formulas and proofs below will also use a vector δ satisfyinga condition symmetric to (1), namely

(Ω + µµ′ )δ = µ. (3)

It always exists, independently of assumptions [i] or [ii]. If (1) and(2) hold, one can show that

1 ≥ µ′δ > 0, (4)

w′δ = µ′γ, (5)

and that, if there are more than one solution of (1) or (3), they differfrom each other only by investment vectors of “zero” portfolios.

Algebraic formulas for possible solutions γ and δ are γ = (Ω + µµ′ )−w

and δ = (Ω +µµ′ )−µ, where (Ω + µµ′ )

−is some g-inverse of Ω +

µµ′. A g-inverse of a matrix A is, by definition, any matrix, desig-nated A−, which satisfies AA−A = A. It always exists but is notnecessarily unique [Rao(1973), p. 24].

3. Absolutely Efficient Portfolios.

Assumption [ii] stipulates that certain (µ, σ)-combinations can onlybe attained by portfolios with positive net worth. This does not mean


that one can always obtain better (µ, σ)-combinations by spendingmore money. Some portfolios are “absolutely efficient,” i.e. they can-not be improved upon even by investors with an unlimited budget.Absolutely efficient portfolios, which are defined without reference tow, will be shown to be efficient as well. From their properties, two ofthe three parameters can be derived which characterize the efficientset.

DEFINITION : Portfolio t is called absolutely efficient if allportfolios q, regardless of net worth, for which µ(q) ≥ µ(t) and σ(q) ≤σ(t), satisfy µ(q) = µ(t) and σ(q) = σ(t).

If a risk-free portfolio exists with positive expected cash flow,then there are no absolutely efficient portfolios, since one can alwaysimprove a given (µ, σ) combination by adding this risk-free portfo-lio. What risk-free portfolios and absolutely efficient portfolios havein common is that they minimize standard deviation subject to agiven expected payoff. We will call such portfolios “absolute min-imum variance portfolios.” They should not be confused with theknown “global minimum variance portfolios,” which minimize stan-dard deviation subject to a given net worth.

PROPOSITION [2]: Define π ≥ 0 by

π2 =1µ′δ− 1. (6)

All portfolios p satisfyσ2

(p) ≥ π2µ2

(p), (7)

and a portfolio is an absolute minimum variance portfolio if and onlyif (7) is an equality. The investment vectors of absolute minimumvariance portfolios are scalar multiples of solutions of (3).

An explicit formula for these investment vectors is therefore easyto come by:

PROPOSITION [3]: Absolute minimum variance portfolios ex-ist for arbitrary values of µ. If p is an absolute minimum varianceportfolio with expected return µ(p), its investment vector has the form

x(p) = µ(p)(1 + π2)δ + z(p), (8)

where δ is a fixed but arbitrary solution of (3), and z(p) is the invest-ment vector of a “zero” portfolio.


(7), as an equality, can also be written σ(p) = π∣∣µ(p)

∣∣. In themean–standard deviation plane, the “absolute frontier” consists there-fore of the two rays through the origin with slopes ±1/π (which be-come the µ-axis if π = 0). π indicates the lowest attainable “t-ratio”of standard deviation divided by expected payoff. 1/π has been called“performance” in [Sharpe(1966)]. Looking ahead at Figure 1 below,one can discern these two rays as the envelopes of all frontiers. Figure2 has these rays drawn in, truncated at the tangency point with theusual “frontier” of (µ, σ)-combinations of all portfolios with net worthunity.

Proposition [2] used assumptions [i] and [ii] only insofar they im-ply µ 6= o. A fuller use of the two assumptions yields a second param-eter characterizing the efficient set. All absolute minimum varianceportfolios have the same positive rate of payoff ρ or, using the morefamiliar notion of “rate of return,” they all have the same rate of re-turn i > −100%. This rate should be considered the generalization ofthe risk-free rate of return.

PROPOSITION [4]: Define ρ = 1 + i > 0 by

ρ =µ′δ

w′δ. (9)

Every absolute minimum variance portfolio t satisfies

µ(t) = ρw(t). (10)

The relationship between absolute minimum variance portfoliosand absolutely efficient portfolios is simple and familiar: the abso-lutely efficient portfolios are located on the positively sloped portionof the absolute frontier, unless this frontier is vertical.

PROPOSITION [5]: If π = 0, then there are no absolutely effi-cient portfolios. If π > 0, the absolutely efficient portfolios are exactlythe absolute minimum variance portfolios with nonnegative expectedpayoff or, equivalently, nonnegative net worth. All risk-free portfoliosand all absolutely efficient portfolios are efficient.

We defined a portfolio p to be efficient if and only if any other portfolio q with

µ(q) ≥ µ(p), σ(q) ≤ σ(p), and w(q) ≤ w(p) satisfies the equalities µ(q) = µ(p), σ(q) = σ(p),

and w(q) = w(p). Proposition [5] says that for absolutely efficient portfolios, these three

equalities already follow from the first two inequalities. Their (µ, σ) combination “locks

in” their net worth.


↑µ

w = −0.6w = −0.4w = −0.2w = 0w = 0.2w = 0.4w = 0.6w = 0.8w = 1

1

→ σ0

Figure 1. Frontiers for various values of w, dotted where they are not efficient

4. Minimum Variance Portfolios

In this section we will explore the set, in three-dimensional space,of all (µ, σ, w) combinations of “minimum variance” portfolios, i.e. ofportfolios which minimize σ for their given µ and w. This set is thesurface area of an elliptical cone with vertex in the origin, which de-generates into a pair of halfplanes if τ 6= 0 and π = 0, and reducesto its two-dimensional counterpart, a pair of straight halflines ema-nating from the origin, if τ = 0. Its interest lies in its mathematicaltractability and in the fact that the vector space of all minimum vari-ance portfolios is spanned by the efficient portfolios. Figure 1 showsthe intersections of a nondegenerate minimum variance surface withseveral planes w = constant. These intersections, which we call fron-tiers, are shown as solid lines where the efficient portfolios are locatedand as dotted lines otherwise. Since Figure 1 may give a mislead-ing perspectivic impression, Figure 2 shows what this surface (withthe frontiers now as level lines) looks like if one cuts off everything


located above the plane w = 1, and looks at it from straight above.In other words, Figure 2 shows that part of the minimum variancesurface accessible to an investor with total wealth w = 1, who selectshis portfolios subject to an inequality wealth constraint. Figures 1and 2 also differ in that Figure 2 follows the convention of labelingthe axes in terms of rates of return instead of total payoffs.

↑r = µ− 1

•t

•g

w = 1

→ σ0%

–100% • o

Figure 2. View from above of the minimum variance cone-surface,

truncated at the plane w = 1

The shape of the minimum variance set is governed by three “effi-cient set constants.” Two of the constants, π and ρ, have already beenintroduced. The third constant, τ , indicates the rate at which excessvariance can be traded for (squared) excess return, where “excess”means in excess of absolute minimum variance portfolios.

PROPOSITION [6]: Take π ≥ 0 and ρ > 0 as defined in equa-


tions (6) and (9), and define τ ≥ 0 by

τ2 = µ′δ

((w′γ)(µ′δ)

(w′δ)2− 1). (11)

All portfolios p satisfy

τ2(σ2(p) − π

2µ2(p)) ≥ (µ(p) − ρw(p))

2. (12)

If τ > 0, then p is a minimum variance portfolio if and only if itsatisfies (12) with equality. Its investment vector is a linear combina-tion of solutions of (1) and (3). If τ = 0, then the minimum varianceportfolios are exactly the absolute minimum variance portfolios.

Investment vectors have a nice, symmetric formula:PROPOSITION [7]: If τ > 0, minimum variance portfolios p

exist for arbitrary values of w(p) and µ(p). Their investment vectorscan be written in the form

x(p) = [γ δ ][ρ2/τ 2 −ρ/τ 2

−ρ/τ 2 1 + π2 + 1/τ2

] [w(p)µ(p)

]+ z(p), (13)

where γ and δ are, as always, fixed but arbitrary solutions of (1)and (3), and the z(p) are investment vectors of “zero” portfolios. Ifτ = 0, all portfolios have the same rate of payoff ρ, and the investmentvectors of all minimum variance portfolios t with expected payoff µ(t)

(and therefore net worth w(t) = µ(t)/ρ) have the form (8).For τ > 0, (12) can be rearranged to give (14), which is [Buser(1977),

equation (8)]:

σ2(p) ≥

π2ρ2

1 + π2τ2w2

(p) +1 + π2τ2

τ2(µ(p) −

ρ

1 + π2τ2w(p))

2. (14)

From this one can see that the variance of the “global minimum vari-ance portfolio” g with net worth 1 is π2ρ2/(1 + π2τ2), and its expectedpayoff is ρ/(1 + π2τ2). These formulas for g remain valid if τ = 0.

Here is summary of the properties of “absolute” and “global” minimum variance

portfolios.

PROPOSITION [i]: t is an absolute minimum variance portfolio iff σ2(t) =

π2µ2(t) iff it is a minimum variance portfolio with µ(t) = ρw(t). g is a global minimum

variance portfolio iff (1+π2τ2)σ2(g) = π2ρ2w2

(g) iff it is a minimum variance portfolio with

(1 + π2τ2)µ(g) = ρw(g).


In Figures 1 and 2, the efficient set constants have the valuesπ = 2, ρ = 2, and τ = 1/3. Different values would give, besides adifferent scaling of the axes, also a different relative slope betweenthe envelope and the asymptotes of the individual frontiers. If τ = 0,the minimum variance surface degenerates to a line, i.e. each frontierdegenerates to a point. If a riskless asset exists, i.e. if π = 0, then theenvelope of all frontiers is the µ-axis, and every frontier for a givennet worth is a pair of rays emanating from this axis or, if in additionτ = 0, a point on this axis. The values of the efficient set constants donot disclose whether there are any portfolios which are not minimumvariance portfolios. There are always two possibilities: either thereare no other portfolios than the minimum variance ones, or for everyvalue of σ2 which is greater than the minimum, portfolios exist whichhave this variance.

For every triple of numbers π ≥ 0, ρ > 0, and τ ≥ 0 one can specify a vector space

of assets for which these are the efficient set constants, by setting

µ =1

1 + π2

[1ρ

1

], w =

[1ρ2

(τ2 + 11+π2 )

1ρ(1+π2)

], Ω =

[1ρ2

(τ2 + π2

(1+π2)2) π2

ρ(1+π2)2

π2

ρ(1+π2)2π2

(1+π2)2

].

Since in this case γ =

[1

0

]and δ =

[0

1

], this example provides at the same time formulas

for the expected returns, net worths, variances, and covariances of γ and δ.

Merton [Merton(1972)] and Roll [Roll(1977)] use the elements of[w′Ω−1w w′Ω−1µµ′Ω−1w µ′Ω−1µ

], which are called

[c bb a

]in Roll’s notation,

as their efficient set constants for nonsingular Ω. From nonsingularityfollows [Rao(1973), p. 33]

(Ω + µµ′ )−1 = Ω−1 − Ω−1µµ′Ω−1

1 + µ′Ω−1µ, (15)

therefore γ = (Ω +µµ′ )−1w and δ = (Ω + µµ′ )−1

µ are uniquelydefined. Using (15) together with (6), (9), and (11), one obtainsπ2 = 1/a, ρ = a/b, and τ 2 = a(ac − b2)/b2. Inversely, a = 1/π2,b = 1/(π2ρ), and c = (1 + π2τ2)/(π2ρ2). a, b, and c become infiniteas π2 → 0.

5. Which Minimum Variance Portfolios are Efficient?

In terms of single frontiers, i.e. of intersections of the minimum vari-ance set with planes w = constant, the efficient portfolios can be


characterized as follows. If τ = 0, every frontier consists of one pointonly, which represents an efficient portfolio. If τ > 0, one has todistinguish whether the portfolio has negative net worth or not. Aportfolio with nonnegative net worth is efficient if and only if it islocated on the portion of the frontier above and including the port-folio t, whose tangent intersects the µ-axis in the origin. A portfoliowith negative net worth is efficient if and only if it is located on therising portion of the frontier (starting from and including the globalminimum variance portfolio g). This is equivalent to the followingalgebraic condition:

The efficient portfolios must be on the surface looking north (maximize µ), looking

down (minimize w), and looking west (minimizing σ). Looking north and west means: the

level lines must have positive slope, i.e. go from southwest to northeast. Looking down

means: level lines for higher w must be above, i.e. northwest of, those for lower w. For

w > 0, the looking-down condition is satisfied only for those portfolios on the part of the

frontier above its tangency with the envelope of all frontiers. For w ≤ 0, all portfolios on

the rising branch of the fronier are efficient. Efficient portfolios are indicated by solid lines

in Figure 1.

PROPOSITION [8]: Portfolio p is efficient if and only if it isa minimum variance portfolio which satisfies both

µ(p) ≥ ρw(p) and (16)

(1 + π2τ2)µ(p) ≥ ρw(p). (17)

The most obvious modification which the present revision bringsto the usual mean-variance theory is therefore that, in the case ofpositive net worth, some portfolios on the rising branch of the frontierare not efficient. There are other changes. Let us look at the rangeof values for which efficient portfolios exist:

PROPOSITION [9]: If τ = 0, efficient portfolios exist for ev-ery µ, and their standard deviation is σ = π|µ|. If τ 6= 0, efficientportfolios exist vor every µ and σ ≥ 0 satisfying

− σ√1 + π2τ2

≤ πµ ≤ σ. (18)

For every efficient portfolio, the net worth is uniquely determined byits µ and σ:

w =µ

ρ− τ

ρ

√σ2 − π2µ2. (19)


Although it is little advertised, the usual definition does not ex-clude the possibility that two efficient portfolios exist with same µand σ but different net worths. While one of them has the mini-mum net worth for the given (µ, σ) combination, the other one hasthe maximum net worth: for it, the wealth constraint is binding frombelow.

Also the following two propositions hold under the revised defi-nition of efficiency, but not under the usual one:

PROPOSITION [10]: If p and q are efficient portfolios withnonnegative net worths, then their covariance has the lower bound

cov(q,p)≥ π2ρ2w(q)w(p). (20)

Equality holds if and only if both portfolios are absolutely efficient.If p and q were allowed to range over all portfolios on the rising

branch of the boundary, one would have to relax this inequality tocov(p,q)≥ π2ρ2w(p)w(q)/(1 + π2τ2).

In this case, equality would hold if both portfolios were “global” minimum variance

portfolios.

PROPOSITION [11]: Let p be an efficient portfolio with σ(p) >

0, and q an arbitrary, not necessarily efficient portfolio. In the specialcase that p is a “global” minimum variance portfolio with w(p) < 0,cov(q,p) is positive (negative, zero) if and only if w(q) is negative(positive, zero). Otherwise, cov(q,p) is positive (negative, zero) ifand only if µ(q) − θ(p)w(q) is positive (negative, zero), where

θ(p) =

0 if p is an absolute minimum

variance portfolio

ρµ(p) − ρw(p)

(1 + π2τ2)µ(p) − ρw(p)≥ 0 otherwise.

(21)θ(p) is the height at which the tangent to the frontier in p intersectsthe µ-axis, divided by µ(p).

Under the usual definition of efficiency, θ(p) ≥ 0 does not gener-ally hold. Certain portfolios are therefore called efficient, for whichnegatively related portfolios exist with negative net worth and positiveexpected return. Adding a little bit of such a portfolio decreases thevariance, decreases the net worth, and increases the expected returnof the so-called efficient portfolio.


6. The Market Line Equation

The net worth of an efficient portfolio is determined by its expectedpayoff and standard deviation. What can be said about portfolioswhich are not efficient? In the presence of a riskless asset, their pay-off in excess of that of an equal investment in the riskless asset, andtherefore their net worth, can be explained by the contribution whichthey make to the variance of any minimum variance “reference port-folio” p of which they are a part, which is usually taken to be themarket portfolio. The “market line equation” reads

µ(q) − ρw(q) = β(q.p)(µ(p) − ρw(p)) where β(q.p) =cov(q,p)

var(p). (22)

Besides σ(p) > 0 it assumes π = 0; ρ denotes the risk-free rate ofreturn.

The usual generalization of (22) to the case without riskless port-folios replaces ρ by the θ(p) defined in (21), i.e. it replaces the excesspayoff over the riskless portfolio by that over portfolios uncorrelatedwith p. The disadvantage of this generalization is that the point ofreference for the “excess payoff” varies as p varies.

Here is a formulation of this usual generalization: Whenever p is a minimum variance

portfolio but not a global minimum variance portfolio, the following relationship holds for

any q:

µ(q) − θ(p)w(q) = β(q.p)(µ(p) − θ(p)w(p)) where β(q.p) =cov(q,p)

var(p). (i)

We propose here an alternative generalization of (22), which usesthe rate of payoff of absolutely efficient portfolios if there is no risklessrate of payoff. A revision of the definition of the “beta” gives it generalvalidity:

µ(q)−ρw(q) = β(r)(q.p)(µ(p)−ρw(p)) where β

(r)(q.p) =

cov(q,p)− π2µ(q)µ(p)

var(p)− π2µ2(p)

.

(23)(23) holds for arbitrary portfolios q and any minimum variance port-folio p which is not an absolute minimum variance portfolio.

Here is a justification of the proposed new definition of the “beta.”The original β(q.p) in (22) has its name because it is the (true) regres-sion coefficient of q on p, i.e. because µ(q) + β(q.p)(p − µ(p)) is the


best linear predictor of q in terms of p. As long as there is a risklessasset—say u yields a payoff unity with certainty—one can construct aportfolio which makes this prediction: q∗ = µ(q)u+β(q.p)(p−µ(p)u) isthat portfolio using only p and u. In other words, it is that minimumvariance portfolio which is most similar to q. It is unique up to azero portfolio. The criterion for similarity is the mean squared errorE [(q∗ − q)2].

The definition of the revised β(r)(q.p)

in (23) uses the construction ofa most similar minimum variance portfolio as its guiding principle. Ifthere are no riskless portfolios, define u to be the absolutely efficientportfolio with expected payoff µ(u) = 1. Then the linear combinationof p and u which is most similar to q is

q∗ = µ(q)u+ β(r)(q.p)

(p− µ(p)u). (24)

While (22) relates excess payoffs to variances and covariances,it is plausible that, in the absence of a riskless asset, excess payoffsmust be related to excess variances and “excess covariances,” as donein (23). As in the riskless case [Fama(1976), pp. 59/60], the “excesscovariance” can be viewed as the contribution of q to the excess vari-ance of p: If portfolio p = q+ r, then the excess variance of p can bewritten

var(p)−π2µ2(p) =

(cov(q,p)−π2µ(q)µ(p)

)+(

cov(r,p)−π2µ(r)µ(p)

).

(25)In other words, (23) carries all desirable features of (22) over

to the case without a riskless asset—and as a byproduct confirmsthat absolutely efficient portfolios are the natural generalization ofthe riskless asset.

7. Inequality Constrained Utility Maximization

Efficient portfolios are exactly the portfolios chosen by investors whomaximize a mean–standard deviation utility function u(µ′x,

√x′Ωx)

subject to an inequality budget constraint w′x ≤ w, and who wantto spend as little money as possible for their desired (µ, σ) combi-nation. All mean–standard deviation utility functions are assumedto be strictly increasing in mean and strictly decreasing in standarddeviation.


Due to the additional requirement of not spending more thannecessary, not every efficient portfolio that satisfies the budget con-straint will be picked by such an investor. He restricts his search to asubset of those, namely, such portfolios p within his budget set whose(µ, σ) combination cannot be improved by any other portfolio in hisbudget set. We will call this the investor’s choice set. It contains allefficient portfolios p with net worth w(p) = w, and in addition allabsolutely efficient portfolios t with net worth w(t) < w. This is thewhole choice set.

If w = 1, the investor’s choice set consists of all portfolios locatedon the straight line segment from the origin o to t in Figure 2, plusthose on the frontier for w = 1 located above t.

8. Summary and Conclusions

The treatment of mean–variance portfolio theory offered here has thefollowing ingredients: (a) the assumption of unlimited short sales; (b)two very weak basic assumptions regarding the market, one sayingthat one cannot get money for free, and the other that money matters;and (c) the assumption that investors will not invest their full wealthif investing less gives them an equal or better (µ, σ) combination.

The first result derived from these ingredients is something whichis easily overlooked if one works with investment proportions ratherthan the total amounts of investments: In a mean–variance world,less money can sometimes buy better portfolios than more money.Those portfolios whose (µ, σ) combination cannot be improved byportfolios with higher net worth, the “absolutely efficient” portfoliost, take the central role in our discussion which the usual treatments ofmean–variance portfolio theory award to the “global minimum vari-ance portfolio” g. This switch from g to t is necessary as long as onedeals with portfolios with nonnegative net worth. g is not efficient ifit has positive net worth, while absolute minimum variance portfoliosare not efficient if they have negative net worth.

Strictly speaking, this is only true if there is a difference between g and t.

It follows from the axioms that all absolutely efficient portfolioshave the same rate of return, which is higher than −100%, and whichprovides a market parameter that specializes to the risk-free rate ofreturn when a risk-free portfolio exists. This rate of return, together


with two other market constants, characterize the geometry of allefficient combinations of net worth, expected payoff, and standarddeviation. We used a mathematical apparatus tailored to the twoaxioms; its main difference to the usual procedures is that it usesinverses of Ω + µµ′ instead of inverses of Ω.

In this formulation, a generalized inverse is considered an inverse.

Instead of adding more complications, as one might think at first,the revised definition endowes efficient portfolios with intuitive prop-erties, some of which are occasionally (and incorrectly) believed tobe true already under the old definition. Regarding portfolios whichare not efficient, a new formulation of the market line equation workswith a more satisfactory definition of excess payoffs, which it relatesto excess variances and covariances, rather than the variances andcovariances themselves. “Excess” means here always: in excess of therisk-free or absolutely efficient portfolios.

Following the complementary slackness principle, the choice setof an investor with mean–variance utility function splits into two com-ponents: all efficient portfolios that satisfy the wealth constraint withequality, plus all absolutely efficient portfolios for which the wealthconstraint is not binding.

The above ingredients not only lead to a logically more com-pelling theory, but also allow for a mathematical apparatus which,once the shortest proofs are found, promises to be a little less “tortu-ous” and “involved” [Fama(1976), p. 279].

APPENDIXProof that (3) always has a solution, expressible by the solution for-mula given after equation (5): To show that

µ = (Ω + µµ′ )(Ω + µµ′ )−µ, (26)

find an R with Ω = RR′ and apply the identity

A−AA′(AA′)−A = O (27)

to A = [R µ ]. To show (27), postmultipy the lefthand side byits transpose and check that the product is O. Source: [Rao(1973),(1b.5.5) on p. 26].


The proof that γ = (Ω +µµ′ )−w satisfies (1) is a much simpler application of the

definition of a g-inverse: Take any γ(0) satisfying (1), whose existence is assumed: then

(Ω +µµ′ )γ = (Ω +µµ′ )(Ω +µµ′ )−w = (Ω +µµ′ )(Ω +µµ′ )

−(Ω +µµ′ )γ(0) = (Ω +µµ′ )γ(0) = w.

Proof of (4): Since µ′δ = δ′(Ω + µµ′ )δ, it cannot be negative;and if it is zero, then already o = (Ω + µµ′ )δ = µ, which contradicts(2). µ′δ ≤ 1 follows from

µ′δ = δ′(Ω + (Ω + µµ′ )δδ′(Ω + µµ′ )

)δ =

= δ′Ωδ + δ′(Ω + µµ′ )δδ′(Ω + µµ′ )δ ≥ (µ′δ)2.

(5) is trivial: w′δ = γ′(Ω +µµ′ )δ = γ′µ.

Statement about “zero” portfolios after (5): From (Ω +µµ′ )x = o followsx′(Ω +µµ′ )x =

0. If µ′x were not zero, then x′Ωx would have to be negative, which cannot be since Ωis nonnegative definite. By the same argument, Ωx = o. Furthermore, (Ω +µµ′ )x = oimplies w′x = γ′(Ω +µµ′ )δ = 0.

Proof of Proposition [1]: To show the necessity of the algebraicconditions, we assume they do not hold. First assume (1) is notsatisfied, i.e. there is no γ with w = (Ω + µµ′ )γ. Then a vectorz exists with z′(Ω + µµ′ ) = o′ and z′w 6= 0. The existence of zcontradicts assumption [i]: If (Ω +µµ′ )z = o, then Ωz = o andµ′z = 0 hold separately, i.e. both z and −z are investment vectors ofriskless portfolios with nonnegative expected payoff. Assumption [i]requires in this case that w′z ≥ 0 and −w′z ≥ 0, i.e. w′z = 0.

Now assume (1) holds but (2) is not satisfied, so that

µ′γ ≤ 0. (28)

There are two possibilities. If µ = o, then every portfolio with pos-itive net worth is dominated by the null portfolio, which contradictsassumption [ii]. Now assume µ 6= o. This is the only assumptionneeded to prove proposition [2], which we can therefore use in thisproof. We will show that (28) again violates assumption [ii], by con-structing, for every portfolio p with positive net worth, another one,whose net worth is ≤ 0 and which dominates it. If µ(p) < 0, thenp is dominated by the null portfolio. And due to proposition [2], ap with µ(p) ≥ 0 cannot have lower variance than the portfolio withinvestment vector µ(p)(1 + π2)δ, which has same expected payoff as


p, and whose net worth is, because of (28) together with (5), zero ornegative.

To show that conditions (1) and (2) are sufficient, note that (2)implies µ 6= o and therefore again proposition [2], and that our proofof propositions [4] and [5] given below does not use the assumptions[i] and [ii] directly, but only through conditions (1) and (2). But ifpropositions [2], [4], and [5] hold, then assumptions [i] and [ii] holdas well. Assumption [i]: If p is risk-free, then proposition [4] saysthat µ(p) = ρw(p) with ρ > 0; therefore if µ(p) ≥ 0 then also w(p) ≥0. Assumption [ii]: any absolute minimum variance portfolio withpositive expected payoff has positive net worth by proposition [4],and is efficient by proposition [5]. Therefore it is not dominated byanything.

Proof of propositions [2] and [3]: π is well defined because of (4).Let t be a portfolio whose investment vector is a scalar multiple of δ.It can be written x(t) = µ(t)(1 + π2)δ, and

Ωx(t) = (Ω + µµ′ )x(t) − µµ′x(t)

= (Ω + µµ′ )µ(t)(1 + π2)δ − µµ(t) = µµ(t)π2

(29)Now take any portfolio p and choose t such that µ(t) = µ(p). Thenthe “excess portfolio” e = p − t has zero expected payoff, thereforecov(t,e) = 0, and

var(p) = var(t) + var(e) ≥ var(t) = π2µ2(p).

This proves inequality (7), and that all portfolios of the form (8)are absolute minimum variance portfolios. In order to show that allabsolute minimum variance portfolios have the form (8), assume p sat-isfies (7) with equality and define t and e as above. Then cov(t, e) = 0can be written x(t)Ωx(p) = x(t)Ωx(t), therefore (x(t)−x(p))′Ω(x(t)−x(p)) = 0, therefore Ωx(e) = o, therefore (Ω + µµ′ )x(e) = o, i.e. x(e)

is the z(p) in (8).

Proof of proposition [4]: (10) follows from (1), which makes thesolutions of (3) unique except for “zero” portfolios. ρ > 0 follows from(2), (4), and (5).


Proof of proposition [5]: If π = 0, riskless portfolios with positivepayoff exist, which, if added to an arbitrary portfolio, will improve its(µ, σ) combination.

Now assume π > 0. Since all portfolios with negative payoff aredominated by the zero portfolio, every absolutely efficient portfolio tsatisfies µ(t) ≥ 0. And obviously, every absolutely efficient portfolio isan absolute minimum variance portfolio.

Therefore let t be an absolute minimum variance portfolio withµ(t) ≥ 0, and take an arbitrary portfolio p with µ(p) ≥ µ(t) and σ(p) ≤σ(t). Then

π2µ2(t) = σ2

(t) ≥ σ2(p) ≥ π

2µ2(p) ≥ π

2µ2(t),

therefore equality holds throughout, i.e. σ(t) = σ(p) and, since π > 0,also µ(t) = µ(p). This shows that t is absolutely efficient. Since both pand t are absolute minimum variance portfolios with equal expectedpayoff, one can apply proposition [4] to get w(t) = w(p). This provesthat t is also efficient.

To show that every risk-free portfolio is efficient, assume t isrisk-free, and p satisfies µ(p) ≥ µ(t), σ(p) ≤ σ(t), and w(p) ≤ w(t).This makes both p and t risk-free, therefore proposition [4] applies:µ(t) = ρw(t) and µ(p) = ρw(p) for a ρ > 0. All this together givesw(p) = w(t) and µ(p) = µ(t).

For the treatment of minimum variance portfolios we need thefollowing lemma:

PROPOSITION [12]: If γ and δ safisfy (1) and (3), then the

matrix[w′γ w′δµ′γ µ′δ

]is symmetric and nonnegative definite. It is pos-

itive definite if and only if w and µ are linearly independent.Proof: Symmetry and nonnegative definiteness follow from[

w′γ w′δµ′γ µ′δ

]=[w′

µ′

][γ δ] =

[γ′

δ′

](Ω + µµ′ ) [γ δ] .

Now assume

0 = [w µ ][γ ′

δ′

](Ω + µµ′ ) [γ δ]

[wµ

]. (30)

This implies o =(Ω + µµ′ ) [γ δ][wµ

]= [w µ]

[wµ

]. (31)


If w and µ are linearly independent, then for any w and µ, (31)—andtherefore also (30)—implies w = 0 and µ = 0, which means positivedefiniteness. If w and µ are linearly dependent, then so are the rows

of[w′γ w′δµ′γ µ′δ

].

Proof of propositions [6] and [7]: A portfolio cannot have lowervariance than the absolute minimum variance portfolio with equal ex-pected payoff, and if µ(p) = ρw(p), then this absolute minimum vari-

ance portfolio has also equal net worth. If τ = 0, then[w′γ w′δµ′γ µ′δ

]has zero determinant, therefore w and µ are collinear by proposition[12], and µ = ρw due to proposition [4].

Now assume τ > 0. Take any portfolio p and let t be the linearcombination of γ and δ which has same net worth and expected payoffas p. One verifies that t has investment vector

x(t) = [γ δ ][ρ2/τ 2 −ρ/τ 2

−ρ/τ 2 1 + π2 + 1/τ2

] [w(p)µ(p)

].

With the help of portfolio e = p− t one can check, exactly as in theproof of proposition [2], that inequality (12) holds, and that portfoliosof the form (13) are exactly those which minimize standard deviationsubject to a given µ and w.

Here is the proof carried out:

var(t) = x′(t)Ωx(t) = x′(t)(Ω +µµ′ )x(t) − µ2(p)

= x′(t) [w µ ]

[ρ2/τ2 −ρ/τ2

−ρ/τ2 π2 + 1/τ2

][w(p)

µ(p)

]= π2µ2

(p) + (µ(p) − ρw(p))2/τ2,

and the portfolio e = p− t has zero net worth and expected payoff, therefore cov(t,e) =

x′(t)Ωx(e) = x′(t)(Ω +µµ′ )x(e) = 0. This gives

var(p) = var(t) + var(e) ≥ var(t).

This proves inequality (12), and that all portfolios of the form (13) have minimum variance.

In order to show that all minimum variance portfolios have the form (13), assume psatisfies (12) with equality and define t and e as above. Then cov(t,e) = 0 can be written

x′(t)Ωx(p) = x′(t)Ωx(t), therefore (x(t)−x(p))′Ω(x(t)−x(p)) = 0, therefore Ωx(e) = o,

therefore (Ω +µµ′ )x(e) = o, i.e. x(e) is the z(p) in (13).

Proof of the two possibilities for the “feasible set” as laid out in the text after pro-

postion [7]: Every portfolio is a minimum variance portfolio, if for every portfolio p, the


“excess portfolio” e, as defined in the proof of proposition [6], has zero variance. if this is

not the case, then at least one portfolio exists with zero expected payoff and net worth,

but nonzero variance. Such a portfolio has zero covariance with any minimum variance

portfolio, and by adding appropriate multiples of this portfolio to any minimum variance

portfolio one obtains portfolios with arbitrarily higher standard deviation. In other words,

from every point on the minimum variance set, the “feasible set” extends indefinitely

eastward, i.e. in direction of increasing σ.

Proof of proposition [8]: Any portfolio p for which (16) does nothold is dominated by the absolute minimum variance portfolio withexpected return µ(p).

Call this absolute minimum variance portfolio t: µ(t) = µ(p), w(t) =µ(p)ρ < w(p)

because (16) does not hold, and σ(t) < σ(p) because p cannot be an absolute minimum

variance portfolio itself.

For nonnegative w(p) we therefore still have to show: if the min-imum variance portfolio p with nonnegative net worth satisfies (16),then it is efficient. If it satisfies (16) with equality, it is an absoluteminimum variance portfolio, and therefore efficient by proposition [5].Now assume p satisfies (16) as a strict inequality, which is only possi-ble if τ > 0. Therefore p satisfies (12) with equality. Furthermore as-sume portfolio q has expected payoff µ(q) ≥ µ(p), variance σ2

(q) ≤ σ2(p),

and net worth w(q) ≤ w(p). Then we have to show that none of thesethree inequalities is strict. Since ρ > 0, one can concatenate theinequalities:

ρw(q) ≤ ρw(p) < µ(p) ≤ µ(q). (32)

The second and third item can therefore not be farther apart than

the first and fourth:

(µ(p) − ρw(p))2 ≤ (µ(q) − ρw(q))

2. (33)

And since 0 ≤ µ(p) due to w(p) ≥ 0 and (16), it follows from µ(p) ≤ µ(q)

that

µ2(p) ≤ µ

2(q). (34)

Divide (33) by τ 2, multiply (34) by π2, and add the two inequal-ities:

π2µ2(p) +

1τ2

(µ(p) − ρw(p))2 ≤ π2µ2

(q) +1τ2

(µ(q) − ρw(q))2. (35)

By assumption, the left side is σ2(p), and by (12), the right side is

≤ σ2(q). (35) implies therefore σ2

(p) ≤ σ2(q), which together with the


assumption that σ2(q) ≤ σ2

(p) gives σ2(p) = σ2

(q). This means, equalityholds in (35), therefore equality holds in both inequalities which wereadded together to get (35). But if equality holds in (33), (32) becomes

ρw(q) = ρw(p) < µ(p) = µ(q).

The argument for negative w is exactly parallel, one merely hasto interchange the roles of (16) and (17), of w and µ, and conse-quently use the “global” minimum variance portfolio g instead of the“absolute” minimum variance portfolio t.

Here is this parallel proof: If µ(p) ≤ 0, then (16) and (17) hold if and only if (17)

holds. Any portfolio p for which (17) does not hold is dominated by the global minimum

variance portfolio with net worth w(p). Call this global minimum variance portfolio g:

w(g) = w(p), µ(g) = ρ

1+π2τ2w(p) > µ(p) because (17) does not hold, and σ2(g) < σ2

(p), since

p cannot be a global minimum variance portfolio itself.

Finally we have to show: if the minimum variance portfolio p with negative net

worth satisfies (17), then it is efficient. Again we must distinguish two cases. If p satisfies

(17) with equality, then it is, due to (14), a global minimum variance portfolio among all

portfolios with same net worth w(p). If π = 0, it is furthermore risk-free and therefore

efficient by proposition [5]. Therefore assume π 6= 0, and take an arbitrary portfolio qwith w(q) ≤ w(p) (therefore w2

(q) ≥ w2(p)), µ(q) ≥ µ(p), and σ(q) ≤ σ(p). Then

π2ρ2

1 + π2ρ2w2

(p) = σ2(p) ≥ σ

2(q) ≥

π2ρ2

1 + π2ρ2w2

(q) ≥π2ρ2

1 + π2ρ2w2

(p),

therefore equality holds throughout, i.e. σ(p) = σ(q) and, since π > 0, also w(p) = w(q).

From (14) follows therefore (µ(q) − ρ

1+π2τ2w(q))2 = 0, therefore

µ(q) =ρ

1 + π2τ2w(q) =

ρ

1 + π2τ2w(p) = µ(p).

Now assume p satisfies (17) as a strict inequality, and portfolio q has expected payoff

µ(q) ≥ µ(p), variance σ2(q) ≤ σ

2(p), and net worth w(q) ≤ w(p). Again we want to show that

none of these three inequalities is strict. Since ρ > 0, one can concatenate the inequalities:

ρ

1 + π2τ2w(q) ≤

ρ

1 + π2τ2w(p) < µ(p) ≤ µ(q). (ii)

The second and third item can therefore not be farther apart than the first and fourth:

(µ(p) −ρ

1 + π2τ2w(p))

2 ≤ (µ(q) −ρ

1 + π2τ2w(q))

2. (iii)

And since w(p) ≤ 0, it follows from w(p) ≥ w(q) that

w2(p) ≤ w

2(q). (iv)

Again, the strict inequality (17) contradicts (12) unless τ2 > 0, therfore we can

multiply (iii) by 1+π2τ2

τ2 , multiply (34) by π2ρ2

1+π2τ2 , and add the two inequalities:

π2ρ2

1 + π2τ2w2

(p)+1 + π2τ2

τ2(µ(p)−

ρ

1 + π2τ2w(p))

2 ≤ π2ρ2

1 + π2τ2w2

(q)+1 + π2τ2

τ2(µ(q)−

ρ

1 + π2τ2w(q))

2.

(v)


By (14), the left side is σ2(p), and the right side is the variance of the minimum variance

portfolio with same net worth and expected return as q, therefore it is ≤ σ2(q). (v) implies

therefore σ2(p) ≤ σ2

(q), hence σ2(p) = σ2

(q). This means, equality holds in (v), therefore

equality holds in both inequalities which were added together to get (v), therefore equality

holds in (iii). Therefore (ii) becomes

ρ

1 + π2τ2w(q) =

ρ

1 + π2τ2w(p) < µ(p) = µ(q).

In other words, variance, expected payoff, and net worth of q are equal to those of p.

Proof of proposition [9]: By solving (12) for w(p) one sees that the net worth of a

minimum variance portfolio with expected payoff µ and variance σ2 satisfies (19) with a

“±” instead of the minus sign in front of the square root. This formula remains valid for

τ = 0. If the net worth with the plus sign is indeed stricly larger than that with the minus

sign, it cannot belong to an efficient portfolio, since it is dominated by that with the minus

sign. If therefore an efficient portfolio exists with the given µ, σ combination, it must be

the minimum variance portfolio with the minus sign.

The portfolios in the range delineated above clearly satisfy (16), and those with µ ≥ 0

therefore also (17). If µ < 0 then (18) implies σ2

1+π2τ2 ≥ π2µ2, therefore√σ2 − π2µ2 ≥

π4τ2µ, which gives (17).

Proof of proposition [10]: If τ > 0, then it follows from (13),premultiplied by Ω + µµ′, that every minimum variance portfolio psatisfies

Ωx(p) = [w µ ][ρ2/τ 2 −ρ/τ 2

−ρ/τ 2 π2 + 1/τ2

] [w(p)µ(p)

](36)

Therefore

cov(q,p) = x′(q)Ωx(p)

= π2µ(q)µ(p) +1τ2

(µ(q) − ρw(q))(µ(p) − ρw(p)) (37)

≥ w(p)w(q)π2ρ2.

For this last inequality, (16) was applied four times. If τ = 0, (20)holds with equality, due to (29).

Proof of proposition [11]: First assume τ2 > 0. If p is efficientand not “global” with w(p) < 0, it cannot be global at all, therefore(1 +π2τ2)µ(p)−ρw(p) > 0 and µ(p)−ρw(p) ≥ 0, i.e. θ(p) is well definedand nonnegative. The assertion follows since (37) can be rewritten

cov(q,p) = (µ(q) − θ(p)w(q))(1 + π2τ2)µ(p) − ρw(p)

τ2.


Here is a step between (37) and this “rewritten” version:

τ2 cov(q,p) = µ(q)

((1 + π2τ2)µ(p) − ρw(p)

)− ρw(q)(µ(p) − ρw(p))

Next assume τ2 > 0 and p is global minimum variance, i.e. µ(p) =ρ

1+π2τ2w(p). This together with (36) gives Ωx(p) = π2ρ2

1+π2τ2w(p)w, andif furthermore w(p) < 0, the special case criterion follows.

If τ2 = 0, then µ(q) = ρw(q), µ(p) = ρw(p), and p is an ab-solute minimum variance portfolio. Therefore by (29) cov(q,p) =ρπ2µ(q)w(p). If w(p) > 0, the criterion with θ(p) (which, by definition,is zero in this case) holds. If w(p) < 0, we are in the special case, be-cause p is also a global minimum variance portfolio, and the criterionholds because cov(q,p) = ρ2π2w(q)w(p). The case w(p) = 0 is ruledout by σ(p) > 0.

Proof of Geometric Interpretation of θ(p): Totally differentiating (12) as an equality

with respect to σ and µ gives

τ2σ(p) dσ(p) − π2τ2µ(p) dµ(p) = (µ(p) − ρw(p)) dµ(p).

The slope of the tangent is therefore

∂µ(p)

∂σ(p)=

τ2σ(p)

(π2τ2 + 1)µ(p) − ρw(p),

and the height of the intersection of the tangent with the µ-axis is

µ(p) − σ(p)

∂µ(p)

∂σ(p)= µ(p) −

τ2σ2(p)

(π2τ2 + 1)µ(p) − ρw(p)

by (12)= θ(p)w(p).

Proof of the zero beta market line equation (i): If p is a minimum variance portfolio

which does not have minimum variance among all portfolios with same net worth w(p),

then τ2 > 0 and (1 + π2τ2)µ(p) − ρw(p) 6= 0. (36) can be written as

Ωx(p) =1

τ2

((1 + π2τ2)µ(p) − ρw(p))(µ− θ(p)w). (vi)

Premultiply (vi) once by x′(q) and once by x′(p), to get

x′(q)Ωx(p) = (µ(q) − θ(p)w(q))1

τ2

((1 + π2τ2)µ(p) − ρw(p)

)(vii)

x′(p)Ωx(p) = (µ(p) − θ(p)w(p))1

τ2

((1 + π2τ2)µ(p) − ρw(p)

)Since the assumptions imply that var(p) 6= 0, we can divide to get

β(q.p) =µ(q) − θ(p)w(q)

µ(p) − θ(p)w(p), therefore (i).


Proof of the market line equation (23): The condition that p isa minimum variance portfolio but not an absolute minimum varianceportfolio presupposes τ2 > 0. Define e to be the “excess” portfoliowith x(e) = x(p) − µ(p)(1 + π)2δ, which already played a role in theproof of proposition [2]; then

Ωx(e) = Ωx(p) − π2µ(p)µ =1τ2

(µ− ρw)(µ(p) − ρw(p)). (38)

Premultiply (38) once by x′(q) and once by x′(p) to get

cov(q,e) = cov(q,p)− π2µ(q)µ(p) =1τ2

(µ(q) − ρw(q))(µ(p) − ρw(p))

cov(p,e) = var(p)− π2µ2(p) =

1τ2

(µ(p) − ρw(p))2.

Since the assumptions imply that var(p) > π2µ2(p), we can divide to

get

β(r)(q.p)

=µ(q) − ρw(q)

µ(p) − ρw(p), therefore (23).

Proof that (24) is the most similar minimum variance portfolio:For every pair of scalars α and γ, we need to show E [(q − q∗ − αu−γp)2] ≥ 0. This is true if the cross product terms E [(q−q∗)u] = 0 andE [(q−q∗)p] = 0 or, since E [q−q∗] = 0, it suffices that cov(q−q∗,u) =0 and cov(q − q∗,p) = 0. This can be written

cov(q − µ(q)u,u) = β(r)(q.p)

cov(p− µ(p)u,u)

cov(q − µ(q)u,p) = β(r)(q.p)

cov(p− µ(p)u,p).

Using (29) one sees that the first equality holds with both sides zero,and that the second can be reduced to the definition of β(r)

(q.p).

Equation (25) follows from

cov(p,e) = cov(q,e) + cov(r,e).

Proof that the set described in Section 7 is indeed the investor’s choice set: First

consider the case w ≤ 0. Since there are no absolutely efficient portfolios with negative net

worth, the set described above consists of all efficient portfolios with net worth w (i.e. it

is equal to the choice set of an investor with equality wealth constraint). We have to show


that the (µ, σ) combination of every portfolio q with w(q) < w can be improved by an

efficient portfolio with net worth w. If µ(q) <ρ

1+π2τ2 w, then the global minimum variance

portfolio g with net worth w has expected payoff µ(g) = ρ

1+π2τ2 w > µ(q) and variance

σ2(g) = π2ρ2w2/(1 + π2ρ2) < π2ρ2w2

(q)/(1 + π2ρ2) ≤ σ2(q). If µ(q) ≥ ρ

1+π2τ2 w, then

necessarily τ > 0, and we will show that q has higher variance than the minimum variance

portfolio p with w(p) = w and µ(p) = µ(q). From µ(q) ≥ ρ

1+π2τ2 w > ρ

1+π2τ2w(q) follows

(µ(q) − ρ

1+π2τ2 w)2 < (µ(q) − ρ

1+π2τ2w(q))2, and from 0 ≥ w > w(q) follows w2 < w2

(q),

therefore by (14)

σ2(p) =

π2ρ2

1 + π2τ2w2 +

1 + π2τ2

τ2(µ(q) −

ρ

1 + π2τ2w)2 <

<π2ρ2

1 + π2τ2w2

(q) +1 + π2τ2

τ2(µ(q) −

ρ

1 + π2τ2w(q))

2 ≤ σ2(q).

Now look at the case w > 0. We will show that every portfolio p with w(p) ≤ w,

which does not lie in the choice set, is “dominated” (with respect to µ and σ, though

not necessarily w) by a portfolio in this choice set. Here we have to distinguish whether

µ(p) < 0, 0 ≤ µ(p) < ρw, or µ(p) ≥ ρw. First case: If µ(p) < 0, then p is “dominated” by

the null portfolio. If π > 0, the null portfolio lies in the choice set, and if π = 0, then the

null portfolio itself is “dominated” by the riskless portfolio with net worth w, which lies in

the choice set. Second case, 0 ≤ µ(p) < ρw: If π = 0, then p is again “dominated” by the

riskless portfolio with net worth w. If π > 0, it is “dominated” by the absolute minimum

variance portfolio with same expected payoff as p, whose net worth is µ(p)/ρ < w. Third

case (which can only happen if τ > 0): If µ(p) ≥ ρw, then µ(p) ≥ ρw ≥ ρw(p) gives

(µ(p) − ρw(p))2 ≥ (µ(p) − ρw)2. Hence by (12), σ2

(p) ≥ π2µ2(p) + 1

τ2 (µ(p) − ρw(p))2 ≥

π2µ2(p) + 1

τ2 (µ(p) − ρw)2. The right side is the variance of a minimum variance portfolio

with expected payoff µ(p) and net worth w. Therefore p is dominated by this efficient

portfolio, unless it is an efficient portfolio with w(p) = w itself.

References[Buser(1977)]S. A. Buser. Mean-variance portfolio selection with either a singu-

lar or nonsingular variance-covariance matrix. Journal of Finan-cial and Quantitative Analysis, 12:347–361, Sept. 1977.

[Ehrbar(1990)]H. G. Ehrbar. Mean-variance efficiency when investors are notrequired to invest all their money. Journal of Economic Theory,50(1):214–18, Feb. 1990.

[Fama(1976)]E. Fama. Foundations of Finance. Basic Books, New York, 1976.

[Markowitz(1987)]H. M. Markowitz. Mean-Variance Analysis in Portfolio Choiceand Capital Markets. Basil Blackwell, New York, 1987.

[Merton(1972)]R. C. Merton. An analytic derivation of the efficient portfoliofrontier. Journal of Financial and Quantitative Analysis, 7:1851–1872, 1972.


[Rao(1973)]C. R. Rao. Linear Statistical Inference and Its Applications. Wiley,New York, second edition, 1973.

[Roll(1977)]R. Roll. A critique of the asset pricing theory’s tests, part I: Onpast and potential testability of the theory. Journal of FinancialEconomics, 4:129–176, 1977.

[Sharpe(1966)]W. F. Sharpe. Mutual fund performance. Journal of Business, 34(1/II):119–138, 1966.

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