Mean variance analysis for indivisible a

OMEGA The Int. JI of Mgmt Sci., Vol. 9, No. I, pp. 77 to 88

0305.0483/81/0101-0077502.00/0

© Pergamon Press Ltd 1981. Printed in Great Britain

Mean-Variance Analysis for
Indivisible Assets
M CHAPMAN
RICHARD
JONATHAN

FINDLAY

III

D McBRIDE
S YORMARK

University of Southern California, USA

STEPHEN

D MESSNER

University of Connecticut, USA
(Received June 1979; in revisedform April 1980)
In this paper we shall demonstrate classic mean-variance analysis for universes of indivisible assets.
Our discrete mean-variance model, the frontier generation mechanism, and a brief description of the
quadratic integer programming algorithm are presented first. We then discuss some computational
strategies important in practical applications, and provide a numerical example adapted from real
estate investment analysis. Finally, we present some observations on the nature of the efficient set
and some general conclusions.

IN LIEU OF an extensive survey of the finance
literature, we shall simply outline the premises
upon which our analysis is based. First, we
assume that the investor does indeed confront
a portfolio problem. This implies the assumption that the various intermarket arbitrage
pricing models which have been proposed do
not extend to the hypothecated market. One

example of such a model is time-state preference theory [1,5], which has never been
claimed operational in any market to our
knowledge. Another would be the capital asset
pricing model E37,39,43], which has been
claimed operational in some markets.1
Second, we shall ignore the broader mixed

asset nature of the investor's portfolio problem
to concentrate solely upon fixed assets and
riskless lending. Although it would certainly be
desirable to solve the more general problem
(and it appears that the algorithm to be discussed here may be ultimately capable of doing
so), a direct solution is currently beyond the
state of the art. Indirect efforts at solution [10]
have often employed a diagonal multiple index
model [4]. These efforts have proved to be less
than satisfactory in many situations because of
a lack of good indices or historical return data
(and, hence, the inability to regress one on the
other) and numerous other estimation and statistical problems. (This is particularly true in the

real-estate market which is the application
demonstrated in this paper.)
Third, we shall employ a full Markowitz
mean-variance approach, developing a discrete
analog
of
Markowitz's
mean-variance
approach to portfolio selection [23] which is
valid when assets are indivisible. We shall
maximize expected portfolio rate of return on

1The model to be presented here could be easily
extended to capital budgeting by the firm if one is prepared
to grant that (a) the firm has a portfolio problem and (b)
may also operate under a funds constraint. Advocates of
the capital asset pricing model would, strongly dispute such
contentions, although, for some reason, the first item seems
to generate a great deal more wrath than the second. One
of the reasons we chose a real estate application of the

model was to avoid such a controversy.
o.M.L. 9/I--F

77

Findlay, McBride, Y ormark, M essner-- Mean- Variance Analysis

78

investment (there may be several interpretations of an appropriate measure of return)
while minimizing the variability of this rate,
under the restriction that borrowing is not
allowed. It is assumed the investor accepts
portfolio variance of the rate of return as a
measure of risk.

the dollar expected return from asset j and
~jxj
j=l

is the portfolio dollar return. Dividing by the
portfolio dollar outlays
CjXj

j=l

I. PORTFOLIO SELECTION FOR
INDIVISIBLE ASSETS
In the quarter century since Markowitz first
proposed the foundations of modern portfolio
theory [22], an immense amount of related
research has appeared (see [23, 37, 44] for highlights). Despite the widespread interest in Markowitz's approach for portfolios of continuously divisible assets, application of this technique for indivisible ('all or nothing') assets is
surprisingly sparse. Weingartner [41] more
than a decade ago presented a comprehensive
survey of a wide range of discrete asset models
and illustrated the applicability of the meanvariance approach in discrete asset selection.
The technology of parametric discrete optimization at that time fell far short of the applicability. It has remained so until recently. When
investments are not continuously and perfectly
divisible, the computational' difficulties become
a significant issue despite the availability of

modern computing machinery. Fortunately, we
may now apply new advances in discrete optimization [24, 25] to provide the solution methodology we seek.
Let xj = I(0) be a decision variable denoting
selection (rejection) of the jth proposed asset
from a universe of n. A convenient form of the
mean-variance approach seeks:
z(e)= min

x~c~tr~p~Fjcix
j
i

j=l

cjxj

/~jxj
j=l

(2)

(3)

z(e)= min ~ Z XiCiUiPij~TjCjXj b2

(1)

J=

c~xj/>e
j

~ cjxj 0.

(4)

Each candidate has a cost c~ and associated
expected rate of return rj and standard deviation of expected return aj. Then p'j = rjcj is

(II)

\i=1 j=l

)/

(6)

subject to:
~0 LJ = t

~ cjxj + x o = bo
)=1

(8)

Omeoa, Vol. 9, No. 1
xi = 0, 1 for j = 1.2..... n,

(9)

x0 />03 for all e>/0.

)

j=l

#)xj + ro bo -

1=

cjxj

>1e.

Recalling that/a~ = rjcj, this restriction reduces
to
~t'[x)=ro + 1 [ ~
(ri-ro)CiXi]>~ e.
uO L J =I

But in order to satisfy (10), we still require condition (3).
The alert reader will immediately recognize

that the term in brackets in this last expression
is simply the excess dollar return over the riskfree rate, and is the direct discrete analog of a
similar construct fundamental to modern portfolio theory [20]. The indivisible assets compete for funds against the risk-free alternative.
We have then the 'risk premium' for the jth
candidate asset.
If we now make the following simplifying
changes of variable,
~i =

(ri -

y =

bo(e

to)C1

ro),

model II now reduces to finding:

z(Y)= {min-bl [,=Z ~ x,cia,P,flicixil
(III)

(11)

subject to"
n

~(x) = ~ ujxi/> 3'

tl2t

CjXj ~ b 0

(13)

j=l
n

j=l

. . . . . nt

forall

y~O.

(14)

The family of efficient portfolios in (III) is
identical to that for (II). Model (III) thus
becomes the main focus of the ensuing discussion, and our model is now pure integer.
Note that evaluation of z(y) requires the use of
parametric techniques in integer optimization.
Unfortunately, little technology exists for
actually computing such functions (see [13, 31]
for the most recent survey). The standard grid
evaluation approach common to the literature
can be found in Peterson and Laughhunn [29]:
approximate z(y) by evaluatint (III) for a fixed
sequence of predetermined values of y. Typically this involves dividing the range of admissible y values into equal intervals and solving
(III) explicitly at the gridpoint y values. Asymmetric spacing of the y values will also work.
Unfortunately, while the portfolios so obtained
are indeed among the efficient set, this method
only bounds the frontier. Since we cannot discriminate among appropriate y values a priori,
this method skips over potentially interesting
portfolios arbitrarily. Furthermore, a 'coarse'
grid size, while computationally economical,
generally produces a poor bound; too 'fine' a
grid can repeat portfolios and may require
extensive calculation--and we still cannot
guarantee that we have the true frontier. Fortunately, we have an alternative.

II. G E N E R A T I O N O F T H E E F F I C I E N T
FRONTIER
Our alternative scheme, outlined briefly in
[24], is based on the strengths of the PetersonLaughhunn idea, but circumvents its weaknesses. The algorithm can be described as follows :2

and
-

j=l,2

(10)

Note that we have now included a continuous
variable, x0. The bracketed term in (7) is the
dollar return from investment in both the indivisible and riskless assets. The equality (8)
also contains more than one asset class. Thus
we might normally treat (II) as a mixed-integer
optimization problem (see [11]). However,
since x0 acts only as a slack variable in the
budget constraint, we can use (8) to eliminate
x0 from the formulation via substitution. The
objective (6) is unaffected, but (7) becomes

V(x) = bo

xi=O,l,

79

2 We ignore the trivial solution xj = 0 for all j, when
y = 0. The empty portfolio is always a member of the efficient set.

0. Set k = 1 and let yk = min {#j}.
1. Find z(yk), the associated minimum variance
p o r t f o l i o .~k = (2k, 2 k ,, . x,),
~k and the resulting portfolio yield p(:~k). If no such ~k exists,
go to 3. Else, set E -- g, the largest common
divisor of the excess yields/~ and,
2. Set k = k + 1 and let yk = p(£k-1) + e. G o
to 1.

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Findlay, McBride, Yormark, Messner--Mean-Variance Analysis

3. Stop. All of the efficient portfolios have been
found.
This procedure always yields the complete
exact efficient frontier so long as ~ is no larger
than the largest common divisor of the #j. The
method has proven to be an effective general
computational tool for generating discrete efficient frontiers. It has probably been considered
by others, but to our knowedge has never
appeared before in the finance literature.
Furthermore, an important feature of this
method is that it is independent of the algorithm chosen for the optimization in step 1.
Also, because we are using the most recent efficient portfolio and its associated yield #(x) to
drive us to the next efficient solution, we never
skip over any efficient points. Note carefully,
however, that since we niust execute step 1 as
many times as there are efficient portfolios, the
key to this approach still lies with finding a
rapid method of solving the integer program
(III) under a fixed y.
From a practical standpoint, the computational expense of the above procedure depends
directly upon the efficiency of the basic integer
optimizing vehicle. The standard linearization
method [40] common to the literature expands
an n variable problem with two constraints like
(III) into a problem with n + n(n - 1)/2 integer
variables and 2 + n ( n - 1) constraints. For
n = 50 candidates this translates into 1275
integer variables and 2452 constraints; such a
problem is still largely beyond the routine
capabilities of current technology. Alternative
linearizations can yield quite compact formulations [14], but have proven disappointing in
practice [25]. Other alternatives can be considered [15, 18, 21, 28, 38]. However, McBride
and Yormark [24] have recently announced
the development of a new and significantly
more efficient parametric quadratic integer
programming algorithm. Their approach combines a unique implementation of the foregoing
parametrization scheme and a new quadratic
integer optimization package [25] along with
several methods for accelerating the computations. The result is an efficient code for generating discrete asset efficient portfolios.
The McBride-Yormark algorithm is built
around a natural nonlinear extension of the
implicit enumeration concept [12] which has
to date become the most successful and wide-

spread computational approach in linear
integer programming. The method implicitly
searches the set of potential solutions to (lib
by utilizing a combination of portfolio improvement criteria and bounding information
to eliminate sub-optimal solutions. The
primary bounding mechanism is based on the
'natural relaxation' to (III), obtained by simply
replacing (4) with the continuous analog:
0~