Application of the special constrained multiparametric linear program to portfolio selection decisions.
Shurden, Michael ; Little, Philip ; Wibker, Elizabeth 等
INTRODUCTION
Individual investors who are faced with portfolio selection
decisions may choose from diversification techniques which range from
the most simple and naive forms to more highly complex schemes requiring
quadratic programming. Regardless of the technique employed, it is
widely recognized that the foremost goal of most investors is a common
one: to achieve the maximum rate of return at the level of risk
considered appropriate for the individual investor's portfolio.
While the risk to total return tradeoff is important, investors do
exhibit individual references for other investment features as well.
This is evident when one considers the many types of investments
available from which to choose. Some investments offer maximum growth
potential in capital appreciation; some periodic income distributions;
some liquidity; and still others some combination of all the above.
Given the myriad of investment opportunities available to the
individual investor, the need exists for a "tool" which will
permit the investor to combine the different types of investments in
such a way as to minimize investment risk and maximize investment
returns while satisfying individual preferences. The purpose of this
research is to demonstrate how the Special Constrained Multiparametric
Linear Program (SCMLP) model can facilitate this complex decision
process in such a manner as no other method can.
The next section of this research explores some of the widely
accepted theoretical and practical means that have been devised to make
portfolio selection decisions. The third section presents the SCMLP and
related theory. The fourth section demonstrates how the SCMLP may be
employed in the portfolio selection decision process. Finally, the last
section provides a summary and the conclusions of this research.
THEORETICAL BACKGROUND: PORTFOLIO SELECTION TECHNIQUES
The classical model for portfolio analysis was developed by
Markowitz (1952). The theoretical substance of the Markowitz model is
brilliant and is rightfully accepted in finance literature. However, its
practical usage is limited by the following factors:
(1) The quadratic programming solution procedure requires a
considerable amount of computer time and space.
(2) The complexity of the model makes it difficult to explain to an
individual user.
(3) The Markowitz model focuses only on risk and total returns and
does not consider other individual preferences.
Various attempts have been made to overcome the practical problems
posed by the Markowitz model. For instance, Sharpe's single index
model greatly simplifies the solution procedure and the time and cost of
obtaining a solution. However, this approach may not lead to the optimal
mean--variance portfolio and does not take into consideration other
preferences of individual investors (Sharpe, 1963).
A variation of Sharpe's single index model, developed by
Elton, Gruber, and Padberg (1976), uses the risk measure from
Sharpe's model (BETA) in a returns to risk ratio which is then used
to determine the proportion of the portfolio to be invested in each
asset. In fact, Burgess and Bey (1988) demonstrate that the Elton,
Gruber, and Padberg procedure is effective in estimating the Markowitz
efficient portfolio and can be an effective screening procedure for
large number of securities. Again, however, regardless of the
simplification and effectiveness of the Elton, Gruber, and Padberg
approach, individual references for investment factors other than risk
and total returns are not taken into consideration in the portfolio
selection process.
The literature is also replete with articles on linear programming
approaches to portfolio selection and optimization which do take into
consideration the individual preferences of investors (e.g., Lee &
Lerro, 1973; Beazer, 1976; Mitchell & Rosebery, 1977; Courtney,
1979; Avery; Hodges & Schaefer, 1977). One major drawback of the
linear programming procedure in portfolio selection is that the
individual investor must place weights on his preferences.
Though not a linear programming approach, Smith presents an
arithmetic algorithm as a suggested approach to a quantitative solution
of the problem of individual preferences in portfolio selection. As an
example, he suggests that an individual investor first weight four
attributes (liquidity, income, capital appreciation, and safety)
according to his preferences. Then the investor examines four asset
types (savings account, corporate bonds, common stock, and real estate)
as to their suitability with respect to each attribute. The final step
is to weight the suitability measures for each asset with the
preferences of the individual, attribute by attribute, in order to get
an overall measure for that asset type. Each of the four attributes is
desirable to the individual, hence, the weighted-suitability measure for
each asset is an overall index of suitability based on the unique set of
preferences. However, by the author's own admission, many
individuals may not feel comfortable with a precise set of relative
weights reflecting their preferences (Smith, 1974).
A more workable approach for making multiasset portfolio decisions
based on individual preferences can be derived using the SCMLP model. In
contrast to the absolute numerical weights necessary in the approach
used by Smith, the SCMLP formulation requires only that the investor
order his preferences with respect to the attributes and that he define
a ranking as to the suitability of each asset type with respect to each
attribute. The manner in which the SCMLP model works after these
individual preferences have been ranked is dealt with in the following
sections of this research. The next section presents the theoretical
development of the SCMLP followed by a demonstration of how the SCMLP
could be applied to the portfolio selection problem.
THE SPECIAL CONSTRAINED MULTIPARAMETRIC LINEAR PROGRAM
A major task in the development of realistic linear programming
models is determining and stating the exact numerical value of the
objective function coefficients. Obviously, many situations exist where
the interdependencies in an economic model as described by a linear
programming problem are not fixed or known with certainty. In practice,
the available data is often too vague or unreliable to be applicable in
a strict delineation of the model. The inexactitude of information
concerning the objective function coefficients necessitates deriving
some method to handle the more general and realistic situation. This
formulation will be termed the Special Constrained Multiparametric
Linear Program (SCMLP).
In this model, the objective function coefficients in a linear
program are constrained in a manner similar to the decision variables.
These constraints may be such that they bound the coefficients in some
fashion, or they may simply be an ordering of the values. Thus, the
model of interest can be expressed as
Maximize: [c.sup.T].X
subject to: Ax [less than or equal to] b x [greater than or equal
to] 0 Gc [less than or equal to] d c [greater than or equal to] 0
The unknown column vectors c and x are of dimension n. The matrix A
of constant coefficients is of order (m,n), and G is an (s,n) matrix of
known constants. The vectors b and d are assumed to have nonnegative constant elements of which at least one is non-zero with b having
dimension m and d being an s-component vector.
It should be noted that the SCMLP can be formulated as a quadratic
programming problem. Letting [t.sup.T] = ([x.sup.T], [c.sup.T]) the
model can be expressed as follows:
O I
Maximize: 1 t[t.sup.T] t
2
I 0
A O b
subject to: t[less than
or equal to]
0 G d
t[greater
than or
equal to]0
where I is an (n, n) identity matrix.
This particular formulation seems to present a viable solution
method. Unfortunately, the SCMLP is not a convex programming problem.
Thus, any quadratic programming solution procedure will be very
complicated and time-consuming to employ and will not guarantee
convergence to the global optimum. Consequently, an alternate procedure
for solving the problem shall be derived that employs the far simpler
and relatively efficient simplex method.
The purpose of the algorithm is to simultaneously determine the
optimal objective function coefficients and decision variables that
solve the SCMLP. The process begins by determining an initial feasible
vector of coefficients [c.sub.i]. A vector of decision variables,
([x.sub.i]) is then derived such that it maximizes the objective
function with the set ([c.sub.i]). If the function is not at the global
maximum, a new (c) can be found that increases the value of the
objective function at ([x.sub.i]). The procedure continues in this
manner until the point is reached where there no longer exists any other
vector of objective function coefficients that will increase the value
of the objective function with the associated decision variables and
vice-versa and simultaneously satisfy the constraint set.
In the next section, the SCMLP model is demonstrated in the
portfolio selection process. A more detailed exemplification of the
solution method is presented there.
A DEMONSTRATION OF THE PORTFOLIO SELECTION PROCESS USING THE SCMLP
MODEL
Earlier in this research, the Smith (1974) model for investment
selection was discussed. In that discussion it was claimed that the
SCMLP model could best achieve the desired objectives of the Smith model
and applications of linear programming procedures to the portfolio
selection process without the necessity of having the individual
investor place absolute weights on his investment preferences. The
following example demonstrates how the SCMLP model would work using the
same individual preferences (liquidity, periodic income, capital
appreciation, and safety) and asset types (savings accounts, corporate
bonds, common stock, and real estate) used by Smith in his research.
The process begins by questioning the individual as to his relative
preferences concerning the four attributes. Define [c.sup.T] =
([c.sub.1], [c.sub.2], [c.sub.3], [c.sub.4]) to be a vector of weights
or rankings on the attributes liquidity, income, appreciation, and
safety respectively such that the sum of the ([c.sub.i]) is one. The
weights may be of the form [c.sub.1] [less than or equal to] 2[c.sub.2],
or [c.sub.1] [greater than or equal to] .1, or [c.sub.3] + [c.sub.4]
[less than or equal to] .3, or they may be some simple ordering such as
[c.sub.1] [less than or equal to] [c.sub.2] [less than or equal to]
[c.sub.3] [less than or equal to] [c.sub.4], which implies that the
investor prefers safety to appreciation, appreciation to income, and
income to liquidity. These relationships form a set of constraints of
the form Gc [less than or equal to] d.
Next the investor must examine the four asset types as to their
suitability with respect to each attribute. These variables are defined
as follows:
Liquidity Income Appreciation Safety Asset
[w.sub.1] [x.sub.1] [y.sub.1] [z.sub.1] Savings Account
[w.sub.2] [x.sub.2] [y.sub.2] [z.sub.2] Corporate Bonds
[w.sub.3] [x.sub.3] [y.sub.3] [z.sub.3] Common Stock
[w.sub.4] [x.sub.4] [y.sub.4] [z.sub.4] Real Estate
where [w.sub.i] = 1, [x.sub.i], = 1 [y.sub.i] = 1,
and [z.sub.i] = 1.
As before, the individual may use some simple ordering of his
preferences or concepts with respect to some attribute and how the four
asset types correspond. For example, an individual may feel that a
savings account is more liquid than corporate bonds implying that
[w.sub.1] [greater than or equal to] [w.sub.2]. Proceeding in this
manner defines a set of constraints of the form At [less than or equal
to] b, with the inclusion of the norming restrictions, where [t.sup.T] =
([w.sup.T], [x.sup.T], [y.sup.T], [z.sup.T]).
Since the preference orderings are independent, the portion of the
investor's nonconsumed wealth that he will want to hold in each
asset type will be as follows:
Savings account: [w.sub.1][c.sub.1] + [x.sub.1][c.sub.2] +
[y.sub.1][c.sub.3] + [z.sub.1][c.sub.4] = SA
Corporate bonds: [w.sub.2][c.sub.1] + [x.sub.2][c.sub.2] +
[y.sub.2][c.sub.3] + [z.sub.2][c.sub.4] = CB
Common Stock: [w.sub.3][c.sub.1] + [x.sub.3][c.sub.2] +
[y.sub.3][c.sub.3] + [z.sub.3][c.sub.4] = CS
Real Estate: [w.sub.4][c.sub.1] + [x.sub.4][c.sub.2] +
[y.sub.4][c.sub.3] + [z.sub.4][c.sub.4] = RE
Thus, SA is the percentage of one's nonconsumed wealth that
should be invested in a savings account given the individual's
preference for the four attributes and his perception of how a savings
account corresponds with respect to these attributes. The quantities CB,
CS, and RE can be defined similarly.
Next, an initial set of values are derived for [c.sub.1],
[c.sub.2], [c.sub.3] and [c.sub.4] which satisfies the Gc [less than or
equal to] d constraints. The values chosen for [c.sub.1], [c.sub.2],
[c.sub.3] and [c.sub.4] serve only as a starting point to derive an
initial set of coefficients for the objective function. The summation of
these values must equal one. The w's, x's, y's and
z's ([w.sub.1], [w.sub.2], [w.sub.3], [w.sub.4] [x.sub.1],
[x.sub.2], [x.sub.3], [x.sub.4], [y.sub.1], [y.sub.2], [y.sub.3],
[y.sub.4], [z.sub.1], [z.sub.2], [z.sub.3], [z.sub.4]) are the sixteen
decision variables presented in the initial formulation of the linear
programming models. The initial set of coefficients ([c.sub.01]) for the
decision variables is derived by multiplying [c.sub.1], [c.sub.2],
[c.sub.3] and [c.sub.4] by each of the expected returns associated with
the four investment categories as follows:
[c.sub.1][r.sub.1] = [c.sub.01] [c.sub.1][r.sub.2] = [c.sub.02]
[c.sub.1][r.sub.3] = [c.sub.03] [c.sub.1][r.sub.4] = [c.sub.04]
[c.sub.2][r.sub.1] = [c.sub.05] [c.sub.2][r.sub.2] = [c.sub.06]
[c.sub.2][r.sub.3] = [c.sub.07] [c.sub.2][r.sub.4] = [c.sub.08]
[c.sub.3][r.sub.1] = [c.sub.09] [c.sub.3][r.sub.2] = [c.sub.010]
[c.sub.3][r.sub.3] = [c.sub.011] [c.sub.3][r.sub.4] = [c.sub.012]
[c.sub.4][r.sub.1] = [c.sub.013] [c.sub.4][r.sub.2] = [c.sub.014]
[c.sub.4][r.sub.3] = [c.sub.015] [c.sub.4][r.sub.4] = [c.sub.016]
where: [r.sub.1] = expected return on a savings account
[r.sub.2] = expected return on corporate bonds
[r.sub.3] = expected return on common stock
[r.sub.4] = expected return on real estate holdings
Next, the original Gc [less than or equal to] d constraints must be
reformulated in terms of [c.sub.0] by substituting [c.sub.0]/r in place
of c. Accordingly, the Gc [less than or equal to] d constraints are
redefined by [Wc.sub.0] [less than or equal to] h. The [Wc.sub.0] [less
than or equal to] h relationships constrain the coefficients
([c.sub.oi]) in the objective function. The At [less than or equal to] b
relationships constrain the decision variables ([w.sub.1], [x.sub.1],
[y.sub.1], [z.sub.1]). The formulation of the SCMLP is presented:
Maximize [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
subject to At [less than or equal to] b t [greater than or equal
to] 0 [Wc.sub.0] [less than or equal to] h [c.sub.0] [greater than or
equal to] 0
The simplex method is used to solve the SCMLP problem. Initially,
the optimal [w.sub.i], [x.sub.i], [y.sub.i], and [z.sub.i] values are
derived that satisfy the constraints at At [less than or equal to] b.
Next, the optimal [w.sub.i], [x.sub.i], [y.sub.i], and [z.sub.i] values
are used as the objective function coefficients to find the optimal (c)
values that satisfy the constraints Gc [less than or equal to] d. The
next (c) values are then used to find the optimal [w.sub.i], [x.sub.i],
[y.sub.i], and [z.sub.i] values subject to the original At [less than or
equal to] b constraints. This process will continue back and forth until
the solution to the simplex method yields the (c) values and [w.sub.i],
[x.sub.i], [y.sub.i], and [z.sub.i] values that will maximize total
portfolio returns. These values are then substituted in the formula
(shown earlier in this section) to solve for the portion of the
individual's nonconsumer wealth to hold in each asset type (for
example: SA = [w.sub.i][c.sub.i] + [x.sub.i][c.sub.2] +
[y.sub.i][c.sub.3] + [z.sub.i][c.sub.4]).
SUMMARY AND CONCLUSION
This research suggests that the SCMLP model is superior to other
approaches to the portfolio selection process when the individual
preferences of investors extend beyond the simple risk to total return
tradeoff. Additionally, the unique feature of the SCMLP model that
allows for a simple ranking of individual preferences versus the
necessity of assigning absolute weights is highlighted. Problems
associated with applying portfolio selection techniques ranging from the
Markowitz Quadratic Programming Technique to the Smith Simple Arithmetic
Algorithm are discussed. This research shows how the problems of the
other approaches are mostly overcome by the SCMLP model.
The other approaches that are used in the portfolio selection
process are wrought with one or more of the following four problems:
(1) Costly to implement.
(2) Complex and difficult to explain.
(3) Do not take into consideration the individuals preferences of
investors.
(4) Required absolute weights to places on individual preferences.
The SCMLP model, which is explained in this research in a portfolio
selection environment, certainly overcomes the problems associated with
individual preferences, is not costly to implement, and is not as
complex and difficult to explain (as the Markowitz Quadratic Approach).
Finally, the question remains: How can the SCMLP model be applied
to the "real world" of portfolio selection? As an example; a
retail investment broker, trained in the use of the SCMLP, could
interview a client with respect to the client's investment
objectives and preferences for various investment features plus
empirical data for various types of investments, a portfolio could be
designed. This portfolio design could be the general framework from
which the client's available funds would be allocated. As
conditions in the market or the client's individual preferences
change, the SCMLP model could be reformulated to determine if
adjustments need be made in the portfolio makeup. The decision process
would certainly be enhanced by the use of the SCMLP model.
REFERENCES
Avery, M.L. A Practical LP Model for Equity Portfolios. Omega,
41-48.
Beazer, W.F. (1976). Applying Linear Programming to Domestic Bank
Portfolios. Euromoney, December, 70.
Burgess, R.C. & R.P. Bey. (1988). Optimal Portfolios: Markowitz
Full Covariance Versus Simple Selection Rules. The Journal of Finance
Research, Summer, 153-163.
Courtney, J.F. (1979). Differentiating Capital Appreciation and
Income in Portfolio Selection/Revision. Journal of Bank Research,
Summer, 111-118.
Elton, E.J., M.J. Gruber & M.W. Padberg. (1976). Simple
Criteria for Optimal Portfolio Selection. Journal of Finance, December,
1341-1357.
Hodges, S.D. & S.M. Schaefer. (1977). A Model for Bond
Portfolio Improvement. Journal of Financial and Quantitative Analysis,
June, 243-260.
Lee, S.M. & A.J. Lerro. (1973). Optimizing the Portfolio
Selection for Mutual Funds. Journal of Finance, December, 1087-1101.
Markowitz, H. (1952). Portfolio Selection. Journal of Finance,
March, 77-91.
Mitchell, K. & D. Rosebery. (1977). Problems of Investment
Appraisal. Accountancy, June, 108.
Sharpe, W.F. (1963). A Simplified Model for Portfolio Analysis.
Management Science, January, 277-293.
Smith, K.V. (1974). The Major Asset Mix Problem of the Individual
Investor." Journal of Contemporary Business, Winter, 49-62.
Michael Shurden, Lander University
Philip Little, Western Carolina University
Elizabeth Wibker, Louisiana Tech University