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