Credit union portfolio management: an application of goal interval programming.

Sharma, Hari P. ; Ghosh, Debasis ; Sharma, Dinesh K. 等

ABSTRACT

The uncertainty of cash flows, cost of funds and return on investments in ever changing financial markets require financial institutions to develop mathematical models for managing portfolios effectively and efficiently. In this paper, we propose a goal interval programming (GIP) technique for credit union portfolio management. In this technique, penalty functions are introduced in a goal programming (GP) formulation. We present how a system of penalties acts in a GIP to obtain efficient portfolios for credit union problems that have multiple goals and constraints. A comparison of GIP with penalty function and lexicographic goal programming (LGP) has been used to test the results for the same data. We observed from our results that the GIP provides better results in terms of portfolio allocation with maximum returns compared to traditional LGP model.

INTRODUCTION

Credit unions are chartered by their respective states or by the federal government in the United States under provisions of the Credit Union Act of 1934. In 1970, the Bureau of Federal Credit Unions became an independent federal agency with the establishment of the National Credit Union Administration (NCUA). The same year, credit unions, without the use of federal tax dollars established the National Credit Union Share Insurance Fund (NCUSIF) to protect credit union deposits against loss. According to the published data by USA Federal Credit Union, currently, there are more than 85 million credit union members in the United States, with deposits worth over $520 billion (National Credit Union Administration, 2006). Credit unions are financial institutions that specialize in consumer credit and residential mortgages. The primary function of credit unions is similar to other financial institutions and involves generating funds from their members by selling shares and savings deposits to its members and then lending the funds to members in the form of personal or consumer loans (Taylor, 1971).

Financial institutions are fundamentally different from other corporations. When a corporation goes bankrupt, shareholders, debtors and creditors suffer financial losses. The overall impact of the failure; however, is limited to stakeholders directly. In contrast, the failure of a financial institution can be potentially much more harmful. Financial institutions play a special role of intermediation e.g. payment flows across customers and maintain markets for financial instruments. This role can also make failure of financial institution much more disruptive for the economy than the failure of other entities. Credit unions are generally local and relatively small institutions whose failure is unlikely to destabilize financial markets. However, their failure can affect the growth and sustainability of such institutions as they are usually composed of persons from the same occupational group or the same local community. Credit unions have a comparatively weak governance structure compared to shareholder-owned financial institutions in the sense that no private individuals has the financial incentive to intervene strongly to discipline the management when the credit union's policies or performance go astray (Rasmusen, 1988). The Unions are operated as member-owned, tax-exempt, not-for-profit financial cooperatives and are democratically governed by a volunteer member-elected board of directors (Walker and Chandler, 1976 and 1977). The tax exemption status of credit unions, which is regulated at the federal level by the NCUSIF arises neither from the credit unions' limited fields of membership nor from the type of services they offer. Moreover, in 1951 and 1999, Congress reaffirmed credit union tax exemption status because credit unions operate "without capital stock" and are "organized and operated for mutual purposes without profit." The single exemption of state chartered credit unions is the 5% state corporate income tax. Federally chartered credit unions are exempt from both the State Sales and Corporate Income taxes (Tax Equalization Act, 1951 and Credit Union Access Act, 1999). As credit unions' are exempt from certain taxes, they are able to provide earnings back to members in the form of lower loan rates on loans and higher deposit rates. Banks are insured by the Federal Deposit Insurance Corp. (FDIC), a Federal government agency, and are taxed because they are designed as profit-making corporations that disburse profits to stockholders.

While management of credit union portfolio is less complex than with commercial banks, it still requires some level of quantitative skills and tools that can assist in making better investment decisions. Credit union management typically involves several conflicting objectives such as the maximization of returns, minimization of risk, expansion of deposits and loans. A credit union must determine its trade-off between risk, return and liquidity in managing its portfolio in light of uncertainty in cash flows, cost of funds and return on investments. Mathematical models that use multiple criteria decision-making (MCDM) techniques can assist to achieve these multiple goals. The MCDM approach differs from the mathematical programming model primarily in that it strives to optimize several objective functions simultaneously as opposed to just one. The MCDM modeling process can be divided into multi-attribute decision making (MADM) and multi-objective decision making (MODM). The former is often applicable to problems with the alternatives in a probabilistic environment, while the latter is generally applied to deterministic problems. Lexicographic goal programming (LGP) falls in the category of MODM (Messac et al., 1996).

LGP is one of the most widely used tools for solving MODM problems (Romero, 1986) developed to handle multi-criteria situations within the general framework of linear programming (LP). The LP assists only in modeling a single objective function while the LGP approach is the most popular for handling multiple objective problems in LP framework. The resulting LGP model yields what is usually referred to as an efficient solution because it may not be optimum, with respect to all the conflicting objectives of the problem (Romero, 1991). In the LGP model, a decision-maker associates a fixed target with each attribute in order to achieve the target. The objectives, as well as the structural constraints, are considered as goals by introducing under- and over-deviational variables to each of them. In this formulation, the model penalizes any deviational variable with respect to its target value according to a constant marginal penalty where the fixed target level of a goal is not achieved. However, in a real world situation the objective usually may not be achieved precisely and some degree of deviations from the fixed target will satisfy the needs of decision-makers. The introduction of penalty functions within GP has removed this difficulty and allows decision-makers to use the percentage target achievement where the goal achievement can lie within a certain target interval (Kvanli, 1980). A decision maker can set penalties according to the importance of the changes considering the marginal changes in the achievement of the target. Romero (1991) demonstrated various types of penalty functions and explained if the marginal penalty increases monotonically with respect to the targets than the V- shaped penalty function turns into U-shaped penalty function. This approach is known as goal interval programming (GIP).

In this paper, we present how the system of penalties acts in a GIP to obtain an efficient portfolio that has multiple goals and constraints. In our model for credit union portfolio selection, we have used U-shaped penalty functions. The GIP model allows credit union portfolio managers to allocate funds to maximize returns given the several constraints including regulatory requirements. A comparison of GIP with penalty function and LGP has been used to test the results for the same data. The remainder of the paper is organized as follows. Section 2 gives a brief review of literature on the related problem. Section 3 presents a mathematical model for the problem. Section 4 demonstrates the application of the model. Section 5 presents the results of the application. Finally, Section 6 gives the concluding remarks.

REVIEW OF LITERATURE

The history of GP modeling techniques goes back to 1955 when Charnes et al. (1955, 1961) published the first application of GP. Since then, several studies have been published using LGP for financial decision-making problems (Lee, 1972; Lee and Lerro, 1973; Kumar et al., 1978; Lee and Chesser, 1980; Levary and Avery, 1984; Schniederjans et al., 1992; Sharma et al., 1995; Cooper et al., 1997; Dominiak, 1997; Leung et al., 2001; Pendaraki et al., 2004 & 2005). Lee and Lerro (1973) developed a LGP portfolio selection model for mutual funds. Kumar et al. (1978) developed a conceptual LGP model for portfolio selection of dual-purpose funds. Lee and Chesser (1980) demonstrated how linear beta coefficient from finance theory reflecting risk in alternative investments could be incorporated into a LGP model. Levary and Avery (1984) also introduced a LGP model representing the investor's priorities and compared the use of LP to GP for the selection of optimal portfolio. Schniederjans et al. (1992) illustrated the use of LGP as an aid to investors planning investment portfolios for themselves. Sharma et al. (1995) presented LGP as an aid for investors or financial planners planning investment portfolios for individuals and/or companies by using beta coefficients and other important parameters. Recently, Pendaraki et al. (2004) applied LGP on a sample of Greek mutual funds.

According to the survey, the lexicographic goal programming (LGP) technique has become a popular technique for solving multi-decision making problems (Sharma et al. 1999; Tamiz and Jones, 1995; Romero, 1986 & 1991). Charnes and Collomb (1972) introduced the idea of goal interval programming. Kvanli (1980) incorporated penalty functions into GP model by analyzing a financial planning problem considering target intervals. Can and Houck (1984), Rehman and Romero (1987), Romero (1991), Ghosh et al. (1993), Sharma et al. (2003), and others have also applied GP with penalty functions.

Most of the studies have focused on commercial banking and other financial institutions. Kusy and Ziemba (1986) applied a stochastic LP model for the Vancouver City Savings Credit Union portfolio. Walker and Chandler (1977 & 1978) used GP models for allocation of Credit Union net revenues and net monetary benefits of credit union membership. Sharma et al. (2002) applied GP modeling for best possible solutions for loan allocation problems. However, the GIP technique has not been significantly used to solve credit unions' portfolio management problems.

METHODOLOGY FOR MODEL DEVELOPMENT

The basic LGP model can be expressed as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.1)

where,

[bar.X] = Vector of K priority achievement functions,

[P.sub.k]([bar.d]) = [P.sub.k]([w.sup.-.sub.ik][d.sup.-.sub.ik] + [w.sup.+.sub.ik][d.sup.+.sub.ik]),

[f.sub.i](.) = the [i.sup.th] functions vector, i = 1, 2, ..., m,

[b.sub.i] = the aspiration level of the [i.sup.th] goal,

[w.sub.ik.sup.-], [w.sub.ik.sup.+] = the respective weights associated with the under and over- deviational variables,

[d.sub.ik.sup.-] and [d.sub.ik.sup.+] at the k-th priority level.

In LGP, the goals assigned to different priority levels are not comparable. To make it directly comparable, percentage of achievement is introduced. In this situation, the decision-maker may desire to achieve the goals in terms of percentage in a specified interval rather than achieving goals in fixed targets. In GIP, a system of penalties works for different specified intervals. Introduction of penalty functions according to different intervals of achievement will help to derive a GIP model from a LGP model. In case of five or more sided penalty functions, goals are less or equal to and greater or equal to the target. If the goals are less or equal to the target, the decision maker does not accept an achievement of the goal over [t.sub.R] % of its target and for the deviation between [t.sub.r] - [t.sub.r-1] % with respect to its target, the marginal penalty is [[delta].sub.r]. The total penalties produced are as [R.summation over (r=1)] [[delta].sub.r] [D.sup.+.sub.ir]. On the other hand, if the goals are greater or equal to the target, the decision maker does not accept an achievement of the goal under [t.sub.s]% of its target and for the deviation between [t.sub.s-1] - [t.sub.s]% with respect to its target, the marginal penalty is [[alpha].sub.s]. The total penalties produced can be represented as [S.summation over (s=1)] [[alpha].sub.s] [D.sup.-.sub.is].

The system of goal achievement and marginal penalties is presented in Table 1.

In terms of general GIP model, the equation (3.1) can be expressed as:

Minimize [bar.X] = [[P.sub.1]([bar.D]), ..., [P.sub.k]([bar.D]), ..., [P.sub.K]([bar.D])],

subject to,

[100/[b.sub.i]] [f.sub.i]([bar.x]) + [S.summation over (s=1)] [D.sup.-.sub.is] - [D.sup.+.sub.is] = 100, for s = 1,2, ..., S ; i = 1,2, ... n

[100/[b.sub.i]] [f.sub.i](bar.x]) + [D.sup.-.sub.ir] - [R.summation over (r=1)] [D.sup.+.sub.ir] = 100, for r = 1, 2, ..., R; i = n+1, n+2, ..., m

0 [less than or equal to] [D.sup.-.sub.is] [less than or equal to] [T.sub.s], 0 [less than or equal to] [D.sup.+.sub.ir] [less than or equal to] [T.sub.r] (3.2)

where, r (= 1 ,2, ..., R) and s (= 1, 2, ..., S) are the target ranges of goals.

[T.sub.r], [T.sub.s] Measurement of achievement of the goal in the r-th target range and s-th target range respectively.

[P.sub.k]([bar.D]) = [P.sub.k]([R.summation over (r=1)] [[delta].sub.r][D.sup.+.sub.ir,k] + [S.summation over (s=1)] [[alpha].sub.s][D.sup.-.sub.is,k]),

By introducing the defined penalty scales in (3.2), the modified constraints can be expressed as follows:

[D.sup.-.sub.is] + [[eta].sub.s] - [[rho].sub.s] = [T.sub.s], [D.sup.+.sub.ir] + [[gamma].sub.r] - [[phi].sub.r] = [T.sub.r]

The priority structure can be modified as:

[P.sub.k] ([bar.D]) = [P.sub.k] ([R.summation over (r=1)] [[delta].sub.r] [D.sup.+.sub.ir,k] + [S.summation over (s=1)] [D.sup.-.sub.is,k] + [w.sup.-.sub.sk] [[eta].sub.sk] + [w.sup.+.sub.sk] [[rho].sub.sk] + [w.sup.-.sub.rk] [[gamma].sub.rk] + [w.sup.+.sub.rk] [[phi].sub.rk]), (3.3)

GIP Model of Credit Union

The following notations are defined to formulate the model of the credit union problem:

Indices

l index for the loan type, l = {1, 2, ..., L).

c index for Fed Funds, money market fonds or short term securities, c [member of] {[c.sub.1], [c.sub.2], ..., [C.sub.n]} [subset or equal to] {1, 2, ..., L}

h index for Home Equity loan h [member of] {1,2, ..., L}

m index for mortgage loan, m [member of] {[m.sub.1], [m.sub.2], ..., [m.sub.n]} [subset or equal to] {1, 2, ..., L}

p index for personal loan, p [member of] {[p.sub.1], [p.sub.2], ..., L} [subset or equal to] {1, 2, ..., L}

nm index for new motorcycle loan, nm [member of] {1, 2, ..., L}

um index for used motorcycle loan, um [member of] {1, 2, ..., L}

nv index for new car/truck loan, nv [member of] {1, 2, ..., L}

uv index for used car/truck loan, uv [member of] {1, 2, ..., L}

nb index for new boat, nb [member of] {1, 2, ..., L}

ub index for used boat, ub [member of] {1, 2, ..., L}

v index for visa loan, v [member of] {1, 2, ..., L}

Variables and Parameters

[X.sub.l] = Amount of money invested in loan l,

[A.sub.l] = Annual rate of return from loan l,

R = Total annual return from all loans,

[tau] = Total available funds available,

C = Required cash for processing all loans,

[C.sub.l] = Percentage of loans as a cash reserve for each loan l.

The Goals and Priorities

The decision maker's priorities with different goals are defined as follows:

[P.sub.1] Utilizes total available funds for investment, maximizing annual return and satisfies home equity loans.

[P.sub.2]: Satisfies new and used motor cycle loans, cash and money market funds, visa loan, new and used boat loan, and minimize the portfolio's risk.

[P.sub.3]: Satisfies mortgage and personal loans.

The decision-maker may decide these target goals and priorities depending on their preference.

Goal Constraints

The following goal constraints appear in the general model of the credit union problem to formulate the GIP model.

(1) Available Funds: The objective is to utilize the total available funds. In terms of percentage achievement, the goal equation appears as:

[100/[tau]] [L.summation over (l=1)] [X.sub.l] + [D.sup.-.sub.1] - [D.sup.+.sub.1] = 100 (3.1.1)

(2) Annual Return: The weighted average annual return on the portfolio should be at least a certain percent of total available funds. In terms of percentage achievement, the goal equation can be written as:

[100/R] [L.summation over (l=1)] [A.sub.l] [X.sub.l] + [D.sup.-.sub.2] - [D.sup.+.sub.2] = 100 (3.1.2)

(3) Operating Cost: The weighted average operating costs should be at least a certain percentage ([C.sub.1]) of the portfolio. In terms of percentage achievement, the goal equation can be expressed as:

[100/C] [L.summation over (l=1)] ([C.sub.l] % * [X.sub.l]) + [D.sup.-.sub.3] - [D.sup.+.sub.3] = 100 (3.1.3)

(4) Diversification: In order to reduce risk through diversification, the decision maker may prefer to invest minimal amount of money in several different types of investments/loans, but at the same time establish a maximum amount that can be invested in any particular investment/loan. The restriction and limits on different types of investments/loans and other securities by a credit union are also predefined by senior management. The percentage achievement is not applicable to any goal constraint where the target value is zero. These restrictions can be defined into the following categories:

(i) Cash/Short Term Securities: To ensure liquidity of funds, a percentage of amounts (a % [tau] (=[kappa], say) are required to be invested in short-term securities such as Fed Funds, money market funds etc. In terms of percentage achievement, the goal equation appears as:

[100/[kappa] [[C.sub.n].summation over (c=[c.sub.1])] [X.sub.c] + [D.sup.-.sub.4] - [D.sup.+.sub.4] = 100 (3.1.4)

(ii) Home Equity Loans: The home equity loans ([X.sub.h]) for a year must be at least a percentage (y) of all mortgage loans (m = [m.sub.1], [m.sub.2], ..., [m.sub.n]). The goal equation can be written as:

[X.sub.h] + [d.sup.-.sub.5] - [d.sup.+.sub.5] = y % of [[m.sub.n].summation over (m=[m.sub.1])] [X.sub.m] (3.1.5)

(iii) Mortgage Loans: A mortgage ([X.sub.mq [member of] {m1, m2, ... mn}]) loan must also be at least a percentage (y) of all other mortgage loans. The goal equation can be expressed as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.1.6)

Also, total allocated mortgage loans must not exceed a percentage (y) of total loan amount. In terms of percentage achievement, the goal equation can be written as:

[100/(y % of [tau])] [[m.sub.n].summation over (m=[m.sub.1])] [X.sub.m] + [D.sup.-.sub.6] - [D.sup.+.sub.6] = 100 (3.1.7)

Similarly, other adjustable rate and unimproved property restrictions for each loan type may also be considered.

(iv) Personal Loans: Personal loans are usually unsecured loans with higher interest rate. In order to minimize risk, there should be a limit on those loans. The total amount of personal loans ([X.sub.p]) for a year must be at least a percentage of total amounts allocated for all other loans. The goal equation can be defined as:

[[P.sub.n].summation over (p=[p.sub.1])] [X.sub.p] + [d.sup.-.sub.7] - [d.sup.+.sub.7] = y % of [L.summation over (l=1,(l[not equal to]p))] [X.sub.l] (3.1.8)

Also, personal loan must exceed a certain percentage of total loan amounts. In terms of percentage achievement, the goal equation can be written as:

[100/(y % of [tau]) [[P.sub.n].summation over (p=[p.sub.1])] [X.sub.p] + [D.sup.-.sub.7] - [D.sup.+.sub.7] = 100 (3.1.9)

(v) Vehicle Loans: The goal constraints for vehicles loans can be written as:

(a) New Vehicles: New motorcycle ([X.sub.nm]) and used motorcycle ([X.sub.um]) loans may not exceed the new car/truck loans ([X.sub.nv]). Again, used car/truck loan ([X.sub.uv]) must not exceed 70% of the new motorcycle loans. Mathematically, this can be represented as:

[X.sub.nm] + [X.sub.um] + [d.sup.-.sub.8] - [d.sup.+.sub.8] = [X.sub.nv] (3.1.10)

[X.sub.nm] + [X.sub.um] + [d.sup.-.sub.8] - [d.sup.+.sub.8] = [X.sub.nv] (3.1.11)

(b) Recreational Loans: New boat ([X.sub.nb]) and used boat ([X.sub.ub]) recreational vehicle loans must be at least a percentage (y) of the total loan amount. In terms of percentage achievement, the goal equation can be written as:

[100/(y % of [tau])] [X.sub.nb] + [X.sub.ub]] + [D.sup.-.sub.10] - [D.sup.+.sub.10] = 100 (3.1.12)

(vi) Visa Loans: Visa loan ([X.sub.v]) amount for a year must be at least a percentage (y) of total amounts allocated for all other loans. The goal equation can be defined as:

[X.sub.v] + [d.sup.-.sub.11] - [d.sup.+.sub.11] = y % of [L.summation over (l=1,(l[not equal to] v)] [X.sub.1] (3.1.13)

For example, after introducing penalties to equation (3.1.7), the goal equation with penalty functions can be written as:

Minimize [2.summation over (r=1)] [[delta].sub.r] [D.sup.+.sub.6,r]

[100/(y % of [tau]) [[m.sub.n].summation over (m=[m.sub.1])] [X.sub.m] + [D.sup.-.sub.6] - [D.sup.+.sub.6,1] - [D.sup.+.sub.6,2] = 100

[D.sup.+.sub.6,1] [less than or equal to] [T.sub.1], [D.sup.+.sub.6,2] [less than or equal to] [T.sub.2] (3.1.14)

The other goal equations may be defined in a similar way.

To demonstrate the use of the proposed GIP model with penalty functions, the following application of Credit Union is presented.

APPLICATION OF GIP TO THE CREDIT UNION

This section considers an application of the GIP model to the portfolio problem of a credit union. We have used the estimated data of a credit union that has $300 million available funds for a given planning year. The objective of decision-maker is to construct a diversified portfolio that provides the maximum total annual return by allocating the funds among twenty different types of loans and investments. Table 2 contains the rates of interest on various loans and investment choices and Table 3 contains the data for the penalty scales for different loans and investments.

The complete GIP problem with penalty functions is as follows:

Minimize ([D.sup.-.sub.1] + [D.sup.+.sub.1] + 2[D.sup.-.sub.2] + 2[d.sup.-.sub.3], [d.sup.+.sub.16] + [d.sup.+.sub.17] + [d.sup.+.sub.18] + [D.sup.-.sub.21,1] + 2[D.sup.-.sub.21,2] + [[rho].sub.5] + [[rho].sub.6] + [d.sup.-.sub.19] + [d.sup.-.sub.20] +

[d.sup.+.sub.22] + [d.sup.-.sub.23] + [D.sup.-.sub.24], 2[d.sup.-.sub.4] [d.sup.-.sub.5] [d.sup.-.sub.6] + [D.sup.+.sub.7,1] + 2[D.sup.+.sub.7,2] + [[eta].sub.1] + [[eta].sub.2] + [d.sup.+.sub.8] + [d.sup.+.sub.9] + [d.sup.- .sub.10] + [d.sup.-.sub.11] + [D.sup.-.sub.12,1] + 2[D.sup.-.sub.12,2] +

[[rho].sub.3] + [[rho].sub.4] + [d.sup.-.sub.13] + [d.sup.+.sub.14] + [d.sup.+.sub.15])

The goal equations (4.1), (4.2), (4.7), (4.12), (4.21) and (4.24) have been converted in terms of percentage achievement whereas the goal equations (4.7), (4.12) and (4.21) have been presented as per penalty scales given in Table 2. The percentage achievement is not applicable to goal constraints (4.3-4.6), (4.8-4.11), (4.13-4.20), (4.22) and (4.23) because of target value is zero. The goal equations are defined as follows:

(i) To utilize the total $300 million in funds, the goal equation appears as:

0.333 [20.summation over (l-1)] [X.sub.l] + [D.sup.-.sub.1] - [D.sup.+.sub.1] = 100 (4.1)

(ii) The average annual rate of return from loans/investments is at least 10% (assumed).

0.2417[X.sub.1] + 0.2167[X.sub.2] + 0.2127[X.sub.3] + 0.2[X.sub.4] + 0.1793[X.sub.5] + 0.1127[X.sub.6] + 0.2583[X.sub.7] +

0.225[X.sub.8] + 0.2266[X.sub.9] + 0.2167[X.sub.10] + 0.2083[X.sub.11] + 0.2583[X.sub.12] + 0.275[X.sub.13] + 0.2917[X.sub.14]

+ 0.2583[X.sub.15] + 0.275[X.sub.16] + 0.275[X.sub.17] + 0.2917[X.sub.18] + 0.325[X.sub.19] - 0.101[X.sub.20] + [D.sup.-.sub.2] - [D.sup.+.sub.2] = 100 (4.2)

(iii) Home equity loans must be at least 15% of all mortgage loans.

[X.sub.1] + [d.sup.-.sub.3] - [d.sup.+.sub.3] = 0.15[6.summation over (l=2)]X.sub.l] (4.3)

(iv) (a) 30-year fixed mortgage must be 25% of fund invested in all other mortgage loans.

[X.sub.2] + [d.sup.-.sub.4] - [d.sup.+.sub.4] = 0.25 [6.summation over (l=3)] [X.sub.l] (4.4)

(b) 20-year fixed mortgage must be 20% of fund invested in all other mortgage loans.

[X.sub.3] + [d.sup.-.sub.5] - [d.sup.+.sub.5] = 0.20 [6.summation over (l=2,(l [not equal to] 3)] [X.sub.l] (4.5)

(c) 15-year fixed mortgage must be at least 30% of all other fixed rate mortgage loans.

[X.sub.4] + [d.sup.-.sub.6] - [d.sup.+.sub.6] = 0.30 ([X.sub.2] + [X.sub.3]) (4.6)

(d) Total allocated mortgage loans must not exceed 40% of total loan amount.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.7)

(e) The adjustable rate mortgage must not exceed 30% of the of fixed rate mortgage loans.

[X.sub.5] + [X.sub.6] + [d.sup.-.sub.8] - [d.sup.+.sub.8] = 0.30[4.summation over (l=2)][X.sub.l] (4.8)

(f) 1-year adjustable rate mortgage loans must not exceed all other mortgage loans.

[X.sub.6] + [d.sup.-.sub.9] - [d.sup.+.sub.9] = [5.summation over (l=2)][X.sub.l] (4.9)

(g) 10-year unimproved property loans must exceed 10% of total mortgage loans.

[X.sub.7] + [d.sup.-.sub.10] - [d.sup.+.sub.10] = 0.10[6.summation over (l=2)][X.sub.l] (4.10)

(v) (a) Personal loans may exceed 10% of the fund invested in all other loans.

[X.sub.8]+ [X.sub.9] + [[X.sub.10] + [d.sup.-.sub.11] - [d.sup.+.sub.11] = 0.10 [20.summation over (l=1([not equal to]8,9,10))] [X.sub.l] (4.11)

(b) Personal loans must exceed 8% of total loan amount.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.12)

(c) Personal loans may exceed Visa loan.

[X.sub.8] + [X.sub.9] + [X.sub.10] - [X.sub.19] + [d.sup.-.sub.13] - [d.sup.+.sub.13] = 0 (4.13)

(d) Personal unsecured loans must not exceed 30% of all other personal loans.

[X.sub.9] + [d.sup.-.sub.14] - [d.sup.+.sub.14] = 0.3 ([X.sub.8] + [X.sub.10]) (4.14)

(e) Personal secured loans must not exceed 20% of all other personal loans.

[X.sub.8] + [d.sup.-.sub.15] - [d.sup.+.sub.15] = 0.2 ([X.sub.9] + [X.sub.10]) (4.15)

(vi) (a) Motorcycle loans may not exceed the car/truck loans.

- [X.sub.11] - [X.sub.12] + [X.sub.17] + [X.sub.18] + [d.sup.-.sub.16] - [d.sup.+.sub.16] = 0 (4.16)

(b) Used motorcycle loans must not exceed 40% of total loans allocated for motorcycle.

- 0.4[X.sub.17] + 0.6[X.sub.18] + [d.sup.-.sub.17] - [d.sup.+.sub.17] = 0 (4.17)

(c) Used car/truck loans must not exceed 40% of total allocated amount for car/truck loans.

- 0.4[X.sub.11] + 0.6[X.sub.12] + [d.sup.-.sub.18] - [d.sup.+.sub.18] = 0 (4.18)

(vii) Cash/Money market funds may exceed 20% of visa loans.

[X.sub.20] + 0.2[X.sub.19] + [d.sup.-.sub.19] - [d.sup.+.sub.19] = 0 (4.19)

(viii) Visa loans may exceed 30% of the total fund allocated to all other loan types.

[X.sub.19] + [d.sup.-.sub.20] - [d.sup.+.sub.20] = 0.30 [20.summation over (l=1([not equal to]19))] [X.sub.l] (4.20)

(ix) a) New and used boat loans must exceed 15% of total loan amount.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.21)

b) Used boat loans must not exceed 40% of total loans allocated for boat loans.

-0.4[X.sub.13] + 0.6[X.sub.14] + [d.sup.-.sub.22] - [d.sup.+.sub.22] = 0 (4.22)

(x) New and used motorcycle loans may exceed new car/truck loans.

-[X.sub.11] + [X.sub.17] + [X.sub.18] + [d.sup.-.sub.23] - [d.sup.+.sub.23] = 0 (4.23)

(xi) The credit union has to maintain at least 5% of loans as a cash reserve with the federal bank.

0.33 [20.summation over (l=1)] [X.sub.l] + [d.sup.-.sub.24] - [d.sup.+.sub.24] = 100 (4.24)

RESULTS AND DISCUSSION

The case analysis has been performed using LGP and GIP models. A computer code based on Ignizio's (1976) algorithm has been used to run both models. A sensitivity analysis on the total available amount has been performed to identify the best allocation of loans and investments. The results are summarized in the Table 3. The goal achievement in all priorities for both the techniques has been fulfilled. However, according to the result, GIP provides better loan allocation compared to LGP in all Runs. Run-1 requires $250.7663 millions in LGP while in GIP $250.5100 millions to achieve the same required rate of return. Similarly, Runs-2, 3 and 4 demonstrates improved results in GIP over LGP techniques.

The results show that the attainment of minimum budget requirement is reflected in Run-2. Here, the allocation of amounts to different types of loans will satisfy the primary purpose of management for the planning year. With this allocation all the priorities have been achieved. Again, it is observed that, using LGP, no loan amount is allocated for 3-year adjustable rate mortgages, 1year adjustable rate mortgages, personal unsecured loans, new vehicles (car/truck), new recreational vehicles, used recreational vehicles, and used motorcycles whereas using GIP, no loan amount is allocated for 3-year adjustable rate mortgages, 1-year adjustable rate mortgages, new recreational vehicles and used recreational vehicles. However, this does not mean that the categories that have an allocation of zero amount will not be considered in practice. These results serve as a guide to the decision-makers based on priorities in a given situation. The LGP and GIP models allocate the total loan amount of $300 million as given in Table 4.

CONCLUSION

Lexicographic goal programming (LGP) and goal interval programming (GIP) provide a basis for handling conflicting objectives in investment decision making and helps provide means for achieving objectives with respect to desired targets. In this study, we have presented the capabilities of LGP and GIP techniques and have applied them to a case study of credit union. From the case example, we have demonstrated that the management decision processes can considerably be enhanced through the application of LGP and GIP models. Our results demonstrate that the GIP model has improved results as we have observed that the desired required rate of return can be achieved by allocating a lower amount in GIP as compared to LGP. The application of these techniques is quite subjective for the decision making process. The improvements of results in a GIP model over LGP depends on the proper selection of penalty values, priority levels, and targets set by the decision- maker in a given situation.

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Hari P. Sharma, Virginia State University

Debasis Ghosh, National Informatics Centre

Dinesh K. Sharma, University of Maryland Eastern Shore

Table 1: Goal Achievement and Marginal Penalties

 Attributes Unit (%) Marginal Penalty

Goal achievement greater Below [t.sub.S] [infinity]
or equal to the target ... ...
 [t.sub.s-1] - [t.sub.s] [[alpha].sub.s]
 ... ...
 Over 100 0

Goal achievement less or Below 100 0
equal to the target ... ...
 [t.sub.r] - [t.sub.r-1] [[delta].sub.r]
 ... ...
 Over [t.sub.R] [infinity]

Table 2: Credit Union Data

Variable Loan Types Annual Rate
 of Return (%)

[X.sub.1] Home Equity Loan (Fixed Rate) 7.25
[X.sub.2] 30-Year Fixed Rate Mortgage 6.50
[X.sub.3] 20-Year Fixed Rate Mortgage 6.38
[X.sub.4] 15-Year Fixed Rate Mortgage 6.00
[X.sub.5] 3-Year Adjustable Rate Mortgage 5.38
[X.sub.6] 1-Year Adjustable Rate Mortgage 3.38
[X.sub.7] 10-Year Unimproved Property 7.75

[X.sub.8] Personal Secured Loan 6.75
[X.sub.9] Personal Unsecured Loan 6.80
[X.sub.10] Personal Computer Loan 6.50

[X.sub.11] New Vehicles (car/truck) 6.25
[X.sub.12] Used Vehicles (car/truck) 7.75

[X.sub.13] New Boat 8.25
[X.sub.14] Used Boat 8.75

[X.sub.15] New Recreational Vehicle 7.75
[X.sub.16] Used Recreational Vehicle 8.25

[X.sub.17] New Motorcycle 8.25
[X.sub.18] Used Motorcycle 8.75

[X.sub.19] Visa loan 8.75

[X.sub.20] Cash/money Market Funds 3.05

Table 3: Penalty Scales for Three Different Loans Attributes

 Attributes Units Marginal Penalties

Personal Loan and Recreational Below 80% [infinity]
 Vehicle Loan 80% - 90% 2
 90% - 100% 1
 Over 100% 0

Mortgage Loan Below 100% 0
 100% - 110% 1
 110% - 120% 2
 Over 120% [infinity]

Table 3: Sensitivity Analysis on Available Loan Amount

 RUN-1 RUN-2

 LGP GIP LGP GIP

Loan Target 250.0000 250.0000 300.0000 300.0000
Achievement 250.7663 250.5100 300.9194 300.0145
[X.sub.1] 9.2590 09.4000 11.1108 11.1108
[X.sub.2] 12.8260 13.6500 15.3911 24.4499
[X.sub.3] 9.9905 05.5100 11.9887 06.1534
[X.sub.4] 8.9103 06.9000 46.6923 07.8554
[X.sub.5] 0.0000 01.4800 00.0000 00.0000
[X.sub.6] 0.0000 00.0000 00.0000 00.0000
[X.sub.7] 4.6478 03.7600 05.5774 06.6891
[X.sub.8] 4.5006 06.9000 05.4008 06.8718
[X.sub.9] 0.0000 06.9300 00.0000 09.9906
[X.sub.10] 2.5034 18.5400 27.0040 26.4303
[X.sub.11] 0.0000 00.0000 00.0000 00.0034
[X.sub.12] 1.1125 75.2500 85.3350 85.3350
[X.sub.13] 22.5000 17.7500 27.0000 26.0980
[X.sub.14] 15.0000 15.0200 18.0000 16.4532
[X.sub.15] 0.0000 00.0000 00.0000 00.0000
[X.sub.16] 0.0000 00.0000 00.0000 00.0000
[X.sub.17] 7.1112 07.3300 08.5335 08.5335
[X.sub.18] 0.0000 00.0000 00.0000 00.0037
[X.sub.19] 7.0040 28.0600 32.4048 21.3454
[X.sub.20] 5.4008 34.0300 06.4809 42.6908

 RUN-3 RUN-4

 LGP GIP LGP GIP

Loan Target 350.0000 350.0000 400.0000 400.0000
Achievement 351.0727 350.4279 401.2259 400.1190
[X.sub.1] 12.9626 14.2900 14.8144 16.6900
[X.sub.2] 17.9564 38.5700 20.5215 40.6500
[X.sub.3] 13.9867 07.1510 15.9848 07.9120
[X.sub.4] 54.4745 08.6600 62.2565 09.3260
[X.sub.5] 00.0000 00.0000 00.0000 00.0000
[X.sub.6] 00.0000 00.0730 00.0000 00.0780
[X.sub.7] 06.5070 09.6540 07.4366 13.4510
[X.sub.8] 06.3009 07.0600 07.2011 07.3800
[X.sub.9] 00.0000 12.0290 00.0000 16.5000
[X.sub.10] 31.5047 34.1260 36.0054 43.7390
[X.sub.11] 00.0000 00.0050 00.0000 00.0240
[X.sub.12] 99.5575 90.2520 113.780 97.4610
[X.sub.13] 31.5000 34.7600 36.0000 43.9336
[X.sub.14] 21.0000 19.1700 24.0000 22.8560
[X.sub.15] 00.0000 00.0203 00.0000 00.0340
[X.sub.16] 00.0000 02.2400 00.0000 02.3200
[X.sub.17] 09.95575 09.2400 11.3780 09.8700
[X.sub.18] 00.0000 00.0056 00.0000 00.0064
[X.sub.19] 37.8056 14.7800 43.2064 09.8620
[X.sub.20] 07.5611 48.3420 08.6413 58.0260

Table 4: Allocation of total loan amounts

 Type of Loan LGP Model GIP Model
 Amount Amount
 ($Million) ($Million)

[X.sub.1] Home Equity Loan (Fixed Rate) 11.1108 11.1108
[X.sub.2] 30-Year Fixed Rate Mortgage 5.3911 24.4499
[X.sub.3] 20-Year Fixed Rate Mortgage 11.9887 06.1534
[X.sub.4] 15-Year Fixed Rate Mortgage 46.6923 07.8554
[X.sub.5] 3-Year Adjustable Rate Mortgage 00.0000 00.0000
[X.sub.6] 1-Year Adjustable Rate Mortgage 00.0000 00.0000
[X.sub.7] 10-Year Unimproved Property 05.5774 06.6891
[X.sub.8] Personal Secured Loan 05.4008 06.8718
[X.sub.9] Personal Unsecured Loan 00.0000 09.9906
[X.sub.10] Personal Computer Loan 27.0040 26.4303
[X.sub.11] New Vehicles (Car Truck) 00.0000 00.0034
[X.sub.12] Used Vehicles (Car Truck) 85.3350 85.3350
[X.sub.13] New Boat 27.0000 26.0980
[X.sub.14] Used Boat 18.0000 16.4532
[X.sub.15] New Recreational Vehicle 00.0000 00.0000
[X.sub.16] Used Boat Recreational Vehicle 00.0000 00.0000
[X.sub.17] New Motorcycle 08.5335 08.5335
[X.sub.18] Used Motorcycle 00.0000 00.0037
[X.sub.19] Visa (Classic, Gold, Student) 32.4048 21.3454
[X.sub.20] Cash/money Market Funds 06.4809 42.6908