Interactivity and soft computing in portfolio management: Should farmers own food and agribusiness stocks?

Yann Duval

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This article proposes a fully integrated and interactive elicitation-optimization procedure for portfolio management. A soft computing approach based on fuzzy logic is developed as an alternative to the traditional mean variance model and compromise programming approach. The models are applied to farmers to examine whether they should buy publicly traded food and agribusiness firms stocks rather than invest in a broader market stock portfolio. Results suggest that investments in publicly traded food and agribusiness stocks allow farmers to capture additional benefits beyond those of simply diversifying in the broader market.

Key words: compromise programming, diversification, fuzzy logic, risk preference, stock market.

The objective of this article is to develop a methodology that shifts part of the responsibility involved in choosing an optimal portfolio of assets to individual investors. Shifting the decision process to investors is consistent with marketing concepts such as one-on-one marketing and product customization. Such a technique would allow farmers, for example, to choose their own asset allocation and investors to interactively choose the portfolio that is optimal for them. Ultimately, such a technique would also allow individuals to learn about their risk/return trade-off.

This article presents a portfolio optimization technique that is intuitive enough for individuals with little or no knowledge of economic theory to systematically determine their own optimal portfolios. A fuzzy logic (soft computing) methodology is developed that allows the processing of words describing an ideal portfolio into fuzzy constraints. This can then be used to solve a fuzzy optimization problem resulting in the portfolio that most closely matches the linguistic description. This intuitive approach allows for complete integration of the preference elicitation and optimization process. The article provides a link between the fuzzy approach presented and the traditional mean-variance (EV) model via a third approach called compromise programming.

The soft computing approach is used to investigate whether investment in food and agribusiness stocks would provide a hedge motive for farmers by capturing some of the benefits associated with true vertical integration. Brealey and Myers argue that the issue of risk and return is not fully resolved because the capital asset pricing model assumes that all investors have similar tastes and, therefore, hold the same portfolio. In fact, Merton has extended capital asset pricing theory to accommodate the hedging motive.

Consideration of farmers investing in food and agribusiness stocks provides an empirical example to examine how interactive portfolio optimization may be implemented. Enhancement of the risk-return position for a sample of Kansas farmers is considered through non-farm equity investment using the soft computing approach. Results are compared with those of more conventional EV and compromise programming models.

The Fuzzy Logic Approach

Traditional optimization techniques are “crisp,” in that they distinguish in a two-valued way between feasible and infeasible, and between optimal and nonoptimal solutions (Zimmerman). They do not allow for a gradual transition between these categories. This limitation is often referred to as the problem of artificial precision in formalized systems (Geyer-Schulz). Bellman and Zadeh were the first to suggest modeling goals and/or constraints as fuzzy sets to account for uncertainty and the fuzziness of the environment.

Fuzzy set theory is a generalization of traditional set theory in that the domain of the characteristic function is extended from the discrete set {0, 1} to the closed real interval [0,1]. Formally, a fuzzy set A of some universe X is represented by a generalized membership function [M.sub.A]: X [right arrow] [0,1]. Fuzzy sets and fuzzy logic allow one to mimic human decision-making process by modeling lexical uncertainty associated with using words rather than numbers to reach a solution (Von Altrock).

Economic research relying on fuzzy set theory or fuzzy optimization techniques is scarce, (but see Billot; Greenhut, Breenhut, and Mansur; Mansur). Recent applications include an investigation of willingness to invest in Sweden (Lindstrom), an analysis of investment cost data (Dohnal, Fraser, and Kerkovsky), and an application to efficiency and productivity analysis (Sengupta). Two fuzzy logic applications in agricultural economics are an analysis of goals and objectives of organic producers in Canada (Molder, Negrave, and Schoney) and a production planning model for tomato packing (Miller).

Fuzzy logic provides a decision-making framework for an individual. A typical individual expresses objectives and constraints using linguistic variables. These linguistic variables are characterized by imprecision and uncertainty. Fuzzy set theory and fuzzy arithmetic have been developed to model lexical uncertainty and mathematically represent words or linguistic variables. Thus, fuzzy logic helps to determine the portfolio that would best satisfy an individual describing his or her ideal portfolio with terms such as “low risk” and “high return.”

The Portfolio Choice Models

The traditional EV objective function is

(1) [Max.sub.x] R(x) – [theta]V(X)

where R(x) is expected portfolio return, V(x) is portfolio variance, and [theta] is the tradeoff between mean and variance (Markowitz). Theta may be interpreted as 1/2 of the Pratt-Arrow risk aversion coefficient when the utility function is negative exponential and the returns are normally distributed. Solutions to (1) satisfy the following first-order condition:

(2) [R.sub.x] = [theta][V.sub.x]

where the subscript denotes partial differentiation.

As illustrated in figure 1(a), the EV model results in a traditional approach to elicitation and optimization that is sequential and unidirectional. As shown in figure 1(b), the fuzzy optimization procedure results in the individuals self-eliciting their preferences to be used in the optimization process. The optimization process results in a suggested optimal portfolio as well as a membership value, [lambda], indicating the extent to which the portfolio satisfies the self-elicited preferences. The individuals can then revise their self-elicited preferences until they obtain a portfolio that satisfies their true preferences. This can be accomplished through the compromise programming and the fuzzy logic approaches.

The compromise-programming approach to an EV portfolio optimization problem yields. (1)

(3) [Min.sub.x] [L.sub.[alpha]] = [[([w.sub.R] [R.sup.+] – R(x)/[R.sup.+] – [R.sup.-]).sup.[alpha]]

+ [([W.sub.v] [V.sup.-] – V(x)/[V.sup.-] – [V.sup.+]).sup.[alpha]]].sup.1/[alpha]]

s.t. x [member of] S [alpha] = 1, 2,…, [infinity]

where [alpha] indicates the distance measure, (2) [R.sup.+] is the maximum portfolio return possible, [R.sup.-] is the minimum return possible, [V.sup.-] is the minimum portfolio variance possible, [V.sup.+] is the maximum variance possible, and [W.sub.R] and [W.sub.V] are weights (or coefficients) on the return and the safety objectives, respectively. This framework is very simple because [R.sup.+] is, in fact, the return on the asset with the highest mean return, [R.sup.-] is the return on the asset with the lowest mean return, and [V.sup.+] is the variance of the asset with the greatest variance. Note, however, that [V.sup.-] is not the asset with the minimum variance but the minimum variance of a portfolio determined by minimizing portfolio variance V(x). All compromise solutions are bounded by the solution to the [L.sub.1] and the [L.sub.[infinity]] problem and form the compromise set. (3)

Some intuition on the meaning of the [L.sub.1] and the [L.sub.[infinity]] bound is given below. The [L.sub.1] portfolio minimizes the sum of: (1) the weighted distance between the maximum return [R.sup.+] and the actual portfolio return; and (2) the weighted distance between the minimum portfolio variance [V.sup.-] and the actual portfolio variance. In contrast, the [L.sub.[infinity]] portfolio minimizes the larger of the two weighted distances until they become equal. The [L.sub.[infinity]] problem can be interpreted as a minimax solution or Chebyshev’s criterion. Hence, the [L.sub.[infinity]] solution is the most well-balanced portfolio of the compromise set (in terms of the risk and return trade-off) given the preference of the individual as indicated by the objective weights. In contrast, the [L.sub.1] solution is much more sensitive to the objective weights chosen by the individual. (4) The [L.sub.[infinity]] portfolio balances risk and return as much as possible given the risk and return weights. The [L.sub.[infinity]] solution may be more appropriate when the individual is unsure about preferences.

Because of the complexity and uncertainties associated with a typical optimization problem, the compromise programming method “concentrates” on eliminating “obviously bad” solutions rather than on identifying the best ones. Once the “obviously bad” solutions have been eliminated, the decision maker chooses a solution within the smaller set of solutions (i.e., the compromise set). Note that the decision maker need only specify a variance (or safety) weight and a return weight between zero and one to determine his or her compromise set.

The soft computing or fuzzy logic approach provides an alternative to specifying numerical weights. In the context of our portfolio problem, we can define two linguistic variables: “low risk” and “high return,” which are the two linguistic variables that every individual might include in the description of their optimal portfolio. Each linguistic variable can be defined by a fuzzy set and its associated membership function as follows:

(4) [[micro].sub.LR](x) = [V.sup.+] – V(x)/[V.sup.+] – [V.sup.-]

(5) [[micro].sub.HR](x) = R(x) – [R.sup.-]/[R.sup.+] – [R.sup.-]

where [[micro].sub.LR] (x) is the membership function of “low risk” and [[micro].sub.HR] (x) is the membership function of “high return.” The membership functions associated with “low risk” (LR) and “high return” (HR) are represented in figure 2. The higher the expected return of the portfolio the greater the membership value associated with [[micro].sub.HR](x). The lower the variance of the portfolio, the greater the membership value associated with [[micro].L.R](x). Also, the portfolio with the highest expected return is given a “high return” value of 1 and the portfolio with the lowest expected variance is given a “low risk” value of 1. Clearly, the optimal portfolio for an individual wanting a “portfolio with high return and low risk” is one that will maximize membership values.

An individual expressing his or her desire of a “portfolio with high return and low risk” might put equal emphasis on return and safety. The optimal portfolio is, thus, characterized by the following fuzzy optimization problem:

(6) Max [lambda]

s.t. [lambda] [less than or equal to][[micro].sub.LR](x), [lambda] [less than or equal to][[micro].sub.HR](x)

where [lambda] is the degree of membership of the optimal portfolio to the fuzzy sets that characterize our linguistic variables.

The fuzzy portfolio optimization method could account for additional linguistic terms. Not every individual would describe his/her optimal portfolio as low risk and high return. Some may qualify the two basic linguistic variables. For example, one individual may describe his optimal portfolio as very low risk and high return while another may describe her optimal portfolio as low risk and fairly high return. Zadeh has originally suggested the modeling of adverbs by fuzzy set operations. Wenstop provides a detailed implementation.

Shmucker separates adverbs into three groups depending on how they modify the original membership function: concentration, dilution or intensification of the original membership function. Adding very in front of “low risk” has a concentration effect on the membership function “low risk,” i.e., [[mu].sub.VLR](x) = [[mu].sub.LR][(x).sup.2]. The curved dotted line below the linear “low risk” membership function in figure 2 is the membership function resulting from the concentration effect. This new membership function implies that all portfolios are now perceived as riskier because they have lower membership values. Adding fairly in front of “low risk” has a dilution effect on the membership function “low risk,” i.e., [[mu].sub.FLR](x) = [[mu].sub.LR][(x).sup.0.5]. The resulting membership function is now the curved dotted line above the linear “low risk” membership function in figure 2. All portfolios would now have higher “low risk” membership values.

For example, the optimal portfolio associated with “very low risk” and “high return” can be found by solving the following maximization problem:

(7) Max [lambda]

s.t. [lambda][less than or equal to][[micro].sub.LR][(x).sup.2], [lambda] [less than or equal to] [[micro].sub.HR](x)

The solution to this problem is on a path characterized by [[micro].sub.LR][(x).sup.2] = [[micro].sub.HR](x) = [lambda]. In this case, all portfolios are now perceived, as relatively less “low risk” because of the concentration effect, and the optimal portfolio suggested will now have a lower variance. A membership value [lambda] is associated with every optimal portfolio. A low value of [lambda] indicates that the linguistic description of the individual’s portfolio does not accurately describe the optimal portfolio.

From Traditional EV to the Fuzzy Logic Approach

It is important to establish a theoretical base between the fuzzy logic approach and the expected utility hypothesis. The first step is to establish the relationship between the traditional EV model and the compromise programming solutions. Investigation of the [L.sub.1] bound provides a direct and intuitive link between the compromise programming and the EV model. Indeed, the [L.sub.1] compromise programming problem (3) can be reformulated into a traditional EV problem plus a constant C as follows:

(8) Max [L.sub.1]=R(x)

-([w.sub.v]([R.sup.+] – [R.sup.-])/([W.sub.R]([V.sup.+] – [V.sup.-])) V(x)+C

Solutions to (8) satisfy the following first-order condition:

(9) [R.sub.x] = [PHI] [V.sub.x], [PHI] = [w.sub.V]([R.sup.+] – [R.sup.-]/[w.sub.R]([V.sup.+] – [V.sup.-].

Equations (2) and (9) are equivalent when [PHI] = [theta]. The equivalence between (2) and (9) demonstrates that all [L.sub.1] solutions lie on the EV frontier. By varying the weights, [w.sub.R] and [w.sub.V], we can trace out the EV efficient set as shown in figure 3. Under the traditional assumptions of the EV model, an individual who puts equal weight on the risk and return factors would have a risk aversion coefficient equal to half the ratio of return range over the variance range in the portfolio of assets under investigation. The risk aversion coefficient implied by the compromise programming approach changes from one portfolio optimization problem to another for the same individual, recognizing the complexity of individual risk perceptions.

We now turn to the relationship between the [L.sub.[infinity]] bound and the EV solution. Following Zeleny (1982), the solution to the [L.sub.[infinity]] problem is characterized by:

(10) Min[[w.sub.R]([R.sup.*] – R(x)/[R.sup.*] – [R.sup.-]), [w.sub.V]([V.sup.*] – V(x)/[V.sup.*] – [V.sup.+])].

No general mathematical relationship may be established between [theta] and the [L.sub.[infinity]] bound. However, Ballestero and Romero observed that the solution to the [L.sub.[infinity]] problem is the intersection between the EV efficient set and the [L.sub.[infinity]] path characterized by:

(11) [w.sub.R] ([R.sup.+] – R(x)/[R.sup.+] – [R.sup.-]) = [w.sub.V]([V.sup.-] – V(x)/[V.sup.-] – [V.sup.+]).

In other words, the EV frontier can be completely traced out by varying the weights, [w.sub.R] and [w.sub.V]. All the solutions of the compromise set are thus pareto-optimal portfolios.

The last step is to establish the relationship between fuzzy optimization and the compromise programming approach. Equation (10) can be rewritten as:

(12) Min[1 – [[micro].sub.LR](x), 1 – [[micro].sub.HR](x)] which is equivalent to the basic fuzzy optimization problem in (6). Thus, the solution to the fuzzy optimization problem is identical to that of the L[infinity] compromise programming problem. The solutions to the fuzzy logic approach are also frontier portfolios and lie on the EV frontier.

This last result implies that all three models always yield EV frontier portfolios. All three approaches also rely on expected return (mean) and variance-covariance data to determine the optimal portfolio because V(x) enters into the solution process as a variance-covariance matrix of investment choices. This implies that correlation between asset returns plays an important role in all three choice models. In fact, the compromise programming and the fuzzy logic approach are a generalization of the traditional EV models to cases where the relationship between the portfolio mean and variance is nonlinear. Thus, the major differences between the three models lie in the underlying measure of the risk-return trade-off and whether the measure is intuitive enough to allow self-elicitation and interactivity.

It is not possible to show which self-elicitation procedure is superior to the other. However, the compromise programming approach and the fuzzy approach may have a significant advantage over the traditional EV approach for some individuals because they express the risk-return trade-off in more intuitive terms. The risk aversion coefficient of an individual may be elicited using implied preferences between a set of linked gambles. However, this would be a lengthier elicitation process that would not intuitively represent the risk-return trade-off of the individual. More importantly, it would not allow for interactivity in the optimization process. For this reason, the compromise programming and fuzzy approaches are best suited for the development of an interactive approach in which individuals can adjust their risk return trade-offs until they find a satisfactory frontier portfolio. The soft computing approach is particularly well suited for an interactive process because it indicates how well the suggested p ortfolio matches an individual’s linguistic definition of the portfolio (see figure 1b).

Application

The three techniques discussed above are used to determine the incentive for farmers to invest in publicly traded food and agribusiness company stocks. As discussed above, Brealey and Myers argue that the issue of risk and return is not fully resolved because the capital asset pricing model assumes that all investors have similar tastes and therefore hold the same portfolio. In particular, there are many low “beta” stocks that return too much and high “beta” stocks that return too little (Black, Jensen, and Scholes). Merton develops an intertemporal version of the capital asset pricing model and finds important differences when compared to the static capital asset pricing model. In particular, Merton’s intertemporal capital asset pricing model suggests a demand for an asset that serves as a vehicle to hedge against unfavorable shifts in the investment set. The key to the hedging motive is the existence of instantaneous correlation between asset returns. This hedging motive provides an explanation for the mark et anomalies found by Black, Jensen, and Scholes.

Retained ownership of raw product as it progresses through the value chain offers the potential for those correlations. This is especially true given the apparent motivation of many retailers is to take the variability out of prices to the final consumer. A fundamental result of Merton’s intertemporal capital asset model is that consumers do not favor unanticipated variability, or that consumers favor a riskless asset as opposed to a risky asset, ceteris paribus. Thus, an incentive exists for the variability in prices to be removed before final consumption. The removal of variability will create intertemporal correlation that can be exploited via vertical integration or “retained” ownership. The hypothesis is that investment in food and agribusiness stocks would provide farmers an opportunity to hedge within the value chain and capture some of those benefits beyond that of offfarm investment in broad market indices.

The hedging motive for investment within the value chain is tested below by computing investment portfolios for a number of Kansas farmers given the choice between retaining investment in their farms and adding financial assets represented by investment in a broad stock market index (ordinary diversification) and a number of food and agribusiness stocks (value chain hedging). If food and agribusiness stocks enter the optimal portfolio, the farmer would be at least as well off investing in these as investing in the market portfolio. The individual stocks contain more risk (systematic and unsystematic) than the diversified portfolio (only systematic risk).

The dataset includes 98 Kansas farms and 16 publicly traded food and agribusiness companies for which the annual return on equity (ROE) was available from 1973 to 1998. The Kansas farm data are from the Kansas Farm Management Associations and the individual firm data and the market index are from the Center for Research in Security Pricing. During this period, the geometric average annual farm ROE in our sample is 5.7% with a mean standard deviation of 16.66%. (5) The top third of the farms achieve average ROEs ranging from 6.7% to 18.7%. The bottom third of farms achieved ROEs between 3.5% and -1.4%. The publicly traded firm with the lowest average annual ROE is Fleming Companies (FLM), a major food retailer with an ROE of 6.1%. The one with the highest ROE is Kroger (KR), a major food retailer with an ROE of 24.9%. The average annual ROE for the 16 food and agribusiness companies is 16% with a standard deviation of 32%. The CRSP value-weighted U.S. market stock index averaged a 12.8% annual ROE with a stan dard deviation of 17% for the 1973-1998 period.

The median farm in our sample is considering whether to buy stock from one or more publicly traded food and agribusiness companies or to invest in the U.S. CRSP stock market index. They log on to the state extension website, where the stock choices and ROE are already stored. They are then asked to enter their historical series of their farm ROE and select the food and agribusiness stocks they would be willing to invest in. They select the sixteen food and agribusiness companies and the broad market index. At this point, the minimum variance of the portfolio is determined and the set of efficient portfolios (EV frontier) is computed and shown on a graph resembling figure 3. Next the farmer is asked to pick safety and return coefficients on a scale similar to that of figure 4. Once the farmer has selected weights, the compromise programming problem is solved and the farmer is given the solution to the [L.sub.1] and the [L.sub.[infinity]] problems (see table 1 which assumes equal weights) as well as an updated EV frontier showing the entire compromise set.

The results of the [L.sub.1] and the [L.sub.[infinity]] problems are reported in table 2 for a variety of safety and return coefficients. For comparison purposes, the value of 0 associated with the [L.sub.1] bound portfolio is also reported. Figure 3 confirms the fact that the EV frontier and the efficient set are one and the same. It also shows the relative position of EV solutions for different relative risk aversion coefficients in compromise sets. We find that, by equation (9), an individual with a negative exponential utility function choosing the pair of weights (0.2, 0.8) and preferring the [L.sub.1] bound has a relative risk aversion coefficient of 2.773 (this coefficient may be interpreted as “relatively strong risk aversion”). However, an individual choosing the pair of weights (0.7, 0.3)–to indicate that, even though they are risk averse, they give relatively more importance to return than to safety–has a coefficient of relative risk aversion of 0.297. (6)

The results from the compromise programming approach suggest that a value of [theta] greater than 7 may be somewhat unrealistic. Indeed, the [L.sub.1] bound portfolios when the return weight is set to 0.1 and the risk weight is set to 0.9 are equivalent to optimal EV portfolios with [theta] between 5 and 7, depending on the farm selected. Even for extreme values of risk aversion coefficients ([theta] = 100), farming never represents more than two-thirds of the resulting portfolio for any of the farms. At lower levels of relative risk aversion ([theta] = 1), farming rapidly disappears from the optimal portfolio. Indeed, the minimum ROE required for farming to marginally enter the portfolio of the median Kansas farmer is 15.1% when the farmer weights return and safety equally ([theta] = 0.693).

As an alternative to choosing coefficients of safety and risks, the farmer could be asked to describe his or her portfolio using words such as low, high, very, fairly, risk, and return. The farmers’ description would then be automatically processed into fuzzy constraints so that the appropriate fuzzy optimization problem may be solved. Table 3 shows the return and standard deviations of portfolios suggested by the fuzzy logic approach that corresponds to various linguistic descriptions. For example, the fuzzy approach suggests that an individual willing to invest in a very low risk, but fairly high return portfolio should invest in a portfolio with an expected return of 17.6% and a standard deviation of 10.4%. The membership value [lambda] associated with this description is 0.852, which can be roughly interpreted as suggesting that the optimal portfolio fits the linguistic description “very low risk and fairly high return” at the 85.2% level.

The same portfolio is suggested for someone looking for a “very very low risk and high return” portfolio but the membership value has a much lower value of 0.616 (table 3). The suggested portfolio fits the linguistic description at a 62% level only. This is because the individual wants a portfolio with both relatively low levels of risk and relatively high levels of return, which is less feasible because risk and return are positively related. Low membership values indicate that the set of assets at hand cannot result in a portfolio that is likely to fully satisfy the individual. In our example, an individual describing their portfolio as “fairly low risk and fairly high return” is the most likely to be completely satisfied ([lambda] = 0.989). The farmer can pick the suggested portfolio or change its portfolio description until they find a suitable portfolio. Using this interactive approach allows farmers to reach their own personal decision while insuring the decision remains efficient from a mean variance p erspective. Because of all the uncertainty associated with reaching a portfolio decision, the interactive aspect of portfolio optimization is very important.

The fuzzy approach was used to determine the optimal portfolio for each of the 98 farms and for sixteen different linguistic descriptions. This analysis resulted in 1,568 individual portfolios. For ease of interpretation, farms were ranked according to their average ROE. The average individual portfolios of farms that ranked in the first tier of the sample, the second tier, and the third tier are reported in tables 4 and 5.

In table 4, the average and standard deviations of the expected returns, standard deviations, and degree of membership are presented for each tier and for all 16 linguistic descriptions. Optimal portfolio returns, standard deviations, and degree of membership for identical linguistic descriptions are relatively homogeneous across tiers, except for the most risk averse linguistic description “fairly high return and very very low risk.” Indeed, the difference in expected return and standard deviation between the first tier portfolios and the third tier farms never exceeds 2.7% and 1%, respectively.

The modest share of farm equity in those portfolios explains the relative homogeneity of optimal portfolios across farms. Table 5 presents the average and standard deviations of the shares of each investment (stocks and farm equity) in all optimal portfolios that include some farm equity–at least four of the seven optimal portfolios possible for each farm do not include any farm equity. Only the most risk averse linguistic descriptions result in the presence of a significant amount of farm equity in optimal portfolios. Regardless of the linguistic description, no portfolios for any farm include more than two-thirds of farm equity. (7) In addition, the two companies present in all optimal portfolios are those of Albertson and Kroger, two major food retailers.

Perhaps the most striking result is that the broad stock market index is not included in any of the optimal portfolios, with the exception of the one corresponding to the most risk averse linguistic description “fairly high return and very very low risk portfolio.” In any case, the share of the stock market index in any of the optimal portfolios never exceeds 15%. These results suggest that the choice of any broad market diversification does not outweigh the benefit from investment strategies focused on food and agribusiness companies for the 1973-98 time period. It is important to realize that the individual company returns contain systematic and unsystematic risk, while the market portfolio has diversified away unsystematic risk. Thus, investment in food and agribusiness stocks is preferred even though they contain undiversified risk.

One concern with the previous analysis may be that the food and agribusiness firms that enter the portfolio are those which have “beat” the market, i.e., are not consistent with a market equilibrium model such as the capital asset pricing model (CAPM). Thus, each individual stock was tested for consistency with the CAPM. Those stocks that did not adhere to the CAPM were eliminated as choices in a revised portfolio. Only three of the stocks were not consistent with the CAPM: Albertsons, Kroger, and Hormel.

All three stocks are featured prominently in the optimal portfolios presented in table 5, possibly overshadowing the stock market index from entering the optimal portfolios. Hence, optimal portfolios were reestimated without these three stocks. (8) Again, the stock market index share does not generally enter the portfolios. It never exceeds 10%, except for some of the third-tier farms where farmers are very risk averse. The share of farm assets in the new optimal portfolios is often smaller than in the portfolios that included company stocks with above-normal rates of returns, but still represents over 50% of the portfolios for some very risk averse farmers. The new optimal portfolios provide additional evidence of the hedging motive suggested by Merton. Thus, farmers may benefit from developing stock investment strategies focused on food and agribusiness companies, rather than the broader stock market.

The optimal portfolios suggest that farmers would want to focus on investment toward the retail end of the food industry, closer to the consumer. Intuitively, because lower prices for agricultural commodities are likely to benefit the food processing and retail industry, negative correlations between a farm return on equity and a number of publicly traded food companies are more likely, making these companies’ common stocks particularly attractive instruments to diversify farm risk (Wilson). Note that Pioneer Hi-Bred International is also represented in most portfolios, suggesting that investment in leading agricultural input suppliers may also be an appropriate strategy.

An interesting and somewhat counterintuitive result is that optimal portfolios of farms with lower ROE include a larger share of farm equity for the same linguistic specification. This result suggests that strongly risk averse under-performing farmers should retain farm equity as a way to diversify risk. It is important to note however, that the linguistic description for farmers with different expected returns, variances, and covariances does not necessarily imply the same level of relative risk aversion.

Implications

The results suggest that farm owners may be better off investing some equity of the farm outside the farm enterprise as others have found (Moss, Baker, and Featherstone; Bjornson and Innes; Arthur, Carter, and Abizadeh; Crisostomo and Featherstone; Hamaker and Patrick; Young and Barry). The uniqueness of the results from this analysis is that investment in a portfolio of food and agribusiness stocks is preferred to equity investment in a well-diversified market portfolio. Investment in individual stocks, even though they contain both systematic and unsystematic risk, is a viable alternative to other value-added investments for Kansas farmers. This result holds for all farms in our sample.

Finally, it should be noted that the interactive portfolio application presented above would benefit from further refinement. Many other factors would need to be considered such as possible measurement error in the return data, the effect of economies of scale, and other nonpecuniary returns that some farmers derive from farming. However, this approach does provide a mechanism for assessing the opportunity cost of staying in farming, while providing a mechanism for analyzing alternative investment opportunities. Investment in publicly traded common stocks of companies along a farmer’s supply chain may provide some of the benefits associated with true vertical integration.

Conclusion

This article has proposed a soft computing approach that would allow for an integrated and interactive elicitation-optimization procedure for portfolio management. The relationship between traditional EV, compromise programming, and fuzzy logic techniques was explored. The techniques presented in this article are found to be consistent with expected utility maximization, but do not require direct assessment of the utility function. Although expected utility maximization is not easily observed, individuals do assign priority weights no matter how imperfectly, fuzzily or arbitrarily. As a result, objective weighting as suggested in compromise programming is an intuitive approach that is consistent but easier to understand than utility concepts. Individuals also use words as subjective categories to process information and reach solutions. If they have difficulty specifying weights, imitating this process using fuzzy logic is easy, intuitive, and yields results consistent with those of compromise programming and the traditional EV model.

The fuzzy logic approach is computationally efficient and consistent with the EV model, but does not require an understanding of expected utility theory, nor does it require determination of risk aversion coefficients using a separate and often lengthy elicitation technique that would prevent interactive decision making. As such, it is suitable for solving risk management problems interactively. The soft computing approach presented here can be easily extended to include more linguistic descriptions as well as additional objectives. This approach remains intuitive even for very complex problems because it does not require direct assessment of weights by the individual, as does the compromise programming approach. Selecting weights becomes more difficult and less intuitive as the number of objectives grows. Comparing the optimal solutions from the fuzzy elicitation approaches introduced in this article to those from conventional elicitation approaches based on lottery choices and expected utility would be of i nterest.

The rapid development of information technology will allow shifting more responsibilities from agricultural economists to end users. Our objective will increasingly be to facilitate decision making and develop procedures and models to make sure that no “obviously bad” decisions are made.

(1.) Compromise programming is a linear multiobjective programming technique originally introduced by Zeleny (1974). For an introduction to compromise programming in the agricultural economics literature, see Romero and Rehman (1984, 1985) and Romero, Amador, and Barco. The literature on portfolio optimization using a compromise programming approach is limited to two articles in the operations research literature (see Ballestero and Romero; Ballestero).

(2.) For example, [alpha] = 1 refers to geometric distance and [alpha] = 2 refers to Euclidean distance.

(3.) The [L.sub.1] is a linear programming problem and [L.sub.infinity] an be easily formulated into one, as we will show below for the case of portfolio optimization (Cohon).

(4.) Note that the size of the compromise set increases as the individual gives more weight to one objective relative to the other. This can be verified in figure 3, by comparing the compromise set for an individual with balanced objectives (0.5, 0.5) with that for an individual more concerned about safety (0.8, 0.2).

(5.) The farm ROE used is pre-tax ROE adjusted for capital gains; to allow comparison with return from publicly traded common stocks. It should also be noted that these are not market-determined rates of return. Therefore, care needs to be taken when interpreting the results realizing that some biases may exist in the farm return series.

(6.)In compromise programming, the individual can be only risk averse ([w.sub.v] > 0) or risk neutral ([w.sub.v]), but never risk lover ([w.sub.v] < 0 not allowed).

(7.) Other course, this fails to account for any nonpecuniary returns to owning farmland, nor for potential biases in the ways returns to farming are computed. It also assumes farmland returns are ant dependent on the scale of investment.

(8.) This table of results is available from the authors.

References

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[Figure 2 omitted]

[Figure 3 omitted]

Table 1.

Bound Compromise Programming (CP) Portfolios for (0.5, 0.5)

Portfolio 1 Portfolio 2

([L.sub.1]) ([L.sub.[infinity]])

Pioneer Hi-Bred 0.0% 12.5%

International

Conagra Inc. 0 0

Hormel Foods 0 0

Seaboard Corp. 0 0

Smithfield Foods Inc 0 0

General Mills 0 0

Kellogg Co. 0 0

Quaker Oats 0 0

Archer-Daniels-Midland 0 0

Pharmacia Corp. 0 0

Deere & Co. 0 0

Fleming Cos 0 0

Albertson’s Inc. 70.6% 69.1%

Kroger Co. 29.4% 18.5%

Winn-Dixie Stores 0 0

McDonald’s Corp. 0 0

Farming 0 0

Stock Index 0 0

Expected return 24.0% 22.8%

Expected standard 21.4% 18.1%

deviation

Table 2

Compromise Programming (CP) Solutions for the Farm with Median ROE

Potential Return

Return Risk [L.sub.1] Distance [L.sub.[infinity]] Distance

Coeff. Coeff. Measure Measure

0.1 0.9 0.160 0.182

0.2 0.8 0.199 0.204

0.3 0.7 0.218 0.215

0.4 0.6 0.228 0.223

0.5 0.5 0.240 0.228

0.6 0.4 0.241 0.233

0.7 0.3 0.242 0.238

0.8 0.2 0.244 0.242

0.9 0.1 0.249 0.245

Return Variability (std)

Return [L.sub.1] Distance [L.sub.infinity] Distance

Coeff. Measure Measure

0.1 0.089 0.116

0.2 0.132 0.139

0.3 0.159 0.154

0.4 0.181 0.168

0.5 0.214 0.181

0.6 0.217 0.194

0.7 0.224 0.207

0.8 0.245 0.222

0.9 0.318 0.254

Value of [theta]

at the [L.sub.1] Bound

Return [L.sub.1] Distance

Coeff. Measure

0.1 6.239

0.2 2.773

0.3 1.617

0.4 1.039

0.5 0.693

0.6 0.462

0.7 0.297

0.8 0.173

0.9 0.077

Note: [theta] is one half of the Pratt-Arrow relative risk aversion

coefficient under normally distributed returns and a negative

exponential utility function.

Table 3.

Correspondence Between Linguistic Terms and Risk and Return of Optimal

Portfolios for the Median Farm

Risk

Return Very Very Low Very Low

Fairly high 0.148 [+ or -] 0.080 0.176 [+ or -] 0.104

[lambda] = 0.722 [lambda] = 0.852

High 0.176 [+ or -] 0.104 0.205 [+ or -] 0.140

[lambda] = 0.616 [lambda] = 0.769

Very high 0.205 [+ or -] 0.140 0.229 [+ or -] 0.184

[lambda] = 0.520 [lambda] = 0.677

Very very high 0.229 [+ or -] 0.184 0.243 [+ or -] 0.234

[lambda] = 0.432 [lambda] = 0.575

Risk

Return Low Fairly Low

Fairly high 0.201 [+ or -] 0.140 0.229 [+ or -] 0.184

[lambda] = 0.948 [lambda] = 0.989

High 0.229 [+ or -] 0.184 0.243 [+ or -] 0.234

[lambda] = 0.896 [lambda] = 0.967

Very high 0.243 [+ or -] 0.234 0.248 [+ or -] 0.294

[lambda] = 0.820 [lambda] = 0.913

Very very high 0.248 [+ or -] 0.294 0.249 [+ or -] 0.317

[lambda] = 0.704 [lambda] = 0.881

Table 4.

Average Optimal Portfolios Associated with Linguistic Descriptions

Risk

Return Very Very Low Very Low

1st Tier farms

Fairly E(return) 15.57 (0.99) 18.22 (0.54)

high Std. Dev. 9.90 (1.28) 11.43 (0.63)

[lambda] 0.743 (.060) 0.871 (0.022)

High E(return) 18.22 (0.54) 20.95 (0.26)

Std. Dev. 11.43 (0.63) 14.47 (0.32)

[lambda] 0.642 (0.029) 0.786 (0.014)

Very E(return) 20.95 (0.26) 23.10 (0.13)

high Std. Dev. 14.47 (0.32) 18.78 (0.29)

[lambda] 0.538 (0.015) 0.686 (0.008)

Very very E(return) 23.10 (0.13) 24.37 (0.05)

high Std. Dev. 18.78 (0.29) 23.83 (0.29)

[lambda] 0.440 (0.008) 0.579 (0.005)

2nd Tier farms

Fairly E(return) 15.04 (0.85) 17.68 (0.53)

high Std. Dev. 8.70 (0.75) 10.88 (0.44)

[lambda] 0.733 (0.042) 0.856 (0.018)

High E(return) 17.68 (0.53) 20.61 (0.24)

Std. Dev. 10.88 (0.44) 14.14 (0.26)

[lambda] 0.621 (0.025) 0.775 (0.009)

Very E(return) 20.61 (0.24) 22.99 (0.09)

high Std. Dev. 14.14 (0.26) 18.49 (0.24)

[lambda] 0.526 (0.010) 0.681 (0.006)

Very very E(return) 22.99 (0.09) 24.31 (0.03)

high Std. Dev. 18.49 (0.24) 23.51 (0.23)

[lambda] 0.435 (0.005) 0.578 (0.003)

3rd Tier farms

Fairly E(return) 12.93 (0.78) 16.45 (0.52)

high Std. Dev. 8.61 (1.35) 10.65 (0.74)

[lambda] 0.718 (0.028) 0.859 (0.022)

High E(return) 16.45 (0.52) 20.06 (0.33)

Std. Dev. 10.65 (0.74) 13.65 (0.32)

[lambda] 0.626 (0.031) 0.785 (0.021)

Very E(return) 20.06 (0.33) 22.78 (0.12)

high Std. Dev. 13.65 (0.32) 17.96 (0.30)

[lambda] 0.536 (0.023) 0.691 (0.013)

Very very E(return) 22.78 (0.12) 24.26 (0.05)

high Std. Dev. 17.96 (0.30) 23.01 (0.25)

[lambda] 0.444 (0.012) 0.586 (0.009)

Risk

Return Low Fairly Low

1st Tier farms

Fairly 20.95 (0.26) 23.10 (0.13)

high 14.47 (0.32) 18.78 (0.29)

0.923 (0.010) 0.990 (0.001)

High 23.10 (0.13) 24.37 (0.05)

18.78 (0.29) 23.83 (0.29)

0.902 (0.005) 0.969 (0.001)

Very 24.37 (0.05) 24.80 (0.00)

high 23.83 (0.29) 29.73 (0.31)

0.823 (0.004) 0.914 (0.004)

Very very 24.80 (0.00) 24.90 (0.00)

high 29.73 (0.31) 31.73 (0.05)

0.708 (0.006) 0.869 (0.053)

2nd Tier farms

Fairly 20.61 (0.24) 22.99 (0.09)

high 14.14 (0.26) 18.49 (0.24)

0.921 (0.008) 0.990 (0.001)

High 22.99 (0.09) 24.31 (0.03)

18.49 (0.24) 23.51 (0.23)

0.898 (0.004) 0.968 (0.001)

Very 24.31 (0.03) 24.80 (0.00)

high 23.51 (0.23) 29.49 (0.10)

0.822 (0.003) 0.914 (0.002)

Very very 24.80 (0.00) 24.90 (0.00)

high 29.49 (0.10) 31.70 (0.00)

0.707 (0.003) 0.883 (0.003)

3rd Tier farms

Fairly 20.06 (0.33) 22.78 (0.12)

high 13.65 (0.32) 17.96 (0.30)

0.926 (0.011) 0.991 (0.002)

High 22.78 (0.12) 24.26 (0.05)

17.96 (0.30) 23.01 (0.25)

0.904 (0.008) 0.971 (0.002)

Very 24.26 (0.05) 24.80 (0.00)

high 23.01 (0.25) 29.24 (0.15)

0.829 (0.007) 0.916 (0.004)

Very very 24.80 (0.00) 24.90 (0.00)

high 29.24 (0.15) 31.70 (0.00)

0.711 (0.006) 0.882 (0.005)

Note: Standard deviations are in parentheses.

Table 5.

Optimal Investment Portfolios of Farmers (Shares are in Percent)

Pioneer Hormel Seaboard Smithfield

“Fairly HR and LR”/

“HR and Very LR”/

“Very HR and Very Very LR”

Portfolios (a)

1st Tier 23.72 0.17 1.50

farms (1.86) (0.41) (1.42)

2nd Tier 25.56 0.78 3.15

farms (1.51) (1.21) (1.35)

3rd Tier 26.61 2.65 4.45

farms (0.52) (1.18) (0.59)

“Fairly HR and Very LR”/”HR

and Very Very LR” Portfolios

1st Tier 23.78 2.23 4.22 0.33

farms (2.37) (1.92) (3.47) (0.55)

2nd Tier 22.51 3.63 3.49 0.06

farms (1.97) (3.30) (1.86) (0.18)

3rd Tier 22.56 5.35 4.36 0.31

farms (1.87) (2.87) (3.14) (0.63)

“Fairly HR and Very Very LR”

Portfolio

1st Tier 20.18 2.82 3.95 1.17

farms (3.47) (3.62) (3.24) (1.83)

2nd Tier 19.33 5.71 4.44 0.71

farms (4.00) (5.91) (3.39) (1.05)

3rd Tier 17.65 5.63 2.76 1.30

farms (3.41) (5.32) (3.77) (1.83)

General

Mills Kellogg ADM Pharmacia

“Fairly HR and LR”/

“HR and Very LR”/

“Very HR and Very Very LR”

Portfolios (a)

1st Tier 4.73

farms (1.41)

2nd Tier 5.96

farms (1.35)

3rd Tier 6.85

farms (0.79)

“Fairly HR and Very LR”/”HR

and Very Very LR” Portfolios

1st Tier 0.25 5.57

farms (0.40) (1.23)

2nd Tier 3.05

farms (1.19)

3rd Tier 12.14 4.05

farms (34.32) (2.19)

“Fairly HR and Very Very LR”

Portfolio

1st Tier 1.40 0.47 4.27 1.20

farms (3.43) (1.14) (2.59) (2.94)

2nd Tier 0.06 1.08

farms (0.18) (1.35)

3rd Tier 0.19 0.30 1.71

farms (0.53) (0.58) (1.90)

Deere Fleming Albertson Kroger

“Fairly HR and LR”/

“HR and Very LR”/

“Very HR and Very Very LR”

Portfolios (a)

1st Tier 59.73 7.37

farms (2.17) (1.82)

2nd Tier 57.59 5.21

farms (2.32) (1.75)

3rd Tier 53.83 3.26

farms (1.55) (0.83)

“Fairly HR and Very LR”/”HR

and Very Very LR” Portfolios

1st Tier 42.48 2.82

farms (2.60) (2.51)

2nd Tier 44.88 2.60

farms (3.19) (1.82)

3rd Tier 0.75 40.66 9.53

farms (2.12) (2.18) (21.71)

“Fairly HR and Very Very LR”

Portfolio

1st Tier 0.22 3.15 26.70 1.77

farms (0.53) (5.18) (3.78) (3.39)

2nd Tier 0.38 32.61 1.29

farms (1.06) (4.87) (2.32)

3rd Tier 3.96 27.49 1.20

farms (6.95) (5.25) (2.02)

Kansas Stock

Farm Market

Equity Index

“Fairly HR and LR”/

“HR and Very LR”/

“Very HR and Very Very LR”

Portfolios (a)

1st Tier 2.78

farms (1.79)

2nd Tier 1.74

farms (1.58)

3rd Tier 2.33

farms (1.72)

“Fairly HR and Very LR”/”HR

and Very Very LR” Portfolios

1st Tier 18.33

farms (4.46)

2nd Tier 19.80

farms (3.49)

3rd Tier 20.33

farms (4.85)

“Fairly HR and Very Very LR”

Portfolio

1st Tier 30.10 2.60

farms (9.69) (3.82)

2nd Tier 35.44

farms (7.88)

3rd Tier 35.69 2.16

farms (9.35) (5.76)

Note: Standard Deviation are in parentheses (in % ); Conagra, Quaker

Oats, Winn-Dixie, and McDonald are not reported because these stocks

are not part of any optimal portfolios; LR stands for Low Risk; HR

stands for High Return.

(a)The portfolios are the same for each of the listed linguistic

descriptions.

COPYRIGHT 2002 American Agricultural Economics Association

COPYRIGHT 2008 Gale, Cengage Learning