Farm Operating Risk and Cash Rent Determination

Transcription

1 Journal of Agricultural and Resource Economics, 17(1): Copyright 1992 Western Agricultural Economics Association Farm Operating Risk and Cash Rent Determination Calum G. Turvey, Timothy G. Baker, and Alfons Weersink This article examines farm operating risks and cash-rent determination through the use of the efficient set mathematics. The efficient set mathematics proves to be a pragmatic approach to characterizing operating risks, and the relationships between operating risks and cash-rent determination. Various separation theorems are used to postulate the relationship between operating risk and cash rents. Preliminary evidence appears to support the theoretical conclusion that operating risk and cash-rent determination are related. Key words: cash rent, efficient set mathematics, EV analysis, risk, separation theorem. Under the assumption of homogeneous risk preferences, classical economic theory suggests that exogenous market forces and land productivity characteristics will set a single cash-rental value of land. There are numerous studies which have examined how land rentals vary within a region according to these exogenous factors (Ketchabaw and van Vuuren). However, if risk preferences are heterogeneous, then it is unlikely that market forces will determine a single cash rent. Although the theory of cash rent for agricultural land generally has not been cast in a stochastic framework, Johnson, and Collins and Barry have done so in the context of a generalized two-parameter single index model. These studies illustrate the fact that including cash rent in an expected value-variance (EV) framework does provide an effective approach to leveraging risk. In particular, it is shown that an EV frontier with riskless cash-rented land is linear and dominates all but one portfolio without cashrented land. However, it is important to recognize that these studies treat the cash rental rates as exogenous, and implicitly assume that the optimum portfolio choice is conditioned on this exogenous rate, an assumption which is sufficient but not necessary. The purpose of this article is to examine the relationship between risky and riskless cash-renting land and farm enterprise selection. Using the efficient set mathematics as an analytical tool, we examine various aspects of the cash-rent problem which extend the traditional Ricardian model to endogenous rent determination under risk, and alternative land tenure arrangements. In the first section, efficient set formulas are described which provide solutions to the primal and dual risk-minimizing problems. 1 The formulas are similar to those found in Roll (1977, 1980) and require only a basic knowledge of matrix algebra. The efficient set framework is then applied to cash-renting land. The results show that an exogenous cash rental rate will determine the portfolio proportions of independent risk preferences while an endogenous cash rent will be determined by the selection of crop mixes which are dependent on risk preferences. In the section preceding the conclusions, a simple econometric model is developed to provide evidence that the fundamental relationships among cash rent, risk, and return hold in practice. Efficient Set Mathematics The EV model used here follows the original Markowitz approach of minimizing the variance of revenues for a given level of expected revenue. The formulation is for a cash crop farm and requires only two constraints. The first constraint restricts expected revenue to a minimum value and the second restricts the use of land to the amount available. Nonnegativity restrictions on the decision variables are not imposed. In addition, the familiar minimization calculus with Lagrangian multipliers can be applied directly. 2 Derivation of the efficient set formulas begins by minimizing the following Lagrangian: (1) Minimize Z = /2X'QX + X,(Kp - C'X) + X(L - e'x), Calum Turvey and Alfons Weersink are assistant professors in the Department of Agricultural Economics and Business, University of Guelph. Timothy Baker is an associate professor in the Department of Agricultural Economics, Purdue University. 186

2 Turvey, Baker, and Weersink Farm Operating Risk 187 where X is an n x 1 vector of farm activity levels (acres); Q is an n x n positive definite variancecovariance matrix; X, and X 2 are scalar Lagrangian multipliers for expected net revenue and land, respectively; Kp is the level of expected revenue of the portfolio; C is an n x 1 vector of activity net revenues; L is the amount of land available; and e is an n x 1 vector of ones. The first order conditions can be manipulated to derive the optimal values of X:, Xl, and X2, (2) X* = Q-l[Ce] [X Premultiplying (2) by [Ce]' and rearranging provides (3) [~,J = A-l[Ce]'X *, where the following A is matrix: where A is the following 2 x 2 matrix: (4) C'Q-CCQ-'e abl (4) A = [e'q-'c e'q-'e] [b c Roll (1977) calls matrix A the "fundamental matrix of information" since it contains all the information about the basic data contained in the means, variances, and covariances of farm activities. This information is sufficient to prove all the important results of the efficient set mathematics. The scalar elements of A are called the "efficient set constants." The EV Efficient Solution Vector X* and Portfolio Variance The most common aspect of EV analysis taught at universities is the trade-off between expected income and risk. Often a figure, such as the curved graph in figure 1, is presented to illustrate how expected income from combined farm enterprises is related to the risk (standard deviation) of the farm portfolio. To illustrate EV analysis beyond this graphical approach requires the use and knowledge of quadratic programming (QP) software. Use of the efficient set mathematics eliminates the need to program computers since simple EV models can be solved by hand or more practically in a spreadsheet program such as QUATTRO Pro or LOTUS 1-2-3, which have matrix capabilities. 3 This section first derives formulas for calculating the vector of activities X*, which is the efficient portfolio for a given level of expected income, and the portfolio variance. Then an example is presented to illustrate the use of these formulas. First, substituting (3) into (2) and rearranging yields the formula for determining the optimal solution for vector X* at given levels of expected net revenue and available land. Scn(5) revnu v ni X* = Q-l[Ce]A-I LI ] Second, revenue variance for portfolio X* is found by substituting equation (5) into the following formula for variance: (6) a 2 = X*'QX* = [Kc - 2KpLb + L 2 a]. ac - b 2 Equations (5) and (6) determine the EV efficient portfolio and variance for a given level of expected net revenues and land. Risk-return trade-offs easily are illustrated using these two formulas for alternative levels of expected income. Because the solutions are solved in a spreadsheet, they are a practical teaching tool since they do not require the use of QP software. The use of equations (5) and (6) is illustrated below using the variance-covariance matrix (Q) and its inverse (Q-'), presented in table 1, for gross revenues of corn, soybeans, and wheat for Wellington County in Ontario. 4 The expected net revenue vector (c') is [ ] for corn, soybeans, and wheat, respectively. Using this information the fundamental matrix is: A_ [C' Q-CQ-el a b [ [e',q-c e'q-'ej [b c [ J-

3 188 July 1992 Journal of Agricultural and Resource Economics P^ E 0 0 c- 0 e O cn a0) xx Standard Deviation ($) Figure 1. Efficient set frontiers Using equation (5) and assigning L a value of 100 acres, the optimal portfolio of activities as a function of expected net revenue is: [ Kp1 X*= Alternative values for expected profit can be entered into the equation once the above relation has been identified to determine the effect on crop mix. For example, if Kp = 12,800, then X* = [ ]. The variance and structural deviation of expected revenues can be determined by using only the efficient set constants from matrix A, a value for Kp, and the number of acres. For example, using equation (6) with expected revenues of$ 12,800, the standard deviation equals 9, The EV frontier can be mapped by using equation (6) with alternative levels of K, and the farm optimum portfolios at each point on the EV frontier can be determined using equation (5). An Economic Interpretation of Negative Activity Levels Negative activity values are the result of excluding nonnegativity constraints from the problem. The economic interpretation of these results relates to an alternative land tenure mechanism similar to shortselling stocks in capital market theory (Fama). For example, at an expected net revenue of $12,800, wheat enters the solution with a negative value of 19.5 acres. The tenure arrangement involves renting 19.5 acres of wheat land from another farmer and allocating this to obtain 69.5 corn and 50 soybean acres. Once crop revenues have been realized, the farmer pays the "lessor" an amount equal to that which would

4 Turvey, Baker, and Weersink Farm Operating Risk 189 Table 1. Variance-Covariance Matrix, Inverse Variance-Covariance Matrix, and Net Revenues for Real Corn, Soybeans, and Wheat-Gross Revenues ( ) Corn Soybeans Wheat Variance-Covariance Matrix (Q) Corn 8, , , Soybeans 5, ,319, Wheat 2, , Inverse Variance-Covariance Matrix (Q- 1 ) Corn Soybeans Wheat Net Revenue Data Expected Gross Revenues ($/acre) Variable Production Costs ($/acre) Expected Net Revenues ($/acre) have been earned had wheat been grown by the lessor. In the above example, based on expected crop revenues defined by the vector C', $14,498 could be earned from corn and soybean production (69.5 acres*$126.16/acre + 50 acres*$114.59/acre), of which $1,698 (19.5 acres*$87.08/acre) is to be paid to the lessor as rent ($14,498 - $1,698 = $12,800). The advantage to the lessee is that the risk associated with $12,800 income is less than that which would have occurred in the absence of any rental opportunities. The advantage to the lessor is less clear. In fact, the lessor's overall risk is not reduced since the rental payment is uncertain, with an expected value of $87.09/acre and a 50% probability of receiving more or less than this. The expected value with uncertainty must therefore at least exceed the riskless cash rental rate in order to compensate the lessor for risk. (In later sections the efficient set mathematics is used to explain the separation theorem, with cash-rental land being included as a riskless enterprise.) The type of arrangement discussed will be of advantage to farmers with varying resources. For example, if the lessee in this example has an excess supply of labor but the lessor is labor constrained, then the rental agreement would be mutually advantageous by freeing up labor for the lessor and utilizing labor for the lessee. Solving the Dual Solution for X, and \2 The Lagrangian multipliers provide important risk opportunity cost information. The value of X, is the marginal change in the variance of the optimal portfolio mix (a 2 ) if Kp is increased or decreased. Similarly, X 2 is the marginal change in a 2 if the amount of available land changes. Note that from (4) both Xi and X 2 can be determined by knowing a, b, c, Kp, and L. Solving equation (4) we find that: ckp - bll [La - bkp (7) XI ' and X2= 7 = ac - b 2 and = - b 2 c - Substituting the values for a, b, c, and L from the above example with Kp equal to $12,800, the value for X, is 18, and the value for X 2 is -1,433,005. (The units of Xi are dollars squared per dollar of expected income and the units of X 2 are dollars squared per acre.) The dual variable X\ is the change in the objective function (one-half of variance) per unit of change in expected income, i.e., (8) XA = da2/2dkp. Therefore, 2X, is the inverse of the slope of the EV frontier at expected income Kp. If income is distributed normally and utility is negatively exponential, then the slope of the EV frontier at an optimal solution is one-half of the coefficient of absolute risk aversion. Under these conditions, individuals who prefer the portfolio corresponding to expected income Kp have absolute risk aversion equal to l/xa (Turvey and Driver 1986). In our example, a farmer choosing the portfolio with an expected income of $12,800 would have an implied coefficient of absolute risk aversion of about (1/18005).

5 190 July 1992 Journal of Agricultural and Resource Economics The dual variable X 2 is the change in portfolio variance with respect to a change in the land resource, i.e., (9) X2 = da/2dl. In general, the shadow price of one constraint can be converted into units of a second constraint, rather than units of the objective function, by dividing by the shadow price of the second constraint (Preckel, Featherstone, and Baker). If X2 is divided by X,, useful units result as follows: (10) 2 = (dao/2dl)/(da2/2dkp) = dkp/dl. Xl Thus, the ratio X2/X1, which is the marginal value per acre of land in dollars of expected income, is expressed in dollars per acre. In the example, the marginal value of land is $79.56 per acre. The Global Minimum Variance Portfolio The portfolio which has the least risk attainable, given the variance-covariance structure described by Q. and using all of the fixed resource given by L, is defined as the global minimum-variance portfolio (see fig. 1). Risk programming applications in agriculture usually consider only portfolios above the global minimum variance. Portfolios lying below the global minimum-variance portfolio have decreasing expected revenue for increasing variance and, therefore, would not be selected by a rational risk-averse farmer. Since nonnegativity restrictions on the choice variables are not imposed in the efficient set mathematics, portfolios below the global minimum-variance portfolio are attainable. It is therefore useful to derive expressions for the global minimum variance, ao, the global minimum-variance portfolio, X,, and the expected revenue associated with X,, Ko. The expected revenue of the global minimum-variance portfolio is found by setting the partial derivative of a 2 [equation (7)], with respect to Kp, equal to zero, and solving for the Kp associated with the minimumvariance portfolio, which we label Ko. Lb (11) Ko= c Substituting Ko for Kp in (6) gives the global minimum variance ao, L 2 (12) o= and the solution vector, Xo, is found by substituting Ko into equation (5): (13) Xo= Q-e-. Using Ko equals $9,331.40, a, equals $4,971.2, and the acreage vector XO is [ ]. In figure 1, the minimum variance portfolio is located at the point (Ko = 9,331.4, a, = 4,971). The selection of portfolios below the global minimum-variance portfolio would not be rational since an alternative portfolio exists with the same amount of risk but with greater expected revenue. The calculation of Ko, ao, and Xo therefore is used as a benchmark for defining a minimum expected income within the feasible set. Separation Theorem The above discussion provided formulas for determining the primal and dual solutions as well as the variances of the risk-minimizing solution. This section describes the separation theorem as it applies to farm operating decisions. The concept is important since it shows how a riskless farm enterprise, cashrent land, can be combined with risky portfolios to decrease risk (Johnson; Collins and Barry; Turvey and Driver 1987). The separation theorem for farm operating decisions is that "the optimal strategy for combining risky enterprise options is independent of the ratio of the amount of land in risky enterprises to the amount of land owned" (Johnson, p. 615). Figure 1 is used to introduce this concept. The parabola represents efficient portfolios when only risky enterprises are available, and the line represents efficient portfolios when riskless cash rental land is available. The two EV frontiers are tangent. At the tangency portfolio, all land is used in the production of risky farm enterprises; no land is cash rented. All other points on the line represent

6 Turvey, Baker, and Weersink Farm Operating Risk 191 Table 2. Efficient Portfolios with and without Cash-Rented Land Std. Deviation Cash- Expected of Net Rented Net Revenues Revenues Corn Soybean Wheat Land ($) ($) (acres) (acres) (acres) (acres) Portfolios Including Only Risky Enterprises 6,500 8,000 8, , ,331a 4, ,800 5, ,012b 6, ,600 7, ,800 9, Portfolios Including Cash-Rented Land 6,500a ,000 2, ,331 3, ,800 5, ,012b 6, ,600 7, ,800 8, Note: Positive values indicate cash-renting land out to others and negative values indicate renting land from others. a Global minimum-variance portfolio. b Tangency portfolio. portfolios which include rental land. For points to the left (right) of the tangency portfolio, the farmer becomes a landlord (tenant) by renting out part of the land base and planting the rest of the land to risky crops in the same proportions as the tangency portfolio. The separation theorem suggests that the selection of the crop mix does not depend upon the decision maker's risk preferences, since it is constant along the expanded EV frontier. Instead, the amount of land rented in or out is the variable affected by risk preferences. Additionally, the availability of a riskless enterprise improves the risk-return possibilities over utilizing only risky assets. For a given level of expected income, farmers can decrease risk by renting land in or out. The resulting operating decision dominates farm plans lying on the original EV frontier. When cash renting is added, equation (5) can be rewritten in terms of n risky activities and one riskless activity as: (14) X* = Q-'[Ce]A- l -L^ where hats (^) denote the fact that a riskless enterprise is included as part of the farm operation. The variance-covariance matrix Q can be represented as: (15) = [ where Q is the n x n matrix of variances and covariances, 0 is an n x 1 null vector, and e is a very small number close to zero. The use of E implies that the riskless enterprise has a variance of E. Setting E equal to a small number, such as 1 or.001, allows Q to be inverted, and ensures that its contribution to the curvature of the linear EV frontier is negligible. 5 To illustrate how the separation theorem works, EV portfolios were computed with and without a riskless enterprise. They were computed over a range of expected net revenue from $6,500 to $12,800 using equations (5), (6), and (14). Results are shown in table 2. The exogenous rental value of land was assumed to be $65/acre. Both EV frontiers are plotted in figure 1. The inclusion of cash-rented land as a riskless farm enterprise causes the efficient frontier to become linear (when graphed in expected income and standard deviation space) and dominates the EV frontier o ]

7 192 July 1992 Journal of Agricultural and Resource Economics of risky activities. The minimum-variance portfolio occurs when the 100 acres of land is cash rented out, giving a net revenue of $6,500 with a zero standard deviation (fig. 1). The EV frontier allowing the farmer to rent land in or out will become tangent to the EV frontier of risky activities. The tangency portfolio in figure 1 represents a crop mix of 15.9 acres of corn, 61.2 acres of soybeans, and 22.9 acres of wheat. The expected net return is $11,012 and the standard deviation is $6,275. Every portfolio along the line grows the same proportion of the three crops. The improvement in the risk-return opportunities using the leverage of cash-rented land over risky crops alone is clear from figure 1. For any level of expected income, other than the tangency portfolio, farmers can decrease risk by renting land. For example, in order to obtain expected returns of $11,600, a portfolio comprised of 33.5 acres of corn, 57.5 acres of soybeans, and 9 acres of wheat can be formed. This portfolio has a standard deviation of $7,171. Alternatively, to obtain the same level of expected income, the farmer can become a tenant, renting in 13 acres of land and forming a portfolio comprised of 17.9, 69.2, and 25.9 acres of corn, soybeans, and wheat, respectively. The standard deviation of this portfolio is $7,093 which, for the same level of expected net revenues, is a substantial decrease in risk. For portfolios below the tangency portfolio, the farmer becomes a landlord, renting land out and reducing risk. The above analysis is a generalization of the separation theorems established by Sharpe's (1963) Single Index Model (SIM), the Capital Asset Pricing Model (Sharpe 1964; Lintner; Johnson). It assumes exogenous cash rents which determine optimum portfolios. 6 The following analysis differs from this classical approach since it assumes first that an optimum portfolio is selected, and this determines endogenous cash rents. Thus, negotiated cash rents are a function of expected portfolio returns, portfolio risk, and risk aversion. To calculate endogenous cash rents under uncertainty, the concept of orthogonal portfolios is introduced within the context of a more generalized separation theorem. The separation theorem states that the solution vector of every EV efficient portfolio is a linear combination of the solution vectors of two other efficient portfolios whose means are different (Roll 1977). For example, if two EV efficient portfolios can be identified, these can be combined to construct a third portfolio which is also EV efficient. To calculate endogenous cash rents, it is necessary to find the returns on a portfolio which lies on the EV frontier which is uncorrelated with (orthogonal to) the selected portfolio. Because these two portfolios are orthogonal, linear combinations will lie on a line intersecting the returns axis and tangent to the selected portfolio. As will be shown, the point at which this line intersects the returns axis is the endogenously determined rental rate. For every mean-variance efficient portfolio Xp, other than the global minimum-variance portfolio, there is a unique portfolio Xz, with an expected value Kz, which is also efficient (Roll's corollary 3). The covariance between two efficient portfolios Z and P is given by: (16) apz = X2QXp[KzL]A-'[KpL]. If Z is orthogonal to P, then lap = 0. The value of Kz which satisfies az = 0 is given by: L 2 a - blkp (17) Kz= Lb- ckp This defines the mean revenues of a portfolio orthogonal to P. Kz will be less than K,, the revenues associated with the global minimum-variance portfolio. Since no orthogonal portfolio exists for the global minimum-variance portfolio, it is always true that Kz < K/ < Kp. In expected returns standard deviation of return space the orthogonal portfolio is located by an array tangent to the efficient portfolio P at the point (Kp, ua) and intersecting the returns axis at the point Kz. In figure 1, portfolio Z on the backward bending portion of the EV frontier is orthogonal to the tangency portfolio P. The result of equation (17) is important because it is the link between the uncertainty and classical models of cash rent determination. Recall from earlier discussion that the ratio of shadow prices X 2 /X 1 is interpreted as the VMP of land dkp/dl: ~(1~8) ~X 2 = dkp La - bkp X, dl La-cKp The number Kz is related to Dkp/dL in that dkp dl which establishes that the per acre return on an orthogonal portfolio is identical to the (expected) marginal Kz L'

8 Turvey, Baker, and Weersink Farm Operating Risk 193 value of land which, according to the classical model, determines the rental value of land. The implication of this result is that a theoretical link between the classical profit maximizing model and the uncertainty model exists. But under conditions of risk, a multitude of cash rent values exist and their existence depends on portfolio returns, portfolio risks, and risk preferences. Thus, unlike the classical model, an equilibrium cash rental value for land may not exist. Using previously defined values, the expected values of the marginal product of land for income levels of $12,800, $11,012, and $10,800 are $79.57/acre, $65/acre, and $60.90/acre, respectively. Note that the solution of$ 11,012 is the one implied by the exogenous market rate and is independent of risk preferences. However, this rate would be a bargain to the moderately risk-averse farmer (a = ) who would pay up to $79.59/acre, but the rate may be too dear for the more risk-averse farmer (a = ) who would not be willing to pay more than $60.90/acre for rental land. An alternative way of viewing this is that neither farmer would pay more than their E[ VMP] of land, since doing so would require growing a more risky crop mix which would reduce individual utility. Both, however, would accept an offer to lease their land out at a higher price since this would enable them to grow a more risky crop mix on less acreage, thereby increasing utility. Finally, individuals may be able to take advantage of those who are more risk averse than themselves by offering to rent land at a price slightly higher than the latter's E[VMP]. More risk-averse individuals would increase their utility vis-dvis the above argument. Less risk-averse individuals would increase their utility since more acres can be sown to a less risky crop mix. Some Preliminary Empirical Evidence The underlying assumption in the above example is that cash rents are related to expected return and risk. In this section a simple regression model is presented to evaluate whether or not there is a tendency for this relationship to hold in practice. The assumptions are therefore treated as hypotheses to be tested. A 1989 survey of 432 Ontario cash crop farmers provided average cash rental rates for each of 27 of Ontario's 50 agricultural producing counties (Ketchabaw and van Vuuren). For each county, gross revenues were tabulated for the major crops from 1970 to 1988 and converted to real 1983 dollars using the implicit price deflator. An average county revenue was computed by weighting each crop by the proportionate acreage planted in the county. This average county revenue is used as a proxy for portfolio returns, and the variance of this revenue is used as a proxy of portfolio variance. Ordinary least squares regression of cash rental rates (R) against expected revenue, E[R], and portfolio risk a resulted in the following equation which has an R 2 of.604: (19) R = E[R] a -.004E[R]a. (-2.22) (2.54) (2.02) (2.08) This equation provides some empirical evidence that cash prices respond to risk and returns; all other things being equal, an increase (decrease) in E[R] will increase (decrease) R (since OR/OE[R] = a > 0 for a = 51.54), and an increase (decrease) in a will decrease (increase) R (since dr/8a = E[R] < 0 for E[R] = ). Furthermore, the result that 0 2 R/dE[R]0a = < 0 implies that for an increasing E[R], a simultaneous increase in a will lessen the increase in R, and for an increasing a, a simultaneous increase in E[R] will lessen the decrease in R. Conclusions The intent of this article was to provide a pedagogic review of the efficient set mathematics and the separation theorem as it applies to farm operating decisions. The importance of the separation theorem as it applies to cash-rented land in agriculture was developed and illustrated. Three aspects of this problem were discussed. First, while most risk analyses in agriculture impose nonnegativity on activity levels, the simple model presented here relaxes the restriction and offers an interpretation of negative activity levels analogous to short-selling securities in the capital market. While we are unaware of any real world transactions of this kind, it may be consistent with certain types of risk-sharing lease arrangements. Unlike the above model which implies an uncertain cash rental rate, the remainder of the article assumed the availability of riskless cash-rental land. A second aspect of the cash-rented land problem analyzed related to one theory posited which suggests that an equilibrium or exogenously determined cash rental rate will determine the portfolio proportions independently of risk preferences. In contrast, a second theory and third aspect discussed posits that cash rents are endogenously determined by the selection of a crop mix which reflects risk preferences. In this latter model, for a given crop mix, the farmer would be willing

9 194 July 1992 Journal ofagricultural and Resource Economics to pay no more than the expected marginal value product for cash rent land, whereas the landlord would not be willing to accept less. Yet opportunities exist whereby less risk-averse farmers can take advantage of the low opportunity costs faced by more risk-averse farmers. Finally, some preliminary empirical evidence of a significant relationship between cash rents, risk, and returns was offered. Regression results of cash rents against expected portfolio revenues and risk were consistent with the mean-variance rules presented. [Received February 1991; final revision received August 1991.] Notes 1 While not intending to offer an alternative to computerized quadratic programming models, the approach used here does serve as an interesting and pedagogic approach to presenting the portfolio choice problems since the solutions can be solved manually with matrix algebra, or in a spreadsheet such as QUATTRO Pro which has matrix facilities. 2 More details concerning the derivation of the relationship presented in this article are presented by the authors in a working paper of the same title. 3 Indeed, this is how portfolio analysis is taught to undergraduate and graduate students of agricultural finance at our university. Providing a spreadsheet program with which students can parameterize K, and graph E-a frontiers has been very successful. As mentioned in endnote 1, this is not intended to displace conventional QP software such as MINOS, or GAMS-MINOS. However, it is convenient for an undergraduate course in agricultural finance, which makes exclusive use of spreadsheet technology. Thus the need to teach students how to run unfamiliar software (e.g., MINOS) is eliminated. A macro-driven spreadsheet template (either LOTUS or QUATTRO Pro) which uses all of the efficient set mathematics is available from the senior author on request. 4 The variance-covariance matrix was calculated from historical prices and yields for Wellington County, Ontario. Yields were detrended and prices were deflated using the CPI (1986 = 100). Variable costs were compiled from Ontario Ministry of Agriculture and Food budget estimates for The approach described is somewhat ad hoc but has the advantage of utilizing the previous set of equations. 6 However, this generalization should in no way be predicated on any underlying assumptions of market equilibrium in the CAPM sense. Collins' comments to Hutchison and McKillop make this aspect of the problem evident. Indeed, the mathematics up to this point does not require any formal statement of equilibrium. CAPM models, for example, represent special cases of the efficient set mathematics since they relate to points on a particular mean-variance frontier which dominates all others. In contrast, what is presented here represents a micro analysis whereby the mean-variance frontier is farm specific. The implication is that for two farmers with the same opportunity sets facing the same exogenous cash rent price, but for some reason or another have different risk profiles, the optimal strategy would include different combinations of risky enterprises, even if the proportion of owned to rented land (in or out) was the same. References Collins, R. A. "Risk Analysis with Single Index Portfolio Models: An Application to Farm Planning: Reply." Amer. J. Agr. Econ. 70(1988): Collins, R. A., and P. J. Barry. "Risk Analysis with Single-Index Portfolio Models: An Application to Farm Planning." Amer. J. Agr. Econ. 35(1986): Fama, E. F. Foundations of Finance. New York: Basic Books, Hutchison, R. W., and D. G. McKillop. "Risk Analysis with Single Index Portfolio Models: An Application to Farm Planning: Comment." Amer. J. Agr. Econ. 70(1988): Johnson, S. R. "A Re-Examination of the Farm Diversification Problem." J. Farm Econ. 49(1967): Ketchabaw, E. H., and W. van Vuuren. "A Statistical Overview of Agricultural Land Renting in Ontario." Work. Pap., Dep. Agr. Econ. and Business, University of Guelph, February Lintner, J. "The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets." Rev. Econ. and Statist. 47(1965): Markowitz, H. Portfolio Selection: Efficient Diversification of Investments. New York: John Wiley and Sons, Ontario Ministry of Agriculture and Food. Agricultural Statistics for Ontario. Publication 20 (various issues). Preckel, P. V., A. M. Featherstone, and T. G. Baker. "Interpreting Dual Variables for Optimization with Nonmonetary Objectives." Amer. J. Agr. Econ. 69(1987): Roll, R. "A Critique of the Asset Pricing Theory's Tests: Part 1: On Past and Potential Testability of the Theory." J. Financ. Econ. 4(1977): "Orthogonal Portfolios." J. Financ. and Quant. Anal. 15(1980): Sharpe, W. F. "A Simplified Model for Portfolio Analysis." Manage. Sci. 2(1963): "Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk." J. Finance 19(1964): Turvey, C. G., and H. C. Driver. "Economic Analysis and Properties of the Risk Aversion Coefficient in Constrained Mathematical Optimization." Can. J. Agr. Econ. 34(1986): "Systematic and Nonsystematic Risks in Agriculture." Can. J. Agr. Econ. 35(1987):