Agricultural Finance Review

volume 55 article #1

Static vs. Dynamic Models of Proprietary Capital Structure: Discussion and Preliminary Empirical Evidence

Robert A. Collins and Larry S. Karp

Robert A. Collins is Naumes Family Professor, Institute of Agribusiness, Santa Clara University. Larry S. Karp is an associate professor in the Department of Agricultural and Resource Economics, University of California, Berkeley, and reader, University of Southampton.

Abstract

Data collected during the farm credit crisis are reused in an attempt to confront alternative capital structure models with data. While the equations are misspecified, the results appear to be more consistent with a dynamic optimal control model of capital than a static expected utility model.

Key words: debt, capital structure, econometrics.

Article

Two unconstrained, single-period expected utility maximizing models have been proposed for explaining farm financial structure: a model suggested by Barry, Baker, and Sanint as a model of liquidity management, and a model of capital structure by Collins (1985). These two models, which have a different form but the same substance (hereafter the Barry-Collins model), have been extended, generalized, and applied to other problems. The Barry-Collins model was substantially extended by Featherstone, Moss, Baker, and Preckel, and by Moss, Ford, and Boggess in innovative models evaluating the probability that aggregate agricultural policies increase the risk of farm failures. A series of clever papers by Turvey and Baker (1989, 1990), and Turvey apply the Barry-Collins model to the relationship between capital structure and optimal hedging decisions. Moss used the model in a discussion of environmental regulation. Moss et al. (1989) used the model in an analysis of the taxation of capital gains.

While many excellent contributions to agricultural policy and economic theory have resulted from these articles which reference, apply, and extend the Barry-Collins model, it is possible the model has not been subjected to adequate critical evaluation. The popular appeal of the Barry-Collins model probably arises from its simplicity and its closed-form solution rather than from any obvious adequacy of its theoretical form or any comprehensive empirical validation. While there has been almost no discussion of the theoretical merits of the model, one rather sophisticated econometric study was published by Moss, Shonkwiler, and Ford. The results of their study were generally consistent with the Barry-Collins model, but since the testable implications of the model relate choices to expectations, testing was complicated by what was assumed about the expectations generation process.

An alternative model of farm capital structure recently has been proposed by Collins and Karp. This stochastic optimal-control model sacrifices the simplistic appeal of the Barry-Collins model, but it does consider more of the complexities of the real world and propose several testable hypotheses. This article examines these two models, discusses the potential validity of the underlying behavioral paradigms of these models in the context of the farm capital structure problem, and presents some preliminary empirical analysis of some of the assumptions and conclusions of these models.

Underlying Behavioral Paradigms <top>

The behavioral paradigms assumed by these two models are different in many ways. The assumed characteristics and objectives of the decision maker differ, as well as what is assumed about the scope and character of the decision to be made. Also, the two models assume radically different functional forms for underlying structural relationships. The potential difficulty of empirically evaluating these paradigmatic differences ranges from extremely complex to reasonably straightforward.

The Barry-Collins model assumes that the decision maker is risk averse and myopic. The decision maker considers the effects of financial leverage on the first two moments of the objective for the next planning period, and chooses the leverage level that maximizes single-period expected utility. Both Barry and Collins use a simple mean-variance approach and refer to the Freund paper. While the length of the planning period is not specified, the decision maker presumably repeats this analysis for each planning period and chooses a leverage level depending on what rate of return on assets and business risk is expected for the following period. The model is clearly static in that the decision maker does not plan to change capital structure during the planning period under consideration. A reviewer, however, pointed out that the solution suggested by the static Barry-Collins model is the same as the solution to one of Merton's lifetime portfolio models, as well as the solution to a similar dynamic model by Ramirez. Therefore, if one were to observe behavior that was consistent with these models, one could not differentiate between the alternative underlying theoretical structures.

Both of these models assume that the underlying objective¹ of the decision maker is normally distributed with μ and variance σ², and that the decision maker's utility of wealth function is negative exponential. Since both μ and σ² are increasing functions of debt choice, the decision maker's problem is to trade off increases in the expected outcome with increased variability. Given the financial structure of a particular firm, increasing the variability of either the rate of return on equity or income also implicitly increases the probability of financial failure. However, the primary concept of financial risk used in these models is variability, and failure risk is considered only in an indirect way.

¹The Collins model focuses on the objective of the expected utility of the rate of return on equity. Barry et al. use the expected utility of annual income. The results are identical.

The Collins-Karp model assumes the decision maker is risk neutral, but is concerned about the effect that the leverage decision will have on both the profitability (I) and failure risk over the entire life of the farm. The farmer's objective is maximization of the present

value of expected wealth, the present value of the stream of withdrawals until retirement plus the present value of terminal equity resulting from the liquidation of the farm at retirement (T). Where δ is leverage, w is the proportion of income withdrawn, and E is equity, the objective function is:

__t is the expectations operator conditional on information available at time t, and ρ is the riskless rate of time preference. The constraint is the stochastic differential of equity. The stochastic differential for equity is the (deterministic) retention of earnings plus the stochastic term that indicates whether a disaster occurs:

where Pr(dπ = 1) = γ[δ(t)]dt + o(dt), and Pr(dπ = 0) = 1 γ[δ(t)]dt + o(dt); γ_ > 0,

γ" > 0; and o(dt) denotes terms of order dt. The decision maker is not concerned about year-to-year variation in earnings, but does realize that increased use of financial leverage increases the likelihood of a catastrophic financial failure. A catastrophic financial failure causes no disutility except through the effect the failure has on the expected present value of the farm's earnings and liquidation value.

Which of these sets of behavioral assumptions best fits the real world? This question really has many dimensions, and suggests hypotheses that are not easily tested. What does the farmer think is "risk"?—the year-to-year variation in income, or the possibility that the business might fail before retirement age?

What is the actual planning horizon that a farmer considers when making leverage choices?—just the next period, or the entire remaining life of the business? Does the concept of mathematical expectation adequately characterize the farmer's objective?—or is some type of multi-dimensional utility needed? These questions are not likely to have definitive answers any time soon, but it seems nearly certain that both sets of assumptions must be less-than-perfect approximations of reality. It is likely that farmers really have some concern about both variation in income and the possibility of bankruptcy. While concern about one or the other might be dominant, it is not likely that either one alone can adequately represent the "risk" associated with the use of financial leverage. It also seems likely that representing the time horizon as either one period or the entire life of the farm my be too bold a contrast. Possibly, the consideration of the time horizon depends on the particular decision at hand.

This highlights another oversimplification in these models—that capital structure is, in fact, a single choice. Realistically, the debt structure of farms may include operating loans with a term of less than a year, loans for machinery and equipment with terms of 3_10 years, and loans for land and structures with terms of 15_45 years. Therefore, the overall capital structure of a farm reflects a series of choices about a variety of issues, not a single choice. Neither model captures this dimension of debt choice very well. However, since there are seldom prepayment penalties, and since there is considerable substitutability among types of debt, perhaps this omission is not critical.

In most cases, it is probably pointless to empirically test the appropriateness of these underlying assumptions. It may be more prudent just to admit at the outset that no model captures the complete complexity of the real world in its underlying assumptions, and to focus more on determining which model has operational conclusions that are most consistent with observed behavior. There is, however, one element of the underlying paradigms of these models that is both starkly different and potentially testable. The Barry-Collins model assumes that the rate of return on equity is a convex increasing function of leverage. By contrast, the Collins-Karp model assumes that the rate of return on equity is a concave function of leverage, but independent of scale. The depiction of this function in Figure 1 of Collins and Karp (p. 228) shows it reaching a maximum and declining, but a concave and increasing function also satisfies the assumptions of the model.

Potentially Testable Conclusions of the Barry-Collins Model <top>

The expected utility framework of the Barry-Collins model focuses on the expected rate of return on equity and its variability. Presumably, at the beginning of each period, the decision maker would consider the anticipated density function of the rate of return on assets (ROA) for the next period, and estimate its expectation and its variance (business risk). Then, given the cost of borrowing and the level of risk aversion, the farmer chooses a debt level that maximizes the expected utility of the rate of return on equity for the next period. The potentially testable hypotheses that arise from this model are that an increase in expected ROA will cause an increase in leverage, while an increase in business risk will cause a decrease in leverage.

However, empirical tests of these hypotheses are not straightforward for several reasons. First, we have the difficulty that we can't directly observe expected ROA or its variance, only ex post realizations. Therefore, we have to rely on some assumptions about an expectations generation process, or look at the factors which would reasonably affect ex ante farm profitability. However, if one decides to avoid an expectations generation process, the number of variables that could affect the expected ROA of a farm is almost staggering. One would have to consider expectations of all relevant output prices and input prices, the expected productivity of each input, as well as all the factors that would potentially affect the productivity of inputs. Also, since debt amount is chosen in nominal terms, one must consider the expected rate of inflation in the value of agricultural assets relative to the inflation premium embodied in the borrowing rate. Therefore, the number of factors that must be considered to test these hypotheses is daunting. But in addition, the magnitude, and even the direction of the effects that many of these factors produce, may be unique to each farm. For example, an increase in the expected price of corn would increase the expected ROA for a corn farmer, but decrease it for an animal feeder.

For these reasons, empirical testing of the hypotheses suggested by the Barry-Collins model requires some kind of assumptions about the process by which expectations are generated. Because of the excessive information costs associated with forming a strictly rational expectation, Moss et al. (1990) used an autoregressive process to model expectations. Their results are consistent with the model, and therefore provide support for the model's underlying structure. However, the general proposition of testing micro models which hypothesize that behavior results from expectations is troubling, since such tests must simultaneously test an expectations generation process and the underlying model. It is possible that such models do not really satisfy the general scientific requirement of falsifiability, because there is no data set that would unambiguously establish that the model is incorrect. No matter what behavior is observed, it could have been consistent with expectations, since expectations can't be observed. However, since the relevant hypotheses for the Barry-Collins model involve only factors that affect the anticipated profitability of farm assets, an indirect test of the completeness of the Barry-Collins model is to see if other factors are associated with farm capital structure. If other factors, such as the farmer's age, wealth, or alternative employment opportunities, are associated with farm capital structure, one might conclude that a single-period, unconstrained model— which looks only at the moments of return to farm assets—is an incomplete model of capital structure, even if previous econometric results appear to be generally consistent with its implications.

Potentially Testable Conclusions of the Collins-Karp Model <top>

The Collins-Karp model produces several specific hypotheses about how factors that don't affect the distribution of the return to farm assets could affect the farmer's leverage choice. It proposes the following specific testable implications regarding age, wealth, and opportunity cost that can be tested with cross-section data:²

Leverage depends on age. The model predicts the partial effect of age to be hump shaped. In a cross-section data set, if equity and opportunity cost are held constant, one would expect to find young farmers increasing leverage while older farmers are reducing leverage.

Leverage depends on wealth. The partial effect of wealth is negative. If opportunity cost and age are held constant, one should find that farmers with high equity choose less leverage (lower debt/asset ratios) than low-equity farmers. This suggests that, ceteris paribus, the equity of the farm grows faster than the scale of operation (total assets).

Leverage depends on the opportunity cost of being in business. The partial effect of capitalized alternative earnings on leverage is positive. The more earnings available from alternative employment, the more willing farmers will be to use financial leverage, and risk the consequences. Simply put, the better the alternative earnings are, the more willing the farmer is to take a chance, fail quickly, and go on to the alternative. Since the opportunity cost of being in business is the present value of alternative earnings, it depends on the amount that can be earned in alternative employment as well as the time to retirement and the discount rate.

Wealth affects the retirement decision. The model suggests that richer farmers will retire later.

²The fifth testable implication listed in Collins and Karp relates to effects of time, and cannot be tested with cross-section data.

Empirical Analysis of Micro Farm Data <top>

One of the serious difficulties in empirical examination of farm capital structure models is access to good data. For this analysis, the individual responses to a 1986 survey of Arkansas farmers described in Collins (1987) were used. This survey collected detailed financial data for the purpose of measuring the amount of financial stress in the farm sector in the state of Arkansas, and fortunately, it also contained most of the information necessary to test these hypotheses. The survey consisted of a stratified random sample of 2,500 farms selected from the nine crop and livestock reporting districts in Arkansas. A total of 989 usable forms were returned. The average annual earnings of farm laborers for each county in Arkansas, obtained from the 1982 Census of Agriculture (U.S. Department of Commerce, Bureau of the Census), was used as a proxy for the individual farmer's opportunity cost of being a farmer. A reviewer pointed out that assuming that all farmers in a county have the same opportunity cost is a somewhat crude measure. A preferred approach would be to estimate each farmer's opportunity cost as a function of their individual human capital.

The one underlying assumption to be tested is whether the rate of return on equity (ROE) is a concave or a convex function of leverage, and whether ROE is independent of scale. This was tested by regressing the rate of return on equity (ROE) on the ratio of debt to assets (δ), the leverage ratio squared (δ²), and assets (A) for the 749 farms in the sample that were still operating. The Barry-Collins model and the Collins-Karp model assume that the return on equity increases with leverage and that there are no economies of scale. Therefore, both sets of models call for the coefficient of δ to be positive and the coefficient of A to be zero. However, the two models differ on what is assumed about the sign of the quadratic term. The Barry-Collins model implicitly assumes a convex relationship between leverage and ROE, while the Collins-Karp model specifically assumes a concave relationship. Therefore, the sign of the coefficient of δ² is a relevant test of the difference between the models. The estimated equation (t-values in parentheses) was:

ROE = 0.0068 +	0.311δ	0.069δ²
	(6.63)	(2.26)

	1.1 ´ 10⁹A, (0.09)
	R² = .0781.

However, this model is misspecified since the primary factors that determine the rate of return on equity were omitted. For example, the education level of the farmer, the location of the farm (delta vs. mountains), the quality of the land, and a host of other important factors are unavailable. Nevertheless, there is no reason to believe that these omitted factors are not orthogonal to the size of the farm and the debt/asset ratio and, therefore, cause no specification error. As a result, one would expect unbiased estimates for the coefficients, but a very low R². Therefore, even though the R² = .0781, the significant t-statistic for the quadratic term provides some empirical support for the major structural assumption of the Collins-Karp model.³ Since the δ coefficient is positive and significant, the δ² coefficient is negative and significant, and the coefficient of assets is not significantly different from zero, then the more general hypothesis that the rate of return on equity is a concave function of leverage and independent of scale is supported for this sample. Although the debt/asset ratio must be less than one, the estimates of the coefficients for δ and δ² imply that the rate of return on equity is maximized when δ = 2.26. This suggests that for this sample, the function is increasing and concave over the relevant range 0 < δ < 1. This does not violate the concavity assumption required for the Collins-Karp model.

³Equity appears in the denominator of the dependent variable and in the debt/asset ratio. This does not present a problem of spurious correlation, however, since the hypothesis is in terms of the ratios and not the numerators. Kuh and Meyer (p. 401) write, "The question of spurious correlation quite obviously does not arise when the hypothesis to be tested has initially been formulated in terms of ratios...."

The primary testable conclusions of the model regard the behavior of leverage with respect to age, wealth, and opportunity cost. Is leverage constant with time, decreasing with time, or hump shaped if other factors are held constant? The model also predicts that farmers with more equity will choose a lower debt/asset ratio. This means that the scale of the farm does not expand proportionately with equity, or that the elasticity of debt with respect to equity is less than one. The model also predicts that the derivative of debt with respect to opportunity cost is positive.

Debt (D) was regressed against AGE, AGE², equity (E), and the proxy for opportunity cost (OPP COST), the earnings of farm laborers.⁴ Since debt must be non-negative, Tobit analysis was used. The estimated equation was:

D = 24.2 +	1.28AGE		0.018AGE²	+	0.0504E	+	3.15OPP COST,
	(1.57)		(2.12)		(11.58)		(5.31)

Log likelihood = 2,346.7.

⁴A reviewer correctly pointed out that using proxies for unmeasurable variables has weaknesses similar to using an expectations generating process.

The t-value for opportunity cost strongly supports the assertion that off-farm factors may affect farm leverage choice. Using a one-tailed test, the coefficients of AGE and AGE² are significant at the 6% and 2% levels, respectively, which suggests that the farmer's stage of life may affect leverage decisions. The signs of these coefficients support the existence of a hump-shaped partial effect of age on leverage. If equity and opportunity cost are held constant, the data show that debt would at first increase, then decrease. The estimated coefficients for AGE and AGE² suggest that leverage would increase until age 35. The elasticity of debt with respect to equity was calculated using the method of Thraen, Hammond, and Buxton. The point estimate of this elasticity at the sample mean ($224,000) was 0.07, and reaches only 0.28 for an equity level of $1,000,000. Therefore, the large t-value and the elasticity measures that are substantially less than one support the hypothesis that farmers with higher equity do indeed choose lower debt/asset ratios.

The remaining testable implication of the Collins-Karp model is that older farmers require more equity to keep them from retiring. This minimum equity level was observed from records of farmers who voluntarily quit farming. A sample of such farmers was created by selecting records of farmers with positive equity and good health who indicated they were voluntarily quitting farming. Records of bankrupt farmers and those who indicated they were quitting because of health problems were eliminated from the sample. The level of equity of these farms (n = 36) was regressed against the farmer's age. The estimated equation was:

E = 176,403 +	7,307.6AGE,
	(1.87)

R² = .0907.

Again, it is clear that there are many (hopefully orthogonal) independent variables omitted from this equation and, therefore, the R² = .0907 is not surprising. The age coefficient is significant at the 3.5% level for a one-tailed test. Given the small sample, this provides at least a moderate level of empirical support for the hypothesis that older farmers require more equity to keep them from retiring.

Concluding Comments <top>

The simplicity of the Barry-Collins model is appealing, and the econometric evaluation by Moss et al. (1990) shows that behavior appears to be consistent with the data if expectations are formed by an autoregressive process. However, the basic structure of the model suggests that only factors that affect the anticipated return on farm assets should be associated with leverage choice. The Collins-Karp model, while less intuitively appealing, produces hypotheses that conflict with the implications of the Barry-Collins model, because it suggests that characteristics of the farmer will affect leverage choice, specifically age, wealth, and the opportunity cost of being in business. An econometric analysis of a sample of Arkansas farmers provides some evidence that these characteristics of the farmer are significant variables in explaining farm debt choice. Since the estimated effects of age, wealth, and opportunity cost are consistent with what the model suggests, there is also some empirical evidence to support the Collins-Karp model. The analysis also provides some support for the concavity assumption that is required by the Collins-Karp model. Since we have econometric results that are consistent with both models, further research would be useful in this area, but if it is verified that factors such as age, wealth, and opportunity cost are indeed important in explaining farm capital structure, then we should be suspicious of models of capital structure that are based on maximization of single-period expected utility.

References <top>

Barry, P.J., C.B. Baker, and L.R. Sanint. "Farmers' Credit Risks and Liquidity Management." Amer. J. Agr. Econ. 63(1981):216_27.

Collins, R.A. "Expected Utility, Debt-Equity Structure, and Risk Balancing." Amer. J. Agr. Econ. 67(August 1985):627_29.

"The Financial Condition of Arkansas Farm Survey Respondents by Farm Type and Location: 1986." Bulletin No. 910. Arkansas Agricultural Experiment Station, 1987.

Collins, R.A., and L.S. Karp. "Lifetime Leverage Choice for Proprietary Farmers in a Dynamic Stochastic Environment." J. Agr. and Resour. Econ. Vol. 18, no. 2(December 1993): 225_38.

Featherstone, A.M., C.B. Moss, T.G. Baker, and P.V. Preckel. "The Theoretical Effects of Farm Policies on Optimal Leverage and the Probability of Equity Losses." Amer. J. Agr. Econ. 70(1988):572_79.

Freund, R.J. "The Introduction of Risk into a Programming Model." Econometrica 24(1956):253_63.

Kuh, E., and J. Meyer. "Correlation and Regression Estimation when the Data Are Ratios." Econometrica 23(1955):400_16.

Merton, R.C. "Lifetime Portfolio Selection Under Uncertainty: The Continuous Time Case." Rev. Econ. and Statis. LI(1969):247_57.

Moss, C.B. "Financing Agriculture Under Changing Environmental Regulation: Discussion." Amer. J. Agr. Econ. 73(1991):1405_06.

Moss, C.B., S.A. Ford, and W.G. Boggess. "Capital Gains, Optimal Leverage, and the Probability of Equity Loss: A Theoretical Model." Agr. Fin. Rev. 49(1989):127_34.

Moss, C.B., J.S. Shonkwiler, and S.A. Ford. "A Risk Endogenous Model of Aggregate Agricultural Debt." Agr. Fin. Rev. 50(1990):73_79.

Ramirez, O. "A Stochastic Optimal Control Formulation on Risk Balancing in the Debt Choice Model: A Basis for Generalized Method of Moments Estimation of Risk Aversion Coefficients." Ph.D. dissertation, University of Florida, 1990.

Thraen, C.S., J.W. Hammond, and B.M. Buxton. "Estimating Components of Demand Elasticities from Cross-Sectional Data." Amer. J. Agr. Econ. 60(1978):674_77.

Turvey, C.G. "The Relationship Between Hedging with Futures and the Financing Function of Farm Management." Can. J. Agr. Econ. 37(1989):629_38.

Turvey, C.G., and T.G. Baker. "A Farm-Level Financial Analysis of Farmers' Use of Futures and Options Under Alternative Farm Programs." Amer. J. Agr. Econ. 72(1990):946_57.

. "Optimal Hedging Under Alternative Capital Structures and Risk Aversion." Can. J. Agr. Econ. 37(1989):135_43.

U.S. Department of Commerce, Bureau of the Census. "Geographic Area Series, Part 4: Arkansas State and County Data." Census of Agriculture, vol. 1. Washington, DC: 1982.

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