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Evaluation of Farm Investments: Biases in Net Present Value Estimates from Using Quasi-Deterministic Models in an Uncertain World Robert A. Collins and Claus-Hennig Hanf Collins is Naumes Family Professor, Institute of Agribusiness, Santa Clara University, Santa Clara, California; Hanf is professor, Department of Agricultural Economics, Christian-Albrechts-Universität, Kiel, Germany. The authors started this paper while they were sabbatical visitors at the Department of Agricultural and Resource Economics at the University of California, Berkeley. They gratefully acknowledge the sabbatical support provided by the Berkeley department. This article discusses the impact that some of the recent theoretical developments in capital budgeting have had on the practical evaluation of farm investments. Some well-accepted and widely taught methods have recently been shown to produce biased results in common circumstances. Most of these problems arise from the existence of risk in the planning environment, but they arise even when risk neutrality is assumed and not as a result of some complex utility argument. Some sources of bias discussed include the errors that can result from using average values of random variables, and biases resulting from neglecting consideration of the possible irreversibility of investment decisions and the option to postpone decisions until more information is available. Key words: capital budgeting, investment, option theory, project evaluation, risk. Article <top> The process for practical investment evaluation that has been universally taught in our college classrooms for decades, and is widely applied in farm management, agribusiness, and extension, has recently been shown to be fraught with some previously unknown perils. This methodology for project evaluation, which has near "sacred cow" status among applied economists, is quite simple. The usual suggestion is to calculate the net cash flow that will result in each period of the investment’s life after consideration of taxes, interactions with other investments, etc., and then discount these cash flows at a discount rate that reflects the overall (weighted average) cost of financing the firm. If the net present value (NPV) of the project is positive, the usual wisdom recommends acceptance of the project, and rejection if it is negative. This method has much appeal to practical people because a positive NPV shows that the cash flows from the project are more than adequate to pay the financing costs that the project requires. Robison and Barry provide a comprehensive coverage of standard NPV topics. Adjusting Uncertain Cash Flows <top> If all of the cash flows are certain, and the cost of financing is constant over the life of the investment, use of the conventional method is not controversial. In the perfect certainty case, the NPV criterion is essentially the condition for intertemporal arbitrage, and it would be difficult to find much scholarly or practical opposition to using it in practice. But when the cash flows are risky, the problem becomes incredibly complex, and there is little agreement about the proper method of analysis. Several methodologies are examined below. Using Risk-Adjusted Discount Rates to Adjust Uncertain Cash Flows <top> One school of thought advocates using risk-adjusted discount rates to compensate for the uncertainty of the cash flows. This involves the application of an opportunity cost concept which suggests that an investor should demand a rate of return on a proposed project that is equal to the return available on an alternative investment of equivalent risk. This implies that riskier projects should be discounted at higher discount rates and low-risk projects at low discount rates, and an approach similar to the capital asset pricing model is frequently suggested. This method is commonly recommended in textbooks for the corporate setting, and it has even been adapted for potential use in the noncorporate setting by Collins and Barry. However, an alternative school of thought, initially proposed by Robichek and Myers, is that it may be a mistake to make adjustments for risk through the discount rate. If risky cash flows are discounted at a higher discount rate, distant cash flows are severely affected by the adjustment while cash flows in the near term are hardly affected at all. In other words, this method implicitly assumes that the further a cash flow is in the future, the more severely it should be adjusted for risk. In many cases, it is reasonable to assume that the distant future is exponentially more risky than the near future, and in these instances the primary problem is deciding how big the risk adjustment should be, i.e., 1%, 5%, or 20%. In other cases, however, it may be possible that the distant future is no more uncertain than the near future. In these cases, using a risk-adjusted discount rate does qualitatively the wrong thing. It appears that either scenario could apply to the farm-management setting. Risk-adjusted discount rates do make appropriate alterations for risk when the more distant future is regarded as progressively more risky. This may fit the conditions of agriculture very well in many cases. However, it could be argued that the agricultural producer may actually have a better picture of the long-term outlook than of the immediate future. For example, next year the entire crop of a particular farmer might be totally destroyed by a freak storm or insect infestation, while growing conditions might be quite good in general. Therefore, in a particular year it is certainly possible to have low production accompanied by a low price. Yet, over the long term, a clearer picture may emerge. Given random effects of weather, etc., and the perfectly competitive nature of the industry, over the long term a producer with acceptable efficiency may expect a normal profit. In this situation one might argue that the near term is, in fact, riskier than the long term. If this is true, making adjustments for risk with a risk-adjusted discount rate does exactly the wrong thing. Using Certainty Equivalents to Adjust Uncertain Cash Flows <top> In cases where risk-adjusted discount rates are not appropriate, an alternative has been offered by Robichek and Myers. They show that risk can be accounted for in capital budgeting by finding the certainty equivalent value of each uncertain cash flow. The certainty equivalent of a risky cash flow is often defined as the minimum amount for which the investor would be willing to sell the right to receive the risky cash flow. Since these are equivalent to certain cash flows by definition, the authors recommend discounting them at a riskless discount rate. While this method may have theoretical appeal for some, it presents complications in practice. First, it is not clear how to estimate a certainty equivalent for each uncertain cash flow in a practical setting. In addition, the suggested discount rate for this method is less than the actual cost of financing, a feature which should be disturbing to a decision maker. The implication here is that it would be theoretically possible to accept a project which, even if cash flow expectations were realized, would not generate sufficient cash to pay the costs of financing. This, of course, makes it unacceptable as a general method for incorporating risk into the process of capital budgeting. Other Methods of Dealing with Uncertain Cash Flows <top> Other methods such as stochastic dynamic programming, optimal control theory, and simulation also have been proposed for dealing with specific risky investment decision settings (see Dreyfus and Law; Kamien and Schwartz for examples). While these methods seem to be appropriate for dealing with some specific problems, they do not appear to solve, in general, the problem of evaluating risky investment cash flows. Additionally, these methods are not feasible for the typical agricultural investor. Since neither scholars nor practitioners have agreed on an acceptable method of formally including risk in capital budgeting, there is no universally accepted way to actually evaluate risky cash flows in a practical setting. But since there is nearly universal agreement that the NPV criterion is the ideal tool in the inter-temporal arbitrage condition of perfect certainty, many practitioners find it reasonable to apply this tool in an approximate way to risky cases also. As a result, many practitioners accept the practice of ignoring risk altogether and calculating an expected NPV by simply replacing random variables with estimates of their expected values. Further, perfect capital markets are often assumed to exist, resulting in a zero acquisition-salvage differential of asset prices (Johnson). Many objections against these simplifying methods recently have been raised. One objection is that the risk-neutrality assumption is inconsistent with the empirical evidence that farmers behave in a risk-averse manner (e.g., Robison, Barry, Kliebenstein, and Patrick; Smidts). But even if the assumption of risk neutrality is accepted, simple approaches that base calculations solely on the expected values of the stochastic variables may result in seriously biased estimates of the expected net present value. Another problem is that deviations of stochastic variables from their expected value may cause asymmetric impacts on the outcome variable (Hanf 1986). A further objection to the simple application of the NPV rule is that the possible irreversibility of initial investment expenses is frequently ignored (Dixit and Pindyck; Brandes and Odening). This article elucidates some of the problems that may result from neglecting uncertainty in the evaluation of risky farm investments even when risk neutrality is assumed. It will be demonstrated that improper evaluation of risks can substantially bias estimates of NPV and lead to serious decision errors in practical settings. While many of these errors are discussed in the theoretically or methodologically oriented literature, they continue to be largely ignored in business, extension, and teaching. Sources of Bias in Practical NPV Estimation <top> The estimation of cash flows from farm investments generally is based on relatively simple models that ignore the fact that many of the numbers are uncertain. The usual practice is to just replace the stochastic variables with their expected values (quasi-deterministic models). We show below that this is often not a good idea in practice because it will frequently produce a significant bias in the estimate of expected NPV. Since there is no reason to expect that competing projects will have proportional amounts of bias, these problems also could alter the preference ordering of potential projects.
Replacing Random Variables with Their Expectations <top> The primary reason random variables should not simply be replaced by their expected values in cash flow estimation is because they frequently occur in multiplicative form. In this case, replacing random variables with their expected values is clearly incorrect. For example, if a proposed project has an expected output of 200 units per period at an expected price of $15, is it reasonable to assume that average revenue will be average price times average output, or $3,000? The answer is absolutely not! To illustrate, suppose there are only three states of nature, and the probabilities of the various output and price levels are as shown in Table 1. In this case, the expected price is $15 and the expected quantity is 200, but the expected revenue is not $3,000. The expected revenue is 0.3($2,000) + 0.4($3,000) + 0.3($3,000) = $2,700. Why is expected revenue $2,700 when expected price times expected quantity is $3,000? In general, the expectation of a product of two random variables is not equal to the product of their expected values. When X and Y are random variables, E denotes expected value, and COV is covariance. E(XY) = E(X)E(Y) + COV(X,Y). (In this case the covariance is -300.) Therefore, substituting expected values for products of random variables is correct only when the two random variables are statistically independent. In any case where two random variables are inversely related, a product of their expected values will always overstate the result. If two random variables are positively correlated, however, simply multiplying their expected values would understate the expected value of the product, so the bias could go either way. The simple solution to this problem is to be certain, whenever products of random variables occur in capital budgeting, that one forecasts the product (i.e., revenue in this case) rather than forecasting the random variables individually and multiplying them together. Asymmetric Impacts of Deviations from Expected Value <top> In addition to the general problem of substituting expected values for random variables, the case can be even more confusing when deviations of stochastic variables from their expected value have asymmetric impacts on the decision variable. In these cases, it can be complicated to apply the conventional methods correctly. When the deviations of random variables have asymmetric impacts, replacing them with their expected values may cause bias even when they are not in multiplicative form. Ignoring these effects can cause the value of a project to be either understated or overstated depending on the particular case. Some common situations in agriculture that produce these effects are transactions costs, temporary capacity shortages, and costs originating from the adjustment to new production portfolios. Transactions costs often cause deviations of random variables to have asymmetric impacts in agricultural capital budgeting. For example, transactions costs cause the sale prices of agricultural commodities to differ from the purchase prices. Yield fluctuations due to weather variation may lead a given farm to be a net seller of a crop in average weather and a net buyer in bad weather. This is particularly relevant in less developed countries where many farms are operating near the self-sufficiency level (De Janvry and Sadoulet). For example, producers of both livestock and feed grain may enter the feed grain market either as sellers or buyers, depending on their own harvest. Another example of asymmetric impacts is dealing with temporary shortages of capacities. Many production constraints are affected by stochastic variables, or the available capacity is itself stochastic. If such a constraint is fully binding under average conditions, negative deviations from the expected value imply that the available capacity will temporarily fall short of the required quantity. This temporary shortfall must be equalized by buying additional capacities or by respective short-run adjustments in production. Both possibilities to eliminate the nonfeasible situation result in substantial increases in costs, whereas a random surplus with respect to a production constraint usually will result in a much smaller or no increase in revenues. A typical example is the stochastic annual capacity of a harvest combine. If, due to bad weather conditions, the combine can harvest only a portion of the acreage, the remaining area must be harvested by hiring a contractor at a price which is likely higher than the cost used in the farm budget. If the combine has excess capacity because of good weather conditions, it likely will have no value since all neighboring farms probably have surplus capacity for the same reason. Hence, the contribution of a random capacity surplus to the cash flow is zero, whereas random capacity shortfalls are costly. When the impact of temporary capacity shortfalls on returns in North German farming is taken into consideration, optimal machinery capacity is 30% greater than that suggested from calculations based solely on expected values (Peters; Goetzke; Hanf 1985). Adjustment costs and learning by doing also asymmetrically affect expected outcome, as these effects only occur in the case when a new production activity is incorporated into the farm’s production portfolio—not if an activity is taken out. Changing the production portfolio usually results in adjustment costs, and it should be recognized that mastery of new production processes will require some time. The industrial organization literature attaches a good deal of importance to the phenomenon of learning by doing, and considers a firm’s time of experience with a technology as a key determinant of firm profitability (e.g., Carlton and Perloff, p. 407; Spence). For example, Noell and Diers analyze the net profit development of farms changing from conventional to organic production. The switch results in temporary net revenue losses of up to 20% during a four- to five-year period due to internal adjustment problems. While it is a simple matter to incorporate these startup costs and learning effects, it is important to remember to do so. Evaluating Irreversible Investment Expenditures <top> Most investment expenditures are partially or even completely irreversible. Since capital markets are not perfect, the initial cost of purchasing an investment project is virtually always more than its liquidation value. Further, in many cases, one can choose to undertake an investment immediately or put off the decision until a later period when there is better information about the payoff. In other words, decision makers not only may have to choose between the two alternatives— "invest" or "don’t invest"—as is generally assumed in investment models, but in many cases they are confronted with a third alternative—"wait and then decide." This third alternative is often neglected in investment analysis. The potential value of postponing decision making is not new, and can be found in many operations research textbooks since the 1950s. Flexible planning, or flexible dynamic programming, are the terms used to describe the implementation of "decision postponement" in operations research procedures. However, the notion of applying the theory of option pricing as a method of placing a value on the alternative of waiting is customarily attributed to McDonald and Siegel, and to Pindyck. Dixit and Pindyck argue that consideration of the possibility of waiting to invest may have an important impact on investment decisions. If an investment decision is irreversible, once the decision to invest is made, the option of making the decision at a later time is foregone. If additional information about the payoff of the project is expected in the future, this option of waiting to decide has value. Therefore, one of the costs of proceeding with a project now is the value of the foregone option of waiting for better information. While most of the literature on this subject involves very complicated mathematics, the basic ideas are fairly straightforward and clearly apply to practical decision making. A Numerical Example <top> The following simple example given by Dixit and Pindyck (p. 27) demonstrates these concepts. It shows that an investment which has a positive expected NPV when evaluated by the conventional method has a negative NPV when the option value of waiting for more information is considered. The key assumptions are that (a) the investment expenditure is irreversible, and (b) more information about the potential profitability of the investment can be gained by waiting. These assumptions are frequently appropriate in practical settings. The decision maker is assumed to be risk neutral, and the facility being considered has a resale value equal to only the removal cost if the project is abandoned, and the facility cannot be converted to produce any other product. In order to construct an uncomplicated example, the following simplifications are also assumed: (a) the investment will produce exactly one unit of output per year, (b) the utilization horizon is infinite, (c) there are no operation or repair costs, (d) the initial investment cost is $1,600, and (e) the risk-free rate of interest is 10%. The price of the product is known to be $200 for the first period. To present a simple illustration where waiting provides information about the payoffs of the project, it is assumed that there are changes on the horizon for this product, and a price change is expected to occur in the second period. The current evaluation of the future price of this product is that it will go up or down by $100 in period 2 with equal likelihood, but it is not known at the present moment which way the price will move. Therefore, it is assumed that the product price will change in period 2 to either $100 or $300, and then remain constant at that value for all subsequent periods. The probability that the product price will assume either one of these two values is 0.5. While this example probably does not precisely fit any real-world situation, it does give a simple illustration of the value of the option of waiting to decide. It is clear that waiting for more information often has value in practical investment evaluation. Application of the traditional expected NPV model to this example results in a positive expected net present value, which is usually interpreted as suggesting that the investment should be undertaken. The expected price is $200, and the stream of expected revenue is a perpetual annuity that Dixit and Pindyck assume begins immediately when the investment is made. The present value of this expected stream is the immediate payment of $200, plus the present value of the perpetual annuity beginning one year from now which has an expected present value of $200/0.1 = $2,000, for a total expected value of revenue of $2,200. Therefore, the expected NPV of the investment is: E(NPV) = -$1,600 + $2,200 = $600. Note that the net present value of the project, assuming that the expected price occurs, is the same as the expected value of the two possible cases of NPV that could potentially occur. If the price in period 2 and subsequent periods turns out to be $300, the perpetual annuity of revenues from period 2 on is $3,000, and the expected NPV of the project is: E(NPV|P2 = 300) = -$1,600 + $200 + $3,000 = $1,600. If the future price turns out to be $100, the annuity value is $1,000, and the expected NPV is: E(NPV|P2 = 100) = -$1,600 + $200 + $1,000 = -$400. Since these NPVs are equally likely, the expected NPV is: E(NPV) = 0.5*E(NPV|P2 = 300) + 0.5*E(NPV|P2 = 100) = $600, the same as above. The conventional wisdom of using the expected NPV suggests that we should go ahead with this investment since it would increase our expected wealth. But this is not correct if the investment expenditure is irreversible, even if the investor is risk neutral. If we do proceed with this investment, we forego the possibility of waiting to see what will happen to the price. However, the calculations above show that if the future price is known, the NPV of the project is positive only if the realized price in the second period is $300—so waiting to make the decision clearly has value. If we have the possibility of choosing to wait before making the decision, the value of this choice can be regarded as an option. If we can wait to make the decision, we have the "option" of paying the cost of the project (the strike price) in order to receive the present value of the returns, which is a random variable (analogous to the stock price). By deciding to invest now, we give up the option of doing so later, and the value of this option that is foregone must be subtracted from the expected NPV of the project. The Black-Scholes value of any option is the present value of the expected value of the option at its expiration. The option to wait has two possible values in this case. If, in one year, the price turns out to be $100, the NPV of the project at that time would be: NPV = -$1,600 + $1,100 = -$500. In this case, the option would have no value and would not be exercised. If, however, the price turns out to be $300, the value of the option would be: NPV = -$1,600 + $3,300 = $1,700. Therefore, the expected value of the option at expiration is: 0.5(0) + 0.5($1,700) = $850. Discounting this back to the present, $850/1.1 = $772.73, the amount one would pay for being able to wait to make the decision. If the decision is made to proceed with the investment now, the expected present value of the returns is $600, but we give up the option of waiting until next year, so we must subtract the value of this option: $600 - $773 = -$173. Thus, the value of the option that is destroyed by going ahead now exceeds the value of the project, which means we should not go ahead with this project if the investment expenditure is irreversible. Waiting as an Alternative for Farm Investment Decisions <top> The previous simple example illustrates that waiting and postponing investments may result in higher or lower expected returns to investment capital, and the value of waiting should be considered before adopting a capital project. For this possibility to exist, however, four conditions must be satisfied: (a) the investment is at least partially irreversible, (b) postponing the decision is technically feasible, (c) there are potential outcomes of the random variable which could cause the optimal decision to be to not invest, and (d) the information acquired during the waiting period can be expected to improve the knowledge about the future. In the farming situation, two types of investment decisions are particularly affected by irreversibility: (a) investment in specialized farm buildings, and (b) investment in education, or human capital formation. Specialized farm buildings such as dairy barns and milking parlors cannot be easily utilized for other productive activities, and generally cannot be used on any farm other than the one on which they were originally constructed. Hence the market value of such buildings is very low if they are not used for their intended purpose. While there is some transferability of skills, human capital specific to farm management will have less value outside of agriculture. As a consequence, investments in farming abilities also have to be considered as partly sunk costs. The second condition—feasibility of postponing the decision—is usually fulfilled in the two types of investments mentioned above. While it is reasonable to assume that the third condition is ordinarily satisfied, the fourth condition, i.e., that waiting contributes positively to the assessment of the future, is not always met. Many of the forecasts used in the farming sector are based on econometric models assuming stationary processes, predominantly trend stationary processes. In the case of trend stationary processes, it is assumed that the actual values randomly fluctuate around a trend value. The observed values of one additional period therefore do not contribute to the assessment of the future, or only to a very marginal extent by a possible revision of the estimated statistical parameters of the underlying projection model. The value of an additional period of observation is much more important if the forecast is based on a difference stationary process or a Brownian motion process with drift (Cox and Miller). In this case, the probability distribution of future states strongly depends on the state occurring in any subsequent period. Hence, waiting may provide the possibility of improving the forecast of the future. Prices in a market economy are often difference stationary processes. In fact, Berck and Roberts (p. 8) argue that the empirical evidence suggests nearly all price series are difference stationary. In cases where the forecast of important variables is partly or fully determined by a discrete stochastic variable, waiting may considerably improve insight into future developments (Chavas). Such situations are particularly relevant to the agricultural sector where many economically important variables are strongly affected by policy decisions, which are viewed by farmers as probabilistic events. But, in any case where waiting improves the information available for making an investment decision, the option to postpone investment decisions should be carefully examined. Sunk Costs and Path Dependency <top> As discussed above, a specific feature of many (if not all) investments in capital goods is that at least a part of the cost of investment should be considered as a sunk cost. This is particularly true for investments in farm buildings. The market value of an unused dairy facility is almost zero, even if the building is new. This discrepancy between the depreciated value of the investment and its liquidation value has two effects with respect to the economic appraisal of investments. First, discontinuing production causes an equity loss not generally considered in simple corporate finance models which assume perfect capital markets. Second, and more importantly, the difference between the value of the capacity in production and the sale value causes the internal interest rate of supplementary investments to increase. As a consequence, the farm firm may be caught in a production path which is not profitable from an ex ante viewpoint. Brandes and Odening demonstrate with a simple example that such a lock-in situation might result if two investments dependent on one another are undertaken at different times and have to be repeated in equal cycles. In this case, an alternating path dependency may occur. Path dependency may become so strong that a path must be followed ad infinitum even though the path is not profitable (Arthur). Balman shows that the dominance and persistence of relatively small farms in Germany may be explained by path dependency, as a result of investment in "sunk cost burdened" assets. Johnson and Quance present an early example of the analysis of path dependency in agriculture—in this case the "overproduction trap" of U.S. agriculture. Considering the possible negative effects of path dependency resulting from the acquisition-salvage differential of asset prices (Johnson), the possibility of being "locked in" should be accounted for in investment appraisal. In principle, dynamic stochastic programming allows for incorporation of this risk, providing that reasonable probabilities can be defined (Kuehl, p. 83). However, this incorporation requires a dramatic increase in model size. Simulation experiments might provide a more tractable alternative for approximating these effects (Balman; Lentz). Concluding Remarks <top> This article discusses possible biases of the decision criterion that is commonly suggested for the appraisal of farm investment decisions. We argue that the widely taught and commonly accepted method of analyzing investment projects may result in considerable bias and incorrect decisions. Most of these biases occur from attempting to use a model intended for the certainty case in an approximate way when there is uncertainty. The following practical suggestions for analysts arise from this discussion: When random variables occur in multiplicative form, estimate the mean of the product rather than multiplying the estimates of the individual means together.
References <top>
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