### Introduction

One of the primary responsibilities of a management team is to make decisions during the execution of projects so that gains are maximized and losses are minimized. This is especially important for projects with built-in flexibility, such as those with options to expand operations in response to positive market conditions, to abandon an asset that is underperforming, to defer investment for a period of time, to suspend operations temporarily, to switch inputs or outputs, to reduce operational scale, or to resume operations after a temporary shutdown. By merging decision analysis with the well-known principles used in valuing options on financially traded assets, we can quantify the potential value associated with these types of options on real assets.

### Asset Valuation

For quite some time, discounted cash flow methods (DCF) have been the primary approach used by practitioners for the valuation of projects and for decision making regarding investments in real assets. With the DCF approach, the net present value of a project is calculated by discounting future expected cash flows at a given discount rate. As an example, consider a simple three-period project for which a pro forma cash flow sheet is shown in Figure 1.

This example might be representative of a typical industrial manufacturing application with a three-year production planning cycle under a forecasted market price environment. In the pro forma, the production and price forecast in each period translate to revenue, which can then be netted of production costs to arrive at the expected cash flow in each period. The cash flows should then be discounted at a rate that is commensurate with the riskiness of the project. In practice, this discount rate is often the weighted average cost of capital for the firm (WACC), based on the assumption that both the firm and the project have the same risk level. While this assumption may be valid for projects that mimic the risks associated with the firm as a whole, it may not be appropriate for unusual or innovative investment projects. In such cases, the practitioner must exercise judgment in choosing an appropriate discount rate for the project.[1]

Unfortunately, this approach ignores the significant incremental value that can be derived from management’s response to conditions in the future. For example, if the product unit price in future periods increases or decreases significantly, relative to the expected prices used in Figure 1, it seems untenable to assume that the firm’s management would fail to respond to such a change. If the firm indeed has the flexibility, we might instead expect management to revise the production level accordingly. Thus, we could have different revenues and cash flows than the ones shown in Figure 1, and the resulting present value would change as well.

An approach that treats future decision-making opportunities as options can account for this value, but because these more advanced methods are less familiar to many managers, their widespread use has been slow to arrive in practice. In recent work, Copeland and Antikarov,[2] Copeland and Tufano,[3] and others have sought to increase the application of more advanced valuation approaches by introducing computational methods that are more accessible to practitioners. In this article, we discuss how this work can be further enhanced by applying an intuitive approach based on familiar concepts from the field of decision analysis.

### Option Pricing

Option pricing methods were first developed to value financial options. However, the potential application of these methods to the valuation of options on real assets was soon identified, and given the moniker “real options.” Although hundreds of scholarly papers have been written on this topic, the complex mathematics required for option pricing techniques have unfortunately limited the appeal of these topics for many practitioners. A study conducted earlier this decade indicated that, while DCF valuation methods were used by over three-quarters of corporate finance practitioners surveyed, only about one-quarter used a real options approach.[4]

Unlike the case with DCF analysis, in an option pricing approach, we do not assume deterministic (certain) expected values for the relevant asset or project uncertainty, and must therefore model how the value evolves over time. There are several different types of mathematical models, called stochastic processes, which have been developed for this purpose. To simplify the analysis of option valuation problems, we typically work with a discrete approximation of the selected stochastic process. A discrete model contains a limited number of outcomes for the uncertainty at regularly spaced intervals in time, rather than a continuous distribution of outcomes for all points in time. This way, the firm need only make decisions at the discrete points to optimally respond to the uncertainty as it evolves. These discrete models have been shown to closely approximate the exact solutions derived using stochastic calculus, without the need for advanced mathematics.

An early example of this type of discrete approach was a binomial lattice model developed by Cox, Ross, and Rubinstein[5] to value options to buy or sell financial instruments, such as stock. This model consists of a binomial lattice, which depicts two possible changes in value for a stock in each time period; a move up by a factor *u* or a move down by a factor *d*. An example of this type of binomial lattice is shown in Figure 2, where *S* is the current market price of the asset, *q* is the probability of an upward move, *u* is a factor greater than 1, and *d* is the reciprocal of *u*.

To find the present value with options with such a lattice, we start from the final time period and work backward through time, finding the value from exercise or deferral of the option at each node in each period until we arrive back to the starting point (time zero). At nodes where the value has gone up, the optimal decision for a call option (option to buy the stock), for example, would be to exercise, while at nodes where the value has gone down, the optimal decision would be not to exercise. The opposite policies would generally be true for a put option (option to sell the stock).

Note that we must accurately assess the level of risk associated with the option exercise decision at each node because it dictates how much future cash flows (option payoffs) should be discounted during the backward induction process. This presents a challenge because the risk level is not constant, but is instead specific to each node in the lattice. Option pricing theory provides us with different methods for addressing this problem, and in the next section we will discuss one such method that can be applied in a decision-tree framework.

### Applying Decision Trees to Solve for Option Value

We can construct a binomial tree that is equivalent to the binomial lattice in Figure 2, with the only difference being that branches do not recombine in the binomial tree. Therefore, the multiple paths that lead to the four possible outcomes in Figure 2 are all explicitly shown in Figure 3.

With this type of tree, we can then model decision making about options in discrete time with decision nodes in the manner of standard decision tree analysis (DTA) familiar to many practitioners. Nau and McCardle[6] and Smith and Nau[7] studied the connection between DTA and standard lattice-based option pricing methods and demonstrated that the two approaches yield the same results, as long as the risk level is correctly specified throughout the tree in the DTA approach.

To adjust for the risk level in the DTA approach, we use a different set of transformed probabilities, *p* and 1-*p*, for the up and down outcomes at each chance node, respectively. These are the probabilities that a risk-neutral investor would assign to the two outcomes, therefore they are often called “risk-neutral” probabilities. The value obtained from solving a decision tree that is transformed with risk-neutral probabilities can be interpreted as the value that a rational risk-neutral investor would assign to the project. Under such risk-neutral conditions, the need for estimating the risk level at any point in the tree is eliminated, and we can simply discount all cash flows at the risk-free discount rate.

There are several different ways to estimate the up and down movements and risk-neutral probabilities, all of which incorporate information about the uncertainty, or “volatility” of outcomes associated with the project. Perhaps the most common method is to follow the convention used by Cox, Ross, and Rubinstein, in which the up and down movements at each step are *u = e ^{σ√Δt}* and

*d = 1/u*, respectively, where

*σ*is the volatility of asset returns per time increment in the tree and

*Δt*is the length of the time increment. Once

*u*and

*d*have been determined, the probability for an up move at each node in the tree is then

*p = (1 + rΔt – d)/ (u-d)*

*, while the corresponding probability of a down move is simply 1-*

*p*. These values for

*u*,

*d*, and

*p*are based on the assumption that the value over time evolves according to a Geometric Brownian Motion (GBM), a common stochastic process for modeling financial values. Details associated with the binomial approximation of a GBM stochastic process can be found in Hull.[8]

We emphasize that only three parameters are needed to specify this discrete approximation: the estimate of the current deterministic value of this project (for the starting point of the tree), the estimated volatility of the returns from the project (for the up and down values in the tree), and the risk-free rate (for the probabilities in the tree).

### An Example

If an initial investment of $1 million is required to commence the project shown in Figure 1, the resulting NPV is $55,000, assuming the firm’s WACC or investment hurdle rate is 10 percent. From a deterministic DCF perspective, the expected future cash flows provide an internal rate of return of 13.7 percent on this investment. Since the project NPV is positive and the rate of return exceeds 10 percent, this indicates that the project is a good investment opportunity; however, there may also be many other projects competing for funding under the firm’s capital budget. Thus, it is important to obtain an accurate valuation of each project that includes all sources of value, including the managerial flexibility to optimize outcomes.

Suppose, for example, that instead of being locked in to the forecasted production levels shown in the pro forma cash flow sheet, the firm can expand production in response to changes in the product unit price in years one and two if it so chooses. Specifically, we assume the firm has the option to increase production by 20 percent after year one at a cost of $160,000, and after year two at a cost of $62,500. From the real options perspective, these investment opportunities are analogous to two independent call options on an incremental 20 percent increase in production capacity. We assume that the optional investments at the end of years one and two will only be exercised if they are justified by the price and the estimates of the remaining project value at those points in time, and thus these investments can only add to the project’s deterministic NPV.

To value the expansion options, as suggested by Copeland and Antikarov, we use the present value of the project without options as the underlying asset for the options. We already have the beginning value given in Figure 1 ($1.055 million), and therefore need only to estimate the volatility in order to construct a discrete stochastic model of project value. It is typically not possible to estimate the volatility for real assets using market information; however we can instead simulate the cash flow pro forma sheet to generate a set of synthetic returns for the project, entering the project uncertainties as random variables, rather than deterministic expected values. In this example, we would enter random variables in each period in the Price row in Figure 1 (using functions from simulation software applications such as @RISK™ or Crystal Ball™). Then the return from period 0 to period 1, for example, can be calculated by dividing the present value in period 1 (currently shown as a fixed value of $1.161 million in Figure 1) by the present value in period 0 (fixed value of $1.055 million) and taking the logarithm of this ratio. When the spreadsheet is simulated for a large number (>1,000) of iterations, the different random prices produced in each iteration yield a probability distribution, including a mean and standard deviation, for the return (instead of a single fixed value). The volatility *σ* of the plant’s present value is then equal to the standard deviation of the returns. In many cases, the volatility will change from period to period, so the simulation should include an output for the return in each period, not just the return from period 0 to period 1.[9]

For this illustration, we assume that a simulation of the cash flow pro forma sheet has provided us with a volatility estimate of 30 percent for all periods and also that the risk-free discount rate *r* is 5 percent per year. We will model the project in one-year time increments, therefore *Δt* =1, and we have *u* = e^{0.30√1}=1.35, *d* = 1/1.35=0.74, and *p* = 0.51 as the parameters needed to construct a binomial model for project value.

The resulting three-period (T1, T2, T3) decision tree for the project value, without options and starting from a value of $1.055 million at *t* = 0, is shown in Figure 4. The values displayed above and below each branch in the tree are the discounted present value and cash flow, respectively. For example, the value shown above the up branch of T1 ($1.354 million) is $1.055 million multiplied by *u* (1.35) and discounted at 5 percent, while the value below the branch ($676) is the value above the branch multiplied by the cash flow ratio for period 1 (0.5, as shown in Figure 1). Figure 4 shows that the tree without options can be “rolled back” to verify its starting value.

Next, the real options in the project can be modeled simply by adding decision nodes to the tree shown in Figure 4. Specifically, we insert nodes after time periods one (Opt1) and two (Opt2) for the decisions about whether to expand production. The solution to the tree with these decision nodes added is shown in Figure 5, which indicates that the expected present value of the project with options is $1.105 million, which increases the NPV to $105,000.

We also note that the optimal decision policy is obvious from the graphic view of the solved decision tree, whereas it must be inferred from a binomial lattice representation. Notice, for example, that the production should be expanded if the expected value of the project moves up during the first time period. Additionally, we can see that the only case where the production should not be expanded after the second period is when the project value has decreased in both periods one and two.

### Conclusions

This example shows how an approach using decision-analysis methods provides a straightforward yet flexible way to apply option-valuation techniques. The solution shown in Figure 5 was obtained using the software application DPLâ„¢, but the basic approach can be implemented using virtually any commercially available decision-analysis package. We refer the interested reader to Brandao, Dyer, and Hahn[10],[11] and Smith[12] for more details and other examples of the application of this valuation approach. We believe that decision-analysis techniques provide managers with more intuition for solutions to valuation problems, and ultimately will lead to more utilization of advanced valuation methods. This will be critical in an increasingly competitive business environment where the ability to accurately assess project and asset values, including the incremental value related to a project’s embedded options, will heavily influence difficult investment portfolio decisions.

[1] Grinblatt, M. and S. Titman, *Financial Markets and Corporate Strategy*, (New York: Irwin/McGraw-Hill, 2^{nd} Edition, 2001).

[2] Copeland, T. and V. Antikarov, *Real Options,* (New York: Texere LLC, 2003).

[3] Copeland, T. and P. Tufano, “A Real-World Way to Manage Real Options,” *Harvard Business Review*, 82 No. 3 (2004): 90-99.

[4] Graham, J. and H. Campbell, “Theory and Practice of Corporate Finance: Evidence from the Field,” *Journal of Financial Economics*, 60 (2001): 187-243.

[5] Cox, J., S. Ross, and M. Rubinstein, “Option Pricing: A Simplified Approach,” *Journal of Financial Economics,* 7 (1979): 229-263.

[6] Nau, R. and K. McCardle, “Arbitrage, Rationality and Equilibrium,” *Theory and Decision*, 33 (1991): 199-240.

[7] Smith, J. and R. Nau, “Valuing Risky Projects: Option Pricing Theory and Decision Analysis,” *Management Science*, 14 No. 5 (1995): 795-816.

[8] Hull, J., *Options, Futures and Other Derivatives*, (New Jersey: Prentice Hall, 2003).

[9] See Brandao, Dyer and Hahn (2005b) for a detailed discussion of simulating pro forma cash flow sheets to obtain volatility estimates for real assets with uncertain variables.

[10] Brandao, L., J. Dyer, and J. Hahn, “Using Binomial Decision Trees to Solve Real-Option Valuation Problems,” *Decision Analysis,* 2 (2005b): 69-88.

[11] Brandao, L., J. Dyer, and J. Hahn, “Response to Comments on Brandao, et al (2005),” *Decision Analysis,* 2 (2005a): 103-105.

[12] Smith, J., “Alternative Approaches for Solving Real Options Problems,” *Decision Analysis,* 2 (2005): 89-102.