A better understanding of why NPV undervalues managerial flexibility

A better understanding of why NPV undervalues managerial flexibility

Feinstein, Steven P


The Real Options paradigm addresses the valuation of managerial flexibility in capital budgeting. Despite the great strides achieved by researchers in this field, many financial analysts have chosen not to adopt this new paradigm due to a lack of comfort with the approach and the mathematical complexity of the valuation models. This article shows how some projects with real options can be valued using simple and familiar tools– discounting expected cash flows after adjusting the discount rate. Unless the discount rate is adjusted to account for the impact of real options on risk, a traditional net present value (NPV) analysis misses the value of flexibility. By narrowing the gulf between Real Options analysis and more familiar tools, the weighted average discount rate (WADR) approach introduced in this paper may help novices better understand the Real Options paradigm, which subsequently may gain the wider acceptance it deserves. Though the WADR approach is practical only for simple real options, comfort with the approach may encourage analysts to pursue more advanced and robust real option valuation techniques for more complex applications.


The Real Options paradigm is the application of options pricing theory to capital budgeting. Recent research has made valuable breakthroughs in identifying and pricing a variety of real options embedded in projects1. When managerial flexibility is present, a Real Options analysis is an improvement over the standard net present value (NPV) approach that relies on conventional discounted cash flow valuation. This is because the Real Options approach accounts for the flexibility in a project, whereas an NPV analysis ignores this additional source of value. Since flexibility has value, ignoring it may cause managers to undervalue projects and make serious capital budgeting mistakes.

Managers may reject projects that would create wealth, or accept projects that are sub-optimal. They might make the mistake of deferring a value-creating project and thereby lose out if competitors gain a “first mover” advantage. If firms do act when really they should wait, or vice-versa, the error in timing could be costly in today’s marketplace. If they fail to recognize and value strategic flexibility, managers may choose the wrong project design when one design is cheaper but rigid and an alternative is more costly initially but flexible. By capturing the value of flexibility, the Real Options paradigm allows managers to better value projects and assess project alternatives. Improved capital budgeting decisions allow for more efficient allocation of capital resources, benefiting both the firm and society as a whole.

Given that managerial flexibility can significantly add to project value, financial analysts and strategists need to appropriately account for real option value in their project analyses. But if they are not comfortable applying the more sophisticated option pricing procedures2, the practicality of the recent breakthroughs in real asset valuation will be limited. Consequently, researchers must also focus on designing user-friendly approaches to implementing Real Options analyses. This article aims to make a contribution toward that end. This article shows that for some simple projects, a Real Options analysis can be conducted in much the same manner as a traditional NPV analysis, as long as a modified discount rate is used for the period over which there is flexibility. The modified discount rate is a weighted average of the risk-free interest rate and the project discount rate that would normally be used in a conventional NPV analysis. Just as a weighted average cost of capital (WACC) captures a firm’s leverage, a weighted average discount rate (WADR) can capture a project’s flexibility. Just as the WACC must sometimes be found by iteration when it both determines and depends on the market value of a firm’s equity, a WADR can be solved in a similar iterative fashion. No mastery of option pricing formulas is required for this new tool. By narrowing the gulf between Real Options analysis and more familiar tools, the WADR approach may help novices better understand the Real Options paradigm.

It is no accident that the WADR method gives the same value for real options as do the more direct and well-known option pricing formulas. Modern option pricing methods are based on the concept of replication. If an option can be replicated, then it can be priced by valuing the replicating combination of assets. The most direct replication valuation approach is to identify and value the replicating portfolio. Another method, risk-neutral option valuation, arrives at the same valuation result by replacing the actual binomial probabilities with risk-neutral probabilities, and then discounting the option’s modified expected cash flows at the risk-free rate. A third method leaves the probabilities unchanged, but modifies the expected cash flows to reflect risk-neutral price dynamics. Still another method is to use the actual cash flows and probabilities, but to adjust the discount rate for the appropriate level of risk. All four methods yield identical results. To account for the value of a project’s real option using a discounted cash flow approach, one must either risk-adjust the cash flows, the probabilities of the cash flows, or the discount rate. The WADR method introduced in this article operationalizes the last approach.

The WADR approach may be computationally more cumbersome than the standard and more advanced option-pricing techniques, but it is intuitive and has the benefit of familiarity, and thus can serve as an entry-level approach. If practitioners are comfortable conducting an NPV analysis with a WACC, they should be equally comfortable conducting a Real Options analysis with a WADR. Once familiar and experienced with the fundamentals of the Real Options approach, financial analysts and strategists may be encouraged to pursue mastery of the more advanced option valuation procedures, and the Real Options paradigm may subsequently gain the wider acceptance it deserves.

A limitation of the WADR approach is that it works best for projects with one-period binomial real options-that is, cases where the flexibility lasts for one period and depends on a binomial event3. However, many projects can be realistically modeled as having one imminent flexible period followed by rigidity, and for such projects the WADR approach works well. Though the WADR approach can be generalized to more complex real option scenarios, either multi-period options or multiple embedded options, the resulting algorithm would become extremely cumbersome. Not only would the WADR change each period, but it would also be different at every binomial node within a period. The failure of the WADR approach in these applications, however, is of significant pedagogical value-it illustrates the need for the more sophisticated option valuation models that researchers have recently been developing. Essentially, financial analysts can use the WADR approach for the simplest types of real options, but when faced with more complicated projects, the need for more advanced pricing procedures becomes starkly apparent. When the WADR approach does not work because the discount rate keeps changing or because multiple embedded options interact, it highlights impressively the value of the contributions of the researchers on the cutting edge of this field today.


The following example involves a construction project with an abandonment option. The example is simple and straightforward and is useful for comparing a conventional NPV analysis, a direct Real Options analysis, and the more userfriendly WADR approach. Although simple, this example sufficiently highlights the differences between the valuation models and illustrates the nature of the possible errors that can occur from not correctly accounting for the value of flexibility.

A developer is contemplating the construction of a new hotel. To get the project underway, $1 million must be spent immediately to search for a site, to conduct market research, and to secure any necessary permits. In addition, this hotel investment is a now-or-never decision. That is, there is no option to delay the $1 million investment to wait and see how market conditions develop.

Two project designs are available; one design is rigid and the other is flexible. For both designs, building the hotel requires that an additional $6 million for construction costs be spent one year from today. In the rigid project, once the project is initiated it cannot be halted. Perhaps this rigidity stems from the nature of the building permit the developer obtained from a local municipality, or perhaps it stems from the contracts required to secure property, equipment, and labor. With the flexible project, however, the developer can evaluate market conditions next year prior to investing the additional $6 million. Information that might be of interest at that time, for example, would pertain to local business conditions, such as whether or not a competing hotel is scheduled to be built nearby. With the flexible project design, after evaluating market conditions next year, the developer can choose either to pay the additional $6 million and build the hotel or not to pay the additional construction costs and instead, abandon the project. If the developer chooses to abandon the project, he can let any permits already acquired expire unused and walk away, neither paying nor receiving any cash, and writing off the initial $1 million as a sunk cost.

If business conditions turn out to be favorable, the hotel will generate free cash flow of $2 million per year, forever, beginning in year-2. If business conditions turn out to be unfavorable, the perpetual free cash flow will be only $1 million per year. In this example, the probability of each outcome is 50%.

Now the questions are, “Does the flexible project have additional value above and beyond that of the rigid project?” and, if so, “Can we accurately capture that value in a capital budgeting project analysis?”


So now the question is, “What is the present value in year-0 of the flexible project?” It is not $2 million discounted at 20%. Valuing real options in this manner is common in both the literature and in practice, but is not correct. Though a rigid hotel project has the same systematic risk as an existing hotel of similar type, a flexible hotel project does not. The abandonment option significantly lowers the systematic risk of the flexible project. Only after the decision is made in year-1 does the flexible project become a rigid project and comparable in systematic risk to completed projects. It is, therefore, correct to discount any free cash flows after year-1 back to year-1 at the 20% rate, but discounting their value back to year-0 at the same 20% rate is incorrect. Between year-0 and year-1, the systematic risk of the flexible project is unlike that of the rigid project or similar type completed projects. So the appropriate discount rate must differ, it is not 20%, and, in fact, for this example it is less than 20%.

Binomial option pricing procedures can be applied to correctly value the flexible project. To do so, we construct a replicating portfolio that mimics the binomial behavior of the flexible project. If we can value the component assets in the replicating portfolio, and assuming the Law of One Price holds (meaning that identical goods must have identical values), we can value the replicated project, i.e., the flexible project. It turns out that, in this case, taking an 80% ownership stake in a hypothetical proposed rigid project and investing in oneyear risk-free Treasury bills with a face value of $800,000 is sufficient to replicate the flexible project. The contingent cash flows of this replicating portfolio are the same as the contingent cash flows of the flexible hotel project, as shown below.


The direct Real Options valuation presented requires facility and comfort with binomial option pricing. The WADR method is an alternative way to value the flexible project and the embedded abandonment option, and it is similar to NPV procedures commonly used. The WADR method rests on the fact that the flexible project can be accurately valued by discounting its expected future free cash flows, as long as the correct discount rate is used. As discussed above, because the project is no more flexible than the rigid project after year-1, the discount rate to bring the free cash flows back to year-1 is the same 20% discount rate of a rigid project, or a completed project of similar type. But the appropriate discount rate for year-0 to year-1 differs. Between year-0 and year1, the flexible project has the same risk as a combination of risk free bills and a stake in a rigid project, because over that interval the flexible project is identical to such a combination. Consequently, the discount rate for the flexible hotel project for that period is a weighted average of the risk-free rate and the rigid project’s discount rate.

The portfolio replicating the flexible project as of year-0 is akin to a $1 million (0.8 * $1.25 million) investment in a rigid project and a $761,905 ($800,000/1.05) investment in risk-free bills. The total value of the replicating portfolio in year-0 is thus $1,761,905. The weight of the stake in the rigid project in year-0 is $1 million divided by $1.76 million, or 0.568, and the weight of the bills is $0.76 divided by $1.76, or 0.432. The weighted average discount rate of the replicating portfolio for the interval from year-I back to year-0 is therefore (56.8% x 20%) plus (43.2% x 5%), which equals 13.51%. After year1, after the flexibility of the project disappears, the appropriate discount rate will be the same 20% that is appropriate for other rigid hotel projects. However, between year-0 and year-1, when there is flexibility, the appropriate discount rate is considerably lower, at 13.51%.

Without the flexibility, the discount rate is 20%. With the flexibility the discount rate is only 13.51 %. The flexibility in the project significantly reduces the project’s discount rate in the first year. Now valuing the flexible project by discounting the $2 million expected year-1 value back to year-0 at the 13.51% WADR gives a present value of $1.76 million. The WADR Real Options method finds the identical value for the flexible project as did the method of summing the values of the components of the replicating portfolio.


Once the WADR is known, valuing the flexible project proceeds in the same manner as a conventional NPV analysis. So if a project analyst knows that flexibility lowers the discount rate for the year-1 free cash flows to 13.51%, all she would need to do to correctly value the project with the embedded abandonment option is to discount the future free cash flows back to the present at the correct rate for each year. That is, she would discount the year-2 through infinity free cash flows to year-1 at the usual 20% rate and then discount the year-1 values one more year at the lower 13.51% rate.

In our example above, however, we computed the WADR only after first valuing the flexible project. But if the flexible project had already been valued, there would be no need to backtrack and determine the WADR. If there were some way to find the WADR before valuing the project, then one could value the project inclusive of its real option in a very familiar way – simply discounting the project’s cash flows at each year’s discount rate. What is needed is an algorithm to approximate the WADR first, without yet knowing the flexible project’s value.

Note that over the period in which the project has flexibility, the range of possible future values is narrower than the range of possible future values of the rigid project. The flexible project’s year-1 value ranges between $4 million and $0. The rigid project’s year-1 value ranges between $4 and -$1 million. Thus, the range of flexible project values is $4 million while the range of rigid project values is $5 million. It is this narrowing of the range of future values that adds safety to a flexible project, thereby lowering its discount rate.

The flexible project’s range is 80% ($4 million/$5 million) of the rigid project’s range. In binomial option pricing, this “range ratio” is known as the replicating portfolio’s delta. This delta corresponds to the percentage of the underlying security, in this case a rigid project, held in the replicating portfolio. That is, it is the same as N^sub s^ in the previous system of simultaneous equations. Since the flexible project must be worth more than the rigid project, an 80% stake in a rigid project must represent less than 80%10 of the value of the flexible project. This relationship provides an upper bound for the weight of the rigid project stake in the replicating portfolio-it must be less than 80%. From the section on WADR Real Option Analysis above, we know that the weight of the rigid project stake in the replicating portfolio is actually 56.8%. The other component of the replicating portfolio is the risk-free bills. If the rigid project stake comprises less than 80% of the replicating portfolio’s value, the investment in bills must comprise more than 20%. Again, we know it is actually 43.2%. Lastly, the rigid project’s discount rate must be greater than the discount rate of the risk-free bills4. This means that the WADR must be less than 80% of the rigid project’s discount rate (0.8 x 20%) plus 20% of the risk-free rate (0.2 x 5%). That is, the WADR must lie between the risk-free rate and the rigid project’s discount rate and be less than 17%.

Discounting the flexible project’s expected future value in year-1 ($2 million) at this 17% WADR upper bound produces a present value lower bound of $1.71 million. Even if the analyst were to stop here, this valuation would be closer to the $1.76 million correct value than the $1.67 million found using a traditional NPV analysis.

An even closer approximation, and even the exact value, can be found with an iterative algorithm. If the replicating portfolio holds an 80% stake in a rigid project, which is worth $1 million, but the flexible project is worth at least $1.71 million, then we now know that the rigid project weight must be less than $1/$1.71, or 58.5%. The risk-free bills’ weight must therefore be at least 41.5%. Applying these new approximations of the weights gives a WADR of 13.78%. Now valuing the flexible project at this 13.78% WADR produces a flexible project value of $1.76 million. Just one iteration results in a project value that is within 1% of the true value. Iterating repeatedly will eventually give the exact project value, and in turn the true value of the abandonment option. The steps of the iteration process are listed in EXHIBIT 2.

Iterating to find a model-consistent discount rate is not unfamiliar to project analysts. The same recursive procedure is commonly applied in an NPV analysis to find the WACC when leverage is unknown, because in such cases, the WACC both depends on and determines the equity value.



STEP 1: VALUE THE PROJECT AS IF IT HAD NO EMBEDDED OPTIONS. Follow the usual steps of a conventional NPV analysis. That is, project all free cash flows for years 1 and beyond, contingent on each of the binomial paths. Then compute the unconditional expected free cash flows assuming no flexibility. Finally, discount the unconditional expected free cash flows at the rigid project discount rate.


Determine whether or not the flexible project ought to be continued or abandoned in year-1, contingent on each binomial outcome. To determine this, first conduct a conventional NPV analysis as of year-1, for each of the two binomial outcomes. Then discount the projected free cash flows for years 1 and beyond back to year-1 at the rigid project discount rate, and sum them. If for a particular binomial outcome the year-1 project value is greater when continued than when abandoned, the correct choice is to continue. If abandonment gives a higher value, then abandonment is optimal. The year1 value of the flexible project for each outcome is the value maximizing choice.


Calculate the ratio of the range of potential year-1 flexible project values to the range of potential year-1 rigid project values.


Compute a WADR from the rigid project discount rate and the risk-free rate. Use the range ratio as the rigid project weight and 1 minus the range ratio as the risk-free bills weight.


Value the flexible project by discounting the flexible project’s expected year-1 value at the approximated WADR.


Multiply the range ratio by the rigid project value found in STEP 1. This gives the value of the rigid project stake in the replicating portfolio. Divide that value by the approximated flexible project value found in STEP 5. This quotient is a better estimate of the rigid project weight in the replicating portfolio. Subtract the new rigid project weight from I to better estimate the risk-free bills’ weight. If these improved weights are substantially different from the weights previously used in STEP 4, then return to STEP 4 to compute a new WADR. Stop iterating when the estimated weights, WADR, and flexible project value settle down to consistent values.


Research in the Real Options approach to capital budgeting has shown that, when managerial flexibility is present, a traditional NPV can noticeably undervalue even simple projects. Ignoring the value of flexibility may lead managers to make serious capital budgeting mistakes such as rejecting projects that would create wealth, implementing sub-optimal investment timing, or choosing the wrong project design.

The free cash flows in an NPV analysis can be easily modified to reflect dynamic decision-making. But a traditional NPV analysis will still use the rigid project’s discount rate, because the discount rate is not adjusted to reflect any operational flexibility. In the example presented, the investor has the ability to “walk away” if the information received in year-1 is “bad news”. This flexibility reduces the project’s systematic risk and so also reduces its discount rate, and therefore increases the project’s total value. Of course, the flexible project’s discount rate differs only during the period in which there is flexibility. Afterwards, the flexible project becomes a rigid project and comparable in systematic risk. Thus the discount rate reverts to the rigid project rate.

The WADR approach presented shows that a flexible project, complete with its real options, can be valued in a manner very similar to a traditional NPV analysis, as long as a WADR is applied. The WADR method uses an initial approximation and a recursive search algorithm to identify the risk-adjusted discount rate. A simple recursive algorithm for finding the WADR was given for the case of an abandonment option. The WADR approach is limited to oneperiod binomial options. More complex projects and options require the more advanced analytical tools developed to deal with greater complexity.

Of all the option pricing methods, the WADR method is most similar to tools already commonly used in project analysis, and consequently should be of interest to researchers and practitioners who wish to introduce the Real Options approach to a wider audience. It is hoped that by relating the Real Options paradigm to tools with which analysts are already familiar, the leap to the superior technology will not be a broad one, and the new paradigm will more quickly gain the widespread acceptance among financial analysts and strategists it deserves.


We are grateful for helpful comments and suggestions offered by Stein Frydenberg, Kathleen Revert, Nalin Kulatilaka, Allen Michel, Israel Shaked, Lenos Trigeorgis, participants at the 1999 Financial Management Association conference, participants at the 1999 Multinational Finance Society conference, and participants at the 27th Euro-Working Group on Financial Modeling conference.


1. A notable example for this paper is SACHDEVA, K., and P.A. VANDENBERG, 1993, “Valuing the Abandonment Option in Capital Budgeting – An Option Pricing Approach,” Financial Practice and Education (Fall), pp. 57-65. In

this paper, the authors value an abandonment option using the binomial option pricing model.

2 LANDER, D.M., and G.E. PINCHES, 1998, “Challenges to the Practical Implementation of Modeling and Valuing Real Options,” Quarterly Review of Economics and Finance, Volume 38, Special Edition, pp. 537-567.

3. The WADR method is used in this paper to value a project that has an abandonment option. The method, however, easily generalizes to oneperiod binomial expansion options, or options to start projects. This application is essentially the real options analogue to put-call parity. A project with an option to expand can be recast as a project designed to expand combined with the option to abandon the expansion. An option to initiate a new project can be viewed as a plan to undertake a new project plus the option to abandon it. In stock option pricing: C = P + S – PV (X), where C is the value of a call with strike price X, P is the value of a put with the same strike price, S is the value of the underlying stock, and PV (X) is the present value of the strike price. Similarly, the value of an expansion (or start) option is given by the same formula, where C is the value of the expansion option, S is the value of a rigid project committed to expand, P is the value of the option to abandon the expansion, and PV (X) is the present value of the cost of expanding. Consequently, if we can use the WADR approach to value a project that has an abandonment option, we can also use it to value a project that has an expansion option.

4. This holds true except for the unusual case in which the project has a negative beta.



University of Southern Maine


STEVEN P. FEINSTEIN is an associate professor of finance at Babson College in Wellesley, Massachusetts. He is also the director of the Stephen D. Cutler Investment Management Center at Babson College, and a consultant with the Michel/Shaked Group, a Boston-based financial consulting firm. Professor Feinstein holds a Ph.D. in economics from Yale University, a B.A. in economics from Pomona College, and the Chartered Financial Analyst designation granted by the Association for Investment Management and Research. Prior to entering academia, he served as an economist at the Federal Reserve Bank of Atlanta. His primary areas of research are financial valuation and the use and pricing of derivatives. He is a member of the American Finance Association, the Financial

Management Association, and is on the education committee of the Boston Security Analysts Society.

DiANE M. LANDER is an assistant professor of finance at the University of Southern Maine. She holds a B.S. degree from the University of California at Davis, an MBA from the University of North Texas, and a Ph.D. in finance from the University of Kansas. Before joining the University of Southern Maine, Professor Lander taught finance, general business, and mathematics at Collin County Community College, the University of North Texas, the University of Kansas, and Babson College. She taught data processing while a member of Electronic Data Systems’ corporate education division. She has also done extensive work as a programmer and systems programmer/analyst, managed a systems group, and owned and managed her own retail business. Professor Lander has gained practical perspective and insights into business situations from her corporate background and from the experience of owning her own business, which she applies to both her teaching and research. Her research focuses on the practical implementation of the Real Options approach. She is a member of the Financial Management Association, the Midwest Finance Association, the Northeast Decision Sciences Institute, the Academy of Business Education, and the Euro Working Group on Financial Modeling.

Copyright Institute of Industrial Engineers 2002

Provided by ProQuest Information and Learning Company. All rights Reserved