The Analogy between financial options and corporate investments that create future opportunities is both intuitively appealing and increasingly well accepted. But for many nonfinance managers, the journey from insight to action, from the puts and calls of financial options to actual investment decisions, is difficult and deeply frustrating.

Experts do a good job of explaining what option pricing captures that conventional discounted-cash-flow DCF and net-present-value NPV analyses do not. Moreover, simple option pricing for exchange-traded puts and calls is fairly straightforward, and many books present the basics lucidly. But at that point, most executives get stuck.

Their interest piqued, they want to know How can I use option pricing on my project? Unfortunately, how-to advice is scarce on this subject and mostly aimed at specialists, preferably with Ph. As a result, corporate analyses that generate real numbers have been rare, expensive, and hard to understand.

The framework presented here bridges the gap between the practicalities of real-world capital projects and the higher mathematics associated with formal option-pricing theory. It produces quantitative output, can be used repeatedly on many projects, and is compatible with the ubiquitous DCF spreadsheets that are at the heart of most corporate capital-budgeting systems. What this framework cannot supply is absolute precision: In such cases, forgoing some precision in exchange for simplicity, versatility, and explicability is a worthwhile trade.

Instead of looking only at the differences between the two approaches, we will also look for points of commonality. Recognizing the differences adds extra insight to the analysis, but exploiting the commonalities is the key to making the framework understandable and compatible with familiar techniques. In fact, most of the data the framework uses come from the DCF spreadsheets that managers routinely prepare to evaluate investment proposals.

And for option values, the framework uses the Black-Scholes option-pricing table instead of complex equations. A corporate investment opportunity is like a call option because the corporation has the right, but not the obligation, to acquire something—let us say, the operating assets of a new business. If we could find a call option sufficiently similar to the investment opportunity, the value of the option would tell us something about the value of the opportunity.

Unfortunately, most business opportunities are unique, so the likelihood of finding a similar option is low. The only reliable way to find a similar option is to construct one. By mapping the characteristics of the business opportunity onto the template of a call option, we can obtain a model of the project that combines its characteristics with the structure of a call option. The option we will use is a European call, which is the simplest of all options because it can be exercised on only one date, its expiration date.

Many projects involve spending money to buy or build a productive asset.

Summary of Real Options - Luehrman. Abstract

Spending money to exploit such a business opportunity is analogous to exercising an option on, for example, a share of stock. The present value of the asset built or acquired corresponds to the stock price S.

Finally, the time value of money is given in both cases by the risk-free rate of return r f. By pricing an option using values for these variables generated from our project, we learn more about the value of the project than a simple discounted-cash-flow analysis would tell us. Traditional DCF methods would assess this opportunity by computing its net present value.

NPV is the difference between how much the operating assets are worth their present value and how much they cost: When NPV is positive, the corporation will increase its own value by making the investment. When NPV is negative, the corporation is better off not making the investment. At that time, either. To reconcile the two completely, we need only observe that when NPV is negative, the corporation will not invest, so the project value is effectively zero just like the option value rather than negative.

In short, both approaches boil down to the same number and the same decision. When Are Conventional NPV and Option Value Identical?

Conventional NPV and option value are identical when the investment decision can no longer be deferred. This common ground between NPV and option value has great practical significance. It means that corporate spreadsheets set up to compute conventional NPV are highly relevant for option pricing. Any spreadsheet that computes NPV already contains the information necessary to compute S and X , which are two of the five option-pricing variables.

Accordingly, executives who want to begin using option pricing need not discard their current DCF-based systems. When do NPV and option pricing diverge? When the investment decision may be deferred. The possibility of deferral gives rise to two additional sources of value. First, we would always rather pay later than sooner, all else being equal, because we can earn the time value of money on the deferred expenditure.

Specifically, the value of the operating assets we intend to acquire may change. If their value goes down, we might decide not to acquire them.

That also is fine very good, in fact because, by waiting, we avoid making what would have turned out to be a poor investment. We have preserved the ability to participate in good outcomes and insulated ourselves from some bad ones. For both of these reasons, being able to defer the investment decision is valuable.

Traditional NPV misses the extra value associated with deferral because it assumes the decision cannot be put off. In contrast, option pricing presumes the ability to defer and provides a way to quantify the value of deferring. So to value the investment, we need to develop two new metrics that capture these extra sources of value. The first source of value is the interest you can earn on the required capital expenditure by investing later rather than sooner.

How much money is that? It is the discounted present value of the capital expenditure. To compute PV X , we discount X for the requisite number of periods t at the risk-free rate of return r f:. The extra value is the interest rate r f times X , compounded over however many time periods t are involved. Alternatively, it is the difference between X and PV X.

We have seen that NPV can be expressed in option notation as: Note that our modified NPV will be greater than or equal to regular NPV because it explicitly includes interest to be earned while we wait. It picks up one of the sources of value we are interested in.

Modified NPV, then, is the difference between S value and PV X cost adjusted for the time value of money. Modified NPV can be positive, negative, or zero. However, it will make our calculations a lot easier if we express the relationship between cost and value in such a way that the number can never be negative or zero.

S divided by PV X. By converting the difference to a ratio, all we are doing, essentially, is converting negative values to decimals between zero and one.

When modified NPV is positive, NPVq will be greater than one; when NPV is negative, NPVq will be less than one. Anytime modified NPV is zero, NPVq will be one. Substituting NPVq for NPV We can rank projects on a continuum according to values for NPVq, just as we would for NPV. When a decision can no longer be deferred, NPV and NPVq give identical investment decisions, but NPVq has some mathematical advantages.

That possibility is very important, but naturally it is more difficult to quantify because we are not actually sure that asset values will change or, if they do, what the future values will be.

Fortunately, rather than measuring added value directly, we can measure uncertainty instead and let an option-pricing model quantify the value associated with a given amount of uncertainty.

The only way to measure uncertainty is by assessing probabilities. How can we quantify this uncertainty? Perhaps the most obvious measure is simply the range of all possible values: But we can do better than that by taking into account the relative likelihood of values between those extremes. Variance is a summary measure of the likelihood of drawing a value far away from the average value in the urn.

The higher the variance, the more likely it is that the values drawn will be either much higher or much lower than average. In other words, we might say that high-variance assets are riskier than low-variance assets. We have to worry about a time dimension as well: For business projects, things can change a lot more if we wait two years than if we wait only two months.

So in option valuation, we speak in terms of variance per period. This sometimes is called cumulative variance. An option expiring in two years has twice the cumulative variance as an otherwise identical option expiring in one year, given the same variance per period. Cumulative variance is a good way to measure the uncertainty associated with business investments.

The probability distribution of possible values is usually quite asymmetric; value can increase greatly but cannot drop below zero. Returns, in contrast, can be positive or negative, sometimes symmetrically positive or negative, which makes their probability distribution easier to work with. Second, it helps to express uncertainty in terms of standard deviation rather than variance. Standard deviation is simply the square root of variance and is denoted by o. It tells us just as much about uncertainty as variance does, but it has the advantage of being denominated in the same units as the thing being measured.

In our business example, future asset values are denominated in units of currency—say, dollars—and returns are denominated in percentage points. Standard deviation, then, is likewise denominated in dollars or percentage points, whereas variance is denominated in squared dollars or squared percentage points, which are not intuitive. Since we are going to work with returns instead of values, our units will be percentage points instead of dollars.

Third, take the square root of cumulative variance to change units, expressing the metric as standard deviation rather than variance. They capture the extra sources of value associated with opportunities.

And they are composed of the five fundamental option-pricing variables onto which we mapped our business opportunity. NPVq is actually a combination of four of the five variables: Finally, each of the metrics has a natural business interpretation, which makes option-based analysis less opaque to non-finance executives.

Linking Our Metrics to the Black-Scholes Model Our two new metrics together contain all five variables in the Black-Scholes model. Combining five variables into two lets us locate opportunities in two-dimensional space. NPVq is on the horizontal axis, increasing from left to right. As NPVq rises, so does the value of the call option.

What causes higher values of NPVq? Higher project values S or lower capital expenditures X. Note further that NPVq also is higher whenever the present value of X is lower.

Higher interest rates r f or longer time to expiration t both lead to lower present values of X. Any of these changes lower X or higher S, r f , or t increases the value of a European call. Locating the Option Value in Two-Dimensional Space We can locate investment opportunities in this two-dimensional space. Cumulative volatility is on the vertical axis of the graph, increasing from top to bottom.

Plotting projects in this two-dimensional space creates a visual representation of their relative option values. No matter where you start in the graph, call value increases when you move down, to the right, or in both directions at once.

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Locating various projects in the space reveals their value relative to one another. How do we get absolute values? That is, how can we get a number? Having gotten this far, we find that getting a number is easy. For example, for a project whose NPVq equals 1.

Any European call option for which NPVq is 1. Option values in the table are expressed in relative terms, as percentages of S , rather than in absolute dollars, to enable us to use the same table for both big and small projects. The Black-Scholes model is used once, to generate the table itself.

The key is to remember that extra refers to a comparison between option value and net present value NPV , not to a comparison between option value and present value S.

In this example, we are not expecting the option value to be greater than S ; we are expecting it to be greater than NPV, which is S minus capital expenditures X. Since NPVq equals 1. Then for NPVq to equal 1. To illustrate how to apply the framework, consider this example of a hypothetical, but representative, capital investment. They plan to build a new, commercial-scale plant immediately to exploit innovations in process technology.

The initial investment is obviously strategic because it creates the opportunity for subsequent growth. This is important because the expenditures in the third year are large—three times the initial investment.

It takes practice to recognize the options that may be buried in conventional projects. However, there are at least two easy ways to see the option in our example. It surely says something about the two-phased nature of the program by way of justifying the large outlays in year 3.

The other is to examine the pattern of the project cash flows over time. The cash flows in the chart are very uneven: A graph of the capital-expenditures line would clearly show the spike in spending in year 3.

Such a large sum is almost surely discretionary. That is, the company can choose not to make the investment, based on how things look when the time comes. This is a classic expansion option, sometimes called a growth option.

The option here is a call option, owned by the company, with three years to expiration, that can be exercised by investing certain amounts in net working capital NWC and fixed assets. Viewing the project in this way, we want to evaluate the following: Phase 1 refers to the initial investment and the associated cash flows.

It can be valued using NPV as usual. Phase 2 refers to the opportunity to expand, which may or may not be exploited in year 3.

To value phase 2, we will use the framework outlined above to synthesize a comparable call option and then value it.

This mapping will create the synthetic option we need and indicate where in the DCF spreadsheet we need to go to obtain values for the variables. The value of the underlying assets S will be the present value of the assets acquired when and if the company exercises the option.

The exercise price X will be the expenditures required to acquire the phase 2 assets. The time to expiration t is three years, according to the projections given in the DCF analysis, although we might want to quiz the managers involved to determine whether the decision actually could be made sooner or later.

The three-year risk-free rate of interest r f is 5. Returns on broad-based U. For some businesses, we can estimate volatility using historical data on investment returns in the same or related industries. Alternatively, we might compute what is called implied volatility using current prices of options traded on organized exchanges. Today we can get implied volatility for shares of a very large number of companies in many industries. The quality and availability of such data have improved enormously in the past ten years.

Once you have the synthesized distribution, the computer can quickly calculate the corresponding standard deviation. These tools also have become far more widely available and much easier to use in recent years.

I generally find it easier to work on S and X first. That requires making a judgment about what spending is discretionary versus nondiscretionary or what spending is routine versus extraordinary. It also requires making a similar judgment about which cash inflows are associated with phase 1 as opposed to phase 2. In this project, expenditures on net working capital and fixed assets obviously are lumpy. The very large sums in year 3 clearly are discretionary and form part of the exercise price X.

The smaller sums in other years are plausibly routine and may be netted against phase 2 cash in-flows, ultimately to be discounted and form part of S , the value of the phase 2 assets. Sometimes it is easy to separate phase 1 cash flows from phase 2 cash flows because whoever prepared the DCF analysis built it up from detailed, phase-specific operating projections. At other times, we have to allocate cash flows to each phase.

A common expedient is simply to break out the phase 1 cash inflows and terminal value. Then, phase 2 cash inflows and terminal value are whatever is left over. Note that when we discount cash flows for the two phases separately, we obtain the same NPV as before. Already, we have a quantitative option-related insight. The DCF valuation contains a common mistake. Overdiscounting future discretionary spending leads to an optimistically biased estimate of NPV. Having reformulated the DCF spreadsheet, we can now pull values for S and X from it.

X is the amount the company will have to invest in net working capital and fixed assets capital expenditures in year 3 if it wants to proceed with the expansion: S is the present value of the new phase 2 operating assets. The other option-pricing variables have already been mentioned: If we discount phase 2 spending at 5.

Accordingly, the dollar value of the option is 0. Recall the value of the entire proposal is given by: Filling in the figures gives: Yet the option-pricing analysis uses the same inputs from the same spreadsheet as the conventional NPV.

What looks like a marginal-to-terrible project through a DCF lens is in fact a very attractive one. Few projects look so good. What should you do next? All the things you would usually do when evaluating a capital project. Check and update assumptions. Examine particularly interesting or threatening scenarios. Compare and interpret the analysis in light of other historical or contemporary investments and transactions.

They also include checking for clear disadvantages associated with deferring investment, such as competitive preemption, which would offset some or all of the sources of value associated with waiting. Some of these concerns can be handled by straightforward modifications to the framework.

Others require more sophisticated modeling than either this framework alone or conventional NPV generally can provide. Real corporate projects will present immediate challenges to some of the simplifications underlying this framework.

Can the framework be souped up to handle more complex problems? Or does its very simplicity present insurmountable limitations? When might it generate seriously misleading information?

Bells and whistles can be added to the framework fairly easily, although they require extra data, and the details are beyond the scope of this article. But what if they are not? The framework can be adapted to handle those circumstances, but the adaptation helps only if we can describe the uncertainty.

Specifically, we need to know the probability distribution of X and the joint probability distribution of S and X. That is, it matters whether X tends to be high when S tends to be high; whether the opposite is true that is, X tends to be high when S is low and vice versa ; or whether they are both not only uncertain but also unrelated.

As another example, suppose the uncertainty or variance associated with a project changes over time. Once again, if we know how the variance changes over time, or if we can make plausible guesses, the framework can be adapted without much trouble. Even when we know that we lack necessary data, the framework can help by showing us what the effect on value would be if the data were one thing or another. Some other real-world complications are thornier.

The framework does a good job of capturing the extra value associated with deferring an investment decision. But what if there are particular costs associated with deferral? For example, companies trying to be first to market with the next generation of a hot product will incur large costs if deferral allows a competitor to preempt them. Anytime there are predictable costs to deferring, the option to defer an investment is less valuable, and we would be foolish to ignore those costs.

If such additional costs were the only issue, we could easily handle them in our framework as long as we knew when the decision to invest or not to invest finally would be made.

Companies may not be compelled to invest at a certain moment, but rather may have discretion to time their investments. So the problem is to decide not only whether to invest but also when.

In effect, many real options are American rather than European. American options can be exercised at any time prior to expiration; European options may be exercised only at expiration. The option-pricing table embedded in this framework prices European options.

American options are more valuable than European options whenever the costs associated with deferral are predictable. That is not quite as easy as merely rewriting the pricing table because it would take a three-dimensional or possibly four-dimensional table to accommodate the extra variables needed. But it can be set up in a spreadsheet. But the framework needs to be augmented, not scrapped. Finally, the Black-Scholes option-pricing model that generated the numbers in the table makes some simplifying assumptions of its own.

They include assumptions about the form of the probability distribution that characterizes project returns. They also include assumptions about the tradability of the underlying project assets; that is, about whether those assets are regularly bought and sold.

And they include assumptions about the ability of investors to continually adjust their investment portfolios.

When the Black-Scholes assumptions fail to hold, this framework still yields qualitative insights but the numbers become less reliable.

Consequently, it may be worthwhile to consult an expert about alternative models to improve the quantitative estimates of option value. Graduate-level corporate-finance textbooks cover the basics of option pricing, beginning from first principles:. Zvi Bodie and Robert C. Merton, Finance Upper Saddle River, N. Brealey and Stewart Myers, Principles of Corporate Finance, fifth edition New York: Westerfield, and Jeffrey Jaffe, Corporate Finance, fourth edition Chicago: Other books go beyond the basics to treat specialized problems and present more advanced models for option pricing.

Hull, Options, Futures, and Other Derivatives, third edition Upper Saddle River, N. Paul Wilmott, Jeff Dewynne, and Sam Howison, Option Pricing Oxford, England: Oxford Financial Press, A few books focus on real options in particular. Each takes a somewhat different approach to modeling corporate opportunities:. Martha Amram and Nalin Kulatilaka, Real Options: Managing Strategic Investment in an Uncertain World Boston, Mass.: Harvard Business School Press, forthcoming Lenos Trigeorgis, Real Options: Managerial Flexibility and Strategy in Resource Allocation Cambridge, Mass.: Dixit and Robert S.

Models, Strategies, and Applications Westport, Conn.: Does the framework really work?

Strategy as a Portfolio of Real Options - Article - Harvard Business School

Even though we have taken some liberties, we know more about our project after using it than we did before. And if it seemed worthwhile, we could further refine our initial estimate of option value.

But the key to getting useful insight from option pricing sooner rather than later is to build on, rather than abandon, the DCF-based NPV analysis your company already uses.

Had we set out to value the option from scratch, it would have been more difficult and taken longer. It also would have been hard to tell how well we had done and when to stop working on it. Option pricing should be a complement to existing capital-budgeting systems, not a substitute for them.

The framework presented here is a way to start where you are and get somewhere better. There are other mathematical advantages, beyond the scope of this article, associated with using the quotient instead of the difference. This form of option-pricing table is nearly as old as the Black-Scholes model itself. I first ran into it as an M.

Luehrman is a professor of finance at Thunderbird, the American Graduate School of International Management, in Glendale, Arizona. Your Shopping Cart is empty. July—August Issue Explore the Archive. Mapping a Project Onto an Option A corporate investment opportunity is like a call option because the corporation has the right, but not the obligation, to acquire something—let us say, the operating assets of a new business. Mapping an Investment Opportunity onto a Call Option.

Take a n educated guess. Graduate-level corporate-finance textbooks cover the basics of option pricing, beginning from first principles: Luenberger, Investment Science New York: Oxford University Press, Pindyck, Investment Under Uncertainty Princeton, N.

Princeton University Press, A version of this article appeared in the July—August issue of Harvard Business Review. About Us Careers Privacy Policy Copyright Information Trademark Policy Harvard Business Publishing:. Harvard Business Publishing is an affiliate of Harvard Business School.