## Real Option Pricing Methods

In their book Investment Under Uncertainty, Dixit and Pindyck use a simple example to illustrate the difference between NPV and option pricing methods.2

1 Empirical evidence indicates that executive stock options are exercised suboptimally. This must be taken into account in their valuation.

2 A. Dixit and R. Pindyck, Investment Under Uncertainty (Princeton, NJ: Princeton University Press, 1994).

Suppose you are deciding whether to invest \$1,600 in a new project that makes widgets. The cash flow per widget is \$200 but will change to either \$300 or \$100 at the end of the year with equal probability. After that it will stay at its new level forever. Note that the expected future cash flow is \$200, the weighted average of the risky outcomes, \$300 and \$100. The cost of capital is 10 percent. Assuming that one widget can be sold immediately, and one per year thereafter, the net present value of the project would be estimated as follows:

The NPV approach discounts the expected project cash flows at the weighted average cost of capital. The decision rule is to take the maximum of the discounted expected cash flows or zero (meaning don't do the project). The NPV rule is the maximum (determined today) of the expected values. It also makes the implicit assumption that the project should be undertaken immediately or not at all, because the maximum must be decided right now. This assumption rules out the possibility of deferring the investment one year until the uncertainty about the price per widget is resolved. If we look at the project given the option to defer, the economics look better:

With the option to defer you can wait one period to invest, then decide whether to do so, based on the arrival of information about the long-term cash flow per widget. If the cash flow is only 100 per unit you will not exercise the option to invest, but if the cash flow is 300 per unit, you will. Although the NPV of investing immediately is \$600, the NPV should you decide to defer is even higher at \$733. Therefore, you will defer.

The value of this call option (with an exercise price of 1,600, a one-year life, a variance determined by the cash-flow spread of \$200 per unit, and an underlying risky asset that has a value without flexibility of \$600) is the difference between the value of the project with flexibility and its value without flexibility, \$733 - \$600 = \$133. Note also that the NPV is the maximum, decided today, of the expected discounted cash flows or zero, while the option value is the expected value of the maximums, decided when information arrives, of the discounted cash flows in each future state of nature, or zero:

Option vtiLue = Expected

The two methods use information quite differently. NPV forces a decision based on today's expectation of future information, while option valuation allows the flexibility of making decisions in the future contingent on the arrival of information. Option-pricing methods capture the value of flexibility while NPV does not. The value of a project using option pricing will always be greater than the value of the project using NPV. Sometimes the difference in value between the two approaches is small. This is usually the case when the project has such a high NPV that the flexibility is unlikely to be used, or conversely when the NPV is very negative. The biggest differences (see Exhibit 20.1) occur when the NPV is close to zero, that is, when the decision about whether to undertake the project is a close call. We have found differences in value of more than 100 percent in such situations. These were cases where senior management had often overruled the NPV results and accepted the project for ''strategic reasons." As you begin to feel

Exhibit 20.1 When Is Managerial Flexibility Valuable?

Expected cash flows Cost of capital

( Cash flow given info

m { Cost of capital

Likelihood of receiving new information

Low Uncertainty High

 Moderate flexibility value High flexibility value Low flexi billty value Moderate flexi billty value

Flexibility value greatest when:

1. High uncertainty a bout the future Very likely to receive new information over time

2. High room for managerial flexibility Allows management to respond appropriately to this new information +

3. NPV without flexibility near zero

If a project is neither obviously good nor obviously bad, flexibility to change course is more likely to be used and therefore is more valuable

In every scenario flexibility value is greatest when the project's value without flexibility is close to break even

Under these conditions, the difference between options valuation and other decision tools is substantial more comfortable with real options, you will see that the concept fits intuition better than the rigid assumptions of NPV.

To extend Dixit and Pindyck's example, let's see what happens if the variability of cash flows per unit increases from \$300 versus \$100 to \$400 versus \$0. Notice that the NPV is the same because the expected cash flows remain unchanged and we also assume that the new risk is uncorrelated with the economy, so the capital asset pricing model beta and the cost of capital are unchanged. But the value of the deferral option will increase because it is based on decisions that are contingent on the way that uncertainty is resolved. The algebra looks like this:

Now the value of flexibility has increased to \$673, because the amount of uncertainty has increased. The value of an option increases as the variability in the value of the underlying risky asset (the cash flow per unit) increases. As with financial options, the value of a real option depends on five parameters: the market value of the underlying asset on which the option is contingent; the exercise price of the option; the time remaining until the maturity of the option; the volatility of the underlying asset, and the risk-free rate of interest. These are all clearly defined for financial options, but require better understanding for real options. A sixth parameter is the amount of dividends paid by the underlying risky asset. We shall come back to it a little later.

The parameters that affect the value of a real option are summarized in Exhibit 20.2. Note that an important difference between financial and real options is that management can affect the value of the underlying risky asset (a physical project under its control) while financial options are side bets owned by third parties that cannot affect the outcome of the underlying asset (e.g., a share of IBM).

Without training, executives often fail to recognize real options and how valuable they can be. One example is a story about the life insurance industry. In the early 1970s, it was common to be able to buy a whole life policy with a clause that allowed you to borrow against the cash value of the policy at, let's say, a 9 percent interest rate, for the life of the policy. At the time, interest rates were about 4 percent on long-term government bonds. No one expected that they would go as high as 9 percent. But the

Exhibit 20.2 Option Value Is Determined by Six Variables

Exhibit 20.2 Option Value Is Determined by Six Variables

life of a policy can be long—as long as the life expectancy of the policyholder. What kind of option was imbedded in the contract, and what were the parameters? The policyholder had a long-term call option on the right to borrow (i.e., on the value of a bond issued by the insurance company) against the cash value (which determines the amount of the loan) of the policy at a fixed rate, for the life of the policy. The underlying uncertainty is the variance of interest rates. In 1981-1982 when interest rates went to double-digit levels, policyholders began to borrow at 9 percent, invest the money in government bonds at double-digit rates, and keep the difference. The losers were the life companies that had to borrow at double-digit rates and receive 9 percent. Several life companies went bankrupt as a result. The executives who originally sold life policies with these valuable options imbedded in them had not clearly understood the value of the options in the life contracts they were writing.

### Taxonomy of Options

To identify potential operating flexibility and strategic factors, we can classify asset options into five mutually exclusive (but not exhaustive) categories.

Abandonment option. The option to abandon (or sell) a project—the right to abandon an open pit coal mine—is formally equivalent to an American put option on a stock.3 If the bad outcome occurs at the end of the first period, the decision maker may abandon the project and realize the expected liquidation value. Then, the expected liquidation (or resale) value of the project may be thought of as the exercise price of the put. When the present value of the asset falls below the liquidation value, the act of abandoning (or selling) the project is equivalent to exercising the put. Because the liquidation value of the project sets a lower bound on the value of the project, the option to liquidate is valuable. A project that can be liquidated is worth more than the same project without the possibility of abandonment.

Option to defer development. The option to defer an investment to develop a property is formally equivalent to an American call option on the stock. The owner of a lease on an undeveloped oil reserve has the right to acquire a developed reserve by paying a lease-on-development cost. The owner can defer the development process until oil prices rise. In other words, the managerial option implicit in holding an undeveloped reserve is a deferral option. The expected development cost may be thought of as the exercise price of the call. The net production revenue less depletion of the developed reserve is the opportunity cost incurred by deferring the investment. If this opportunity cost is too high, the decision maker may want to exercise the option (that is, develop the reserve) before its relinquishment date.

Option to expand or contract. The option to expand the scale of a project is formally equivalent to an American call option on the stock. Management may choose to build capacity in excess of the expected level of output so that it can manufacture at a higher rate if the product is more successful than was anticipated. The expansion option gives management the right, but not the obligation, to make additional follow-on investment (for example, to increase the production rate) if project conditions turn out to be favorable. The option to contract the scale of a project's operation is formally equivalent to an American put option on stock. Many projects can be engineered so that output can be contracted. Foregoing future spending on the project is equivalent to the exercise price of the put.

Option to extend or shorten. It is possible to extend the life of an asset or a contract by paying a fixed amount of money—an exercise price. Conversely, it is possible to shorten the life of an asset or a contract. The option to extend is a call, and the option to shorten is a put. Real estate leases often have clauses that are examples of the option to extend or shorten the lease.

Option to scope up or scope down. Scope is the number of activities covered in a project. Its optionality is expressed in terms of the ability to switch among alternative courses of action at a decision point in the future. Scope is like diversification—it is sometimes preferable to be able, at a

3 An American option can be exercised at any time up to the maturity date of the option. A European option can only be exercised on the maturity date.

higher exercise cost, to chose among a wide range of alternatives. Buying the option to have greater scope is a call.

Switching options. The option to switch project operations is a portfolio of options that consists of both calls and puts. Restarting operations when a project is shut down is equivalent to an American call option. Shutting down operations when unfavorable conditions arise is equivalent to an American put option. The cost of restarting (or shutting down) operations may be thought of as the exercise price of the call (or put). A project whose operation can be turned on and off (or switched between two distinct locations, and so on) is worth more than the same project without the flexibility to switch. A flexible manufacturing system with the ability to produce two products is a good example of this type of option, as is peak-load power generation and the ability to exit and reenter an industry.

Compound options. These are options on options. Phased investments are a good example. You may have a factory that can be built as a sequence of real options, each contingent on those that precede it. The project can be continued at each stage by investing a new amount of money (an exercise price). Alternatively, it might be abandoned for whatever it can fetch. Other examples are research and development programs, new product launches, exploration and development of oil and gas fields, and an acquisition program where the first investment is thought of as a platform for later acquisitions.

Rainbow options. Multiple sources of uncertainty produce a rainbow option. Most research and development programs have at least two sources of uncertainty—technological and product-market uncertainty. The latter is represented by the evolution of the uncertain price of the product from a value that is relatively well known today, to less certain values that are affected by the state of the economy as well as other uncertain influences in the future. Thus, product- market uncertainty increases through time. Technological uncertainty, on the other hand, is reduced over time by conducting research until we learn what the product is and what its capabilities are. A similar type of rainbow option is exploration and development of natural resources like oil reserves.

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