Options on Real Assets The valuation of stock options is rather complex, but with the assis-

Options on Real Assets The valuation of stock options is rather complex, but with the assis-

tance of some well-accepted models such as the Black-Scholes model, we can estimate the value of an option. For example, in the Black-Scholes option pricing model discussed in Appendix A, there are five factors that are important in the valuation of an option: 6

1. The value of the underlying asset, P

2. The exercise price or strike price of the option, E

3. The risk free rate of interest, r

4. The volatility of the value of the underlying asset, σ

5. The time remaining to the expiration of the option, T In Chapter 9 we examine the relation between each of these factors

and the value of a stock option. Our focus here is to map these factors onto a real asset option. Like other options, real options may be a call option (the option to buy an asset), a put option (the option to sell an asset), or a compound option (an option on an option). And, like other options, real options may be a European option (an option that can only

be exercised on the expiration date) or an American option (an option that can be exercised at any time on or before the expiration date). In general terms, the relation between the factors that affect the value of a stock option and those that affect a real option correspond as follows:

6 Fischer Black and Myron Scholes, “The Pricing of Options and Corporate Liabili- ties,” Journal of Political Economy (May/June 1973), pp. 637–659.

Capital Budgeting and Risk

Parameter Option on a Stock Option on a Real Asset

X The stock’s price The present value of cash flows from the investment opportunity S

The exercise price of the The present value of the delayed capital option

expenditure or future cost savings r

The risk-free rate of interest The risk-free rate of interest

s Volatility of value of the Uncertainty of the project’s cash flows underlying asset t

The time to maturity The project’s useful life Of course, the factors that correspond to a specific option can be

better described when we examine the particular option. Consider the option to abandon. In this case, the underlying asset is continuing oper- ations and so the value of the underlying asset is the present value of the cash flows associated with the asset. The strike price or exercise price for this option is the exit value or salvage value of the asset. A number of common real options are described in Exhibit 14.5.

EXHIBIT 14.5 Examples of Real Options

Option Type*

Value of Underlying Asset

Exercise Price

To abandon American The present value of the The exit or salvage put

cash flows from the

value

abandoned assets

To defer an American The present value of The deferred invest- investment

call

completed project’s net

ment outlay

operating cash flows

To abandon Compound The present value of the The investment outlay during

necessary for the next construction

option

completed project’s cash

stage To contract

flows

European The present value of The costs of rescaling the scale

the project of a project To expand

put

potential cost savings

European The present value of The additional invest- call

incremental net operating

ment outlay

cash flows

To switch inputs American The present value of the The cost of retooling or outputs

put incremental cash flows from production or distri- the best alternative use

bution *A put option is an option to sell the underlying asset; a call option is an option to

buy the underlying asset. An American option is one that can be exercised at any time up to and including the expiration date; a European option is one that can only be exercised at the expiration date.

LONG-TERM INVESTMENT DECISIONS

Identifying the options associated with an investment opportunity is the first step. The second step is to value these options. Consider an investment opportunity to defer an investment. This investment oppor- tunity is similar to what a firm experiences in their investment in research and development: An expenditure or series of expenditures are made in research and development and then sometime in the future, depending on the results of the research and development, the actions of competitors, and the approval of regulators, the firm can then decide whether to go ahead with the investment opportunity.

Real Options: An Example Let’s put some numbers to the analysis of a project with a real option.

Suppose that research and development is $2 million initially and $2 million more for each of the next three years. And suppose that at the end of the fourth year the firm has an option to either go ahead with the product or simply abandon it. If the firm goes ahead with the develop- ment of the product, this will require an investment of $100 million at the end of the fourth year. To make the analysis simpler, let’s assume that we can sell the investment in the product to another party—that is,

cash out—at the end of the fourth year for $120 million. 7 And, because we know that all of this is uncertain, let’s attach probabilities of this being a marketable product and, hence, one that the firm is able to cash out. Let’s assume that there is a 60% chance that the firm can cash out for $120 million and a 40% change that the firm cannot cash out at all (and will, therefore, not make the investment).

Given this scenario, it means that: ■ If the R&D is successful and the firm is able to cash out, the value at

the end of the fourth period is $120 million – $100 million = $20 million.

■ If the R&D is not successful and the firm is not able to cash out, the value at the end of the fourth period is $0.

Before we can value the project with or without the option, we need to estimate the cost of capital. The cost of capital is the sum of the risk- free rate of interest and the risk premium. 8 The risk premium is deter- mined relative to the market’s risk premium. Suppose the risk-free rate

7 We are simplifying this example. More realistically, we would estimate future cash flows from the successful project beyond the fourth year and discount these to the

end of the fourth year—and then use this value in place of the $120 million. 8 To be consistent with the Black-Scholes option-pricing model, we’ll use continuous-

ly compounded cost of capital throughout our example.

Capital Budgeting and Risk

of interest is 5%, the market risk premium is 4%, and the volatility is 5 times that of the market. If the market’s volatility (i.e., the standard deviation of expected cash flows) is 15%, the cost of capital is: 9

Cost of capital = Risk-free rate of interest + Risk premium = 5% + ( 5 4% ) = 25%

Using a continuously compounded discount rate of 25%, the present value of the research and development costs is –$5.72 million:

in millions 0 1 2 3

Research and development –$2.00 –$2.00 –$2.00 –$2.00 Present value of research and development –$2.00 –$1.56 –$1.21 –$0.94 Total present value of R&D

Putting the R&D together with the value of the investment four years from today,

$20 million  NPV = – $5.720 million +  ( 0.60 ) ------------------------------ ( 4 0.25 ) + [ ( 0.40 ) $0 ( ) ] 

= – $5.720 million + 4.415 million = – $1.305 million This NPV represents the cost to the firm if the firm makes the decision

today to commit to both the R&D and the investment at the end of the fourth year.

Using the traditional capital budgeting NPV technique, this suggests that we should reject the project because its net present value is less than $0. But wait—we have not considered the valuable option of the deferred investment because the firm can wait until the end of the fourth year to decide on the investment.

We can see the value of the option by estimating how much value the option itself adds to the project. First, we estimate the parameters of the option pricing model. Then we see how the value of the option, when considered along with the present value of the cost of acquiring the option (that is, the present value of the research and development), can make an unattractive project into an attractive project.

9 This means that the volatility of the project’s cash flows are 5(15%) = 75%. This is the estimate of volatility that we include in the valuation of the project’s option.

LONG-TERM INVESTMENT DECISIONS

The value of the underlying asset is the discounted value of the probability-weighted possible outcomes:

 Value of the underlying asset = 0.6  $120 million ---------------------------------  + [ ( 0.40 ) $0 ( ) ]  e ( 4 0.25 ) 

= $26.487 million

Therefore, the value of the parameters in the option valuation are as fol- lows:

Value of the underlying asset $26.487 million Strike price

$100.000 million Risk-free rate of interest

Number of periods to exercise 4 Using the Black-Scholes option pricing formula, the value of this

option is $7.774 million. Therefore, the value of the project is:

Project NPV = Present value of the R&D + Value of the option

= – $5.72 million + 7.774 million Project NPV = $2.054 million

Another way of looking at this is to estimate the value-added of the deferral option:

Value-added of the option = Project NPV Static NPV – = $2.054 million – ( – 1.305 million ) Value-added of the option = $0.749 million

Hence, the project has a positive NPV considering the valuable option to defer investment.