# Real options

We have been almost entirely concerned with the valuation of derivative dependent on financial assets. Real assets include land, building, plant, and so on.Often there are embedded options.Valuation is difficult because market prices are not readily available.

We start by explaining the traditional approach to evaluating investments in real assets and their shortcomings.

CAPITAL INVESTMENT APPRAISAL

The traditional approach to valuing a potential  capital investment project is known as the “net” present value” or NPV  approach. The NPV of a project is the present value of its expected future incremental cash inflows and outflows. The discount rate used to calculate the present value is a “risk-asjusted” discount rate, chosen to reflect the risk of the project. As the riskness of the project increases, the discount rate also increases.

As an example consider an investment that costs $100 million. If the risk-adjusted discount rate is 12%,the net present value of the investment is(in million of dollars) $-100+\frac{25}{1.12}+\frac{25}{1.12^{2}}+\frac{25}{1.12^{3}}+\frac{25}{1.12^{4}}+\frac{25}{1.12^{5}}=&space;-9.88$ A negative NPV, such as the one we have just calculated,indicates that the project will reduce the value of the company to its shareholder and should not be undertake. A positive NPV indicates that the project should be the undertaken because it will increase shareholder wealth. The risk adjusted discount rate should be the return required by the company, or the company’s shareholder, on the investment. This can be calculated in a number of ways. One approach often recommended involves the capital asset pricing model. The steps are as follows. 1) Take a sample of companies whose main lines of business is the same as that of the project being contemplated. 2) Calculate the betas of the companies and average them to obtain a proxy beta for the project. 3) Set the requested rate of return equal to the risk-free rate plus the proxy beta times the excess return of the market portfolio over the risk-free rate. One problem with the traditional NPV approach is that many projects contain embedded options. Consider, for example, a company that is considering building a plant to manufacture a new product. Often the company has the option to abandon the project if things do not work out well. It may also have the option to expand the plant if demand for the output exceeds expectations. These options usually have quite different risk characteristics from the base project and require different discount rate. This involved a stock whose current is$20. In three months ‘ time the price will be either $22 or$18. Risk neutral valuation shows that the value of a three-month call option on the stock with a strike price of 21 is 0.633.

The expected return required on the call option is 42.6%.In practice it would be very difficult to estimate these expected returns directly in order to value the option on real assets.

There is no easy way of estimating the risk-adjusted discount rates appropriate for cash flows when they arise from abandonment, expansion , and other options. This is the motivation for exploring whether the risk-neutral valuation principal can be applied to options on real assets as well as options on financial assets.

Another problem with the traditional NPV approach lies in the estimation of the appropriate risk-adjusted discount rate for the base project. The companies that are used to estimate a proxy beta for the project in the three-step procedure above have expansion options and abandonment options of their own. Their betas reflect these options and may not therefore be appropriate for estimating a beta for the base project.

EXTENSION OF THE RISK-NEUTRAL VALUATION

Consider an asset whose price, f , depends on variable 0 and time t. Assume that the process followed by 0 is.

$\frac{d\theta&space;}{\theta&space;}=m&space;dt+s&space;dz$

Where dz is a Wiener process. The parameters m and s are the expected growth rate in $\Theta$ and t. The variable $\Theta$ need not be a financial variable. It could be something as far removed from financial markets as the temperature in the Centre of New Orleans.

The asset price f follows a process of the form

df= $\mu&space;f&space;dt+\sigma&space;fdz$

Any solution to equation for s is a solution to equation for $\theta$, and vice versa , when the substitution

$q=&space;r-m+&space;\lambda&space;s$

Using risk- neutral valuation. This involves setting the expected growth rate of a equal to r-q and discounting expected payoffs at the risk-free interest rate. It follows that we can solve by setting the expected growth of $\theta$ equal to

$r=&space;(r-m+&space;\lambda&space;s)=&space;m-\lambda&space;s$

and discounting expected payoffs at the risk-free interest rate.

APPLICATION TO THE VALUATION OF A NEW BUSINESS

Traditional methods of business valuation, such as applying a price earnings multiplier to current earnings, do not work well for new business. Typically a company ‘s earnings are negative during  it’s really years as it attempt to gain market share and establish relationship with customers. The company must be valued by estimating future earnings and cash flows under different scenarios.

The company ‘ future cash flows typically depend on a number of variables such as sales, variable costs as a percent of sales, fixed costs, and so on. Single estimates should be sufficient for outlined in the previous two sections. A Monte Carlo simulation can then be carried out to generate alternative scenario for the net cash flows per year in a risk-neutral world.  It is likely that under some of these scenarios the company does very well and under others it becomes bankrupt and ceases operations.

The simulation must have a built-in  rule for determining when bankruptcy happens. The value of the company is the present value of the expected cash flow in each year using the risk -free rate for discounting.