## Comparing Approaches

In this section, we compare three decision methodologies: net present value (NPV), decision tree analysis (DTA), and option pricing methods. We also introduce the fundamental concept behind option pricing. This is that a replicating portfolio of priced securities can be found that has the same payouts as the option and therefore has the same market value. This is also called a zero-arbitrage condition, or the law of one price, because assets with the same payouts should have the same prices in the absence of arbitrage profits.

We use a simple deferral option to illustrate. Suppose that you have the opportunity to invest $115 at the end of the year in a project that has a 50-50 chance of returning either $170 or $65 in cash flows. The risk-free rate, r, is 8 percent. You have found a perfectly correlated or twin security that has payouts of $34 and $13 and is trading in the market for a price of $20 per share. Note that the payouts of the twin security are exactly one-fifth of the payouts on our project in each state of nature. There are two ways to use the twin security to help value our project. First, we can estimate the cost of capital for the twin security and apply it to the expected cash flows of our project—a traditional approach. The cost of capital is calculated as the rate that equates the present value of the expected cash flows with the present value of the twin security as follows:

Since the twin security has perfectly correlated payouts, it has the same risk as our project and we can use the same risk-adjusted discount rate, 17.5 percent, to discount the expected cash flows on the project. The value of the project is therefore,

A second approach is to create a replicating portfolio—one that uses the expected cash flows of the priced securities to replicate the cash flows of our project. Suppose we choose N shares of the twin security and B dollars of a risk-free bond to compose our replicating portfolio. In the favorable state of nature this portfolio must yield $170,

In the unfavorable state of nature, this portfolio must yield $65: NSU+ B(l + rf) = SA5

Together, we have two equations and two unknowns. The solution is N = 5 and B = 0. Using this result plus the fact that one share of the twin security is worth $20, our project must have the same value as the replicating portfolio:

The net present value of the project, given that we must make the decision to invest today, is the present value of the cash inflows. These have been found to be $100, minus the present value of the cash outflows. This is calculated from the certain outlay of $115 next year, discounted at the riskless rate (8 percent)—a present value of -$106.48:

Having maintained the assumption that we must decide now whether to invest at year's end, our decision would be not to invest. But the answer changes if we have a deferral option that allows us to decide next year, after observing which of the two outcomes have occurred. If we were using a decision-tree analysis (DTA), we would observe (from Exhibit 20.3) that the net cash flows in the favorable state are $170 - $115 = $55 because we would decide to invest. In the unfavorable state we would simply not invest,

Exhibit 20.3 Decision Tree Analysis (DTA)—Flexibility "Valued"

thereby giving us net cash flows of $0. Discounting the expected cash flows at the cost of capital gives us the result of the DTA approach:

The value of the deferral option is the difference between the estimated value of the project with flexibility and its value without flexibility, or $23.4 - (-$6.5) = $29.9.

The problem with this DTA approach is that we used the cost of capital for the underlying project without flexibility to value the deferral option, a real option that has different payouts and therefore different risk than the underlying project. The DTA approach uses an ad hoc discount rate that is incorrect for the riskiness of the cash flows being evaluated.

The option pricing methodology uses the replicating portfolio approach. As before, we construct a portfolio consisting of N shares of the twin security and B dollars of risk-free debt. In the up state, the twin security pays $34 for each of the N shares and the bond pays the face value of the bond, $B, plus interest equal to rp. These payouts must equal $55. A similar construction applies to the down state. The result is two equations and two unknowns:

The solution is that N = 2.62 and B = - $31.53. The value of the project with the flexibility of deferring is:

Option viiluc - W (Price of twin security) - B

The value of the deferral option itself is the difference between the value of the project with flexibility and its value without flexibility $20.86 - (-$6.48) = $27.4. This is the correct, arbitragefree solution. If we were using the implied risk-adjusted discount rate, it would be 31.9 percent (not 17.5 percent):

The risk of an option on an underlying risky asset is always greater than the risk of the asset itself. The project has a present value of $100 and a 50-50 chance of going up to $170, a 70 percent increase, or down to $65, a 35 percent decrease. The project with the option is worth $20.86 and has a 50-50 chance of paying off $55, a 164 percent increase, or zero, a 100 percent decrease. This greater risk helps explain why the risk-adjusted discount rate for the project with the option is 31.9 percent.

Exhibit 20.4 summarizes the results. The NPV approach undervalues the project because it does not take into account the value of flexibility. The DTA approach overestimates the value of flexibility because it uses the project risk-adjusted discount rate to discount the cash flows of the deferral option— cash flows that are much riskier.

The option pricing approach gives the correct value because it captures the value of flexibility correctly by using an arbitrage-free replicating portfolio approach. But where does one find the twin security? We can use the project itself (without flexibility) as the twin security, and use its NPV (without flexibility) as an estimate of the price it would have if it were a security traded in the open market. After all, what has better correlation with the project than the project itself? And we know that the DCF value of equities is highly correlated with their market value when optionality is not an issue. We shall use the net present value of the project's expected cash flows (without flexibility) as an estimate of the market value of the twin security. We shall call this the marketed asset disclaimer.

If we use the project itself as the twin security, then the replicating portfolio has the following payouts in the up and down states, given a risk-free rate of 8 percent:

Exhibit 20.4 Comparison of the Approaches

Solving the two equations above for the unknowns, we have

Given that the present value of the project without flexibility is $100, the value of the replicating portfolio is also the value of the project with flexibility, namely

It is no accident that this approach gives the same answer as the twin security approach—the outcomes are perfectly correlated.

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