see note 4.

tional NPV rule, we can calculate the present value of any branch in the event tree as the expected payouts discounted at the 12 percent risk-adjusted rate. Let's take the uppermost branch in the fifth time period. Its present value is:

A similar calculation will produce any of the values in the event tree. If we have to decide on the project today, we would reject it because the investment required is $105 and the present value of the project inflows is only $100.

There is an equivalent approach. Instead of discounting the expected payouts at a risk-adjusted rate, we could first risk-adjust the payouts (by using risk-adjusted probabilities, or as they are often called, risk-neutral probabilities) and then discount them at the risk-free rate. If we define the risk-adjusted probability of an up movement as q and the risk-adjusted probability of a down movement as 1 - q, then the present value of a branch can be written as:

Solving for the risk-adjusted (or risk-neutral) probability, we have:

Using the risk-neutral probabilities to calculate a risk-adjusted expected payoff that is then discounted at the risk-free rate gives the same answer. The present value calculation for the uppermost branch in the fifth time period is:

Risk-adjusted expiated payouts .7283(1211.711) + .2717(5156^3)

Next we turn the event tree into a decision tree, and in so doing introduce real options. Decision Trees

When decision nodes are added to an event tree, it becomes a decision tree. In this section we will illustrate the value of flexibility that is created if it becomes possible to expand, contract, or abandon the project. Suppose that it is possible to expand our simple project and its payouts 20 percent by spending an additional $15, and the expansion is an American option that can be exercised any time during the life of the project. The resulting decision tree is given in Exhibit 20.6. The payouts on the tree have to be solved by working from the final branches backward through time. Take the upward-most branch in period 5. On the upward limb the payout in the absence of expansion would have been $211.70, but with expansion, it is 1.2($211.70) -$15 = $239.04. Since the value with expansion is higher, we

Exhibit 20.6 Option to Expand f

= Decision to expand t= 5

= Decision to expand t= 5

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