## SIMPLE APPROACH FOR DECIDING: The Interest Rate Option

Assume that each state is equally likely under the risk neutral probability measure, or equivalently, that the Arrow-Debreu state price of each pure state claim is the same. If r is the current risk free short rate and rc is the current risk free consol rate, then, under the risk neutral measure, the value today of the consol, ^-, is just its discounted expected value in one period plus one coupon payment:

Note that the project has positive NPV whenever the current short rate is below 53%. However, if the investment decision is delayed, then one of two possibilities will occur. Either the short rate goes to 20% and the project is never taken on, or the short rate drops to 10% and the project is taken on immediately. Therefore, the NPV of the investment opportunity, if it is delayed, is the expected discounted value of the investment opportunity in each state:

which is positive whenever r < 13%. Thus, applying the NPV rule using the mortgage rate as the discount rate provides the correct rule for deciding when to invest in this project in this interest rate environment.
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This example illustrates the underlying intuition in this paper. The reason why it is optimal to delay taking on the positive NPV investment is that by not taking it on, the firm does not give up the opportunity to capture an even bigger NPV should rates drop to 10%. Ideally, though, the firm would like to have it both ways, that is, take on the project today without giving up the option to capture the bigger NPV should rates drop to 10%. The way the firm can do this is to finance the investment by borrowing in a mortgage bond. If rates drop, then the larger NPV is captured by simply calling the bond — exercising the prepayment option by returning the principle. Of course, this strategy can only work if financing the investment at the mortgage rate does not involve an additional expense, or to put this another way, if the investment has non-negative NPV when discounted at the mortgage rate. In the next section we demonstrate that this result holds generally.