## SIMPLE APPROACH FOR DECIDING: The Interest Rate Option 3

Another example would be projects in which all cash flow uncertainty is idiosyncratic (cov (e(t), “(t)) = 0, Vt). In both cases the assumption is satisfied trivially because ж is always zero. A non-trivial example of a set of primitives that provides (1) with ж non-zero is if

Let rm be the first time (4) is satisfied, that is, the first time it is optimal to invest in the project. It is suboptimal to wait longer, say to time r . To see why, assume that when the investment is made at time rm, it is fully financed by borrowing $1 in a mortgage. Because (4) is satisfied, the cash flows of the project exceed the interest owed on the mortgage, so this strategy generates a non-negative cash flow stream. The key insight is to notice that by calling (repaying) the mortgage at time r , the firm can duplicate exactly the strategy of investing at time r (in both cases the firm will need to pay $1 and will receive с forever). However, by investing at time ra the firm is worse off because it misses the non-negative cash flow stream from rm + 1 to ra, which implies that it is suboptimal to wait beyond r .

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The reason why it is suboptimal to invest in the project before rm is that before rm, the interest paid on the mortgage always exceeds the cash flow of the project (because at any time before rm, (4) is not satisfied). So before rm, the firm is always better off investing the $1 in the mortgage rather than the project.

Another way to understand why this proposition holds is to recognize that the callable bond or mortgage is a portfolio which consists of a long position in a non-callable consol bond and a short position in an American call option on the bond. Thus a portfolio that consists of a long position in the non-callable consol and a short position in the mortgage is exactly a long position in the call. The investment should be made when the NPV exceeds the value of this call or equivalently, the portfolio of a long position in the consol and a short position in the mortgage. However, since the underlying project is essentially a non-callable consol bond, the non-callable consol appears in both expressions. Canceling leaves the simpler expression that price of the mortgage must exceed the net investment, which is (4).