A General Rule

The example demonstrates that by simply replacing the riskless rate with the mortgage rate, the simple NPV rule can be used to determine when to invest. Since mortgage interest rates are as easily observable as riskless rates themselves, this result has the potential to be as useful as the simple NPV rule itself.

Consider the decision, which can be costlessly delayed, to take on a project at date t which, for expositional simplicity, is assumed to provide a perpetual cash flow stream. Let r(t) be the (date-t) yield of a riskless bond that pays $1 forever (hereafter, the consol rate), and let rm(t) be the yield of an equivalent consol bond that is callable at par at any time, that is, at any time it allows the issuer (i.e., the borrower) the option to absolve himself of all future interest obligations by returning the face value of the bond. At time r, the callable consol bond issued at time t with current price PTO(r) that is callable at Pm(t) is defined to have yield
examples of callable riskless bonds. In light of this we will refer to rm(t) as the mortgage consol rate, or more simply as the mortgage rate.

Let “(t) be the pricing operator (kernel) in the economy. Using standard finance arguments (see Duffie (1996)) this operator can be used to derive the price of any asset. Explicitly, the (date-t) price of any cash flow c(r), where r > t, is given by @ [^ryc(r)]. So, for example, the price of the consol bond is ^ = Q= t @ ].
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We will limit attention to potential projects that require an investment of $1 at time t and, if this investment is made, deliver a risky cash flow stream c(r) = с + e(r) for all r > t, where с is constant and @[e(r)] = 0. As was pointed out in the introduction, to derive a general result that can be applied to all projects an assumption on the nature of the resolution of cash flow uncertainty is required. What is needed is that certainty equivalent of every cash flow be constant. This assumption requires that for some constant ж and for all r > t,
Under this assumption, the project can be valued by discounting the certainty equivalent of each cash flow at the consol rate. To see why, let P(t) be the time-t value of the cash flow stream. Then, using the pricing operator to value the cash flows and (1),
We will henceforth take (1) to be the definition of ж, the certainty equivalent of the uncertain part of the project’s cash flows.

Since the certainty equivalent of any cash flow is determined endogenously, it is important that we establish that assumptions on primitives exist that will provide cash flows with constant certainty equivalents. Clearly, as was pointed out in the introduction, attention must be restricted to projects for which waiting provides no new information about the distribution of the cash flows. Perhaps the most straightforward example of such projects that also satisfy (1) are projects with riskless cash flows (e(t) = 0, Vt).

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