In the general case with uncertain cash flows, the above proposition derives the optimal investment rule using certainty equivalents. The next proposition derives the appropriate discount rate if the simple NPV rule is used instead (i.e., the expected cash flows rather than their certainty equivalents are discounted). Let the risk adjusted discount rate (i.e., the discount rate that is used in a simple NPV analysis) be defined to be the rate that discounts the expected cash flows to give the current price.

Corollary 2.1 An investment opportunity with risky cash flow c(r) = c + e(r),r > t, that requires an initial investment of $1 and has a risk adjusted discount rate of R(t) should be taken on at time t if, and only if, it has positive NPV when the expected cash flows are discounted at R(t) x ^.

This proposition provides the simple rule for deciding whether to invest in a risky project that can be delayed — apply the standard NPV rule, but discount the expected cash flows of the project at the discount rate multiplied by the ratio of the mortgage rate to risk free rate. Note that for projects with riskless cash flows (or cash flows with only idiosyncratic risk), R(t) = r(t), and so the rule provides the result obtained in the example — simply discount at the mortgage rate.
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Even in the case when there is no option to delay investment, the simple NPV rule gives a theoretically correct answer only under certain conditions. For instance, when the riskiness and expected value of a project’s cash flows varies stochastically the simple NPV rule fails because the present value of the cash flows is obtained by discounting the current expected cash flow at a single discount rate that is not a function of the cash flow (or interest rate) dynamics. To apply the simple NPV rule, such dynamics must be ruled out. One way to do this is to assume that the certainty equivalent of each cash flow is constant. Since our object is to provide an adjustment to the simple NPV rule it is not surprising that we have made assumptions at least as restrictive as in the standard case.

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