There is No Hidden Reinvestment Assumption in Discounting Formula and IRR : Logical and Mathematical Arguments

There is an ongoing dispute about the assumption of reinvesting intermediate cash flows at cost of capital in calculation of NPV and at IRR in calculation of implied yield to maturity. The lack of conclusive evidence lends support to confusion around the issue. This paper provides logical and mathematical proofs that there is no implicit reinvestment assumption embedded in present value formula and in IRR formula. We prove this conclusion by rewriting NPV and IRR equations in expanded form in order to discriminate between the discount rate for the initially evaluated cash flow occurring at time t and the discount rate for reinvested intermediate cash flows occurring after time t. In so doing we pay attention not only to reinvestment rate for intermediate cash flows, but also emphasize and justify that discount rate for intermediate cash flows over life of the project is not the same as discount rate of the project's expected cash flows. The revised formulas do not prohibit reinvestment of intermediate cash flows; however the discounting rate for reinvested intermediate cash flows should be the same as the reinvestment rate. This means that reinvestment rate can be arbitrary and not necessarily the cost of capital in the case of NPV. In much the same way, in the case of IRR the reinvestment rate for intermediate cash flows is not necessarily be equal to IRR. The conclusion is that reinvestment assumption is a wrong belief. We also show that MNPV and MIRR formulas contain erroneous restrictive assumptions about the discount rate for interim cash flows and can be highly misleading measures. We provide corrected formulas for MNPV and MIRR and demonstrate that MNPV and MIRR strongly depend on the life of the project, which makes estimates unstable and unreliable. MNPV and MIRR admix effects of expected reinvestment of interim cash flows to the outcomes of initial investment being evaluated. In fact the initial investment and reinvestments of interim cash flows, if they actually have to take place, are different projects and should be measured separately