A Resolution to the Problem of Multiple IRR: A Modified Capital Amortization Schedule (MCAS) Method for Non-Normal Cash Flows (NNCF) to Obtain a Unique IRR

Abstract

The problem of multiple IRR remained unresolved for almost a century. This problem is associated only with some of the non-normal net cash flow (NNCF) that wrongly includes reinvestment income as income or benefit stream. The reinvestment income, which is not a benefit from the investment or project under analysis, causes the multiple IRR problem. This is often misinterpreted as problem of IRR but its neither a problem with IRR nor NPV. It is a problem associated with some NNCF data and the failure to update the discounted cash flow (DCF) or capital amortization schedule (CAS) methods to handle such problem. Using NNCF data, analyses are conducted with special emphasis on topics such as: a. A modified CAS (MCAS) method that eliminates multiple IRR associated with NNCF data; b. Multiple IRR problem and the Descartes rule of sign and Norstrom’s criteria; c. A NNCF data with a unique IRR under DCF/CAS methods vs IRR by MCAS method; d. Resolving the problem of multiple IRR by MCAS Method Versus MIRR; and e. A critical review of the GIRR and AIRR Methods to Estimate NNCF. The salient findings of the present analysis are: a. The MCAS method, presented in this paper, identifies and eliminates the reinvestment income associated with NNCF investments (with positive opening balance in one or more years in the CAS) from the benefit stream; b. This new method overcomes the multiple IRR problem and leads to a unique and real IRR; The effectiveness of MCAS to handle the NNCF data is illustrated with numerical analysis; c. The assumption of reinvestment at IRR or at hurdle rate in NPV are false assertions in the cases of normal NCF and some of the NNCFs. However, such reinvestment is evident only with NNCFs with positive opening balance in one or more years under the CAS. d. The reinvestment income under the benefit stream causes multiple IRRs and multiple NPVs too. As NPV is a static point estimate (at hurdle rate) the multiple NPVs are not exposed. Without eliminating the reinvestment income, none of the criterions viz. NPV, IRR or MIRR, is useful as a decision criterion. Neither NPV or MIRR is a preferred criterion, under such circumstances, as recommended in some published works. e. The MCAS method is appropriate for both normal NCF and NNCF as illustrated in this paper. CAS or DCF method is appropriate only for normal NCF investments. f. Even when there is no multiple IRRs with some NNCFs under DCF/CAS method, the MCAS method estimated IRR or NPV, without reinvestment income, are different from that of the DCF/CAS estimated IRR and NPV. For a consistent estimate of IRR and NPV, the MCAS method is most appropriate both for NCF and NNCF investments. g. The generalized IRR (GIRR) and the Average IRR (AIRR) are also not appropriate estimates for NNCF and they are not NCF consistent as discussed in this paper. The problem of multiple IRR associated with the popular cases of NCF investments used in GIRR and AIRR, are also resolved now. In conclusion, the MCAS method resolves the problem of multiple IRR and leads to a unique IRR that is real and NCF-consistent. Neither the NPV nor the MIRR could resolve the problem of multiple IRR.