WOLFRAM|DEMONSTRATIONS PROJECT

A Model Illustrating Multiple Interest Rate Analysis (MIRA)

​
r
0.05
Cash flows
I
0
-100
c
1
10.
c
2
10.
c
3
10.
c
4
10.
c
5
10.
c
6
10.
c
7
10.
c
8
10.
c
9
10.
c
10
10.
c
11
10
c
12
100
defaults
Output show lines
NPV = 38.748
*
D
= -9.004
m
1
= 0.043
Net present value (NPV) and internal rate of return (IRR) are criteria commonly employed by organizations to appraise investment projects. NPV is the present value of an investment's cash flows
I
0
,
c
i
for
i=1
to
n
, discounted at the cost of capital
r
, where
I
0
is the investment’s initial outlay at time zero (negative), and
c
i
is the investment’s net cash flow at time
i
(revenues minus costs; usually positive but sometimes negative). IRR is the value of the discount rate setting NPV to zero.
When deciding the viability of a single project, the criterion that its NPV be positive usually gives the same investment decision as the criterion that its IRR exceed the cost of capital
r
. Exceptions exist when a project’s cash flows yield either no real-valued IRR (an anomalous situation) or multiple real-valued IRRs (an ambiguous situation). Additionally, when ranking multiple projects, the rank order determined by how far their IRRs exceed the cost of capital sometimes agrees with the rank order determined by how far their NPVs exceed zero, but sometimes it does not; projects’ cash flows can be such that the rank orders conflict (another ambiguity). Modern textbooks label these anomalies and ambiguities the "IRR pitfalls" (for example, see [1]). Given the pitfalls, conventional academic opinion is that NPV-based decisions are better than IRR-based decisions because the former possess "clarity and uniformity" [2].
Generations of business students have learned the academic preference for NPV; nevertheless, practitioners routinely ignore the advice. Research finds that managers in corporations [3], financial institutions [4], and government agencies [5] continue to use IRR.
The model described here employs the fundamental theorem of algebra to clarify the relationship between NPV and IRR, this clarification providing novel, cogent support to the academic preference for NPV. The fundamental theorem implies that an
th
n
degree polynomial in
x
solves for
n
values of
x
(see Related Link). Multiple interest rate analysis (MIRA) applies the fundamental theorem to the time value of money (TVM) polynomial. MIRA demonstrates that any TVM equation possesses a dual formula containing every interest rate (real and complex) solving its conventional counterpart, thereby employing interest rates that have been ignored by financial economists for centuries (see [6]).
The dual formula for NPV per dollar outlay,
NPV/
I
0
, is equal to the product of markups of every IRR over the cost of capital
r
, that is, the product of
m
j
for
j=1
to
n
, where
m
j
is defined by the expression
1+
m
j
(1+r)=1+
IRR
j

. This product conveniently divides into two parts, one familiar and the other unfamiliar. The familiar part is the markup,
m
1
, based on a selected value of IRR, usually the one produced by a financial calculator or spreadsheet, here designated
IRR
1
, meaning
m
1
=(
IRR
1
-r)/(1+r)
. The unfamiliar part is the product of the (previously ignored) unconventional markups,
m
j
for
j=2
to
n
. MIRA demonstrates that this unfamiliar component contains valuable information; the product is equal to a statistic summarizing structure in the cash flows (duration, as defined by Macaulay [8]).
This result has implications. First, multiple solutions for IRR do not constitute a pitfall to be avoided; rather, all solutions convey information, and therefore they are to be embraced. Second, NPV per dollar outlay is a superior criterion to the markup of IRR over the cost of capital because it contains the additional information about project cash flow structure conveyed by the unconventional markups. Third, inconsistent rankings of multiple investment projects by the two criteria are wholly explained by differences in the projects' cash flow structures.