Aggregate Return On Investment for investments

under uncertainty

Carlo Alberto Magni

University of Modena and Reggio Emilia, Department of Economics “Marco Biagi”

CEFIN − Center for Research in Banking and Finance

viale Berengario 51, 41100 Modena, Italy

tel. +39-059-2056777, fax +39-059-2056937, E-mail: magni@unimo.it.

Webpage: <http://morespace.unimore.it/carloalbertomagni/>

forthcoming in International Journal of Production Economics

Aggregate Return On Investment for investments

under uncertainty

Abstract. This paper deals with capital budgeting decisions under uncertainty. We

present an Aggregate Return On Investment (AROI), obtained as the ratio of total (undis-

counted) cash flow to total invested capital and show that it is a genuine rate of return

which, compared with the risk-adjusted cost of capital, correctly signals wealth creation.

For choosing between two mutually exclusive projects, we derive an incremental AROI and

an incremental risk-adjusted cost of capital, by means of which two unequal-risk projects

can be correctly compared. Iterating the incremental procedure, we show that the AROI

approach correctly ranks any bundle of different-risk competing projects. Relations with

other criteria such as Modified Internal Rate of Return, Average In ternal Rate of Return,

Cash Multiple, and Profitability Index are provided.

Theoretically, the AROI approach constitutes a link between arbitrage choice theory

and corporate investment theory, and shows that explicit discounting is not necessary for

measuring economic profitability. Practically, the AROI is a user-friendly, easy-to-compute

rate of return derived from the same set of data required by the net present value (NPV).

Also, it does not incur the difficulties met by the Internal Rate of Return: in particular,

it is unique and it is based on economically significant capital values (i.e., market-driven

values). As such, the AROI significantly expresses the efficiency of the project’s invested

capital.

JEL Codes. G11, G12, G31, D24, D81, E22.

Keywords. Return On Investment, net present value, uncertainty, ranking, rate.

2

1 Introduction

A theoretically correct procedure for investment appraisal and decision under uncertainty

is the well-known Net Present Value (NPV ). NPV is an absolute measure of worth,

expressing the investor’s wealth increase in monetary amounts (Peterson and Fabozzi,

2002; Hartman, 2007; Berk and DeMarzo 2011). Brealey, Myers and Allen (2001, ch.

34) places the NPV as the first one in the list of the seven most important ideas in

finance. In real-life applications, relative measures of worth are also often required. The

reason why a relative measure of worth is often searched is that a percentage return is

easily understood and felt as an intuitive measure by most investors (Evans and Forbes,

1993). Furthermore, a rate of return supplies information on the efficiency of capital

that the NPV cannot supply. For example, consider firm A investing in a one-period

investment of 100 at a rate of return of 25%, and let 5% be the cost of capital. The NPV

is 125/1.05 − 100 = 19.05. Consider firm B investing 1000 in a one-period investment

at a rate of return of 7% with the same cost of capital. Then, the NPV is the same:

19.05 = 1070/1.05 − 1000, but firm A is more efficient, since, for every euro invested,

investors earn an active return of 0.25−0.05 = 0.2, whereas firm B’s investment generates

an active return of only 0.07 − 0.05 = 0.02 (the poorer efficiency is compensated by a

greater investment scale).

Among the relative measures of worth, the most widely used is the internal rate of

return (IRR). Both NPV and IRR are massively employed in industry (Remer and Nyeto,

1995a, 1995b; Slagmulder, Bruggeman and van Wassenhove 1995; Graham and Harvey,

2001; Sandahl and Sjogren, 2003). In particular, the use ofNPV is particularly widespread

in industry and engineering (Gallo and Peccati, 1993; Naim, 1996; Van der Laan, 2003; Giri

and Dohi, 2004; Borgonovo and Peccati, 2004, 2006). The IRR is often employed as well,

not only in industry and engineering, but also in real estate and investment performance

measurement (Jaffe, 1977; Graham and Harvey, 2001; Geltner, 2003). These two criteria

are often used together, and other criteria are also employed such as the profitability index,

3

residual income (e.g., EVA), return on investment, payback period (Remer, Stokdyk and

Van Driel 1993; Lefely, 1996; Graham and Harvey, 2001; Sandahl and Sjogren, 2003;

Lindblom and Sjogren, 2009; Magni, 2009; Hahn and Kuhn, 2012).

Unfortunately, IRR often conflicts with NPV and suffers from many weaknesses, some

of them only recently discovered (see Magni 2013). Among the difficulties, particularly

compelling is the fact that IRR is not capable of correctly ranking competing projects.

Some scholars advocate the use of an incremental IRR for this kind of problems, but the

difficulties of IRR reverberate on incremental IRR: the incremental IRR may not exist

or multiple incremental IRRs may exist. Most importantly, the incremental IRR is not

applicable when two (or more) projects have different risks, not even if it exists and is

unique: there are two (or more) risk-adjusted costs of capital, one for each project, so it is

not clear which one should be compared with the incremental IRR in order to determine

the preferred alternative.

In this paper we consider investments under uncertainty and describe a simple, intu-

itive, metric to capture a project’s economic profitability. For this purpose, we build upon

Magni (2011) and make use of an Aggregate Return On Investment (AROI), which is a

modification of the average internal rate of return (AIRR) introduced in Magni (2010).

In particular, resting on arbitrage choice theory, we show that the comparison of AROI

and the risk-adjusted cost of capital (COC) signals wealth creation. Contrary to IRR, the

AROI exists and is unique; consistently with basic tenets of corporate financial theory, it

does not make any assumption on reinvestment of cash flows. This differentiates AROI

from the well-known Modified Internal Rate of Return (MIRR): AROI is a project rate of

return, MIRR is a rate of return which is an average of the project’s rate of return and the

rate of return of the reinvested cash flows. This also implies that MIRR, as opposed to

AROI, is not really unique, because its value depends on the way the project’s cash-flow

stream is modified. We also show that AROI, while based on undiscounted values, does

take time value of money into account, though in a new, indirect way. It is just this

feature that enables AROI to give economic significance to some naıve approaches used

4

by real-life practioners, such as the cash multiple and the undiscounted profitability index.

We show that an incremental procedure can be applied to AROI in order to correctly

rank projects under uncertainty: an incremental AROI is derived, which is compared with

an incremental (risk-adjusted) cost of capital (COC), so as to obtain a ranking that is

equal to the NPV ranking. Both incremental AROI and incremental COC are weighted

averages of the two projects’ AROIs and COCs, respectively.

The remainder of the paper is structured as follows. Section 2 introduces a replicating

strategy whereby the investor can construct a benchmark asset which replicates the free

cash flows of the project. We show that the value of such a benchmark coincides with

the capital infused into the project. Section 3 is divided into two subsections: the first

one defines the AROI as total return on total capital and shows that it coincides with

the ratio of total cash flow on total capital. This implies that the NPV can be framed

as an aggregate excess return, namely the product of invested capital and the AROI, net

of the risk-adjusted cost of capital. The AROI acceptability criterion is stated. In the

second subsection, it is clarified that the AROI approach takes account of the time value

of money by incorporating it implicitly. Section 4 deals with choice between unequal-risk

mutually exclusive alternatives and project ranking: an incremental procedure is supplied

which takes account of the risk of the incremental project. The procedure guarantees

that the ranking of projects is equivalent to the ranking via NPV . An illustrative ex-

ample is presented in the following section. Some concluding remarks end the paper

and an Appendix is devoted to describing some relations between the AROI and (i) the

Average-Internal-Rate-of-Return approach, (ii) the Modified Profitability Index, (iii) the

Modified Internal Rate of Return, (iv) two rules of thumbs, the (Cash Multiple and the

undiscounted Profitability Index ). The latter metrics, sometimes used by practitioners but

considered inappropriate by academics, are resurrected (to some extent) thanks to the

AROI approach.

5

2 The replicating strategy and the benchmark asset

Consider a firm facing a project with initial cost c0 and let ft be the (random) free cash

flow generated by the project at time t = 1, 2, . . . , n, where n is the maturity date of the

last nonzero cash flow (i.e., ft = 0 for all t > n); let ft = E(ft) be its expected value

and let sn the project’s residual (scrap) value. The random variables ft, t = 1, 2, . . . , n,

sn, and r are assumed to be mutually independent. The expected cash flow vector is

~f = (−c0, f1, . . . , fn + sn) ∈ Rn+1 where sn = E(sn) is the expected residual value.

Assume that the capital market is complete and in equilibrium (i.e., no arbitrage exists)

and denote as r the risk-adjusted cost of capital (COC), which expresses the minimum

acceptable rate of return. The market value of the project is then

V0 =n∑

t=1

ft(1 + r)−t + sn(1 + r)−n

and represents the price the project would have if it were traded. Hence, the project NPV

is the difference between value and cost:

NPV0 = V0 − c0 =n∑

t=1

ft(1 + r)−t + sn(1 + r)−n − c0,

which measures the investor’s wealth increase. More generally, the time-t NPV is denoted

as NPVt := NPV0(1 + r)t.

Consider now a shift in perspective: assume an investor constructs an equal-risk portfolio,

denoted as p, which warrants the same payoffs ft of the project, t = 1, . . . , n and ask what

the expected terminal value of this portfolio should be in order to get the price of p equal

to c0. Letting s∗n be such a terminal value, the no-arbitrage principle implies that the

following equality must hold:

c0 =

n∑t=1

ft(1 + r)−t + s∗n(1 + r)−n.

6

This equality shows that p is constructed in such a way that r is its expected rate of

return. More precisely, r is the internal rate of return of p and, at the same time, the

risk-adjusted COC (i.e., discount rate) for the project. Portfolio p’s NPV is zero: in a

normal market where no-arbitrage pricing holds, all assets have zero NPV : “The insight

that security trading in a normal market is a zero-NPV transactions is a critical block

in [. . . ] corporate finance. Trading securities in a normal market neither creates nor

destroys values.” (Berk and DeMarzo 2011, p. 68.) Being a zero-NPV alternative to the

project, portfolio p acts as a benchmark asset, with which the project is compared to assess

value creation. Note that p replicates the project’s free cash flows from time 0 to time n,

while leaving a terminal value s ∗n which is, in general, different from sn. Given that the

two alternatives share the same free cash flows, acceptance of the project depends on the

difference between the expected terminal values, s∗n − sn: this amount just represents the

opportunity cost of investing in the project: the project is worth undertaking if and only

if s∗n − sn < 0 (see also Remark 1).

Now, consider that V0 − NPV0 represents the market value of the project net of in-

vestors’ wealth increase; as we know, this is just equal to c0 (by definition of Net Present

Value), which is the capital invested into the project at time 0. We then generalize this

equality and give the following definition of invested capital.

Definition 1. At time t < n, the capital ct invested in a project is given by the difference

between the market value of the project and the wealth increase:

ct := Vt −NPVt.

Let V ∗t be the expected time-tmarket value of portfolio p. In every period, the following

recursive equation holds:

V ∗t = V ∗t−1 · (1 + r)− ft (1)

where V ∗0 = c0. Armed with the above definition, we show the following result.

7

Proposition 1. For every t = 0, 1, . . . , n − 1, the capital invested in the project is equal

to the market value V ∗t of portfolio p:

ct = V ∗t (2)

Proof. From (1), V ∗t = c0(1+r)t−∑t

h=1 fh(1+r)t−h. Given that Vt =∑n

h=t+1 fh(1+r)t−h

and NPVt =∑n

h=1 fh(1 + r)t−h − c0(1 + r)t, one gets

Vt −NPVt = c0(1 + r)t −t∑

h=1

ft(1 + r)t−h

which is just (2).

Essentially, the firm infuses the capital ct−1 in the project at the beginning of the

interval [t− 1, t], t = 1, 2, . . . , n and receives an end-of-period payoff equal to the free cash

flow ft plus the project end-of-period capital ct. The sequence of the amounts of cpaital

invested by the firm in the various periods is ~c = (c0, c1, c2, . . . , cn−1).

The sum C := c0 + c1 + . . .+ cn−1 is the aggregate capital infused into the project and

represents the project’s invested capital. In the next section we show that a simple ratio

of aggregate cash flow to invested capital constitutes a unique, NPV -consistent rate of

return.

3 The Aggregate Return on Investment

3.1 AROI and value creation

A rate of return is, literally, an amount of return per unit of invested capital. Therefore,

it is obtained by dividing return by capital. Let It be the return generated by the project:

It = ft + ct − ct−1 t = 1, 2, . . . , n− 1

In = fn + sn − cn−1.(3)

8

Hence, consider the aggregate return I :=∑n

t=1 It and the invested capital C introduced

above. We call

i =I

C(4)

the Aggregate Return On Investment (AROI).1 We now show that this rate of return is

consistent with the NPV : the comparison of i and r signals value creation. First, note

that

I =n∑

t=1

It =n−1∑t=1

(ct − ct−1 + ft) + sn − cn−1 + fn = −c0 +n∑

t=1

ft + sn,

so that AROI can be computed as the ratio of total cash flow to total capital:

i =F

C(5)

where F :=∑n

t=1 ft+sn−c0. Also, NPVn = sn+∑n

t=0 ft(1+r)n−t−c0(1+r)n = sn−s∗n.

But

sn − s∗n = F + c0 −n∑

t=1

ft − s∗n

= F +[c0 −

( n∑t=1

ft + s∗n)]

= F −[n−1∑t=1

(ft + V ∗t − V ∗t−1) + (fn + s∗n − V ∗n−1)]

= F −n∑

t=1

rV ∗t−1 = F −n∑

t=1

rct−1

(6)

Hence, one finds that the investor’s wealth increase is

NPVn = C · (i− r) (7)

1Note that this is a modified AIRR with cost of capital equal to zero (see Appendix).

9

or, as NPV0 = NPVn(1 + r)−n = (sn − s∗n)(1 + r)−n,

NPV0 = PV [C] · (i− r) (8)

where PV [C] = C(1+r)−n is the present value of C. We have then proved the consistency

of the AROI with the NPV criterion, as the following proposition states.

Proposition 2. The investors’ wealth increase is measured by the product of the AROI,

net of the cost of capital, and the invested capital C (eq. (7)). Therefore, the project is

worth undertaking if and only if the AROI exceeds the cost of capital: i > r.2

Note that the value created per unit of invested capital is

NPVnC

=NPV0PV [C]

,

which is an adjusted profitability index (API) which signals that value is created if and

only if it is positive.3

Example 1. Consider a project whose initial cost is 100 and the free cash flows are f1 =

20, f2 = 50, f3 = 80, with an expected scrap value equal to s3 = 10. The risk-adjusted

COC is r = 5%. The infused capital amounts are c0 = 100, c1 = 100 · 1.05− 20 = 85, c2 =

85 · 1.05− 50 = 39.25. Therefore, the project’s invested capital is C = 100 + 85 + 39.25 =

224.25. The total cash flow, inclusive of the scrap value, is F = −100+20+50+80+10 = 60.

The project’s AROI is then i = 60/224.25 = 26.76%, which is greater than 5%: thus, the

project is worth undertaking. The NPV is obtained as NPV0 = (1.05)−3 ·224.25·(0.2675−

0.05) = 42.14.

Table 1 and Figure 1 show the relations among AROI, IRR, and COC. We also con-

sider the well-known Modified Internal Rate of Return (MIRR) in the reinvestment ap-

2If C < 0, then the project is a net financing, and the AROI represents a borrowing cost while the costof capital expresses the maximum acceptable borrowing rate: the project is then worth undertaking if andonly i < r.

3“Adjusted” as it takes account of the aggregate capital rather than the initial capital. The API is justequal to the residual rate of return i− r.

10

proach (see appendix). It is worth noting that, while IRR is obviously constant, AROI

is decreasing with respect to r, whereas MIRR is increasing. In particular, using the

45◦-degree line, it can be seen that when COC is below the IRR of 21.76%, then one

obtains AROI > IRR > MIRR > COC and when IRR is below COC, then one obtains

COC > MIRR > IRR > AROI.

Example 2. Consider a project whose initial cost is 1000 and the free cash flows are f1 =

3900, f2 = −5030, f3 = 2000, and the residual value is s3 = 145. Assume the risk-adjusted

COC is r = 35%. The equation −1000+3900(1+x)−1−5030(1+x)−2+2145(1+x)−3 = 0

has three solutions: IRR1 = 10%, IRR2 = 30%, IRR3 = 50%. Conversely, the AROI is

unique and easily computed: the total cash flow F = −1000 + 3900 − 5030 + 2145 = 15,

and the infused capital is

c0 = 1000

c1 = 1000 · 1.35− 3900 = −2550

c2 = −2550 · 1.35 + 5030 = 1587.5

whence the project’s invested capital C = 37.5. Therefore, the AROI is F/C = 15/37.5 =

40%; the AROI is greater than the cost of capital, so the project is worth undertaking.

The NPV is obtained as (1.35)−3 · 37.5 · (0.4− 0.35) = 0.76.

Remark 1. The use of a benchmark asset and arbitrage pricing for deriving the AROI

approach is particularly compelling for two reasons: first of all, it frames the NPV criterion

as a choice between a project and a zero-NPV alternative which warrants the same cash

flows as the project. So, if the investors invest in the project, the expected cash-flow stream

is ~f = (f0, f1, . . . , fn−1, fn + sn); if they invest in the benchmark asset, the expected cash-

flow stream is ~f∗ = (f0, f1, . . . , fn−1, fn +s∗n). Given that cash flows are equal, choice only

depends on the difference between the two terminal values, s∗n − sn. In other words, the

project is just equal to the replicating asset, net of the opportunity cost of investing in

the project: ~f = ~f∗ − ~∆s∗ where ~∆s∗ := (0, 0, . . . , s∗n − sn) is the excess-terminal-value

11

vector. If s∗n−sn is positive, the cost of investing in the project exceeds the benefits, so the

project destroys value; if s∗n − sn is negative, then value is created. Thus, s∗n represents a

benchmark terminal value for wealth creation: it is the minimum required terminal value

to be achieved by the project for wealth increase to occur.

Further, an important byproduct of this arbitrage-based analysis is that it makes clear

that the NPV criterion and the AROI approach do not presuppose any reinvestment of

cash flows: the NPV only measures the (present value of the) difference between the cash-

flow stream generated by project ~f and the cash-flow stream generated by the benchmark

asset ~f∗. Therefore, one does not need estimate the reinvestment rate of the interim cash

flows. This is consistent with corporate financial theory, according to which a project

value is not affected by how interim cash flows are spent in the future:

The value of a project does not depend on what the firm does with the cashflows generated by that project. A firm might use a project’s cash flows to fundother projects, to pay dividends, or to buy an executive jet. As a result, thereis generally no need to consider reinvestment of interim cash flows”. (Ross,Westerfield and Jordan 2011, p. 250).

3.2 AROI and the time value of money

It may come as a surprise the fact that, in the definition of AROI (eq. (4)), interests and

capital values are aggregated with no capitalization process (i.e., with no compounding

nor discounting). However, owing to (8), its NPV -consistency is quite natural, because

NPVn (NPV0) is the product of the adjusted profitability index, i − r, and the invested

capital, C (PV [C]). This means that time value of money is indeed taken into account,

somehow. We now dig deeper into the issue and explain how the time value of money is

implicitly taken into account in the AROI approach.

First of all, let yt, t = 0, 1, . . . , n be any measure of capital satisfying y0 = −f0 and

yn = 0 and let

it =ft + yt − yt−1

yt−1(9)

12

be the return on investment. The product yt−1(it − r) is called Residual Income, which

we denote as RIt. It is well known that the economic value created can be written as the

sum of capitalized residual incomes:

NPV0 =

n∑t=1

RIt(1 + r)−t ⇐⇒ NPVn =

n∑t=1

RIt(1 + r)n−t

(see Peasnell 1981, 1982; Martin and Petty 2000; Lundholm and O′Keefe 2001; Fernandez

2002; Martin, Petty and Rich 2003; Ohlson 2005; Ben-Horin and Kroll 2010). Consider

now the excess return defined as ξt := it ·yt−1−r ·V ∗t−1. This difference expresses the period

return generated by the project over and above the return generated by the benchmark.

We first study the relations between RIt and ξt and then the relations between ξt and

AROI. Note that RI1 = ξ1, since V ∗0 = −f0. As for t > 1, note that (9) can be rewritten

as ft = yt−1(1 + it)− yt, whence

V ∗t−1 = V ∗0 (1+r)t−1−t−1∑k=1

fk(1+r)t−1−k = V ∗0 (1+r)t−1−t−1∑k=1

(yk−1(1+ik)−yk)(1+r)t−1−k.

Manipulating algebraically, V ∗t−1 = yt−1−RI1(1+r)t−2−RI2(1+r)t−3− . . .−RIt. Hence,

ξt can be written as

ξt = RIt + rt−1∑k=1

RIk(1 + r)t−1−k. (10)

This shows that the return over and above the benchmark return incorporates the residual

income along with the interest on past accumulateed residual incomes. In other words, ξt

itself takes time into account. By induction upon (10), the following holds:

t∑k=1

ξt =

t∑k=1

RIk(1 + r)t−k

13

for every t ≥ 1. Picking t = n,

n∑k=1

ξk =

k∑k=1

RIk(1 + r)n−k = NPVn. (11)

This just explains why, in order to get the created economic value, the excess returns

ξt’s can be summed with no need of compounding: they already take into account the

time value of money in an implicit way, so the sum of such excess returns is just the

sum of the accumulated residual incomes. But if such excess returns do not need any

capitalization, cash flows (and capital values) do not need either. The reason is that∑nk=1 ξk =

∑nk=1(ik · yk−1) −

∑nk=1(rV

∗k−1) and,

∑nk=1(ik · yk−1) =

∑nk=1 fk + f0 = F

whatever the choice of yk, k = 1, 2, . . . , n− 1, Therefore,

NPVn =n∑

k=1

ξk = F −n∑

k=1

rV ∗k−1 = F − rn∑

k=1

ck−1 (12)

(see eq. (6) above). Dividing by C,

NPVn =

(F

C− r)· C = (i− r) · C

which is just (7). So, the AROI approach does take time into account, though in a new,

indirect way which enables one to get a rate of return where capitalization does not appear

explicitly.

4 Project ranking under uncertainty

Consider two unequal-risk projects, labeledA andB, and let ~f j = (f j0 , fj1 , . . . , f

jnj ) ∈ Rnj+1

be project j’s vector of expected cash flows, j = A,B, where nj is the maturity date of

the last nonzero cash flow of project j. Denote as rj the risk-adjusted cost of capital for

14

project j. From (8), the two projects NPV s are given by

NPV j0 = PV [Cj ] · (ij − rj) j = A,B. (13)

One can rank the two projects by making use of the incremental project, labeled A− B,

whose cash flows are obtained as the difference between the two projects’ cash flows:

~fA−B = (fA−B0 , fA−B1 , . . . , fA−Bn ) with fA−Bt := fAt −fBt and n = max[nA, nB]. Therefore,

~fA = ~fB+ ~fA−B, so project A can be viewed as a portfolio of project B and the incremental

project A − B. Therefore, alternative A is preferred to alternative B if and only if the

incremental alternative is acceptable according to the acceptability criterion stated above.

However, in general, A − B has a risk which differs from both A’s and B’s. Neither

rA nor rB can be used as a cutoff rate for A− B. Intuitively, the risk of the incremental

project should somehow be a combination of the two risks. Evidently, even abstracting

from problems of existence and uniqueness, an IRR-based incremental procedure is not

applicable. In contrast, the AROI model successfully copes with this kind of problems, as

we show below.

The question is: how can one find the appropriate incremental rate of return iA−B and

risk-adjusted cost of capital rA−B? The answer derives from eq. (13): by value additivity,

NPV A−B0 = NPV A

0 −NPV B0 = PV [CA] · (iA − rA)− PV [CB] · (iB − rB). (14)

We then use the invariance requirement

PV [CA]·(iA−rA)−PV [CB]·(iB−rB) = PV [CA]·(iA−B−rA)−PV [CB]·(iA−B−rB) (15)

and solve for iA−B:

iA−B =iA · PV [CA]− iB · PV [CB]

PV [CA]− PV [CB]. (16)

The incremental AROI, iA−B, represents the incremental rate of return of A over B, which

15

is formalized as a weighted mean of the two alternatives’ AROIs.4

However, to get this incremental return, the firm incurs an incremental opportunity

cost (the incremental capital PV [CA] − PV [CB] might be invested in the market). To

quantify such an opportunity cost, one must take account of the incremental risk incurred

by undertaking alternative B as opposed to undertaking alternative A. To this end, we

impose the invariance requirement

PV [CA]·(iA−rA)−PV [CB]·(iB−rB) = PV [CA]·(iA−rA−B)−PV [CB]·(iB−rA−B) (17)

and solve for rA−B:

rA−B =rA · PV [CA]− rB · PV [CB]

PV [CA]− PV [CB]. (18)

The incremental COC, rA−B, is then a weighted mean of the two projects’ COCs. The

result is rather intuitive: the incremental alternative can be interpreted as a portfolio

consisting of a (long) position in A and a (short) position in B. Therefore, the risk is

a combination of the risks of the two positions, so the AROI is the combination of the

two positions’ expected rates of return, and the cost of capital is the combination of the

respective required rates of return.

An equivalent way of deriving the incremental AROI is as follows. For a generic project,

the AROI is i = FC . Multiplying and dividing the right-hand side by (1 + r)−n one gets

i =PV [F ]

PV [C](19)

where PV [·] denotes discounted value (from n to 0). Hence, applying (6) to A−B,

NPV A−Bn = FA − FB − rA · CA + rB · CB

whence

NPV A−B0 = PV [FA]− PV [FB]− rA · PV [CA] + rB · PV [CB].

4More precisely, it is an affine combination of the AROIs.

16

Applying (19) to the incremental rpoject A−B, one finds

iA−B =PV [FA−B]

PV [CA−B]=PV [FA]− PV [FB]

PV [CA]− PV [CB](20)

so the incremental NPV can be written as

NPV A−B0 = PV [CA−B] · (iA−B − rA−B) (21)

where PV [CA−B] := PV [CA] − PV [CB] is the aggregate incremental capital. We have

then proved the following NPV-consistent decision criterion.

Proposition 3. Consider two mutually exclusive project, A and B. A is preferred to B

if and only if the incremental AROI exceeds the incremental risk-adjusted cost of capital:

iA−B > rA−B.5

Note that the incremental COC can be computed with the following shortcut:

rA−B = iA−B − NPV A−B0

PV [CA−B](22)

and that NPV A−B0 /PV [CA−B] = iA−B − rA−B is the API of the incremental project

(difference between incremental AROI and incremental COC), whose sign directly signals

wealth creation per unit of invested capital.

Practically, the steps to follow when a pair of competing projects is to be ranked are:

(i) compute the incremental AROI via (16) or (20),

(ii), compute the incremental COC via (18) or (22),

(iii) compare the incremental AROI with the incremental COC (or, equivalently, com-

pute the sign of the incremental API)

Choice between mutually exclusive projects is but a particular case of project ranking

5If PV [CA−B ] < 0, then one can consider the incremental project B − A and apply the inequalityiB−A > rB−A.

17

where the number of competing projects is m = 2. For m > 2, the incremental technique

can be iterated via pairwise comparisons.

Project ranking. In a bundle of m > 2 alternative, the ranking is determined by

(a maximum of) m!2·(m−2)! pairwise comparisons of the incremental AROIs (ij−l) and the

incremental costs of capital (rj−l), j, l = 1, 2, . . . ,m, j < l.

5 A numerical example

Consider three projects A, B, and C, described in Table 2, such that nA = nB = 4, nC = 3.

They are assumed to have different risks, and their risk-adjusted COCs are assumed to

be equal to rA = 10%, rB = 14%, rC = 8%. These projects are acceptable, as can be

gleaned by both the NPVs and the comparisons of respective AROIs and COCs. Ranking

is determined by the incremental analysis, which brings about three pairwise comparisons.

Consider the comparison between A and B: the former can be seen as a portfolio of B

and the incremental project A−B, which has a rate of return of 7.7%. The required rate

of return turns out to be 2.7%, which signals wealth increase. Therefore, the portfolio

is acceptable or, in terms of ranking, A is ranked above B. The incremental NPV (=3)

is obtained by multiplying the invested capital (=58.8) by the excess return rate (=5.5),

which coincides, as expected, with the difference between the two NPV s (=17.5− 14.6).6

Likewise, C is found to be preferred to B as well as to A. So, the ranking is

C � A � B,

which is the same ranking as that supplied by the NPV s.

6Numerical discrepancies are due to rounding errors.

18

6 Conclusions

This paper constitutes a link between corporate investment theory and arbitrage choice

theory: a suitable portfolio replicating the project’s cash flow is used for deriving a

project’s Aggregate Return on Investment (AROI).

The AROI can be used for uncertain as well certain cash-flow streams. Mathematically,

it is easy to compute: it is the ratio of total cash flow to total invested capital, which is

equal to the market value of the replicating portfolio. AROI is NPV -consistent, in that it

correctly captures wealth increase via comparison with the risk-adjusted cost of capital.

Contrary to IRR and MIRR, it is unique. The AROI does not discount nor compound

cash flows (nor capital values): the time value of money is taken into account in an

indirect way by incorporating it in the excess return, i.e., the return over and above the

benchmark’s return. Consistently with NPV and unlike the MIRR, AROI does not make

any assumption on reinvestment of interim cash flows, so abiding by the basic principles

of corporate financial theory and providing a genuine rate of return for the project under

scrutiny.

The AROI correctly ranks projects under uncertainty: its ranking is consistent with the

NPV ranking. In particular, for choosing between two unequal-risk competing projects

j and k, an incremental technique can be employed: j is preferred to k if and only if

the incremental AROI exceeds the incremental (risk-adjusted) cost of capital. Both the

incremental AROI and the incremental cost of capital are obtained as weighted means

of the projects’ AROIs and COCs, respectively. In general, for a bundle of competing

projects, a pairwise incremental analysis is employed, leading to a ranking which is always

equal to the ranking of the NPV approach.

Acknowledgments. The author wishes to thank two anonymous reviewers for their

useful suggestions.

19

Table 1. Behavior of AROI, MIRR, IRR, and COC (Example 1)

𝑟 (COC) 0.0% 2.5% 5.0% 7.5% 10.0% 12.5% 15.0% 17.5% 20.0% 21.76% 22.5% 25.0% 27.5% 30.0%

IRR 21.76% 21.76% 21.76% 21.76% 21.76% 21.76% 21.76% 21.76% 21.76% 21.76% 21.76% 21.76% 21.76% 21.76%

MIRR 16.96% 17.59% 18.06% 18.61% 19.16% 19.71% 20.27% 20.82% 21.37% 21.76% 21.93% 22.48% 23.04% 23.59%

AROI 28.57% 27.64% 26.76% 25.91% 25.10% 24.33% 23.60% 22.90% 22.22% 21.76% 21.58% 20.96% 20.37% 19.80%

Figure 1. Behavior of AROI, MIRR, IRR, and COC (Example 1).

21.76% 21.76%

0.00%

5.00%

10.00%

15.00%

20.00%

25.00%

30.00%

0.0% 2.5% 5.0% 7.5% 10.0% 12.5% 15.0% 17.5% 20.0% 21.76% 22.5% 25.0% 27.5% 30.0%

Cost of Capital-COC IRR MIRR AROI

IRR IRR

Table 2. Ranking of three alternatives

Time

Cash flow

(𝒇𝒕𝒋)

Invested capital

(𝒄𝒕𝒋)

Incremental cash flow

(𝒇𝒕𝒋−𝒌

)

Incremental invested capital

(𝒄𝒕𝒋−𝒌

)

A B C A B C

A−B B−C A−C A−B B-C A-C

0 −130 −50 −60 130.0 50.0 60.0

−80 10 −70 80 −10 70

1 65 0 10 78.0 57.0 54.8

65 −10 55 21 2.2 23.2

2 50 35 50 35.8 30.0 9.2

15 −15 0 5.8 20.8 26.6

3 40 −10 35 −0.6 44.2 0.0

50 −45 5 −44.8 44.2 −0.6

4 25 75 − 0.0 0.0 −

−50 75 25 0.0 0.0 0.0

𝑃𝑉[𝐶] 166.1 107.3 98.4

𝑃𝑉[𝐶] 58.8 8.8 67.7

NPV 17.5 14.6 19.9

NPV 3.0 −5.3 −2.4 AROI 20.6% 27.6% 28.2%

AROI 7.7% 20.6% 9.4% 𝒓𝒋 10.0% 14.0% 8.0%

𝒓𝒋−𝒌 2.7% 80.8% 12.9%

acceptable acceptable acceptable preference 𝐴 ≻ 𝐵 𝐶 ≻ 𝐵 𝐶 ≻ 𝐴

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26

A Relations of AROI with other criteria

In this appendix we describe some relations the AROI approach bears to some recent notions such as Aver-

age Internal Rate of Return (AIRR) (Magni 2010) and Modified Total Invested Capital (MTIC) (Ben-Horin

and Kroll 2010), as well as to such rules of thumb as the Cash Multiple (CM) and the undiscounted Prof-

itability Index (UPI), which are sometimes used by practitioners. Finally, we shed light on the differences

between AROI and the Modified Internal Rate of Return (MIRR).

A.1 Average Internal Rate of Return

Magni’s (2010) AIRR approach shares with the AROI approach a simple, intuitive product structure which

links the rate of return and the NPV. The latter is reframed as a product of a capital base and an adjusted

profitability index, which is equal to the difference between the project rate of return and the cost of

capital. In particular, Magni (2010, Theorem 2, eq. (6)) shows that, for any vector ~y = (y0, y1, . . . , yn−1)

of capital values (i.e., for any depreciation schedule), NPV can be computed as

NPV0 =k(~y)− r

1 + r·

n∑t=1

yt−1 · (1 + r)−(t−1) (23)

where

k(~y) =i1 · y0 + i2 · y1(1 + r)−1 + . . .+ i2 · yn−1(1 + r)−(n−1)

y0 + y1 · (1 + r)−1 + . . .+ y1 · (1 + r)−(n−1)(24)

represents the project rate of return associated with the capital stream ~y (see also Magni 2010, eq. (7);

Magni 2013, eq. (18b)).7 This rate of return is called by the author Average Internal Rate of Return

(AIRR).

Suppose now an IRR exists and let yIRRt :=

∑nh=t+1 fh(1 + IRR)t−h be the capital induced by the

IRR, also known as project balance (Teichroew, Robichek and Montalbano 1965a,b) or unrecovered balance

(Hajdasinski 2004). Hence, choosing yt = yIRRt in (24), one finds that the corresponding AIRR is just the

IRR: k(~y) = IRR (see Magni 2010, Theorem 3; Magni 2013, Figure 2), so that eq. (26) becomes

NPV0 =IRR− r

1 + r·

n∑t=1

yIRRt−1 · (1 + r)−(t−1) (25)

(see Hazen 2003, Theorem 1; Ben-Horin and Kroll 2010, Theorem 1).

7Magni (2013) suggests that the choice of the appropriate capital stream ~y is a matter of informedjudgment: market values for financial portfolios, outstanding balances for loans, book values for corporateprojects (where pro forma financial statements are available) etc.

27

Note that (23) can be rewritten as

NPV0 =(k(~y)− r

)·

n∑t=1

yt−1(1 + r)−t (26)

which says that the economic value created is the product of a capital base and an excess return rate. If

one picks yt := yIRRt , the capital base is IRR-induced and (23) becomes

NPV0 =(IRR− r

)·

n∑t=1

yIRRt−1 · (1 + r)−t (27)

(see Hazen 2003, Theorem 1). The amount∑n

t=1 yt−1(IRR) · (1 + r)−t is the aggregate project balance,

which Ben-Horin and Kroll (2010) call Modified Total Invested Capital (MTIC) (see Ben-Horin and Kroll’s

2010 eq. (17)).

The AROI can be derived from AIRR in the following way: consider an AIRR where the measure of

capital selected is market-based: yt := ct; then, (24) becomes

k(~c, r) =

∑nt=1 It · (1 + r)−(t−1)∑n

t=1 ct−1 · (1 + r)−(t−1)

where dependence on r is highlighted. If the values are not discounted, then the result is just the AROI:

i =∑n

t=1 It∑nt=1 ct−1

= IC. As previously seen, subtracting the COC and multiplying by the capital base C, one

gets the product structure NPVn = C · (i− r) or, equivalently, NPV0 = PV [C] · (i− r). Taking the ratio

of the created economic value and the capital base, one gets the API:

API = i− r =NPVn

C. (28)

A.2 Modified Profitability Index

Eq. (28) is conceptually analogous to Ben-Horin and Kroll’s (2010) Modified Profitability Index (MPI),

which is equal to IRR − r. However, it is substantially different for the following reasons: (i) given that,

in general, C 6= MTIC, the API is different from the MPI: they express two profitability indexes referred

to different capital bases; (ii) no problem of multiplicity exists for API, as AROI is unique; (iii) the capital

base C consists of interim values which are market-based (i.e., ct = ct−1(1 + r) − ft), whereas MTIC is

derived from IRR-based interim values (i.e., cIRRt = cIRR

t−1 (1 + IRR)− ft)), (iv) the API and the MPI can

signal value creation in symmetric way; for example, if MTIC < 0 < C, then the AROI is an investment

rate whereas the IRR is a borrowing rate, so the API is a profitability index on a net investment, whereas

MPI is a profitability index on a net borrowing (the opposite holds if C < 0 < MTIC).

Remark 2. From (13), the choice between A and B can be framed as (assuming both capital bases are

28

positive)iA − rA

PV [CB ]>iB − rB

PV [CA]. (29)

Formally, the same structure obtains with the AIRR paradigm:

kA(~yA)− rA

Y B>kB(~yB)− rB

Y A. (30)

where Y j :=∑n

t=1 yjt−1(1 + r)−(t−1), j = A,B. If the evaluator chooses the project balance as the relevant

capital value (i.e., if one picks yt = yIRRt ), then Y j = MTIC and (30) becomes

MPIA

MTICB>

MPIB

MTICA(31)

where MPIj = IRRj−rj , j = A,B. Equation (31) is Ben-Horin and Kroll’s (2010) equation (19). Despite

(29) and (31) share the same formal structure, not only the depreciation method is different (market-based

and IRR-based, respectively) but also, as already seen, the capital bases can have different signs and,

therefore, different economic meaning. Further, the aggregation of the capital values is symmetric: in

the AROI approach capital values are first summed and then discounted, whereas in the AIRR approach

capital values are first discounted and then summed.

Evidently, eqs. (29)-(31) do not provide any clue on the incremental rate of return and on the incre-

mental cost of capital, which are valuable pieces of information: the incremental rate of return says how

much project A is more profitable than project B in relative terms, and the incrermental cost of capital

indirectly measures the incremental risk associated with project B.8

A.3 Cash Multiple and Undiscounted Profitability Index

Finally, we show that the AROI approach can give economic significance to some rules of thumb sometimes

used by real-life practitioners who do not take time into account:

A naıve approach that is often used is to divide the inflows by the outflows [. . . ] Thisformula does not work in a multiperiod setting. [. . . ] The naıve approach − because it doesnot properly take into account the timing of the cash flows − is not a correct measure ofreturn (Rao, 1992, pp. 74−75)

This index is often called Cash Multiple (CM).

8For example, using the Capital Asset Pricing Model, one can use the security market line, rA−B =rf + βA−B · (rm − rf ) where rf is the risk-free rate and rm is the expected market rate of return. Aftercomputation of rA−B , one reverses the equality to get the incremental risk:

βA−B =rA−B − rfrm − rf

.

29

Real-world practitioners often use [. . . ] the cash multiple (or multiple of money) as alternativevaluation metrics. [. . . ] The cash multiple (also called the multiple of money or absolutereturn) is the ratio of the total cash received to the total cash invested. [. . . ] The cashmultiple is a common metric used by investors [. . . ] It has an obvious weakness: The cashmultiple does not depend on the amount of time it takes to receives the cash (Berk and DeMarzo, 2011, p. 663).

Letting f−t denote an outflow and f+

t denote an inflow, the CM is then formalized as

CM =f+

f− = 1 +f+ − f−

f− .

Another similar index is the undiscounted profitability index (UPI), that is, a profitability index where

cash flows are not discounted:

UPI =

∑nt=0 ft

c0=f+ − f−

c0.

This index is considered incorrect as well:

A few companies do not discount the benefits or costs before calculating the profitabilityindex. The less said about these companies the better (Brealey, Myers and Allen 2011,p. 143).

Evidently, UPI and CM provide analogous pieces of information, as referred to the initial cash investment

c0 or the total cash investment f− respectively.9

We can rewrite UPI as an adjusted AROI:

UPI =

∑nt=0 ft

C

C

c0= i ·

(C

c0

).

Therefore, exploiting Proposition 2, we can state a correct acceptability criterion:

undertake the project if and only if UPI > %

where % := r · Cc0

is an adjusted cost of capital which adjusts for investment scale. Analogously, the CM can

be retrieved in the same way. Assuming c0 is the only outflow, CM = 1 + UPI, the correct acceptability

criterion becomes

undertake the project if and only if CM > 1 + %.

Therefore, the AROI approach is able to rescue two rules of thumb and give them economic significance:

CM and UPI reveal a different way of accounting for profitability (see section 3.2); the (non-trivial) caveat

is that they cannot be compared directly with r, but with a cost of capital % which adjusts for the capital

base.

9Note that if f−=c0 (i.e., the only cash investment is made at time 0), then CM = 1 + UPI.

30

A.4 Modified Internal Rate of Return

Another methodology often used by practitioners and sometimes suggested in the literature is the Modified

Internal Rate of Return (MIRR), which consists of modifying the cash flows of the project in such a way

that the IRR of the modified cash-flow stream exists and is unique. Contrary to IRR, existence of MIRR is

guaranteed, but there are some disadvantages that are worth underlining and that differentiate the MIRR

methodology from the AROI approach:

(i) Ambiguity. It is not clear how the original cash-flow stream should be modified. Ross, Westerfield,

and Jordan (2011) (henceforth RWJ) classify the procedure into three classes: the discounting

approach, which consists of discounting back the negative cash flows; the reinvestment approach,

which consists of compounding all cash flows except the first out to the end of the project’s life;

the combination approach, where negative cash flows are discounted back and positive cash flows

are compounded to the end of the project. Evidently, there are many other (indeed, infinite) ways

of adjusting a series of irregular cash flows into a regular one which supplies a unique IRR (the so-

called Sinking Fund Methods represent other ways of obtaining MIRRs. See Herbst 2002, ch. 11).

Furthermore, consider that the prospective reinvestment rate might be different from r. If this is

the case, a further two-rate approach is generated, based on two rates: the cost of capital r is used

for discounting the negative cash flows, and the reinvestment rate is used for positive cash flows

(see Hartmann 2007, p. 397). Essentially, MIRR is not a metric, but a methodology comprising a

vast class of metrics: there are many different ways of adjusting the cash-flow stream, so there are

many different MIRRs, and “there is no clear reason to say one of our three methods is better than

any other” (Ross, Westerfield and Jordan 2011, p. 250). As a result, the MIRR is not really unique:

there are as many MIRRs as are the ways of modifying the cash-flow stream.10

(ii) Meaning. The MIRR is not the project ’s rate of return: “it’s a rate of return on a modified set

of cash flows, not the project’s actual cash flows” (Ross, Westerfield and Jordan 2011, p. 250).

Indeed, to take reinvestment of interim cash flows into consideration means to compute a rate of

return which is an average of the project rate of return and the rate of return of the reinvestments

(which have not to do with the project). As Brealey, Myers and Allen (2011, p. 141) put it: “The

prospective return on another independent investment should never be allowed to influence the

investment decision”.

(iii) Structure. The natural product structure we have previously described does not hold for MIRR.

More precisely, it is true that one could well reverse the equality x · (MIRR− r) = NPV0 for x and

10Evidently, this non-uniqueness is different from the non-uniqueness of IRR: “multiple IRRs” means“multiple solutions of a polynomial equation”, whereas “multiple MIRRs” means “multiple ways of modi-fying the project’s cash flow-stream”.

31

solve for x = NPV0/(MIRR− r) to find the capital base associated with MIRR, but the solution is

not economically significant, for it has nothing to do with the amounts of capital actually invested in

the project nor with the reinvested cash flows (conversely, MTIC and C are economically significant:

MTIC is the aggregate invested capital induced by the IRR, and C is the aggregate capital induced

by the market).

(iv) Reinvestment assumption. As seen in Remark 1, the NPV decision criterion is not based on any

reinvestment assumption, whereas the MIRR methodology makes explicit use of reinvestment (ex-

cept the discounting approach). This means that the MIRR is based on assumptions which are

different from the NPV, so providing non-equivalent information. Also, if the reinvestment rate

differs from r, then the MIRR cannot even guarantee consisteny with NPV .

(v) Incremental MIRR. An incremental procedure for the MIRR is not possible if the two projects have

unequal risk, for it is not clear how the incremental cost of capital should be computed.

AROI does not suffer from the above mentioned drawbacks: (i) given that it does not modify cash flows,

it expresses, genuinely, a project’s rate of returnt; (ii) no (conceptual nor) computational ambiguity arises,

so it is unique ; (iii) it fulfills the significant product structure according to which economic value created

is the product of a capital base and a residual rate of return (project rate of return minus COC); (iv) it

does not make any assumption on reinvestment, consistently with basic principles of corporate finance and

with the NPV, (v) the incremental cost of capital can be easily calculated, as shown in section 4.

32