Breadcrumbs Section. Click here to navigate to respective pages.

Chapter

Chapter

# The relationship between book (accounting) rates of return and the cost of capital for ﬁrms and capital projects

DOI link for The relationship between book (accounting) rates of return and the cost of capital for ﬁrms and capital projects

The relationship between book (accounting) rates of return and the cost of capital for ﬁrms and capital projects book

# The relationship between book (accounting) rates of return and the cost of capital for ﬁrms and capital projects

DOI link for The relationship between book (accounting) rates of return and the cost of capital for ﬁrms and capital projects

The relationship between book (accounting) rates of return and the cost of capital for ﬁrms and capital projects book

## ABSTRACT

Note here that At−1 − At is the amount by which the accountants have written down the book value of the capital project over the period from time t − 1 until time t. It is, in other words, the estimate that the accountants have made of the capital project’s deprecation over this period. Hence, we can subtract the depreciation from the cash ﬂow Ct generated by the capital project over the period from time t − 1 until time t to determine the proﬁt Pt that the accountants attribute to the capital project for this period. It then follows that the cash ﬂow can be restated in terms of the capital project’s proﬁt and book values as follows:

Ct = Pt + (At−1 −At)

Given this, we can replace the cash ﬂow variable in the expression for V0 and thereby restate the present value in terms of the capital project’s proﬁtability and book values as recorded by the ﬁrm’s accountants:

V0 = N∑ t=1

νtCt + νNRN −C0 = N∑ t=1

νt(Pt +At−1 −At)+ νNRN −C0

or

V0 = N∑ t=1

νtPt + (

νtAt−1 − N∑ t=1

νtAt

) + νNRN −C0

§ 6-3. Now, if we expand the two terms within parentheses in the above expression, we ﬁnd

νtAt−1 = ν1A0 + ν2A1 + ν3A2 + . . .+ νNAN−1

and

νtAt = ν1A1 + ν2A2 + ν3A3 + . . .+ νN−1AN−1 + νNAN

so that

νtAt−1 − N∑ t=1

νtAt

= ν1A0 + ν2A1 − ν1A1 + ν3A2 − ν2A2 + . . .+ νNAN−1 − νN−1AN−1 − νNAN We now add and subtract ν0A0 from the above expression, to give

νtAt−1 − N∑ t=1

νtAt

= (ν1A0 − ν0A0)+ (ν2A1 − ν1A1)+ (ν3A2 − ν2A2)+ . . .+ (νNAN−1 − νN−1AN−1) + ν0A0 − νNAN

But this latter result may also be stated as

νtAt−1 − N∑ t=1

νtAt = N∑ t=1

(νt − νt−1)At−1 + ν0A0 − νNAN

Note here, however, that the term within parentheses in this expression can be restated as follows:

νt − νt−1 = 1 (1+ r)t −

(1+ r)t−1 = 1− (1+ r) (1+ r)t =

−r (1+ r)t = −rνt

It then follows that

νtAt−1 − N∑ t=1

νtAt = − N∑ t=1

rνtAt−1 + ν0A0 − νNAN

Substituting this result into the expression for the present value of the capital project’s future cash ﬂows then gives

V0 = N∑ t=1

νtPt + (

νtAt−1 − N∑ t=1

νtAt

) + νNRN −C0

= N∑ t=1

νtPt − N∑ t=1

rνtAt−1 + ν0A0 − νNAN + νNRN −C0

Collecting terms and simplifying then leads to the following expression for the present value of the capital project’s future cash ﬂows:

V0 = N∑ t=1

νt(Pt − rAt−1)+ νN (RN −AN )− (C0 − ν0A0)

Now, the variable Pt − rAt−1 is the proﬁt attributed to the capital project over the period from time t − 1 until time t less the cost of capital multiplied by the book value at time t − 1. Moreover, if one thinks of rAt−1 as the proﬁt to be expected from the capital project over the period from time t − 1 until time t then Pt − rAt−1 is the unexpected or abnormal proﬁt that has accrued over this same time period (as in §5-10 of Chapter 5). Pt − rAt−1 is normally called the abnormal (or residual) earnings relating to the period, since it represents the proﬁt that remains after paying a normal return to the ﬁrm’s investment in the capital project as represented by its book value (again as in §5-10 of Chapter 5).