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).
