2.1 Case with Unique Life-Spans
Deriving the long-run profitability of the firm will first be tackled under the conventional assumption of unvarying life-spans for the individual capital investments underlying the cash outflows and cash inflows making up the firm.
To accomodate growth in the analysis assume that the capital
expenditures in the firm grow at a constant long-run rate denoted by
g. Denote by the capital
expenditures in year t. Then
To functionally link the revenues with the corresponding capital
expenditures assume that a capital expenditure in year t-n induces a revenue of
in year t. Figure 1
illustrates the case of a single capital investment.
Call the contribution coefficient
with a lag of n years. The revenue
in year t is constituted by the individual contributions from all
the relevant capital expenditures. Hence
Figure 2 illustrates the composition of a revenue when
= 0
and
=
= ... = 0.
If it is assumed, as is customary, that the contribution
distribution is the same for all the
capital investments
undertaken by
the firm a common internal rate of return results. This IRR is
defined in this paper as the long run profitability we wish to solve
and then estimate from company data. At this stage denote the IRR by
x. In principle x is the solution of
Equation (3) follows directly from our concepts and the well- known definition of the internal rate of return.
Estimating the contribution coefficients directly is not sound, since the data required is not
easily available, and even if it were one would quickly run out of
degrees of freedom in estimation. Therefore, a contribution
distribution with a limited number of parameters has to be specified
for estimation._2 In Salmi (1982) the following contribution
distribution was applied based on Ruuhela (1975) with the
consequences described in Introduction.
Instead specify the contribution distribution as
where N is the life-span of a capital investment. We call this
contribution distribution an Anton distribution. It is shown in
Appendix A that the IRR x = i when
is specified as in (5).
Our choice of the contribution distribution is based on the
following considerations. The commonly applied uniform contribution
distribution
= i(1 + i)
/[(1 + i)
-1] is both unrealistic and leads to analytically
inconvenient results. A uniform cash-flow pattern does not reflect
the decrease of revenues from a capital investment. This feature due
to the ageing of fixed assets is evident in business practice. The
Anton distribution is a linearly decreasing distribution as can be
seen from the fact that
Futhermore, the Anton distribution is the well-known_3 special case which eliminates the discrepancy between the concepts of the straight-line depreciation and annuity depreciation. The Anton distribution has a direct connection with the empirically available life-span information of fixed assets, and it is superior to the uniform contribution distribution in the other respects.
By substituting definition (5) of the contribution distribution into
(2) the revenues in year t can be
written as
It is shown in Appendix B that profitability i is then given by
where
It is easy to see from (l) and (2) that the capital expenditures
and the revenues
grow at the same rate g. Denoting
profitability estimation formula (8) becomes
Formula (11) can be used for estimating the profitability of business companies, since the empirical counterparts of the components of (9) and (11) are readily defined for company data.
2.2 Case with Varying Life-Spans
The fixed assets of a business firm consist of several categories such as land and water, buildings, machinery and equipment, and other fixed assets. The typical life-spans vary considerably between the categories.
For more reliable estimation of profitability from the financial
data of business enterprises we accomodate different life-spans in
the model. For this purpose assume K different classes of capital
investments. Denote by the average
life-span of a capital investment belonging to class k.
Furthermore, denote by the capital
expenditures in year t in class k. As before in (1) assume that the
capital expenditures grow at the constant long run rate g. Thus
The revenues in year t are made up
by the individual contributions brought about by the capital
expenditures in each class. Hence, corresponding (2), we now have
For each class k of capital expenditures specify a contribution distribution with a common profitability parameter i.
Analogously with Appendix B it can be shown, after substituting (14) into (13), that profitability i is given by
where by definition (9)
The model under observation is a constant-growth model. It follows
from (12) and (13) that the capital expenditures in each class as well as the revenues
grow at rate g. Denote
The formula applicable for estimating the profitability of business enterprises becomes then
__________
3 See Solomon (1971, p. 168, footnote) for references.
2 Estimation of the contribution coefficients utilizing lag
parameters has recently been considered by Tamminen (1977) and
(1979), Appendix I), and Laitinen (1980, pp. 125-141). See also
Sampson (1969).
The next section (3 Equivalence of Depreciation
Methods)
The
previous section (1 Introduction and Review of the
Problem)
The contents section of Ruuhela, Salmi, Luoma and Laakkonen (1982)
Other scientific publications by Timo Salmi in WWW format