2 PROFITABILITY ESTIMATION

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 Ft the capital expenditures in year t. Then

Image: Formula (1)

To functionally link the revenues with the corresponding capital expenditures assume that a capital expenditure F(t-n) in year t-n induces a revenue of b(n)F(t-n) in year t. Figure 1 illustrates the case of a single capital investment.

Image: Figure 1.

Call bn the contribution coefficient with a lag of n years. The revenue Qt in year t is constituted by the individual contributions from all the relevant capital expenditures. Hence

Image: Formula (2)

Figure 2 illustrates the composition of a revenue Qt when b0 = 0 and b5 = b5 = ... = 0.

Image: Figure 2.

If it is assumed, as is customary, that the contribution distribution bn is the same for all the capital investments Ft 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

Image: Formula (3)

Equation (3) follows directly from our concepts and the well- known definition of the internal rate of return.

Estimating the contribution coefficients bn 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.

Image: Formula (4)

Instead specify the contribution distribution as

Image: Formula (5)

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 bn is specified as in (5). Our choice of the contribution distribution is based on the following considerations. The commonly applied uniform contribution distribution bn = i(1 + i) ^N /[(1 + i) ^N -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

Image: Formula (6)

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 Qt in year t can be written as

Image: Formula (7)

It is shown in Appendix B that profitability i is then given by

Image: Formula (8)

where

Image: Formula (9)

It is easy to see from (l) and (2) that the capital expenditures Ft and the revenues Qt grow at the same rate g. Denoting

Image: Formula (10)

profitability estimation formula (8) becomes

Image: Formula (11)

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 Nk the average life-span of a capital investment belonging to class k.

Furthermore, denote by Fkt 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

Image: Formula (12)

The revenues Qt 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

Image: Formula (13)

For each class k of capital expenditures specify a contribution distribution with a common profitability parameter i.

Image: Formula (14)

Analogously with Appendix B it can be shown, after substituting (14) into (13), that profitability i is given by

Image: Formula (15)

where by definition (9)

Image: Formula (16)

The model under observation is a constant-growth model. It follows from (12) and (13) that the capital expenditures Fkt in each class as well as the revenues grow at rate g. Denote

Image: Formula (17)

The formula applicable for estimating the profitability of business enterprises becomes then

Image: Formula (18)

__________
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).

3 See Solomon (1971, p. 168, footnote) for references.


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Department of Accounting and Finance, University of Vaasa,
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