3 EQUIVALENCE OF DEPRECIATION METHODS

Measuring the yearly income of the firm is closely related to profitability estimation. For determining the annual income of the firm the concept of depreciation must be defined.

We shall demonstrate the following result concerning two central depreciation theories. Furthermore, we shall discuss the estimation of the long-run depreciation policy of the firm in the next chapter on estimating the long-run financing policy of the firm.

If the stream of revenues for the firm is generated according to the Anton distribution under the steady-state growth process assumed earlier, the prevalent straight-line method of depreciation and the concept of economic depreciation (i.e. annuity depreciation or discounted-value depreciation) will lead to identical depreciation figures.

The above property is an important special feature of the Anton distribution. It prompted our selection of the contribution distribution bn in the previous chapter.

The straight-line method of depreciation is discussed first. To begin with consider a single capital expenditure as delineated in Figure 1. The yearly depreciation is given by (1/N)Ftn. Introduce the concept of depreciation coefficient an. For the straight-line method the depreciation coefficient an is specified as

Image: Formula (19)

In the general steady-state growth situation with capital investments taking place each year, depreciation Dt. in year t is given by (20) analogously with (2).

Image: Formula (20)

In the case of straight-line depreciation we have

Image: Formula (21)

Consider the discounted-value depreciation next. Discounted value depreciation (called by various names such as annuity depreciation) is based on the economist's valuation of the firm. The capital stock Ct. of the firm in year t is defined accordingly as the present value of the future net cash flows making up the firm. On the other hand the capital stock in year t is defined by

Image: Formula (22)

In other words the capital stock is increased by capital expenditures (Ft) and decreased by depreciation (Dt).

Consider, again, a single capital investment project. As illustrated by Figure 1 the capital investment in the model involves the capital expenditure at the very beginning of the project and the corresponding revenues during the life-span of the project. For a single capital investment Formula (22) can therefore be rewritten as

Image: Formula (23)

The capital stock Ct in year t is defined for the single-investment case as the present value of the future revenues in accordance with the economist's valuation.

Image: Formula (24)

Substituting the capital stock given by (24) into the depreciation formula (23) gives the discounted-value depreciation as

Image: Formula (25)

Expressing the same matter in terms of the depreciation coefficients an and the contribution coefficients bm we have after changing the indices

Image: Formula (26)

It is shown in Appendix C that the depreciation coefficient avn for discounted-value depreciation is exactly the same as for straight-line depreciation, i.e. that

Image: Formula (27)

provided that the contribution coefficients bm are specified according to the Anton distribution given in (5). Comparing formula (19) for the straight-line method depreciation coefficient with Formula (27) for the discounted-value method depreciation coefficient it is obvious that the two depreciation methods give the same level of depreciation.

The equality of the level of depreciation via the straight-line method and the discounted-value method was derived above for a single capital in vestment project. Generalization into the firm- level model with capital expenditures made each year is straight- forward. Depreciation for the firm level model is given by (21) for both depreciation methods. It is easy to show_4 that in the steady-state growth situation

Image: Formula (28)

where the capital investment ratio F is defined by (10), depreciation ratio D (in general) analogously as D = Dt/ Qt, and hN(g) by (9).

Generalizing the depreciation ratio formula (28) into the case with varying life-spans gives the following estimation formula for theoretical depreciation ratio

Image: Formula (29)

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4 Ruuhela (1981, p. 24).


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Department of Accounting and Finance, University of Vaasa,
Finland

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