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REIJO RUUHELA - TIMO SALMI - MARTTI LUOMA - ARTO LAAKKONEN

Direct Estimation of the Internal Rate of Return from Published Financial Statements*

with results on the long-run growth, profitability, and financing behavior of major Finnish pulp and paper firms

This paper is reproduced at the University of Vaasa in the electronic format with the permission of The Finnish Journal of Business Economics. Copyright © 1982 by The Finnish Journal of Business Economics and the authors.

CONTENTS

1 Introduction and Review of the Problem

2 Profitability Estimation

2.1 Case with Unique Life-Spans

2.2 Case with Varying Life-Spans

3 Equivalence of Depreciation Methods

4 Long-Run Financing Policy

5 Estimation and Empirical Results

5.1 The Estimation Procedure

5.2 Long-Run Profitability, Growth, and Financing Behavior of Eight Major Finnish Pulp and Paper Firms

References

Appendix A

Appendix B

Appendix C

Please use the following reference to this publication: Ruuhela, R., Salmi, T., Luoma, M. and Laakkonen, A. "Direct Estimation of the Internal Rate of Return from Published Financial Statements; with results on the long-run growth, profitability, and financing behavior of major Finnish pulp and paper firms", The Finnish Journal of Business Economics 31:4, 329-345. Also available from World Wide Web: <URL:http://www.uwasa.fi/~ts/dire/dire.html>.

* Presented at Louvain-la-Neuve at the tenth anniversary conference of the European Institute for Advanced Studies in Management

This paper presents further results on measuring profitability by estimating the internal rate of return (IRR) of the capital investments constituting the firm.

Estimating the profitability of a single capital investment project is straight-forward. In external financial analysis, however, only the published financial statements are available. The expenditures and revenues of the capital investments making up the firm are then completely mixed. In order to estimate the long-run profitability of the firm it is customary to assume a single internal rate of return for the series of capital investments constituting the firm. Several authors¹ have tackled this profitability measurement problem, often in the framework of establishing the relation between the internal rate of return and the accountant's rate of return.

Ruuhela (1972) and (1975) presented in Finnish a model for measuring the long-run profitability of the firm by estimating the IRR from published financial statements. A major characteristic of Ruuhela's model is that in assessing the long- run profitability, it takes into account the growth trend of the capital expenditures thus avoiding a potential bias. Salmi (1982) improved and streamlined the derivation of Ruuhela's model. The results and concepts described in Salmi (1982) are the background for this paper.

In this paper we improve Ruuhela's method for estimating the IRR in two important respects. First, we do away with necessity of choosing a depreciation theory by modelling the relationship (called contribution distribution) between the capital expenditures and the corresponding revenues using an Anton distribution. Earlier, the contribution distribution used included a depreciation parameter (denoted by d). This parameter had to be estimated by a cumbersome procedure involving the selection of a depreciation theory (discounted revenue depreciation), choosing another contribution distribution, and equating the cumulative depreciation formulas indicated by each contribution distribution. The application of the Anton distribution was first presented by Ruuhela (1981) in Finnish. The functional form of this contribution distribution is compatible with the pattern of net receipts assumed in Anton (1956, 125).

Second, it was earlier assumed that all the capital investments making up the firm have the same life-span (denoted by N). We develop the estimation method to include the more general case of capital investments with differing life-spans.

If it is accepted that there is a long-run consistency in the financial policy of the firm, the long-run financing behavior can be characterized by average ratios of the different sources and uses of financing. Based on Ruuhela (1981) we also present the derivation of the theoretical depreciation ratio when the Anton distribution is used. For this purpose, naturally, a selection of the depreciation theory has to be made. When compared with the theoretical depreciation ratio the ratio of the depreciation on books describes the depreciation policy of the firm.

Forest industry is a central branch of industry in Finland. The branch's share of Finland's total manufacturing was 23.1 per cent in 1980 in terms of value added. We estimate the growth, profitability, and financing behavior of eight major Finnish pulp and paper firms for the 1970's.

__________
¹ See Salmi & Luoma (1981) for the references.
 

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 F_t the capital expenditures in year t. Then
 
(1)   F_t = F₀(1+g)^t
 
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_nF_t-n in year t. Figure 1 illustrates the case of a single capital investment.
 

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

Figure 2 illustrates the composition of a revenue Q_t when b₀ = 0 and b₄ = b₅ = ... = 0.

If it is assumed, as is customary, that the contribution distribution b_n is the same for all the capital investments F_t 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 b_n 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.

(4)   b_n = (1+i)ⁿd(1-d)ⁿ   n = 0,1,...

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 b_n is specified as in (5). Our choice of the contribution distribution is based on the following considerations. The commonly applied uniform contribution distribution b_n = 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

(6)   b_n+1 - b_n = -i/N

Futhermore, the Anton distribution is the well-known³ 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 Q_t 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 F_t and the revenues Q_t grow at the same rate g. Denoting

(10)   F = F_t/Q_t

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

Furthermore, denote by F_kt 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

(12)   F_kt = F_k0(1+g)^t

The revenues Q_t 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 F_kt in each class as well as the revenues grow at rate g. Denote

(17)   F(k) = F_kt/Q_t

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

__________
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.

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 b_n 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)F_t-n. Introduce the concept of depreciation coefficient a_n. For the straight-line method the depreciation coefficient an is specified as

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

In the case of straight-line depreciation we have

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 C_t 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

(22)   C_t = C_t-1 + F_t - D_t.

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

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

(23)   D_t = C_t-1 - C_t.

The capital stock C_t 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.

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

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

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

provided that the contribution coefficients b_m 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 show⁴ that in the steady-state growth situation

where the capital investment ratio F is defined by (10), depreciation ratio D (in general) analogously as D = D_t/Q_t, and h_N(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

__________
⁴ Ruuhela (1981, p. 24).

As discussed in more detail in Salmi (1980) and (1982) the long-run financing policy of the firm is characterized by a long-run funds-flow statement as well as growth and profitability. It is readily shown that in steady state growth all the components of the funds-flow statement grow at rate g in the model. (The same goes for depreciation.) Consequently, all the components of the statement can be divided by Q_t (funds from operations in empirical application). The financing policy of the firm is thus described by the following constant ratios.

The funds-flow statement is based on the following yearly income statement and change in balance sheet

5.1 The Estimation Procedure

If empirical data is available on yearly capital expenditures in different classes, as well as on the average life-spans N_k of fixed assets in the various classes. Formula (18) is applied in estimating profitability. If either data is unobtainable, (11) is used instead. In that case the annual capital expenditures of the firm are calculated as a lumpsum from the financial statements of the business enterprise considered, and the life-span N is estimated e.g. as an industry average of all fixed assets. Such data is available.⁵ The categories of fixed assets used in the application to be presented shortly are 1) Construction in process, 2) Land, water, bonds and shares, 3) Buildings, and 4-5) Two classes of machinery and equipment.

Growth-rate g has normally been estimated from the empirical time-series of funds from operations in applications of the model.⁶ The seemingly unrelated regression technique can be used in troublesome cases.⁷

The components of the long-run financing policy are computed as the weighted sum of the yearly observations of the empirical counterparts of the items of the funds-flow statement. These empirical counterparts are easily computed from the annual income statements and the balance sheets of a business enterprise. Details of these computations are omitted here.⁸ The estimated growth factors are used as the weights. (For example, if = 0.10, the second set of observations is divided by 1.10, the third by 1.21, etc.) Profit financing, i.e. undistributed profit or change in retained earnings, is defined as the funds provided by operations Q less depreciation D, interest expense I, income taxes T, and dividends V.

The theoretical depreciation ratio D is estimated from (29). Formula (28) is used instead if the more detailed information required by (29) is not empirically available. The depreciation policy of a business enterprise is described by comparing D with the observed ratio of depreciation on books, denoted D^b. A positive difference D^b-D indicates that the firm has utilized accelerated depreciation as a means of generating undisclosed profit financing. A negative difference indicates propped up disclosed profits. It should be stressed, however, that accelerating or decelerating depreciation is by no means the only way of adjusting disclosed profits. Depending on the legislation in the relevant country changes of reserves, deferring taxes, etc. can be used, too. In Finland utilizing an inventory write- off reserve (a kind of stock appreciation), allowed by tax-laws up to 50 per cent of the historical acquisition cost (FIFO), is of special importance in adjusting the disclosed profits in business practice.⁹

5.2 Long-Run Profitability, Growth, and Financing Behavior of Eight Major Finnish Pulp and Paper Firms

The empirical results concerning the eight most important Finnish firms producing mainly pulp and paper are given below. The firms are Enso Gutzeit (sales turnover in 1980 3745.2 million Finnish marks ¹⁰), Kajaani (FIM 804.6 million), Kaukas (1010.0), Kymi Kymmene (2028.2), Metsäliiton Teollisuus (1293.9), Wilh. Schauman (1542.9), Serlachius (1637.6), and Yhtyneet Paperitehtaat (2583.0). The analysis is based on deflated time-series for 1970-1980 derived from the financial statements of these firms. The time series are deflated in order to obtain the growth and profitability estimates in real terms. The information concerning the capital expenditures in the different classes was obtained from the firms. Because of space limitations the details of the statistical analysis of the time-series are not repeated here.¹¹ For the same reason the empirical results are given in Table 1 without further interpretation.¹²

__________
⁵ For information see Artto (1980, pp. 27-29), and Yritystutkimusneuvottelukunta.
 
⁶ See Salmi (1982) for details.
 
⁷ See Luoma & Ruuhela (1980).
 
⁸ See Salmi (1980, pp. 17-20 and 27) for details.
 
⁹ See Jägerhorn & Troberg (1981) for the current accounting practices in publicly traded Finnish companies. For the relevant Finnish legislation a foreign reader is referred to Price & Waterhouse (1979).
 
¹⁰ The average exchange rates in 1980 were 1 USD = 3.730 EIM, 1 GBP = 8.691 FIM, and 1 DEM= 2.0558 FIM.
 
¹¹ For the outlines of the statistical analysis see Salmi (1980, Ch. 3). One detail should be brought up here, in addition. For the capital expenditure classes of land, water, bonds and shares as well as construction in process the life-span is deemed infinite. Changes in net working assets are also treated as capital expenditures. These features are easily accomodated in the estimation by letting the relevant Nk:s go to infinity in formulas (16), (18), and (29).
 
¹² A Finnish reader is referred to the writtings by Timo Salmi in the Finnish Business Daily (Kauppalehti) for analyses of Finnish firms in other branches. Ruuhela presented recently results on Finnish publicly traded companies also in Kauppalehti(16th March, 1982).

Anton, H. R. (1956). Depreciation, cost allocation and investment decisions. Accounting Research (April), 117-131.

Artto, E. (1980). Profitability and cash stream analyses. Helsinki School of Economics D-44.

Jägerhorn, R. & Troberg, P. (1981). Financial reporting practices in Finland 1980. Swedish School of Economics and Business Administration Working Papers 72. Helsinki.

Laitinen, E. K. (1980). Financial ratios and the basic economic factors of the firm: A steady state approach. Doctoral dissertation. Jyväskylä University Library.

Luoma, M. & Ruuhela, R. (1980). Analysing short and long term variations in yearly financial time series of the firm. Proceedings of the Vaasa School of Economics. Research Papers No 70.

Price Waterhouse & Co Oy (1979). Finland: The Finnish companies act of 1978, act on the implementation of the companies act, the Finnish accounting act of 1973. Accounting Ordinance (November).

Ruuhela, R. (1972). Yrityksen kasvu ja kannattavuus. Summary: A capital investment model of the growth and profitability of the firm. Acta Academiae Oeconomicae Helsingiensis, Series A: 8. Helsinki.

-"- (1975). Yrityksen kasvu ja rahoitus. Gaudeamus.

-"- (1981). Yrityksen kasvu, kannattavuus ja rahoitus (KASKARA): Luennot 1981, Osa I; teoria. University of Vaasa.

Salamon G. L. (1982) Cash recovery rates and measures of firm profitability. The Accounting Review (April), 292-302.

Salmi, T. (1980). An empirical analysis of the capital investment and financing behavior of a Finnish firm (an application of a long-run constant growth and profitability model). European Institute for Advances Studies in Management Working Papers 80-1.

Salmi, T. (1982). Estimating the internal rate of return from published financial statements. Journal of Business Finance and Accounting (Spring). (Show abstract)

Salmi, T. & Luoma, M. (1981). Deriving the internal rate of return from the accountant's rate of profit: Analysis and empirical estimation. The Finnish Journal of Business Economics No. 1, 20-45.

Sampson, A. A. (1969). Measuring the rate of return on capital. The Journal of Finance (March), 61-74.

Solomon, E. (1971). Return on investment: The continuing confusion among disparate measures. Accounting in Perspective, ed. R. R. Sterling & W. F. Bentz. Cincinnati, Ohio.

Tamminen, R. (1977). Relationship of cash outflows to cash inflows: Some experiments in the construction of a distributed lag model. Reports on Computer Science, Economics and Statistics, University of Jyväskylä 25/1977.

-"- (1979). Profitability and the financial mechanism of the firm. Report from the Department of Economics and Management, University of Jyväskylä 1/1979.

Yritystutkimusneuvottelukunta (1979). Inflaation huomioon ottaminen yritystutkimuksessa: ohjeelliset käyttöomaisuuden pitoajat AHI sovellutuksessa.

It is shown that the parameter i of the Anton distribution equals the internal rate of return x. In other words it is shown that the root of

is x = i. (A1) results from (3) and (5). (A1) can be rewritten as

Denote

Since (A3) is a geometric series if follows that

provided that . Differentiating (A3) with respect to x gives

On the other hand differentiating (A4) with respect to x gives

Substituting (A8) into (A6), and the result and (A4) into (A2) gives after some algebra

Provided that and it follows from (A9) after some simple, but lengthy algebra that

It is easy to see from (A10) that x = i is the root of equation (A1).

Other roots of equation (A10) are not relevant. It can be proved that no other root exists when N is odd, and that the other root is less than - 1 when N is even.

It is shown that the profitability i is given by (8) and (9).

From (1) we have

(B1)   F_t-n = F_t(1+g)^-n

From (7) and (B1) follows that

Comparing the right-hand side of (B2) and the left-hand side of (A2) it is easy to see, after replacing x by g in (A2), that analogously with (A9) it follows from (B2) that

After some algebra (B3) can be written as

Denoting

(B4) becomes

Solving i from (B5) gives (8).

It is shown that in the discounted-value method the depreciation coefficient equals the straight-line method coefficient 1/N when the contributions follow an Anton distribution. Formula (26) can be rewritten as

Substituting (5) into (C2) we have

Since the first term of (C3) is a geometric series, we have

After some standard algebra it readily follows from the above that

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