Counting since 15.10.2005
This paper is reproduced at the University of Vaasa in the electronic format with the permission of The Finnish Journal of Business Economics. Copyright © 1981 by The Finnish Journal of Business Economics and the authors.
Please use the following reference to this publication: Salmi, T. and 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 30:1, 20-45. Also available from World Wide Web: <URL:http://www.uwasa.fi/~ts/jkay/jkay.html>.
* We wish to acknowledge with gratitude the contribution by Matti Heikkilä, Lic.Sc. (Techn.) and similarly thank Professor Reijo Ruuhela for his useful comments. Any errors and inadequacies are however, our sole responsibility. Kay's method was first brought to our attention by Associate Professor Jouko Ylä-Liedenpohja.
It is an important, but at the same time a complicated task to determine the profitability of a firm. This problem can be approached by considering the firm a series of cash inflows and outflows, or, more specificially, a series of capital investment projects. The task of measuring profitability can then be defined as the problem of estimating, from published financial statements, the internal rate of return for the capital investments making up the firm. There are several papers since the mid 1960's considering the problem of deriving the internal rate of return (IRR) from the return on capital invested (ROI), alternatively called the accountant's rate of return (ARR), accountant's rate of profit (ARP), or book-yield. Such research has notably been done by Harcourt (1965), Solomon (1966), Vatter (1966) in a critique on Solomon, Sarnat and Levy (1969), Livingstone and Salamon (1970), Stauffer (1971), Gordon (1974), and Kay (1976), as well as in the consequent notes and replys by McHugh (1976), Livingstone and Breda (1976), Stephen (1976), Wright (1978), and Kay (1978). Whittington (1979) discusses the use of ARR vs. IRR in empirical research. The problem of estimating the internal rate of return simultaneously with the growth rate of the firm from published financial statements has been tackled by Ruuhela (1972) and (1975). His model includes both capital and net working asset investments. (Also Stauffer (1971) includes this feature.) An improved derivation of Ruuhela's model has recently been published by Salmi (1980a) and (1980b) in the English language.
In this paper we shall review and scrutinize some of the results by Kay (1976) and the discussion on Kay's results by Wright (1978) and Kay (1978). We shall test the model first by applying it on simulated financial statements where the true rate of return is known in advance. (A simulation approach has also been used by Livingstone and Salamon (1970), and Bhaskar (1972).) Then, we shall estimate the long-run profitability of a Finnish business firm with the model for a comparison with the estimation results in Salmi (1980).
We shall argue that the applicability of Kay's method on actual published financial statements is more limited than indicated by Kay, one reason being that the accountant's and the economist's valuation of capital stocks is compatible only if the annuity depreciation method is used by the accountant. Our simulations indicate that Kay's transformation between the two valuations would be unstable in business practice, since the direction of the error in Kay's profitability estimate is not well-behaved for the prevalent straight-line depreciation method. Furthermore, as a technical point, we shall demonstrate that initial instead of average capital stocks and discrete instead of continuous discounting should be used in Kay's profitability estimation equation.
We commence with continuous mathematics using the following definitions in accordance with Kay.
The cash inflows and outflows in the discrete case can be illustrated in the standard manner as below
Returning to the continuous case, the book value v(t) is defined by
i.e. the accumulated capital investments less the accumulated depreciation. It is natural to assume that
(3) v(0) = 0
since this indicates that the book value of the assets of the firm is zero outside its life-span.
The accountant's rate of profit (ARP) is defined by Kay as
In other words it is the revenues less depreciation (= accounting profit) divided by the capital employed at time t. In fact, the discrete counterpart of the above measure is called return on investment (ROI) rather than accountant's rate of profit in business practice. Kay ignores the fact that his definition is not consistent with the accrual basis of accounting measurement. In other words, his f(t) is a cash flow, not a revenue based on the realization principle, as it should be if defined rigorously. This is not, however, a major point in this discussion and we shall therefore accept the above definition of a(t).
The internal rate of return r (IRR) is defined in the continuous case by the equation
The internal rate of return is a natural and well-accepted measure of profitability1. Several problems arise, however, when it is used in estimating the profitability of a real-life business firm.
The IRR-method gives a single profitability figure for the whole life-span of the firm (in the absence of multiple or imaginary roots2). This rises two questions.
Firstly, it is obvious that the firm can be regarded as a series of capital investments. The profitability of the various capital investments usually differs, however. Therefore, on the level of the firm, a single long-run profitability must be assumed for all the capital investments making up the firm. On the other hand this is not quite as restrictive as may seem at first, since the firm can be deemed a single long-range capital-investment-like project. (The applicability of this idea can naturally be criticized especially in the case of conglomerates.)
Secondly, and more critically, the data for the whole life-span of the firm (as indicated by the integration limits in (6)), is seldom available and even if it were the profitability information would hardly be of much interest at that stage (with the potential exception of bankruptcy research). The approach has to be confined to a segment of the total life-span of the firm. Even so, the IRR-profitability of the firm covers several years. Theoretically this is sound, since the underlying capital investments are long-term already by definition. Nevertheless, the management (and other interested parties) of business enterprises need (also) short-term (yearly and shorter) profitability measures for decision making and control activities. The flaw with these short-term profitability measures is that they tend to be rules of thumb rather than based on proper theoretical considerations. On the other hand, it may be contended that such long-term measures as discussed in this paper can be less useful for frequent decision making and control purposes in business practice. E.g. Vatter (1966) strongly criticizes the yearly unvariability. This is naturally a question which could be debated at great length. As a potential compromise between the long-term and short-term measures we suggest rolling the long-term measures. By this we mean estimating the yearly profitability from the change in the IRR-profitability brought about by adding the last year to the data.
A third problem is that f(t) and g(t) in the equation for r are not available in published financial data, as such.
Kay has proved that if the accountant's rate of profit for a project is constant it is equal to the internal rate of return. No assumptions about the shape of the cash inflow f(t), the cash outflow g(t), nor the depreciation d(t) schedules are needed for this result. The result is weaker than it seems at first sight, since it is derived for the total life-span.
Accountant's rate of profit a(t) is not constant in actual practice. To tackle this problem Kay utilizes the weighted average of accountant's rate of profit. As natural weights he selects discounted book values v(t)e-rt. Kay's weights are natural only in the sense that they lead to the results discussed below. They are not, however, based on any actual practice in accountant's profit assessment. According to our interpretation, Kay actually proves that the weighted average accountant's rate of profit a solved from (7) is equal to the internal rate of return r.
In other words, for the life-span of the firm (or a project), the accountant's rates of profit weighted by discounted capital employed yields the IRR sought. To tackle the case of shorter time segments (from T1 to T2 let us investigate the following equation, given by Kay.
As is shown in Appendix I it follows from (8) that
If T1 = 0 and T2 = , the averaged accountant's profitability a, given by (8), will be equal to the internal rate of return. This results from the assumption v(0) = v() = 0, and the definition (6) of the IRR.
Generally a is not equal to r. Kay contends that a as given by (8) will be equal to r if the accountant's book values v(T1) and v(T2) join with their economic values. In fact, Gordon (1974, p. 347) indicates this result. The economic value is defined as the discounted net cash flows:
As is shown in Appendix II by substituting v(Tl) = w(Tl) and v(T2) = w(T2) Kay's contention a = r holds.
At this point two major objections arise in our opinion. First, there is no reason why this r in (10) must be the internal rate of return defined by (6). Actually, this r in (10) is but a rate of interest used for discounting the future cash flows. To elaborate, Kay defines the internal rate of return r by (6), and uses the same r in definition (10). It turns out, however, as indicated by Appendix II, that the results given by Kay are arrived at even if a different r from the r in (6) were used in (10). This means that when the results based on (10) to be given shortly are used in estimating profitability from published financial data, there will be no guarantee that Kay's method will estimate the internal rate of return defined in (6). Our critique is analogous to the critique by Stephen (1976) against the similar results by Gordon (1974). As far as we see Gordon's reply (1977) to Stephen does not resolve the circular reasoning involved in solving IRR from a relation where it is already assumed available beforehand. Second, we agree with Wright (1978) and Whittington (1979), and thus disagree with Gordon (1977), that even "an enlightened accounting profession" does not easily (if at all) accept the idea of the accountant's initial and terminal book values of assets equalling those of the economist's.
Nevertheless, should the economist's valuation of the initial and the ending capital stocks of the period under observation be accepted, equation (8) for profitability can be modified into a discrete form to allow the handling of actual financial data. IRR is then approximated by solving a from equation
where vt-1 is the book value of assets at the beginning of the year t, is the accountant's profit (operating income) in year t [being the discrete counterpart of a(t)v(t) as indicated by 5)], and n is the number of years in the period under observation. Unlike Kay we use the book value of assets at the beginning of the relevant years rather than the average book values. We shall demonstrate later that Kay's use of average book values is erroneous. Furthermore, we use (1+a)t as the discounting factor instead of e-at, since the former conceptually corresponds the discrete capital investment models. The difference is negligible, because . Nevertheless, it must be stressed that we have duly checked all our results using Kay's original formulation of (11) as well.
The following interpretation can be given to (11). In business practice the accountant's yearly profitability at (ROI) before interest and taxes is calculated as the operating income of the year divided by the capital employed at the beginning of the year (or, alternatively, the average capital employed). Thus
In (11) Kay in fact suggests that the internal rate of return should be estimated by weighing the accountant's profits and the capital employed vt by discounting factors over the whole period under observation3. A BASIC computer program for solving a from (11) with the so called secant method is given in Appendix IV.
Kay (1976, p. 449 and 456) contends that his results make no assumptions about the shape of the cash inflow (ft), cash outflow (gt) and depreciation (dt) schedules. This contention must be interpreted cautiously lest it gives an inflated impression about the strongness of Kay's results. As admitted by Kay (1976, p. 449), the depreciation scheme will together with the shape of cash inflow and outflow schedules strictly predetermine the capital stock. In fact, formula (11) for estimating the IRR is valid for the annuity method of depreciation only, as we shall demonstrate shortly. It is also a well-known fact that when the annuity method of depreciation is used, ARR will equal IRR (see Appendix III) and the whole problem does not arise4. The annuity method cannot, however, be applied in business practice, because it requires advance knowledge about the IRR of the capital investment projects constituting the firm. We do not claim that Kay implies otherwise, but these facts should have been clearly brought up by Kay to put the applicability of his results in proper scope.
Kay (1976, p. 455) tackles the case where the accountant's valuation (v) and the economist's valuation (w) of the capital stock do not agree by deriving the following relationship for the internal rate of return.
(13) r = k + (a-k)v/w
In estimation k is the growth rate of the firm (n in Kay's paper) and a is the accountant's rate of profit (where a has to be constant). We interpret this suggestion as follows. When the annuity method of depreciation is not used by the accountant (as always is the case in business practice) the internal rate of return can be estimated from (11) and (13) where v is based on accounting data and w is based on the annuity method of depreciation. (Our Appendix III confirms that using the economist's valuation of the capital stock is equivalent to using the annuity method of depreciation.) Although Kay's method is formally valid, we claim that it is not valid in business practice because of the necessity of estimating w (or v/w). Wright's (1978) criticism of Kay supports our views.
1 See, however, Tamminen (1980) for a discussion on the concept of profitability and the assumptions underlying profitability measurement with the internal rate of return.
2 A fairly recent discussion on the conditions for the uniqueness of the internal rate of return can be traced back from Bernhard (1980).
3 The idea of weighing the accountant's
profitabilities is seen more clearly, if (11) is rewritten as
4 See Lonka (1976, pp. 38-39) and Tamminen (1976, pp. 38-41) for a proof of the equality of IRR and ARR in the continuous case when the annuity method of depreciation is used.
We shall now move over to the testing of Kay's model by applying (11) and (13) on simulated financial data where the true rate of return r is known in advance. The simulation will be carried out for different growth situations, depreciation theories, and cash flow schedules. As expected, it will be seen that Kay's method is applicable only when the unapplicable annuity theory of depreciation is used.
In accordance with Ruuhela (1975) we assume for the simulation a capital investment expenditure gt in year t, and the corresponding revenues fs (s = t,...,t+N). The revenues each year can formally be linked to the expenditure by a contribution distribution bi as shown below.
(14) ft+i = bigt i = 0,...,N
N represents the life-span of the capital investment project. It is easy to see that the internal rate of return r is then defined by the equation
The process generating the financial data is created by repeating the capital investment expenditure yearly increased by a growth rate of k. Thus
(16) gt = g0(1+k)t
Consequently, the revenues each year are made up by the contributions of each capital investment project as delineated in (17).
Since we assume that (15) holds for every capital investment project making up our simulated firm, it is obvious that the profitability of our firm is r.
Three different depreciation methods will be considered and applied, i.e. the annuity method of depreciation, the discounted revenue depreciation method, and the straight-line method. Consider the annuity method of depreciation first. It is a well-accepted definition for the annuity method that the profit (before interest and taxes) is assessed as the interest on the initial capital stock vt-1 in year t, i.e.
In any depreciation method the profit is given by deducting depreciation dt from revenues ft.
Hence, depreciation in the annuity method is given by
(20) dt = ft-rvt-1
In any depreciation method the capital stock vt is arrived at from
(21) vt = vt-1 + gt - dt
This follows from the fact that expenditures gt increase the capital stock while depreciation dt decreases it. [See e.g Ruuhela (1975, p. 11), Kay (1976, p. 449), or Salmi (1980, p. 13).]
The presentation of the simulation model in the case of annuity depreciation is now complete. The input to be given are the initial capital investment g0, the growth rate k, and the contribution coefficients bi (i = 0,...,N). A BASIC computer program performing the simulation is given in Appendix V. The internal rate of return r is solved from (15) by the secant method in the program.
It is shown in Appendix III that (18) (and thus (20) i.e. the annuity method of depreciation) follows from accepting the economist's valuation of the capital stock. In this case the book value vt at the end of year t is defined by
It is also shown in Appendix III that the accountant's yearly rate of profit at then equals the internal rate of return r. (There is nothing novel in this, because these are well-known results.)
The results of our first simulation are given below as an example, where g0 = 40, b1 = 0.7, b2 = 0.6, and k = 0.08. In other words this is a growth situation, a declining contribution distribution, and the annuity method of depreciation.
capital funds from depretiat operating book expendit operations income value T G(T) F(T) D(T) P(T) V(T) 0 40.0000 .0000 .0000 .0000 40.0000 1 43.2000 28.0000 20.0000 8.0000 63.2000 2 46.6560 54.2400 41.6000 12.6400 68.2560 3 50.3885 58.5792 44.9280 13.6512 73.7145 4 54.4195 63.2655 48.5222 14.7433 79.6138 5 58.7731 68.3268 52.4040 15.9228 85.9830 6 63.4749 73.7929 56.5963 17.1966 92.8616 7 68.5529 79.6963 61.1240 18.5723 100.2910 8 74.0372 56.0720 66.0139 20.0581 108.3140 9 79.9601 92.9578 71.2950 21.6628 116.9790 10 86.3569 100.3940 76.9986 23.3958 126.3370 INTERNAL RATE 0F RETURN = 20 %Note that V(T) (that is vt) is the ending book value for each year. Kay's method is run below for a span of six years after the system has reached a steady state, although the conclusions will remain the same if we consider all the years.
KAY'S ALGORITHM BY TIMO SALMI WITH 1+A AS DISCOUNTING FACTOR IDENTIFICATI0N ? GROWTH ANNUITY DEPRECIATION GIVE THE NUMBER OF YEARS, AND THE FIRST YEAR ? 6,3 GIVE THE BOOK VALUES 3 ? 68.2560 4 ? 73.7165 5 ? 79.6138 6 ? 85.9830 7 ? 92.8616 8 ? 100.2910 GIVE THE OPERATING INCOMES 3 ? 13.6512 4 ? 14.7433 5 ? 15.9228 6 ? 17.1966 7 ? 18.5723 8 ? 20.0581 ESTIMATED INTERNAL RATE OF RETURN A = 20 %Kay's method estimates correctly the internal rate of return as 0.2, when initial book values are used in accordance with our reformulation of (11). If average book values are used instead, as suggested by Kay, the internal rate of return will be underestimated as 0.1923. (The results can be verified using the computer program given in Appendix V. For reasons of limited space we shall not reproduce any more of the actual computer runs for the annuity method.) Our further simulations indicate that in growth situations the use of average book values will lead to underestimation, in zero-growth situations there is no bias, and in the rarer cases of decline it will lead to overestimation of the internal rate of return. Our inferences will hold whether 1+a or ea is used as the discounting factor in (11). (This can be verified by rerunning after changing (1 + A)^T to EXP (A + T) in statement 460 in Appendix IV.)
At this point it should be stated that we are well familiar and in agreement with the principle that nothing is ever proved with numerical experiments. But, numerical counter-examples serve well to disprove erroneous suggestions [here, the use of average book values in (11)]. Furthermore, numerical experiments can lend support to a suggestion (here, the use of initial book values and Kay's underestimation, although the former is not a good example of the principle, since the contention follows strictly from the proof in Appendix III, as well).
Consider the discounted revenue depreciation method next. This theoretical depreciation method, which has been called by various names, is based on the following reasoning [adapted from Salmi (1978)]. In order to earn a profit the firm must incur expenditures as a prerequisite of the revenues. Thus, in principle, there is a fundamental association between expenditures and revenues. Expenditures expire (become expenses, i.e. are "depreciated" from the revenues) only when the associated revenues are realized. (Hence the method has also been called "realization depreciation".) Depreciation is directly dependent on the revenues being consequently a function of them. The functional relation is given by the internal-rate-of-return model. In Finland this idea was developed by Saario (1958) and (1961). Saario computed his depreciation in the same way as Bierman (1958) calculated his "basic depreciation". Later, Dixon (1960, p. 592) suggested the same way of calculating depreciation. All these suggestions seem to have been independent of each other. (It is not altogether impossible that the roots of the method would lie in Anton (1956) although no references are made to it by the above authors.)
To illustrate the discounted revenue method, consider the simple numerical example involving an expenditure of 40 at the beginning of the first year, and revenues of 28 and 24 at the end of the first and the second year respectively. (This is actually the first capital investment project in our simulation). The internal rate of return on this capital investment is 20 %, since 28/1.2 + 24/1.22 = 23.33 + 16.67 = 40. The depreciation for the first year is 23.33 and 16.67 for the second in the discounted revenue depreciation method. For a single capital investment the method leads to a declining depreciation pattern, unless the revenues resulting from the expenditure increase at some stage at a rate greater than the internal rate of return. Alike the annuity method of depreciation the discounted revenue depreciation is not applicable in accounting practice, because of the necessity of knowing the internal rate of return in advance. It is, however, interesting to compare Kay's method under these two different theoretical depreciation methods.
It follows from (17) and the definition of the discounted revenue depreciation discussed above that
With the exception of now omitting (18) and (20), and augmenting (23), our simulation model remains the same in the case of discounted revenue depreciation [(15), (16), (17), ,(19), (21) and (23)].
A simulation result, again for g0 = 40, b1 = 0.7, b2 = 0.6, and k = 0.08, is given below.
capital funds from depreciat operating book expendit operations income value T G(T) F(T) D(T) P(T) V(T) 0 40.0000 .0000 .0000 .0000 40.0000 1 43.2000 28.0000 23.3333 4.6667 59.8667 2 46.6560 54.2400 41.8667 12.3733 64.6560 3 50.3885 58.5792 45.2160 13.3632 69.8284 4 54.4195 63.2655 48.8333 14.4322 75.4147 5 58.7731 68.3268 52.7399 15.5868 81.4479 6 63.4749 73.7929 56.9591 16.8338 87.9637 7 68.5529 79.6963 61.5159 18.1805 95.0008 8 74.0372 86.0720 66.4371 19.6349 102.6010 9 79.9601 92.9578 71.7521 21.2057 110.8090 10 86.3569 100.3940 77.4922 22.9021 119.6740 INTERNAL RATE OF RETURN = 20 % GIVE THE BO0K VALUES 3 ? 64.656 4 ? 69.8284 5 ? 75.4147 6 ? 81.4429 7 ? 87.9637 8 ? 95.0008 GIVE THE OPERATING INCOMES 3 ? 13.3632 4 ? 14.4322 5 ? 15.5868 6 ? 16.8338 7 ? 18.1805 8 ? 19.6349 ESTIMATED INTERNAL RATE OF RETURN A = 20.6681 % NUMBER OF ITERATIONS = 4The estimated internal rate of return arrived at above actually equals the accountant's rate of profit. This equality is always true when the firm grows steadily as is easily seen as follows. According to Kay (1976, p. 454) the accountant's rate of profit is constant for a firm in a steady growth. Substituting the accountant's profit a(t) by a constant in (8), we have a = a(t), since the integrals cancel each other, and thus the estimated internal rate of return equals the accountant's rate of profit. This fact does not affect our simulation results or our conclusions, although in the special case of constant growth the application of Kay's method could simply have been substituted by calculating the accountant's rate of profit for any of the years simulated.
The valuation of the capital stock seems more conservative under the
annuity method of depreciation applied earlier. Our simulations
indicate, however, that the difference between the valuations
decrease with increasing growth rate of the firm. In other word's,
the deviation between the economist's and the accountant's
valuations is diminished if the firm is growing rapidly. In the
simulation given above the ratio of the capital stock as valuated by
the economist (annuity depreciation) and the book value in the
discounted revenue depreciation method is v/w 0.947. We see that
(13) holds in the simulation, as it should, since
r = 0.08 + (0.206681-0.08)0.947 0.2000
In harmony with Kay's analysis (1976, p. 456) our further simulations indicated that Kay's method overestimates the internal rate of return r when it exceeds the growth rate k of the firm. The internal rate of return is underestimated when it is smaller than the growth rate.
When the discounted revenue depreciation method is used, the form of the contribution distribution affects the profitability estimate based on (11), i.e. the shape of the cash inflow schedule does affect the estimation results. The following figure illustrates the relationship.
Both the depreciation methods considered this far are theoretic rather than based on business practice. Various declining balance methods (the double-declining-balance method and the years'- digits method for example in the US) are, however, prevalent in business practice. Thus the deviations in Kay's method observed for the discounted revenue depreciation method are indicative of a bias in profitability estimation when Kay's method is applied on published financial.data. (Recall that v/w will hardly be known in the estimation.)
Next, consider the much-applied straight-line depreciation method over the service life of an asset. The depreciation dt in year t in our simulation model is then made up by the depreciations on the individual capital investments. Hence, we have
when the life-span of each capital investment making up our simulated firm is m, and the first depreciation is made a year after the relevant capital expenditure. (It is equally easy to tackle the case where the first depreciation is made already the same year as the capital expenditure in accordance with business practice. The reason for our selection here is simply to make the numerical results comparable with the previous simulations.) Our simulation model is now constituted by (15), (16), (17), (21) and (24). The results of our simulation runs are delineated by the following figure.
As indicated by the figure above, it is not known whether Kay's method will overestimate or underestimate the internal rate of return when the straight-line method of depreciation is applied by a business firm under observation. This contradicts Kay's contention (1976, p. 456) that the accountant's principle of conservatism leads to a predictable direction of bias in estimating the IRR profitability.
Kay's method is now applied on a Finnish Business Firm, Rauma-Repola, in order to estimate its long-run profitability for 1962-1978 by the internal rate of return. Rauma-Repola is chosen for the emlpirical application because a previous estimate of the internal rate of reburn for the period under obser vation is available in Salmi (1980), where Ruuhela's method was used for the estimation.
The book values of assets and operating incomes needed in estimation are assessed as follows from the financial statements in the annual reports of the business firm under observation.
Book value of assets = net current assets + undisclosed reserves + fixed assets ./. cumulative write-ups in fixed assets Operating income = profit (loss) for the period ("the bottom line") + direct taxes + interest and other expenses on debt + disclosed increases in reserves + undisclosed increases in reservesThe net current assets are defined as financial assets plus inventories less short term liabilities involving no interest cost (accounts payable, advance payments received, and adjusting entries). The undisclosed reserves are due to the inventory write-off possibility allowed by the Finnish tax laws up to 50 %. A foreign reader is referred to Jägerhorn (1980) for financial reporting practices in Finland and to the Price & Waterhouse manual (1979) for relevant Finnish legislation or Järvinen and Sihto (1974) for a short review.
* RUN KAY'S ALGORITHM BY TIMO SALMI WITH I+A AS DISCOUNTING FACTOR IDENTIFICATION ? RAUMA-REPOLA GIVE THE NUMBER OF YEARS, AND THE FIRST YEAR ? 17,1962 GIVE THE BOOK VALUES GIVE THE OPERATING INCOMES 1962 ? 208916 1962 ? 21401 1963 ? 235770 1963 ? 22494 1964 ? 266483 1964 ? 22060 1965 ? 275891 1965 ? 24906 1966 ? 279373 1966 ? 23921 1967 ? 276294 1967 ? 25003 1968 ? 305821 1968 ? 45481 1969 ? 392473 1969 ? 65933 1970 ? 502916 1970 ? 71745 1971 ? 752163 1971 ? 72231 1972 ? 851326 1972 ? 95231 1973 ? 1009203 1973 ? 115466 1974 ? 1269257 1974 ? 185345 1975 ? 1482696 1975 ? 128359 1976 ? 1716983 1976 ? 185855 1977 ? 1969858 1977 ? 383909 1978 ? 2180760 1978 ? 377246 ESTIMATED INTERNAL RATE OF RETURN A = 12.2733 % NUMBER OF ITERATIONS = 9 STOP AT 0420 *As can be seen in the computer run given below, Kay's method yields 12,3 % as the estimate of the IRR. Ruuhela's method gives 17,6 % when the depreciation is computed using the discounted revenue depreciation method. When the depreciation shown on books is used in the estimation, Ruuhela's method gives 16,2 %. There is a difference of 3,9 % in the results. Naturally, this can be due to a bias in either (or both) of the methods, since in our empirical application we have no way of knowing the underlying true internal rate(s) of return.
In this paper we scrutinized Kay's method and demonstrated sources of bias in it. Furthermore, we contested its applicability, our empirical estimation notwithstanding. Testing Ruuhela's model in a similar framework remains the subject of future research work. Similarly, testing the cash-flow based procedure suggested in Finland by Artto (1978) and (1980) remains intended further research.
Starting with (8) we have
Partial integration of the left-hand side yields
because the derivative of v(t) is as is seen from (2). (The economic interpretation of is naturally that the change in capital stock (= book value) is cash outflows less depreciation.)
By substituting (5) into the right-hand side of (I1), we have
Equating (I2) and (I3) gives (9).
Applying definition (10), we have
By substituting (II1) and (II2) into (9), and dividing both sides by we have
An obvious solution of (II3) is a = r, because the terms then cancel each other in (II3). This does not, however, indicate that (6) must necessarily hold. Thus r might in this case be any interest rate used for discounting the future net cash flows.
From (22) we have
Substituting (22) in the above we have
(III2) (1+r)vt-1 = ft - gt + vt
Rearranging we have
(III3) vt = vt-1 + gt - (ft - rvt-1)
Comparing (III3) with (21) it is easy to see that (20) and thus (18) holds.
Applying (18) we have
10 REM KAY'S ALGORITHM WITH 1+A AS DISCOUNTING FACTOR 20 DIM V(30), P(30) 30 PRINT 40 PRINT "KAY'S ALGORITHM BY TIMO SALMI" 50 PRINT "WITH 1+A AS DISCOUNTING FACTOR" 60 PRINT 70 PRINT "IDENTIFICATION"; 80 INPUT A$ 90 PRINT 100 PRINT "GIVE THE NUMBER OF YEARS, AND THE FIRST YEAR "; 110 INPUT N, M 120 PRINT "GIVE THE BOOK VALUES" 130 FOR I = 1 TO N 140 PRINT M + I - 1; " "; 150 INPUT V(I) 160 NEXT I 170 PRINT "GIVE THE OPERATING INCOMES" 180 FOR I = 1 TO N 190 PRINT M + I - 1; " "; 200 INPUT P(I) 210 NEXT I 220 REM SECANT METHOD 230 LET A1 = 0 240 LET A2 = .2 250 FOR I = 1 TO 30 260 LET A = A1 270 GOSUB 430 280 LET F1 = F 290 LET A = A2 300 GOSUB 430 310 LET F2 = F 320 LET A = (A1 * F2 - A2 * F1) / (F2 - F1) 330 LET A1 = A2 340 LET A2 = A 350 LET L = I 360 IF ABS(A2 - A1) < .000001 THEN GOTO 400 370 NEXT I 380 PRINT "ALGORITHM FAILED IN 30 ITERATIONS" 390 END 400 PRINT "ESTIMATED INTERNAL RATE OF RETURN A ="; 100 * A2; "%" 410 PRINT "NUMBER OF ITERATIONS ="; L 420 END 430 REM SUBPROGRAM FOR THE FUNCTION OF SUMS 440 LET F = 0 450 FOR T = 0 TO N - 1 460 LET F = F + (P(T + 1) - A * V(T + 1)) / (1 + A) ^ T 470 NEXT T 480 RETURN 490 END
10 REM GENERA3 BY TIMO SALMI. ANNUITY DEPRECIATION 20 REM NO NET CURRENT ASSETS "C" 30 REM GENERATES THE SERIES FOR G(T),F(T),D(T),P(T),V(T) 40 REM G(T) = CAPITAL EXPENDITURES IN YEAR T 50 REM F(T) = FUNDS FROM OPERATIONS IN YEAR I 60 REM I.E. REVENUES DUE TO CAPITAL EXPENDITURES 70 REM D(T) = DEPRECIATION 80 REM P(T) = OPERATING INCOME (ACCOUNTANT'S PROFIT) 90 REM V(T) = CAPITAL STOCK AT THE END OF YEAR T 100 REM 110 REM WHEN GIVEN K,B(N),G(0) 120 REM K = GROWTH RATE 130 REM B(N) = CONTRIBUTION COEFFICIENTS 140 REM G(0) = FIRST CAPITAL EXPENDITURE 150 REM 160 DIM G(60), B(60) 170 PRINT 180 PRINT "FINANCIAL DATA GENERATION FOR SALMI-LUOMA, BY TIMO SALMI" 190 PRINT "ANNUITY DEPRECIATION, NO NET CURRENT ASSETS" 200 PRINT 210 PRINT "IDENTIFICATION"; 220 INPUT A$ 230 PRINT 240 PRINT "GIVE THE NUMBER OF YEARS TO BE SIMULATED"; 250 INPUT T1 260 IF T1 >= 60 THEN GOTO 240 270 PRINT "GIVE GROWTH K"; 280 INPUT K 290 PRINT "GIVE FIRST CAPITAL EXPENDITURE G(0)"; 300 INPUT G(1) 310 FOR I = 1 TO 60 320 LET B(I) = 0 330 NEXT I 340 PRINT "GIVE THE MAXIMUM LAG IN CONTRIBUTION DISTRIBUTION"; 350 INPUT N1 360 IF N1 >= 60 THEN GOTO 340 370 PRINT "GIVE THE CONTRIBUTION COEFFICIENTS B(I)" 380 FOR I = 0 TO N1 390 PRINT "B("; I; ")"; 400 INPUT B(I + 1) 410 NEXT I 420 GOSUB 710 430 REM (App. V cont. ) 440 REM *** COMPUTING THE TIME SERIES ** 450 LET V = 0 460 PRINT 470 PRINT " capital funds from depreciat"; 480 PRINT " operating book" 490 PRINT " expendit operations "; 500 PRINT " income value" 510 PRINT "T G(T) F(T) D(T) "; 515 PRINT " P(T) V(T)" 520 PRINT 530 FOR T = 0 TO T1 540 LET G(T + 2) = (1 + K) * G(T + 1) 550 LET F = 0 560 IF T > N1 THEN GOTO 580 570 LET N2 = T 580 FOR I = 0 TO N2 590 LET F = F + B(I + 1) * G(T - I + 1) 600 NEXT I 610 LET P = R * V 620 LET D = F - P 630 LET V = V + G(T + 1) - D 640 PRINT USING "##"; T; 650 PRINT USING "###########.####"; G(T + 1); 660 PRINT USING "#########.####"; F; D; P; V 670 NEXT T 680 PRINT 690 PRINT "INTERNAL RATE OF RETURN ="; 100 * R; "%" 700 END 710 REM 720 REM INTERNAL RATE OF RETURN SUBROUTINE 730 REM THE SECANT METHOD 740 LET H = 1 750 LET X1 = 0 760 LET x2 = .2 770 FOR J = 1 TO 30 780 LET X = X1 790 GOSUB 930 800 LET Y1 = Y 810 LET X = x2 820 GOSUB 930 830 LET Y2 = Y 840 LET X = (X1 * Y2 - x2 * Y1) / (Y2 - Y1) 850 LET X1 = x2 860 LET x2 = X 870 IF ABS(x2 - X1) < .000001 THEN GOTO 910 880 NEXT J 890 PRINT "IRR-ALGORITHM FAILED IN 30 ITERATIONS" 900 END 910 LET R = x2 920 RETURN 930 REM PRESENT VALUE SUBROUTINE 940 LET Y = -H 950 FOR I = 0 TO N1 960 LET Y = Y + B(I + 1) / (1 + X) ^ I 970 NEXT I 980 RETURN 1000 END
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