Counting since 15.10.2005 <URL:http://lipas.uwasa.fi/~ts/jkay/jkay.html>

Timo Salmi and Martti Luoma

Deriving the Internal Rate of Return from the Accountant's Rate of Profit: Analysis and Empirical Estimation

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.

CONTENTS

1. Introduction
2. The Method for Estimating the Internal Rate of Return
3. Testing the Model with Simulated Financial Statements
4. An Application of Kay's Method on a Finnish Business Firm, and Directions for Further Research

• Appendix I
• Appendix II
• Appendix III
• Appendix IV
• Appendix V
• REFERENCES

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.

f(t) = cash inflow rate at time t (revenue rate)

g(t) = cash outflow rate at time t (expenditure rate)

d(t) = depreciation rate at time t

v(t) = book value of assets (capital stock, capital employed) at time t.

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

and

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(T_l) = w(T_l) 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 v_t-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 a_t (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 v_t 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 (f_t), cash outflow (g_t) and depreciation (d_t) 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 g_t in year t, and the corresponding revenues f_s (s = t,...,t+N). The revenues each year can formally be linked to the expenditure by a contribution distribution b_i as shown below.

(14)   f_t+i = b_ig_t   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)   g_t = 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 v_t-1 in year t, i.e.

In any depreciation method the profit is given by deducting depreciation d_t from revenues f_t.

Hence, depreciation in the annuity method is given by

(20)   d_t = f_t-rv_t-1

In any depreciation method the capital stock v_t is arrived at from

(21)   v_t = v_t-1 + g_t - d_t

This follows from the fact that expenditures g_t increase the capital stock while depreciation d_t 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 b_i (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 %

      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 %

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 = 4

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 d_t 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 reserves

* 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
*

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)v_t-1 = f_t - g_t + v_t

Rearranging we have

(III3)   v_t = v_t-1 + g_t - (f_t - rv_t-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

Anton, Hector R. "Depreciation, Cost Allocation and Investment Decisions," Accounting Research, Vol. 7 (April 1956), 117-131.

Artto, Eero. Kassaperusteinen kannattavuus ja rahoitus. Helsinki: Oy Gaudeamus Ab, 1978.

Artto, Eero. Profitability and Cash Stream Analyses. Helsinki: Helsinki School of Economics, D-44, 1980.

Bernhard, Richard H. "A Simplification and an Extension of the Bernhard-deFaro Sufficient Condition for a Unique Non-negative Internal Rate of Return," Journal of Financial and Quantitative Analysis, Vol. XV, No. 1 March 1980), 201-209.

Bhaskar, K. N. "Rates of Return under Uncertainty," Accounting and Business Research, No. 9 (Winter 1972), 40-52.

Bierman, Jr., Harold. "A Theory of Depreciation Consistent with Decision Making," The Controller, August 1958, pp. 370-1, 393.

Dixon, Robert L. "Decreasing Charge Depreciation-A Search for Logic," The Accounting Review, Vol. XXXV, No. 4 (October 1960), 590-597.

Gordon, Lawrence A. "Accounting Rate of Return vs. Economic Rate of Return," Journal of Business Finance and Accounting, Vol. 1, No. 1 (Spring 1974), 343-356.

Gordon, Lawrence A. "Further Thoughts on the Accounting Rate of Return vs. the Economic Rate of Return," Journal of Business Finance and Accounting, Vol. 4, No. 1 (Spring 1977), 133-134.

Harcourt, G. G. "The Accountant in a Golden Age," Oxford Economic Papers, March 1965, pp. 66-80.

Jägerhorn, Reginald & Troberg, Pontus. "Financial Reporting Practices in Finland 1979," Helsinki: Swedish School of Economics and Business Administration Working Papers 56, July 1980.

Järvinen, Risto & Sihto, Mauno. "The Revision of the Finnish Accounting Legislation," Bank of Finland Monthly Bulletin, No. 9 (1974), pp. 3-6.

Kay, J. A. "Accountants, too, Could Be Happy in a Golden Age: The Accountants Rate of Profit and the Internal Rate of Return," Oxford Economic Papers (New Series), Vol. 28, No. 3 (November 1976), 447-460.

Kay, J. A. "Accounting Rate of Profit and the Internal Rate of Return; A Reply," Oxford Economic Papers (New Series), Vol. 30, No. 3 (November 1978), 469-470.

Livingstone, John Leslie & Salamon, Geral L. "Relationship between the Accounting and the Internal Rate of Return Measures: A Synthesis and an Analysis," Journal of Accounting Research, Vol. 8, No. 2 (Autumn 1970), 199-216.

Livingstone, John Leslie & Van Breda, Michael F. "Relationship between Accounting and the Internal Rate of Return Measures; A Reply," Journal of Accounting Research, Vol. 14, No. 1 (Spring 1976), 187-188.

Lonka, Harri. On the Revenue and Capital Formation of the Firm. Acta Wasaensia No. 6, Vaasa School of Economics, 1976.

McHugh, A. J. "Relationship between Accounting and the Internal Rate of Return Measures," Journal of Accounting Research, Vol. 14, No. 1 (Spring 1976), 181-186. Price Waterhouse & Co Oy. Finland: The Finnish Companies Act of 1978, Act on the Implementation of the Companies Act, The Finnish Accounting Act of 1973, Accounting Ordinance, Translated by Price Waterhouse Co Oy, November 1979.

Ruuhela, Reijo. Yrityksen kasvu ja rahoitus. (English summary: A Capital Investment Model of the Growth and Profitability of the Firm.) Doctoral Dissertation. The Helsinki School of Economics, 1972.

Ruuhela, Reijo. Yrityksen kasvu ja rahoitus. Helsinki: Oy Gaudeamus Ab, 1975.

Saario, Martti. "Om räinteutgifter i bokslutet och kalkylräntan som avskrivningsfaktor," Affärsekonomisk Revy, No. 6, 1958, pp. 157-164.

Saario, Martti. "Poistojen pääoma-arvo ja oikea-aikaisuus," Mercurialia MCMLXI, Helsinki 1961.

Salmi, Timo. "A Comparative Review of the Finnish Expenditure- Revenue Accounting," European Institute for Advanced Studies in Management Working Paper 78-2, Brussels, January 1978.

Salmi, Timo. "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 Advanced Studies in Management Working Paper 80-1, Brussels, 1980 (1980a)

Salmi, Timo. "Estimating the Long-run Internal Rate of Return of the Firm, and Its Financing Behavior from Published Financial Statements," Management Science in Finland Proceedings, ed. Christer Carlsson, Stiftelsens för Åbo Akademi offset- och kopieringscentral, 1980, pp. 349-359. (1980b)

Sarnat M. & Levy H. "The Relationship of Rules of Thumb to the Internal Rate of Return: A Restatement and Generalization," The Journal of Finance, Vol. XXIV, No. 3 (June 1969), 479-490.

Solomon, Ezra. "Return on Investment: the Relation of Bookyield to True Yield," Research in Accounting Measurement, ed. R. K. Jaedicke, Y. Ijiri, and O. W. Nielsen, American Accounting Association, 1966.

Stauffer, Thomas R. "The Measurement of Corporate Rates of Return: A Generalized Formulation," The Bell Journal of Economics and Management Science, No. 2 (Autumn 1971), 434-469.

Stephen, Frank H. "On Deriving the Internal Rate of Return from the Accountant's Rate of Return," Journal of Business Finance and Accounting, Vol. 3, No. 2 (Summer 1976), 147-149.

Tamminen, Rauno. A Theoretical Study in the Profitability of the Firm. Doctoral Dissertation. Acta Wasaensia No. 6, Vaasa School of Economics, 1976.

Tamminen, Rauno. "On the Concept and Measurement of Profitability," Management Science in Finland 1980 Proceedings, ed. Christer Carlsson, Stiftelsens for Åbo Akademi offset- och kopieringscentral, 1980, pp. 361-369.

Vatter, W. J. "Income Models, Book Yield, and Rate of Return," The Accounting Review, October 1966, pp. 681-698.

Whittington, Geoffrey. "On the Use of Accounting Rate of Return in Empirical Research," Accounting and Business Research, Summer 1979, pp. 201-208.

Wright, F. K. "Accounting Rate of Profit and Internal Rate of Return," Oxford Economic Papers (New Series), Vol. 30, No. 3 (November 1978), 464-468.

mjl(ät)uwasa.fi
[ts(ät)uwasa.fi ] [Photo ] [Programs ] [FAQs ] [Research ] [Lectures ] [Acc&Fin  links ] [Faculty ] [University ]
 
[Revalidate]