3 Testing the Model with Simulated Financial Statements

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+1=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

(15) Sum[bi(1+r)^i]=1

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).

(17) ft = Sum [bi gt-i]

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) pit is assessed as the interest on the initial capital stock v(t-1) in year t, i.e.

(18) pi(t) = rv(t-1)

In any depreciation method the profit pit is given by deducting depreciation dt from revenues ft .

(19) pit = ft - dt

Hence, depreciation in the annuity method is given by

(20) dt = ft - rv(t-1)

In any depreciation method the capital stock v(t) 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

(22) vt = Sum (fj-gj)/(1+r)^(j-t)

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 v(t)) 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/ 1.2^2= 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

(23) d(t) = Sum b(i)g(t-i)/1+r)^i

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 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.

Figure 2

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

(24) d(t) = Sum (1/m)g(t-i)

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.

Figure 3

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.


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Department of Accounting and Finance, University of Vaasa,
Finland

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