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 in
year t, and the corresponding revenues
(s = t,...,t+N). The revenues each year can formally be
linked to the expenditure by a contribution distribution
as shown below.
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
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
in year t, i.e.
In any depreciation method the profit is given by deducting depreciation
from revenues
.
Hence, depreciation in the annuity method is given by
In any depreciation method the capital stock
is arrived at from
This follows from the fact that expenditures increase the capital stock while depreciation
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 , the growth rate
k, and the contribution coefficients
(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
= 40,
= 0.7,
= 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
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
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/ = 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
= 40,
= 0.7,
= 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 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.