4.2.1 Effect of Regular Business Cycles
We begin the evaluations by assessing the effect of business cycles on Kay's method. The IRR estimates by Kay's method are presented in Table 4 for the three different levels of amplitudes in the business cycle. The results are presented in Table 4 for the negative binomial contribution distribution which is the most general of the alternative distributions. To see the pure effect of the cyclical component we first omit the random noise term. The results are presented for the four different growth-profitability combinations and the three different depreciation methods "Str" straight-line depreciation, "Decl" double-declining-balance depreciation and "Ann" annuity depreciation.
It is readily seen in the table that the effect of the business cycles is marginal for Kay's IRR estimation method. In the worst case with the strong cycles (A = 1.00) the difference between the IRR estimates 18.9% and 18.6% (16% true profitability and double- declining-balance depreciation) is only 0.3%. The presented result is for the negative binomial contribution distribution. The results for the other two contribution distributions, the uniform distribution and the Anton distribution, indicate a similar insensitivity. (The additional tables are not displayed for brevity.) Hence we can safely conclude that Kay's IRR estimation method is not affected by regular business cycles. This being the case the rest of the analysis of Kay's method can be conducted without a loss of generality using the medium cycle strength (A = 0.50).
4.2.2 Overall Accuracy of the Kay's IRR estimates
We can now analyze the total error in Kay's IRR estimates. Table 5 presents the results for Kay's IRR estimation method under medium business cycles. The noise component is included at this phase. The results are condensed into a single table for the three contribution distributions.
The general impression conveyed by Table 5 is that the level of Kay's IRR estimates is fairly well in line with the true profitability. In particular, when the firm's growth and profitability are near each other, Kay's method performs excellently.
There are, however, situations where Kay's method performs poorly.
The biggest absolute discrepancy in Table 5 in an estimate ( = 19.5% vs. r = 16%) takes place when the
true internal rate of return deviates most from growth, the capital
investments contribute according to the uniform distribution and the
firm uses the double- declining-balance method. Kay's IRR estimate
is off by 3.5% (by a fifth in relative terms). Likewise, at the true
profitability of 4% the IRR estimate, with the uniform contribution
distribution and the double-declining-balance depreciation, is off
by a third (2.6% vs. 4.0%). These are marked deviations.
It is not easy to evaluate how serious the observed errors are from the point of view of decision making. It depends on whether the alternative methods give better estimates. Most importantly, the seriousness of a deviation would depend on what would be the consequences of the management of the firm having erroneous profitability information. Predicting such consequences in quantitative terms is a very involved question and is outside the scope of our research.
4.2.3 Effect of Noise
In Table 4 it was observed that the effect of the business cycles on the estimation error is marginal. To asses the effect of the noise component Table 6 presents Kay's IRR estimates without the noise for a comparison with Table 5.
While the noise term in the capital investment level seems to have more effect on the Kay's IRR estimation than the regular cycles the effect of noise is rather mild. At most the IRR estimate changes from 20.3% to 19.5% (in the case of the 16% true profitability, uniform contribution distribution and double-declining-balance depreciation). The magnitude of the difference is 0.8% compared to a total error of 4.3%. We conclude that noise is not a main source of the estimation errors. Hence the analysis can be founded on the results in Table 5.
At this juncture a general word of caution is in order. It goes without saying that generalizing from the conclusions based on simulation rather than analytical deduction always should be considered with a fair amount of caution.
4.2.4 Effect of Contribution Patterns, Growth-Profitability Relationship and Firm's Depreciation Choice
Our second research question concerns the effect of the type of capital investments available to the firm. Consider Table 5 anew for effect of the alternative contribution patterns. As pointed out earlier, the shape of the contribution distribution of the capital investments is not readily known for real-life firms. Therefore it is of interest to test whether the IRR estimation results are sensitive to this factor. It is seen that under capital investment opportunities that contribute in accordance with the negative binomial distribution, or the Anton distribution, the results are more accurate than under the non-declining uniform contribution distribution.
Our third research question concerns the effect of the discrepancy between growth (k) and profitability (r). It is obvious from the results that a discrepancy between growth and profitability levels is the crucial source of error in the Kay's IRR estimates. It is also noted that when r > k Kay's IRR systematically overestimates the true profitability (the special case of the straight-line depreciation under the Anton contribution distribution will be discussed in a later section). Thus it appears that Kay's method gives even too optimistic IRR estimates to firms with good profitability. For r < k the direction on the estimation error depends on the contribution distribution, the depreciation combination and the irregularities in the capital investments (the noise). Thus it would seem that it is not possible to make any predictions whether Kay's estimates for firms with low profitabilities are optimistic or pessimistic.
Our fourth research question concerns the effect of the depreciation method choice that the firm makes. The effect of the firm's accounting choice appears highly important to the accuracy of the IRR estimates. The error in the estimates in Table 5 is about half or less when the firm applies the straight-line depreciation method instead of the double-declining-balance method. This observation raises interesting accounting issues about the depreciation method choice.
4.2.5 Effect of Major Capital Investment Shocks
Our fifth research question concerns the effect of major capital investment shocks. Figure 9 delineates an example time-series data. Table 7 gives Kay's IRR estimates under a third year shock ("early shock"). Table 8 is for a ninth year shock ("late shock"). To isolate the effect of the shocks, noise has been excluded.
Kay's IRR estimation method seems to be reasonably robust to the capital investment shocks even if there is some disruption in the estimates. The effect of the shock seems to be to decrease the IRR estimates, the more the bigger and later the shock appears. The observed behavior is easy to explain. The one-time investment shock becomes dominating, and its effects are much outside the period under observation.
For high profitabilities relative to growth the shock compensates for the error caused by the growth-profitability discrepancy. For low profitabilities the error from the growth-profitability discrepancy is even aggravated. It can be noted, however, that logically it is not equally likely that major capital investment shocks will appear in corporations with profitability problems than in firms with good profitability prospects. Furthermore, as will be observed in the next section, the introduction of major capital investment shocks will cause deviations from the theoretically expected results.
4.2.6 Theoretical Considerations
There are several theoretical assertions about the relationship between the internal rate of return and the accountants rate of return under the specific growth rates, depreciation methods and contribution distributions presented in earlier literature. Next we consider these assertions, under the more general conditions of business cycles and noise, utilizing our simulation results.
Solomon (1966: 115) posed that when the growth rate and the true internal rate of return are equal, the accountant's rate of return also becomes the same. Consequently, it is theoretically to be expected that if the growth and profitability are exactly equal, Kay's method should give exactly the correct IRR estimate because it is built on the relationship between the IRR and ARR. The equality would be expected to hold over all the contribution distributions and over all the depreciation methods.
Formula (2) generates the capital investments. It added several components to the constant growth model. Consider the presented theoretical contention with the added components. Table 6 confirms that the expected equality holds (within the used numerical precision) not only in the case of constant, exponential growth but also in the case with the business cycles added. However, when irregularities are introduced in terms of the noise (cf. Table 5), the expected theoretical result no more fully holds. The deviation is not marked numerically, but theoretically the assertion breaks. As is natural, the disruptive effect of the one-time capital investments shocks is more marked than that of the noise.
Analytically, the accountant's rate of return and the internal rate of return are equal when the annuity method of depreciation is used (see e.g. Salmi and Luoma, 1981: 28 and Peasnell, 1982a: 364). The simulation results for Kay's method are in agreement with this contention for all the observed combinations of growth vs. profitability and for all contribution distributions even with the irregularities introduced upon the growth-trend and the business cycles in terms of the noise and the capital investments shocks. See the columns marked "Ann" in Tables 5, 7 and 8. The theoretical results about the annuity depreciation are very strong. They are in line with discussions and results in literature about accountant's and the economist's concepts of income.
It is a well-known result that the theoretical annuity depreciation method and the business practice straight-line depreciation method yield the same depreciation if the contribution distribution for the capital investments is the Anton distribution. See Solomon (1971; 168 footnote) for references. Consequently, for the Anton contribution distribution the simulations should produce the same IRR estimate for the straight-line depreciation as it does for the annuity depreciation. Also this theoretical contention is corroborated by the simulation. Compare the columns marked "Ann" and "Str" below "Anton" in Table 5. This result holds even if major capital investment shocks are introduced. (The numerical tables for the Anton distribution with the major investment shocks are not displayed for brevity.)
4.2.7 Conclusions about Kay's Method
The main findings about Kay's IRR estimation method are the following. Under ordinary circumstances Kay's method performs quite well. However, the deviation of Kay's IRR estimates from the true internal rate of return can be considerable if the growth rate of the firm and its profitability are not near each other. This is the main source of error in Kay's method.
Kay's method seems to lead to systematically overoptimistic profitability estimates when the firm's true profitability exceeds the firm's growth considerably. If the true profitability is below the firm's growth the nature of Kay's IRR estimate is ambiguous.
The magnitude of the error caused by a growth-profitability gap is jointly dependent on the contribution pattern of the capital investments, the firm's depreciation choice and the noise in the capital investment time series.
Kay's method is not affected by regular business-cycle fluctuations in the capital investment time series, but it is mildly affected by noise. Kay's method is reasonably robust to major capital investments shocks. The irrelevance of business cycles and the mild effect of noise on the accuracy of the estimates are important advantages in Kay's method.
Kay's method has a firm theoretical background in the theory of accountant's and economist's profit concepts. This fact is reflected in always getting exactly the expected IRR estimates under the theoretical annuity depreciation and getting fairly accurate IRR estimates under the equality of growth and profitability. Furthermore, if the capital investments contribute in accordance to the Anton distribution, the estimates under the firm applying a straight-line depreciation are accurate.