The last method to be analyzed in this paper is the method of using the average ARR as the IRR estimate. The long-standing debate about the relevance of the averaged accountant's rate of return as a surrogate of the economist's theoretical profitability comes down to the question whether the average ARR is a good approximation of the firm's IRR, or whether the more complicated methods are the only avenue to a proper long-term profitability estimation (or if any are). The accountant's way of evaluating annual profits is dominant in business practice. Hence the soundness of extending the ARR concept to long-term profitability estimation is of paramount practical importance and interest.
4.5.1 Closeness of the Average ARR Method to Kay's Method
As for Kay's method the effect of cycles is negligible for the average ARR method. Thus the results of the simulation analysis are not presented for all the cycle alternatives. Table 23 gives the IRR estimates using the average ARR method in the case of medium level of business cycles (A = 0.50)
The IRR estimates produced by the average ARR method in Table 23 are strikingly similar to the simulation results with Kay's method in Table 5. The maximum difference in the estimates is only 0.1 per cent in absolute terms. This closeness is not an unexpected result, since Kay's method in the format in Formula (21) can be interpreted as an iterative weighted-average ARR method. Only if major investment shocks are introduced the average ARR method gives estimates that are markedly different from Kay's estimates. This can be seen by comparing Table 24 for the average ARR method and Table 7 for Kay's method for an early shock. A similar comparison be done for a late shock in Tables 8 and 25. The second entry in each cell gives the deviation between the IRR estimates from the average ARR method and Kay's method.
The tables confirm that under ordinary cyclical conditions the average ARR method and Kay's method give virtually equivalent results. In practical long-run profit evaluation terms of the accountant there is no numerical difference between the two methods. Only with the excessive seventeen-fold capital investment shocks the picture of the equivalence between the two methods changes. The methods start deviating markedly for the disparate growth-profitability combinations. Neither method, Kay's nor the average ARR, consistently outperforms the other when shocks are present. For example, Kay's method fares better for an early seventeen-fold shock in the case high profitabilities, but the situation is reversed for the late shock or low profitabilities.
4.5.2 Theoretical Considerations and Conclusion
Given the close kinship between Kay's method and the average ARR method it is interesting to observe which of the theoretical contentions still hold in the simulation for the average ARR method.
The first theoretical contention discussed in connection with Kay's method was Solomon's position that when the growth rate and the true internal rate of return are equal, the accountant's rate of return also becomes the same. For Kay's method no numerical deviation from this equivalence is observed assuming perfectly regular cycles, no noise and no shocks (see Table 6). For the average ARR method the same observation is made when there are no cyclical fluctuations, no noise and no shocks. However, with the cyclical fluctuations, but no noise in Table 26 the relationship no longer holds accurately. The deviation, however, is marginal. (The maximum deviation 0.1 occurs in the table in the case of negative binomial contribution distribution and double-declining-balance depreciation).
As will be recalled, the next theoretical contention is about the equivalence of the IRR and the ARR under the theoretical annuity depreciation method. The validity of this contention is very strong. In our simulations it holds throughout both for Kay's method and the average ARR method, even under disparate growth-profitability combinations and major capital investment shocks as can be observed from Tables 7, 8, 24 and 25.
If the contributions from the capital investments follow the Anton distribution, the straight-line depreciation method results remain equivalent to the annuity depreciation results. Looking at Table 23 this result is seen to hold even under the ordinary conditions of business cycles and noise. However, with major capital investment shocks this theoretical contention ceases to hold both for Kay's and the average ARR methods.
To conclude about the average ARR method, the simulated IRR results are virtually equivalent to the results with Kay's method with the exception of the effect of excessive capital investment shocks. Therefore much the same numerical conclusions apply which already were discussed in connection of evaluating Kay's method. They are not repeated. The general conclusion about the business-practice based average ARR method is, however, very important. The average ARR method mostly performs as well (or as badly) as any of the sophisticated IRR estimation methods analyzed in our research project. Considering this fact and the average ARR method's practical appeal it is safe to say that for a practitioner it comes out best of the methods analyzed in this paper. The importance of the other IRR estimation methods, especially that of Kay's and Ruuhela's, lies in their merits for the theory of accounting.