4.3 Evaluation of Ijiri-Salamon Method

4.3.1 Exposition of the IRR Estimates with Ijiri-Salamon

The cash-recovery-rate-based Ijiri-Salamon IRR estimation method differs from Kay's method in two respects in the data that it needs. An estimate of the life-span of the firm's capital investments is needed. Furthermore, an estimate of the gross book value of the firm's assets is needed. (The gross assets Vt are the net assets vt plus the accumulated depreciation. Cf. Formula (25).) This fact introduces two additional, potential sources of error to the method: a misestimation of the life-span of the capital investments and a misestimation of the gross book value. In evaluating Ijiri-Salamon method we can utilize the fact that in simulation the life-span (N = 20) and the true accumulated depreciation, and hence the gross book value of the firm's assets are known precisely. An example of the accurate accumulated depreciation Dt and the accurate gross book value Vt was presented in Table 3 in describing the data of the simulation.

Tables 9 to 11 present the IRR estimates with Ijiri-Salamon method with noise. These tables for the three different contribution distributions include the results for three alternative estimates of the capital investments' life-span E(N). The IRR estimates are presented assuming a correctly estimated life-span of 20 years, an underestimate 16 years, and an overestimate 24 years. In other words for the life-span estimates being off the mark by a fourth. Table 12 presents the IRR estimates for comparison without the noise. Table 13 presents the estimates in the case of early, realistic shock. For brevity, only the cases with the negative binomial distribution are displayed by Tables 12 and 13. The full set of the tables can, however, be readily reproduced for verification since the relevant computer source codes have been made available to the interested reader from the World Wide Web: .

The IRR estimation results are presented assuming that the firm either employs the straight-line depreciation ("Str") or the double- declining-balance depreciation ("Decl"). The accumulated depreciation must be estimated from the financial statements. In accounting practice, the accumulated depreciation figure usually is an approximation based on a time series of recent financial statements. We use the estimate given by Formula (27). An example of the gross book value figures can be seen in the last column of Table 3. The accumulated depreciation can also be calculated accurately in the simulation approach. Ijiri-Salamon's IRR estimates with accurate accumulated depreciation is presented in the "Accu" column of the tables. This particular information facilitates a decomposition analysis of the error sources in the IRR estimates.

Image: Table 9.
Estimation of IRR with Ijiri-Salamon method, uniform contribution
distribution, growth rate k = 8%, amplitude A = 50%, noise = 20%, no
shock.

Image: Table 10.
Estimation of IRR with Ijiri-Salamon method, negative binomial
contribution distribution, growth rate k = 8%, amplitude A = 50%,
noise = 20%, no shock.

Image: Table
11. Estimation of IRR with Ijiri-Salamon method, Anton contribution
distribution, growth rate k = 8%, amplitude A = 50, noise = 20%, no
shock.

Image: Table 12.
Estimation of IRR with Ijiri-Salamon method, negative binomial
contribution distribution, growth rate k = 8%, amplitude A = 0.50,
no noise, no shock.

Image: Table 13.
Estimation of IRR with Ijiri-Salamon method, negative binomial
contribution distribution, growth rate k = 8%, amplitude A = 0.50,
no noise, early realistic shock (tau = 24, S = 5.309).

4.3.2 Effect of Various Factors on Ijiri-Salamon IRR Estimates

As is recalled, the first of our research questions concerns the effect of the business cycles on the IRR profitability estimation methods. As for Kay's method our simulations for Ijiri-Salamon method indicate that the method is not sensitive to cycles. For brevity, the numerical IRR estimation results for the different cycle amplitudes are not displayed. Therefore, the cycle amplitude was fixed at A = 0.50 in the tables presented in the previous section. As for Kay's method, the effect of noise is rather mild as can be seen comparing the representative Tables 10 and 12.

Our fifth research question concerns the effect of investment shocks on the profitability estimates given by the various methods. Our simulations indicate that like Kay's method the Ijiri-Salamon method is reasonably robust to capital investment shocks. Compare Tables 12 and 13 for an example effect of the capital investment shock. In fact, an investigation of the two tables shows that the effect of misestimating the life-span of the capital investments is mostly more marked a source of the IRR estimation error than the effect of the capital investment shocks. A comparison of Tables 12 and 13 with the pair of Tables 6 and 7 for Kay's method indicates that while the effect of the capital investment shocks is not destructive on the methods, its effect on Ijiri-Salamon method is more unpredictable.

Overall, Ijiri-Salamon method fares on the average in the simulations comparably to Kay's method. The worst cases in the regular Tables 9 to 11 appear when the profitability is low compared to the growth. Ijiri-Salamon IRR estimate at worst is 50% off the mark in relative terms. However, in the Ijiri-Salamon method there is no clear pattern to the errors. Unlike in Kay's method there are no cases where the error would disappear. Furthermore, there is no clear pattern to the direction and the magnitude of the error.

As has been discussed, the realization of the theoretical assertions concerning the growth-profitability equality conditions, the annuity depreciation and Anton distribution could be checked. However, these assertions do not cover the relationship between the cash recovery rate and the internal rate of return. This state of matters also is clearly reflected in the simulation results as a lack of similar theoretical regularities as were observed in the results for Kay's IRR estimation method. This can be considered a disadvantage.

4.3.3 Decomposition of the Ijiri-Salamon Method Estimation Error

The simulation results for Ijiri-Salamon method seem at rough par with Kay's method. However, a decomposition of the sources of the overall error exposes a more critical picture of the potential quality of the IRR estimates by Ijiri-Salamon method. The total error in Ijiri-Salamon's IRR estimates is made up by several components, which individually can be larger in absolute terms than the total error, but the components of the error compensate each other in the presented simulations. Table 14 gives one example of the decomposition of the total error into three components. The error decomposed is for the IRR estimates listed in Table 10 for the columns of the double-declining-balance depreciation.

Image: Table
14. Decomposition of the estimation error in Ijiri's method. An
example with negative binomial contribution distribution, declining
balance depreciation, growth rate k = 8%, amplitude A = 50%, noise =
20%, no shock.

The total error is made up of the following three components. If the user of Ijiri-Salamon method knew exactly the true life-span of the capital investments and were able to calculate the accumulated depreciation figures accurately, all the error would be attributable to the method's formal derivation. This error is listed in Table 14 in the column "Formula". However, the focus of interest is on deriving the estimates for real-life business firms. Hence the life-span of the capital investments cannot be readily known accurately. The column "Life span esti" displays how much of the total error is due to errors in estimating the life-span. Furthermore, obtaining the accumulated depreciation from a time series of published financial statements is not trivial and involves approximations in actual accounting practice. The column "Cumu depr calc" reflects the resultant error. The column "Tot. err" gives the total error, which is equivalent to the error in Table 10 between the estimated IRR and the true internal rate of return.

4.3.4 Conclusions about Ijiri-Salamon Method

The main findings about Ijiri-Salamon IRR estimation method are the following. Like Kay's method the Ijiri-Salamon method performs quite well in estimating the long-run profitability of the firm. However, the error of the Ijiri-Salamon method is less predictable and thus more risky than in Kay's method, because of the many sources of the error. In Kay's method the main source of error is a discrepancy between growth and profitability. In the Ijiri-Salamon method it is not possible to pinpoint the main source of error because of their complicated interaction.

Ijiri-Salamon method is unaffected by regular business-cycle fluctuations, but it is mildly affected by noise. The method is reasonably robust to one-time capital investments shocks. The other sources of errors dominate the shocks.

Ijiri-Salamon method lacks similar theoretical results as are characteristic of Kay's method. The mathematical derivation of the method is sound. But the method is not based on the linkage between the income determination in accounting and economics. Hence, there are no theoretical expectations for the method's behavior under special circumstances. To sum up, the method fares comparatively well in practice but fares less well in the theoretical background.


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Goto: The previous section (4.2 Evaluation of Kay's Method)
Goto: The contents section of Salmi and Virtanen (1997)
Goto: Other scientific publications by Timo Salmi in WWW format

Departments of Accounting and Mathematics, University of Vaasa,
Finland

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