4. EVALUATION OF THE ESTIMATION METHODS

4.1 Simulation Design and Data Description

To tackle the research questions posed we use the research design delineated by Figure 4. The financial data is generated for the different parameter combinations listed in Table 1. The IRR estimates are obtained for the chosen methods under these different parameter combinations. The obtained IRR estimates are then compared with the true internal rate of return for which the data was generated.

Image: Figure
4. Structure of the Simulation Design

Image: Table 1.
The variation of the parameters in the simulation runs

Our first research question concerns the effect of the business cycles on the robustness of the four IRR estimation methods. For our simulation it is realistic to assume that the long-run average length of a business cycle is six years (C = 6 in Formula (2)). In the simulation the length of the observation period is set at 13 years covering two full business cycles. Three alternative amplitudes of the cycles are used in our simulations. For no cycles we set A = 0.00, for medium cycles we set A = 0.50 and for strong cycles A = 1.00. With an amplitude A = 0.00 there are no business cycles in the capital investments, only the trend and the noise. With A = 1.00 the capital expenditures double from the trend and fall to zero in six year cycles. The amplitude A = 0.50 is between the two. Where the results are found to be insensitive to the cycles, the amplitude is fixed at the average case in the exposition of the results.

The IRR estimation results for the methods under observation will be presented for the different combinations of the essential parameters based on one instance of each combination. One of the components is the random fluctuation in the cyclical level of the capital investments, i.e. the noise term 1+sz in the capital-investment generating Formula (2). We choose a moderate noise level of s = 20% to arrive at a realistic capital investment time series. Only one realization (for each parameter combination) of the randomized time series is picked in our simulation. Stated in terms of statistics and operations research our approach is not a Monte Carlo simulation that would repeat the same parameter combinations with the random term varied. However, to assess the effect due to the noise we conduct for comparison the simulations without the random term (s = 0%). This approach has the advantage of avoiding an exponential amount of further computations without a significant loss of generality.

Our second research question concerns the effect of contribution patterns and the life-span estimates of the capital investments. As was discussed in an earlier section, the underlying contribution pattern of the capital investment process of a real-life business firm cannot be readily, if at all, unraveled from the firm's financial statements. Thus the generic contribution distribution of the firm is not known. Consequently, we simulate the effects of three potential contribution distributions (c.f. Figure 3). The "neutral" uniform contribution distribution, the "growth- maturity-decline" negative binomial contribution distribution and the "steady-decline" Anton contribution distribution are selected. The life-span of the capital investments in the simulation will be set at 20 years. The contribution coefficients for the uniform contribution distribution from Formula (7) for alternative profitabilities become 0.0735 for r = 4%, 0.1018 for 8%, 0.1339 for 12% and 0.1686 for 16%. The negative binomial contribution distribution coefficients from Formulas (9) and (10) are delineated by Figure 5 for an 12% example-level of profitability. Likewise, from Formula (11) the corresponding contribution coefficients for the Anton distribution for the 12% profitability level decline linearly from 0.170 to 0.056.

Image: Figure
5. Negative binomial contribution distribution for profitability of
12%

The life-span of the capital investments affects the numerical values of the chosen contribution distribution and the annual depreciation figures. The life-span of the capital investments is known in the simulation (we have chosen a typical 20 years), but it cannot be accurately known in applications on real-life business firms. This is one of the potential sources of inaccuracy in the IRR estimation methods. The Ijiri-Salamon method and Ruuhela's method require an estimate of the life-span as part of the IRR estimation procedure while Kay's and the average ARR methods do not. The effect of misestimating the life-span in the two susceptible methods will be considered in the analysis section by comparing the IRR estimates with a 20-year life-span to the results with a 16-year and a 24-year life-span.

Our third research question concerns the effect of a disparity between the firm's growth and profitability. As was discussed the earlier literature poses that a growth and profitability equality has a special meaning in the relationship between IRR and ARR. We fix a growth rate of k = 8% in the simulation. The simulated data is generated to produce true profitability figures of r = 4%, 8%, 12% and 16%. The true rates are at and on both sides of the growth rate. Here the relation between the profitability and growth is crucial rather than the absolute levels. Therefore, either growth or profitability could have been fixed for a meaningful simulation and the other varied. We have chosen to fix the growth rate and vary profitability to achieve the cases of low profitability (4%) compared to growth, equal rates (8%) and high profitabilities (12% and 16%). The selected combinations are intended to tally with common growth-profitability combinations of real-life business firms. Figures 6 and 7 present the growth vs. profitability combinations for a sample of 87 U.S. and 244 Finnish firms between 1969-88 and 1965-94 respectively. The data are based on unpublished master's theses written at the University of Vaasa using one of the methods, Ruuhela's method.

Image: Figure
6. Growth vs. profitability; U.S. observations

Image: Figure
7. Growth vs. profitability; Finnish observations

Our fourth question concerns the sensitivity of the methods to the depreciation choice that the firm has used in preparing its financial statements. The simulated time series are produced for three different depreciation methods to evaluate their effect on the results. The first two methods are the straight-line depreciation and double-declining-balance depreciation based on the common accounting practice. The third method to be used in the analysis is the theoretical annuity depreciation. The assumed 20-year life-span of the simulated capital investments means that the annual rate of depreciation in generating the simulated data is 5% in the straight-line method and 10% in the double-declining-balance method. The figures for the theoretical annuity method of depreciation are a function of the true profitability as is seen in Formula (17).

Our last question involves the effect of major irregularities in the level of the capital investments. The robustness of a profitability estimation method can be tested by including capital investment shocks in the model. In business terms such a shock is usually related to a major deviation from the level of capital investment pattern. Experiments are made with different magnitudes and timing of a one-time shock. The shock alternatives simulated are a five-fold shock and a seventeen-fold shock relative to the normal capital investment level in the third or in the ninth year.

Table 2 gives an example of one realization of the time series from the simulated financial statements. The observation period in Table 2 is 13 years from the simulated year 22 to 34 (the lines not denoted by the *). The realization presented in Table 2 is for the case of the negative binomial contribution distribution with a true profitability of 12%, a growth trend of 8%, medium amplitude (A = 0.50) of business cycles, with noise (s = 0.20), no shock, a life-span of 20 years of the capital investments, and a double- declining-balance depreciation of 10%.

Image: Table 2.
Example of simulated observations, negative binomial contribution
distribution, declining-balance depreciation, growth 8%, IRR 12%,
amplitude 50%, noise 20%, no shock

The data are presented graphically in Figure 8. Because of their different scale the book values have not been included in Figure 8. The figure can be visually compared with the corresponding time series of actual business firms. Contrary to the more rigid, steadily growing series of earlier research, the series produced by our simulation model and parameters are realistic in terms of factual observations. This contention is readily corroborated by the empirical time series data gathered in the course of several research projects at University of Vaasa, such as Ruuhela et al. (1982).

As in real-life business firms the simulated time series of the capital investments show a wide fluctuation while the derivative series are much smoother. This results from the fact that the capital investments produce the corresponding cash inflows over a long, lagged period and that similarly the depreciation is extended over the life-span of the capital investments. Furthermore, despite the fluctuations the underlying growth-trend for the firm is a constant in the simulation.

Image: Figure
8. Visualization of simulated observations; the case of negative
binomial contribution  distribution, declining-balance depreciation,
growth 8%, IRR 12%, amplitude 50%, noise 20%, no shock

Capital investment shocks are simulated to test the robustness of the IRR estimation methods. When shocks are included the noise is excluded (see 1+sz and 1+dttS in Formula (2)). This is done in order not to confuse the effect of the irregularities caused by the ordinary noise and the shocks with each other. The time-series data with the one-time shock of a five-fold order relative to the normal capital investment level is presented in Figure 9.

Image: Figure
9. Visualization of simulated observations; the case of negative
binomial contribution  distribution, declining-balance depreciation,
growth 8%, IRR 12%, amplitude 50%, no noise 20%, early five-fold
shock

The Ijiri-Salamon method needs an estimate of the gross book value of assets as is seen in Formula (26). This figure is not routinely available on the balance sheet of a business firm. For obtaining the gross book value an estimate of the cumulative depreciation is needed as is seen in Formula (27). Table 3 displays the cumulative depreciation and the gross book value for the data in Table 2. The numbers are calculated for our error analysis in two different ways. The first two columns are calculated with the exact cumulative depreciation. In a simulation approach this is possible since the engine producing the financial data is known accurately. The two last columns are calculated in line with what could be done with actual data from business firms.

Image: Table 3.
Accumulated, declining-balance depreciation and gross book values;
accurate and estimated figures


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Departments of Accounting and Mathematics, University of Vaasa,
Finland

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