The previous chapter presented our simulation engine. The current chapter presents four IRR estimation methods from earlier literature to be analyzed and evaluated with our simulation approach. The methods are Kay's, Ijiri-Salamon's, Ruuhela's and the accounting-practice compliant average ARR method. Before any IRR estimation method can be applied on the simulated (or actual financial) statements, the IRR estimation method must be made operational for the financial data available. This fact is observed whenever necessary. We do not evaluate market-value-based methods in this paper. However, the market-value-based methods by Lawson and Steele are briefly discussed at the end of the chapter.
3.1 Kay's Method
3.1.1 Presentation of the Method
Kay (1976) presented an iterative method for estimating the IRR. Kay's original presentation used continuous notation. From the accounting point of view, however, a discrete version of Kay's results is needed to make the method applicable on simulated (or real-life) financial statements of a business firm. For Kay's method we have from Kay (1976: 451), Salmi and Luoma (1981: 25) and Peasnell (1982a: 371) the discrete version for IRR estimation
As is recalled, the years in our data-generating simulation engine run from 1 to T while the actual observation period is from T-n-1 to T. For notational simplicity the indexing of the years of the observation period has been adjusted in Formula (20) to run from 1 to n. In this notation, the annual accountant's profit (operating income) pt and the book values of the firm's assets vt at the end of each year are now observed for years 1 to n. Therefore the first vt-1 available is for year t = 2. This fact is duly reflected in the summation notation in Formula (20).
Kay's iterative method is easily coded as a computer program to solve Kay's IRR estimate given the profit and book value observations from the financial statements. For the conditions of convergence of the IRR iteration procedure see Steele (1986: 2-5). The actual programs coded for this paper are Turbo Pascal 7.01 programs for an MS-DOS PC. (The programs are made publicly availableon the World Wide Web at the following address <URL:http://www.uwasa.fi/~ts/smuc/prog/smucprog.html>.)
3.1.2 Discussion of the Method
Rewrite Formula (20) as
It is immediately obvious that Kay's IRR estimate is a weighted average of the accountant's rates of return over the observation period, which would have been
The factors act as Kay's weights. Thus Kay's method
implicates a strong link between the economist's and the
accountant's rate of return concepts.
The following question naturally arises and will be tackled in our simulation evaluation. Is the more complicated IRR estimation Formula (20) decidedly better than the straight-forward Formula (22) which furthermore is based on well-established accounting concepts?
The link between the accountant's and the economist's valuation
concepts are very evident also in Kay's (1976: 455) derivation. Kay
presents the following relationship between the IRR estimate
(),
the true (economist's) internal rate of return (r), growth-rate (k),
the accountant's book value (v) and the economist's valuation of the
firm (w)
In the above the economist's value of the firm w is the present
value of the future net cash flows (c.f. Formula (19)). The
accountant's book value is based on the historical accounting data
of the capital investments and depreciation (c.f. Formula (13)). If
the two valuations agree, then also the accountant's and the
economist's rates of return agree. The first corollary of this fact
is that if the (theoretical) annuity depreciation could be used,
then Kay's method is expected to give the exactly correct IRR
estimate (
= r). See e.g. Salmi and
Luoma (1981: Appendix III) for the proof. The second corollary,
in line with Solomon (1966: 115), is that if the growth and the
profitability agree (r = k) then, again, Kay's method is expected to
give the exactly correct IRR estimate (
= r).
3.2 Ijiri-Salamon Method
As was seen in the previous section Kay's method can be interpreted as a method that seeks the link between the IRR and the ARR. Another route is taken in the Ijiri-Salamon method. Ijiri (1979) presented what Salamon (1982) interpreted and expanded as an IRR estimation method based on the concept of the cash recovery rate, CRR. Ijiri (1979: 259) derived the following relationship between CRR and IRR
When the CRR is known, the corresponding value of IRR can be readily solved by numerical iteration from Formula (24) using e.g. the bisection method. The IRR estimation problem thus becomes a CRR estimation problem. The central idea of the Ijiri-Salamon method is using this surrogate because CRR is easier than IRR to estimate from the financial statements.
The cash recovery rate CRR can be defined as the ratio between the cash inflows from capital investments and the outstanding gross capital investments. Ijiri (1980: 55) presents the calculation of an annual CRR from published financial statements as
In our simulation evaluation the cash recoveries are simply equivalent to f t. The gross assets must be discussed in more detail. The total assets are given directly by the book value vt-1. First, when the total assets have been defined the accumulated depreciation must be assessed to get the gross assets. Second, the beginning instead of the average book values are used in our study.
In financial statement analysis practice the accumulated depreciation is typically obtained by canceling backwards the depreciations for a suitable span of years. In analysis practice the choice of the backwards span tends to be somewhat arbitrary. However, it is mathematically obvious that given the average life-span of the capital investments and a constant level annual depreciations, the accumulated depreciation will be given by accumulating the depreciations from half the average life-span. While this result concerns the straight-line depreciation, the choice will be used as the best approximation for all the depreciation profiles.
Furthermore, Ijiri's approach requires an estimate of the life-span N of the firm's capital investments. This means a potential source of further estimation errors in the method. In simulation the true life-span of the capital investments is known accurately. Hence the effect of the accuracy of estimating the life-span of the capital investments can be examined for Ijiri-Salamon method in our simulation approach. Note that this potential source of error is not present in Kay's method.
Next consider the different conventions in calculating the book values in financial statement analysis. Instead of the often suggested averaging between the annual beginning and ending book values we use the beginning values vt-1. This leads to more accurate results when a discrete instead of a continuous approach is used. This choice is in line with the treatment of Kay's method in Salmi and Luoma (1981) and Peasnell (1982a).
The estimates of the annual cash recovery rates are
calculated from
where Vt denotes the gross assets at the end of year t calculated from
where N is assumed an even integer for notational simplicity.
The calculated values are averaged and the average is substituted as
CRR into Formula (24) in line with Ijiri (1980). Ijiri's IRR
estimate can then be iterated from Formula (24).
Since we are using a simulation approach with a fully known engine to generate the observations, we also have the option to calculate the exact accumulated depreciation. This enables us to differentiate between the sources of the error in the IRR estimate. The components of the error are the error due to Ijiri-Salamon's method and the error due to the approximation of the accumulated depreciation.
3.3 Ruuhela's Method
The third method to be included in our analysis is the IRR estimation component of Ruuhela's "Growth, Profitability and Financing" model. As we have seen in the above, Kay's method is based on a relationship between the ARR and the IRR and Ijiri-Salamon method on the relationship between the CRR and IRR. Ruuhela's method can be considered to fall into a category of direct estimation of the IRR from the financial statements without the intermediate ARR or CRR concepts.
The method was first presented in Ruuhela (1972) and mathematically streamlined by Salmi (1982). The method was restructured in Ruuhela et al. (1982). The explicit estimation of the firm's growth and the assumption of a stable business-culture period are characteristic of Ruuhela's approach.
Ruuhela's IRR estimate is given by
where k is growth-rate trend of the capital expenditures,
is
the annuity factor
and F is defined as the capital investment ratio
Ruuhela's method assumes a constant, exponential growth of the capital-investment g t and the cash-inflow f t time series of the firm. The quotient F of the two time series thus is constant in the method. Ruuhela's method also assumes that the capital investments contribute in accordance to the Anton distribution.
In applying Ruuhela's method an estimate of the common growth rate
of the firm's time series is needed. Most often an OLS estimate of
the growth-trend of the firm's funds from operations corresponding
to f t is used as the estimate. Given the OLS estimate
of the growth trend the capital
investment ratio is estimated from
3.4 Discussion of the Model-Oriented IRR Estimation Methods
Consider the conceptual backgrounds of the three methods presented so far. The IRR estimation formulas of Kay's and Ijiri-Salamon methods draw on the relationship between an income statement variable (a flow variable) and a balance sheet variable (a stock variable). As is seen from Formula (20) Kay's method involves the accounting profit pt and the book value of the firm vt. Conceptually, Kay's presentation leans heavily on exploring the relationship between the economist's and the accountant's rate of profit.
The Ijiri-Salamon method involves the cash inflows f t, the gross assets Vt, i.e. the book value of the assets undepreciated, and the life-span N of the firm's capital investments. This is readily seen from Formulas (26) and (20). The concept of the cash recovery rate is central in the method.
Ruuhela's method is directly based on the conventional internal rate of return model of capital investments. Ruuhela's method consequently directly involves the two relevant flow variables the cash outflows to the capital investments g t and the annual cash inflows f t and the concept of discounting in the form of the annuity factor a N,k. No stock-concept variable is involved. The role of the growth variable k comes from the fact that the consecutive capital investments that produce the corresponding, lagged cash inflows, typically grow in a going concern. Ruuhela stresses that the profitability of the firm is a long-term concept based on business culture of the firm to be able to generate and utilize capital investment opportunities. According to Ruuhela firms usually experience long phases of stable business culture when the long-run profitability stays on a rather fixed level. Profitability can be measured for such stable intervals. In corporate life a change or a discontinuity in business culture often coincides with a change of the top-level management. At such junctures the long-run profitability typically changes and should be estimated anew. Ruuhela prefers to call the profitability of such a stable period the profitability of the business culture rather than the profitability of the legal entity, the firm. In our simulation testing the business culture is taken as unchanged.
3.5 Averaged Accountant's Rate of Return Method
The fourth and last method included into our analysis is based on straight-forward accounting practice. Much of the discussion, ever since Vatter (1966), in the ARR vs. IRR debate has centered around the question whether or not the ARR is a good approximation of the IRR. Instead of reentering the deductive debate we seek a resolution to this question by including the averaged ARR in our simulation and comparison. The inclusion of the average ARR method is prompted by the fact that accounting practitioners routinely use and are comfortable with the concept of annual profits and return on investment. Employing averaged ARR as the IRR estimate can be considered a direct extension of this business practice.
The average ARR is calculated as the arithmetic average of the accountant's annual rate of return from Formula (22). Technically, an average can be calculated as an arithmetic average or a value-weighted average. We use the former for two reasons. First, the arithmetic average is in line with business practice. Second, an average with a large fairly stable denominator is very little affected by the choice of the averaging method. Only in the case of major shocks some differences might exist. The beginning book values are used in the denominator instead of the annual averages in line with our treatment of Kay's and Ijiri-Salamon methods.
Our advance hypothesis is that the average ARR method will not be inferior to the other methods. Our hypothesis is based on the concept of economic Darwinism. Quoting Watts and Zimmerman (1986; 195) "Competition among firms implies that operating procedures ... that are used systematically by surviving organizations are efficient."
3.6 Discussion of Market-Based Methods
The methods discussed so far use pure accounting data from the income statement and the balance sheet. The internal rate of return, however, is based on future cash flows in line with the economist's income concepts and valuation of assets. The question arises if IRR estimation methods based on market values rather than book values should be used. There are several papers putting forward implicit or explicit suggestions of an estimation of the IRR involving the market values of the firm's stock.
Reconsider Kay's method. Formula (23) can be interpreted as a suggestion by Kay to adjust the accounting-based IRR estimate with the market value of the firm wt to arrive at the internal rate of return which would agree with the economist's rate of return.
Lawson (1980) presented a method for estimating the equity, debt and entity rates of return for the firm. To estimate IRR, his cash-flow based method equates the discounted operating cash flow less net capital investment less tax payments less/plus liquidity change to the discounted sum of initial and the ending (market-based) value of the firm.
Steele (1986: 8) suggested in his paper evaluating the derivations in Salmi (1982) and Peasnell (1982a, 1982b) an alternative version of Kay's Formula (20) to include market values into the estimation of the firm's IRR.
Theoretically the idea of basing the IRR estimates on stock prices is sound, because the prices reflect the economist's valuation of the firm's future income in line with the internal rate of return concept. However, there are some serious problems with the practicality of this theoretically well-founded approach. First, firms are not necessarily traded on stock exchanges so genuine market values would not be readily available for a considerable number of firms. Second, it is a well-known fact that stock prices are more volatile than accounting earnings. This indicates a potential, temporal instability in what should be stable long-run profitability estimates. Third, it is not easy to assess whether or not the accounting function of business firms would agree on a measure of income based on market values instead of deep-rooted accounting conventions.
Despite the practical reservations stated in the above, the evaluation of the market-based methods of IRR estimation would be highly interesting. However, we do not pursue this avenue in the paper at hand. The line of enquiry is not readily amenable to the present simulation model. In particular, the problem is establishing a reliable procedure that would give the market values of the simulated firm which would be exactly compatible with the true internal rate of return. Extrapolating the time series into infinity in the simulation is not a viable answer. The results would be too volatile for the simulation evaluation. Furthermore, an extrapolation to infinity would be unrealistic. The business culture of a firm is not preserved to the infinity with an unchanged long-term profitability.
However, it can be noted that there is some recent research information about the IRR estimates arrived at by the accounting-based vs. market based methods. An unpublished master's thesis prepared under our supervision tentatively indicates that the IRR estimates from a sample of real-life business firms derived from Ijiri-Salamon method are much more closely related to the estimates from the accounting-based Ruuhela's method than the from the market-based Lawson's method.