To recount the general, formal definitions, ARR is defined in
literature as
. a(t) = (F(t) - D(t))/K(t),
where F(t) is the funds flows from operations in period t, D(t) is
the depreciation in period t, and K(t) is the net book value of
assets at the beginning of year t. (The average of K(t) and K(t+1)
is also often used.) IRR is naturally defined as r by
n t I(o) = sum R(t)/(1+r), t=1where I(o) is the initial capital investment outlay, R(t) the net cash flow in period t, and n is the life-span of the capital investment. (The existence conditions for a rational solution for r, and the multiple solutions of the polynomial equation have been tackled in the relevant literature but are not reviewed in this paper).
British economists present one tradition of tackling the question of the divergence between the ARR and IRR since Harcourt (1965) put forward his position that the accountant's rate of return is "extremely misleading". Using four different cases of accumulation of assets (growth) he asserts that it is not possible to develop rough rules of thumb to adjust ARR to reflect IRR under different life-spans of investments, the net cash flow patterns generated by the investments, different growth rates, and different depreciation methods. He concludes by an explicit warning about profitability comparison between firms in different industries or different countries if accountants' measurements are used. It can only be deduced that he implicitly gives very little value for the financial statements annually prepared by the accounting profession.
The formal mathematical relationship between the ARR and IRR is independently considered by Solomon (1966). Using both a zero-growth and a growth model he demonstrated that the ARR (book-yield in Solomon's terms) is not a reliable measure of the IRR (true-yield in Solomon's terms). His paper shows that the difference between the two measures involves project lives, the depreciation method, and the lag between the investment outlays and their recoupment. Further numerical examples to illustrate the disparity of accounting and economic profitability measurement are provided in Solomon and Laya (1967). Interestingly these two papers are practically devoid of references to other literature. Vatter (1966) ponders the content of Solomon's paper at great length. He questions both the realism of Solomon's assumptions and the validity of IRR as a practical measure of profitability.
The relationship between the ARR and IRR is also indirectly involved in studies considering the relation between rules of thumb for capital investment decisions (payback reciprocal) and the ARR on the other hand and IRR on the other. See Sarnat and Levy (1969, p. 483).
Livingstone and Salamon (1970) build on Solomon's model and conduct a simulation analysis of the ARR-IRR relationship by extending the assumptions of the previous models into more general cases. They observed under their assumptions that ARR shows a dampening cyclical behavior determined by the project life-spans, pattern of cash flows generated by the projects making up the firm, the reinvestment rate, and the level or IRR. They also include the effect of growth. McHugh (1976) and Livingstone and Van Breda (1976) have an exchange of views about the mathematical derivations and the generality of the results of Livingstone and Salamon (1970).
Stauffer (1971) presents a generalized analysis of the ARR vs IRR relationship using continuous mathematics under several cash profile assumptions. He demonstrates that the depreciation schedule affects the relationship. Also he puts forward that the accounting and the economic measurements (ARR/IRR) are irreconcilable, and that the situation is aggravated by the introduction of taxation into the analysis. From the accounting point of view it is interesting that he points to the task of estimating the real rates of return from historical accounting data.
Also Bhaskar (1972) arrives at the conclusion that "in general ARR does not perform satisfactorily as a surrogate for the IRR". He also points out that the using the annuity method of depreciation makes ARR a more accurate reflection of the IRR, but points out that the annuity method has undesirable side-effects for accounting measurement. Bhaskar augments his deductions with a statistical analysis of his simulation results on ARR and IRR levels. Likewise, for example, Fisher and McGowan (1983) consider economic rate of return (IRR) the only correct measure of economic analysis. They conclude that the accounting rate of return is a misleading measure of the economic rate of return and see little merit in using the former. Long and Ravenscraft (1984) present a critical view on Fisher and McGowan's claim of the prevalence of the IRR, the assumptions in their examples, and their mathematical derivations. Fisher (1984) discards the criticism insisting that ARR does not relate profits with the investments that produce it.
Gordon (1974, 1977) takes a more optimistic view on the potential reconciliation between ARR and IRR. He shows that ARR can be a meaningful approximation of the IRR when "the accountant's income and asset valuations approximate the economic income and asset values". The central condition is linked to the depreciation method. The accountant's accumulated depreciation must approximate the accumulated economic depreciation for the ARR and IRR to converge. Gordon concludes by pointing out that even if no general "cook-book tricks" can be devised for converting the ARR to the IRR, the managers can be able to make sufficient adjustments. To us this view appears logical because it is unlikely that profit oriented business firms could, in the long run, indulge in totally unsound measurement and management practices. Stephen (1976), on the other hand, claims that Gordon fails to resolve the difference between ARR and IRR.
Kay (1976) refutes Salamon's conclusion and contends that IRR can be approximated by the ARR irrespective of the cash flow and depreciation patterns. The crucial requirement is that the accountant's evaluation of the assets (their book value) and the economist's evaluation (the discounted net cash flow) are equal. He also applies his results to estimate the profitability of the British manufacturing industry 1960-1969 from aggregate accountant's data. Key and Mayer (1986) revisit the subject coming to the conclusion that "accounting data can be used to compute exactly the single project economic rate of return". Wright (1978) considers Kay's (1976) view too optimistic and claims that one cannot easily translate ARR into IRR except under special circumstances. Salmi and Luoma (1981) demonstrate using simulated financial statements that applying Kay's results require more restrictive assumptions than originally indicated by Kay (1976). Stark (1982) recounts Key's results by including working capital, loan financing and taxation.
Tamminen (1976) presents a thorough mathematical analysis (with continuous time) of ARR and IRR profitability measurement under different contribution distribution, growth conditions, and depreciation methods. As one result he derives a growth-dependent formula for a conversion between IRR and ARR assuming realization depreciation. (For the definition of the realization depreciation see e.g. Bierman, 1961, and Salmi, 1978). The analysis is conducted under steady-state growth conditions, then extended to structural changes and for under cyclical fluctuations.
Whittington (1979) points out that the ARR vs IRR discussion should also consider whether ARR, instead of IRR, already inherently is a valid and useful variable especially in the positive research approach. He also studied the possibility of extenuating circumstances that could reduce the ARR vs IRR discrepancy in statistical analysis. Peasnell (1982b) goes on to consider the usefulness of ARR as a proxy of IRR for FRA. Applying a standard variation measure he comes to the conclusion that the usage of ARR does not lead to serious valuation errors in FRA provided that the variations in the ARRs are not too great. He presents an iterative weighting scheme for estimating the IRR from the ARR. Peasnell (1982a) also considers economic asset valuation and yield vs accounting profit and return in a discontinuous (discrete time) mathematical derivation framework (while Kay, 1976 used continuous time). He proves that if there are no opening and closing valuation errors of assets with respect to their economic values, and ARR is a constant, then the constant ARR equals the IRR. Under constant growth equal to IRR he proves that IRR can be derived as the mean of ARRs. He also studies the relationship when IRR is not equal to the growth rate of assets.
Luckett (1984) reviews and summarizes the ARR vs IRR discussion. He also stresses the fact that the measures are conceptually different by nature. The IRR is a long-term, average-type ex ante measure, while the ARR is a periodic ex post measure. His main conclusions are pessimistic. He points to the results stating that the annual ARR is a surrogate of the IRR only under very special circumstances. He also claims that it estimating the IRR in actual practice from the annual ARRs is not generally practical. Kelly and Tippett (1991) present a stochastic approach to estimating the IRR and ARR, and find them significantly different in a sample of five Australian firms. Shinnar, Dressler, Feng, and Avidan (1989) estimate the IRR, ARR and the cash flow pattern for 38 U.S. companies for 1955-84.
Jacobson (1987) takes another approach to the IRR vs ARR controversy. He evaluates the validity of ARR as a proxy for IRR by examining the association between corporate level ARR and the stock return for 241 Compustat firms for 1963-82. He concludes that while ARR has serious limitations as a measure of business performance, claiming that ARR has no relevance is an overstatement. However, he does not take on examining the association between IRR and stock returns, possibly because of the difficulty of estimating the IRR from the published data.
The pioneering work from the account's point of view in estimating the internal rate of return from the firms financial statements is Ruuhela (1972). He presents a model of firm's growth, profitability and financing. Assuming constant growth and that the firm is constituted of a series of capital investments, he establishes a general method to estimate the firm's long-run profitability (IRR) from published financial statements. He also points out that the annual income of the firm can be measured from this IRR estimate and the capital stock of the firm. Furthermore, they point out that the long-run financial policy of the firm manifests itself in growth-discounted average balance sheet.
The mathematical derivation of Ruuhela's model is streamlined in Salmi (1982). The IRR estimation procedure is later enhanced in Ruuhela, Salmi, Luoma and Laakkonen (1982). The paper also presents an empirical application to compare the long-run profitability of eight major Finnish pulp and paper firms for 1970-1980. Salmi, Ruuhela, Laakkonen, Dahlstedt, and Luoma (1983a, 1983b, 1984) present hand-book type instructions for IRR estimation from published financial statements for the accounting profession. Jegers (1985), Salmi and Ruuhela (1985), Van der Hagen and Jegers (1993), and Salmi and Ruuhela (1993) exchange views about the validity of the presented IRR estimation methods.
An integral part of in Ruuhela's method is the estimation of the firm's growth rate. Salmi, Dahlstedt and Luoma (1985) consider how the growth estimation can be improved by eliminating cycles from the accounting data. Ruuhela's method requires about 11-13 years of data and thus often covers different phases of business cycles.
Steele (1986) criticizes Ruuhela's model for its strong steady-state assumptions. Based on Kay's model and Peasnell's results he presents an iterative process for estimating the IRR from published financial statements without the steady-state assumption. On the other hand his approach requires market-based values and thus limits the range of firms that can be the target of the profitability evaluation. Brief and Lawson (1991a, 1991b) derive a simplified error term for the IRR estimation. Using simulation they cast doubt especially on the accuracy of IRR estimation for a small number of observations.
Ijiri (1979, 1980) shows that under certain general conditions the recovery rate converges to the "discounted cash flow rate" which is similar to economist's measure of the firm's profitability. The conceptual difference is that the economist's valuation is based on the future cash flows while in the profitability estimation only the historical data is used. By not involving the ARR, this approach circumvents the major ARR vs IRR controversy, that is the disagreement whether ARR can be a proxy of the IRR under any realistic assumptions. Salamon (1982) indicates explicitly that Ijiri's discounted cash flow rate is the firm's IRR. According to Salamon "Ijiri has shown that if the measure of a firm's IRR is desired it can be obtained by analyzing a model of the relationship between the IRR and the firm's cash recovery rate rather than by analyzing a model of the relationship between the IRR and the firm's accounting rate of return". Salamon extends Ijiri's analysis to the case where the firm does not reinvest all its cash flows. Furthermore, Salamon examines the relationship between the firm's CRR and IRR under inflation. Later Salamon (1988) utilizes the CRR method for studying the usefulness of ARR in IRR estimation. He casts doubt on the usefulness of ARR-based measures in economics.
The CRR method does not remain unchallenged. Brief (1985) casts doubt on the practicality of the CRR method. He notes that in the CRR method for IRR estimation requires information about a firm's past as well as its future cash flows. He argues that the CRR papers do not deal with the problem of predicting the future cash flows. Lee and Stark (1987) reject Ijiri's CRR method as "unsound". They put forward mathematically, and using numerical examples, that Ijiri's method can produce investment evaluations which differ from the conventional discounted cash flow approach. The conclusion would be that CRR cannot be used for unique IRR estimation. Also Stark (1987) casts doubt on the operationality of the CRR approach.
Gordon and Hamer (1988) present a more optimistic view on the CRR method. They extend the CRR method to a concave cash flow pattern. Estimating the IRR and CRR profitability from the same sample which Ijiri (1980) and Salamon (1982) used, they come to the conclusion that the rankings given by the two methods are sufficiently consistent. Griner and Stark (1988) develop an alternative approach making explicit predictions of the future cash flows in order to estimate the CRR. They claim using a sample of 307 Compustat firms that their method gives different rankings than Ijiri's method, and that their estimates are better correlated with the economic rates of return. Unfortunately it is not clear to us how the IRR estimates are assessed, and how a circular deduction has been avoided. In a later paper Stark, Thomas and Watson (1992) revisited Griner and Stark (1988) using simulation. Buijink and Jegers (1989) comment on the effects of various depreciation methods on the relationships between IRR, ARR and CRR. Stark (1994) analyzes the consequences for CRR based IRR estimation of incorrectly formulating the outflow/inflow patterns and the effect of growth.
To summarize the section on "Measurement of Profitability", the following main trends are evident in the ARR vs IRR discussion. 1) A prevalent conclusion is that the IRR is a theoretically well-founded profitability concept even if it is pointed out that the ARR can have managerial relevance as a practical profitability concept. 2) The question whether it is possible and sound to calculate the firm's IRR from its ARR (or CRR) remains unresolved. 3) The estimation of the IRR from published financial data is one of the directions for measuring the long-run profitability of the firm.
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