We commence with continuous mathematics using the following definitions in accordance with Kay.
The cash inflows and outflows in the discrete case can be illustrated in the standard manner as below
Returning to the continuous case, the book value v(t) is defined by
i.e. the accumulated capital investments less the accumulated depreciation. It is natural to assume that
and
since this indicates that the book value of the assets of the firm is zero outside its life-span.
The accountant's rate of profit (ARP) is defined by Kay as
In other words it is the revenues less depreciation (= accounting profit) divided by the capital employed at time t. In fact, the discrete counterpart of the above measure is called return on investment (ROI) rather than accountant's rate of profit in business practice. Kay ignores the fact that his definition is not consistent with the accrual basis of accounting measurement. In other words, his f(t) is a cash flow, not a revenue based on the realization principle, as it should be if defined rigorously. This is not, however, a major point in this discussion and we shall therefore accept the above definition of a(t).
The internal rate of return r (IRR) is defined in the continuous case by the equation
The internal rate of return is a natural and well-accepted measure of profitability_1. Several problems arise, however, when it is used in estimating the profitability of a real-life business firm.
The IRR-method gives a single profitability figure for the whole life-span of the firm (in the absence of multiple or imaginary roots_2). This rises two questions.
Firstly, it is obvious that the firm can be regarded as a series of capital investments. The profitability of the various capital investments usually differs, however. Therefore, on the level of the firm, a single long-run profitability must be assumed for all the capital investments making up the firm. On the other hand this is not quite as restrictive as may seem at first, since the firm can be deemed a single long-range capital-investment-like project. (The applicability of this idea can naturally be criticized especially in the case of conglomerates.)
Secondly, and more critically, the data for the whole life-span of the firm (as indicated by the integration limits in (6)), is seldom available and even if it were the profitability information would hardly be of much interest at that stage (with the potential exception of bankruptcy research). The approach has to be confined to a segment of the total life-span of the firm. Even so, the IRR-profitability of the firm covers several years. Theoretically this is sound, since the underlying capital investments are long-term already by definition. Nevertheless, the management (and other interested parties) of business enterprises need (also) short-term (yearly and shorter) profitability measures for decision making and control activities. The flaw with these short-term profitability measures is that they tend to be rules of thumb rather than based on proper theoretical considerations. On the other hand, it may be contended that such long-term measures as discussed in this paper can be less useful for frequent decision making and control purposes in business practice. E.g. Vatter (1966) strongly criticizes the yearly unvariability. This is naturally a question which could be debated at great length. As a potential compromise between the long-term and short-term measures we suggest rolling the long-term measures. By this we mean estimating the yearly profitability from the change in the IRR-profitability brought about by adding the last year to the data.
A third problem is that f(t) and g(t) in the equation for r are not available in published financial data, as such.
Kay has proved that if the accountant's rate of profit for a project is constant it is equal to the internal rate of return. No assumptions about the shape of the cash inflow f(t), the cash outflow g(t), nor the depreciation d(t) schedules are needed for this result. The result is weaker than it seems at first sight, since it is derived for the total life-span.
Accountant's rate of profit a(t) is not constant in actual practice.
To tackle this problem Kay utilizes the weighted average of
accountant's rate of profit. As natural weights he selects
discounted book values .
Kay's weights are natural only in the sense that they lead to the
results discussed below. They are not, however, based on any actual
practice in accountant's profit assessment. According to our
interpretation, Kay actually proves that the weighted average
accountant's rate of profit a solved from (7) is equal to the
internal rate of return r.
In other words, for the life-span of the firm (or a project), the
accountant's rates of profit weighted by discounted capital employed
yields the IRR sought. To tackle the case of shorter time segments
(from to
) let us investigate the following equation, given by
Kay.
As is shown in Appendix I it follows from (8) that
If and
, the averaged accountant's profitability a, given by
(8), will be equal to the internal rate of return. This results from
the assumption
, and
the definition (6) of the IRR.
Generally a is not equal to r. Kay contends that a as given by (8)
will be equal to r if the accountant's book values and
join with their economic values. In fact, Gordon (1974,
p. 347) indicates this result. The economic value is defined as the
discounted net cash flows:
As is shown in Appendix II by substituting and
Kay's contention a = r holds.
At this point two major objections arise in our opinion. First, there is no reason why this r in (10) must be the internal rate of return defined by (6). Actually, this r in (10) is but a rate of interest used for discounting the future cash flows. To elaborate, Kay defines the internal rate of return r by (6), and uses the same r in definition (10). It turns out, however, as indicated by Appendix II, that the results given by Kay are arrived at even if a different r from the r in (6) were used in (10). This means that when the results based on (10) to be given shortly are used in estimating profitability from published financial data, there will be no guarantee that Kay's method will estimate the internal rate of return defined in (6). Our critique is analogous to the critique by Stephen (1976) against the similar results by Gordon (1974). As far as we see Gordon's reply (1977) to Stephen does not resolve the circular reasoning involved in solving IRR from a relation where it is already assumed available beforehand. Second, we agree with Wright (1978) and Whittington (1979), and thus disagree with Gordon (1977), that even "an enlightened accounting profession" does not easily (if at all) accept the idea of the accountant's initial and terminal book values of assets equalling those of the economist's.
Nevertheless, should the economist's valuation of the initial and the ending capital stocks of the period under observation be accepted, equation (8) for profitability can be modified into a discrete form to allow the handling of actual financial data. IRR is then approximated by solving a from equation
where is the book value of
assets at the beginning of the year t, t is the accountant's profit
(operating income) in year t [being the discrete counterpart of
a(t)v(t) as indicated by 5)], and n is the number of years in the
period under observation. Unlike Kay we use the book value of assets
at the beginning of the relevant years rather than the average book
values. We shall demonstrate later that Kay's use of average book
values is erroneous. Furthermore, we use
as the discounting factor instead of
, since the former conceptually
corresponds the discrete capital investment models. The difference
is negligible, because
.
Nevertheless, it must be stressed that we have duly checked all our
results using Kay's original formulation of (11) as well.
The following interpretation can be given to (11). In business
practice the accountant's yearly profitability (ROI) before interest and taxes is calculated as the
operating income of the year divided by the capital employed at the
beginning of the year (or, alternatively, the average capital
employed). Thus
In (11) Kay in fact suggests that the internal rate of return should
be estimated by weighing the accountant's profits and the capital employed
by discounting factors over the whole
period under observation_3. A BASIC computer program for solving a
from (11) with the so called secant method is given in Appendix
IV.
Kay (1976, p. 449 and 456) contends that his results make no
assumptions about the shape of the cash inflow (), cash outflow (
) and depreciation (
)
schedules. This contention must be interpreted cautiously lest it
gives an inflated impression about the strongness of Kay's results.
As admitted by Kay (1976, p. 449), the depreciation scheme will
together with the shape of cash inflow and outflow schedules
strictly predetermine the capital stock. In fact, formula (11) for
estimating the IRR is valid for the annuity method of depreciation
only, as we shall demonstrate shortly. It is also a well-known fact
that when the annuity method of depreciation is used, ARR will equal
IRR (see Appendix III) and the whole problem does not arise_4. The
annuity method cannot, however, be applied in business practice,
because it requires advance knowledge about the IRR of the capital
investment projects constituting the firm. We do not claim that Kay
implies otherwise, but these facts should have been clearly brought
up by Kay to put the applicability of his results in proper
scope.
Kay (1976, p. 455) tackles the case where the accountant's valuation (v) and the economist's valuation (w) of the capital stock do not agree by deriving the following relationship for the internal rate of return.
In estimation k is the growth rate of the firm (n in Kay's paper) and a is the accountant's rate of profit (where a has to be constant). We interpret this suggestion as follows. When the annuity method of depreciation is not used by the accountant (as always is the case in business practice) the internal rate of return can be estimated from (11) and (13) where v is based on accounting data and w is based on the annuity method of depreciation. (Our Appendix III confirms that using the economist's valuation of the capital stock is equivalent to using the annuity method of depreciation.) Although Kay's method is formally valid, we claim that it is not valid in business practice because of the necessity of estimating w (or v/w). Wright's (1978) criticism of Kay supports our views.
__________
1 See, however, Tamminen (1980) for a discussion on the
concept of profitability and the assumptions underlying
profitability measurement with the internal rate of
return.
2 A fairly recent discussion on the conditions for the uniqueness of the internal rate of return can be traced back from Bernhard (1980).
3 The idea of weighing the accountant's profitabilities
is seen more clearly, if (11) is rewritten as
4 See Lonka (1976, pp. 38-39) and Tamminen (1976, pp. 38-41) for a proof of the equality of IRR and ARR in the continuous case when the annuity method of depreciation is used.