2.1. Elementary financial arguments on accrual basis

The simple financial arguments to be applied here consist of the application of elementary probability theory to the rate of return on capital [45,46,47,48]. All financial approaches are applied to simple periodic growth processes [47,48,49]. Even if consequent growth cycles may not be completely similar in real life, the discussion is simplified by the application of the periodic boundary condition. A further simplification for the systems discussed is the absence of intermediate divestment: the periodic growth process is not disturbed by intermediate harvest.

As the discussion regards a growth process, in the absence of intermediate harvest, the capitalization increases along with the growth process. In the determination of ROI or RROI (rate of return on investment), no single time instant is decisive. Consequently, we discuss the expected value of RROI within any growth period. Further, a growth process being discussed, the expected value of the capitalization is not necessarily determined as the accumulated cash flow into investment goods. That is why we now turn the discussion into the expected value of the rate of return on capital (RROC) instead of rate of return on investment (RROI). This is simply produced by computing the expected value of the operating profit rate, divided by the expected value of capitalization.

The elementary financial arguments applied here are formulated as follows. The expected value of the operating profit rate is

(1)

where τ is cycle (or period) duration, is the probability density of time within the cycle, and is any current operating profit rate. On the profit/loss – basis, the profit rate includes value growth, operative expenses, interests, and amortizations, but neglects investments and withdrawals. It is worth noting that the profit rates in Eq. (1) are additive, instead of compounding [26,50]. On the other hand, the expected value of the capitalization is

(2)

where the capitalization K, on the balance sheet basis, is directly affected by any investment and withdrawal. Again, the capitalizations in Eq. (2) are simply additive. Then, the expected value of the rate of return on capital (RROC) is

(3)

As the capitalization in Eq. (2) depends on both accumulated profits and investments, at any time within the rotation cycle b+t the capitalization can be written

(4)

where refers to the rate of investments and divestments. The operating profit rate is written

(5)

2.2. Elementary financial arguments on cash flow basis

In addition to the elementary financial arguments given above, the basic features of the two established families of financial criteria are given as follows. Internal rate of return o refers to a discount rate where discounted cash flow approaches zero, or

(6)

where refers to cash flow rate. In general, Eq. (6) has many solutions for the discount rate, many of them complex [11,12,13]. It is worth noting that in Eq. (6), the discount rates are compound rates [26,50]. Eq. (6) can be discretized to a polynomial, then having as many solutions as is the degree of the polynomial [12,13]. However, most of the solutions have limited application [8,11,51].

The net present value of future cash flows is simply given as

(7)

where q refers to discount rate. It is found that the right-hand sides of Eqs. (6) and (7) are of the same form. Interestingly, unlike Eq. (3), the value of Eqs. (6) and (7) depend on the phase of the period (or the time instant b) where the computation is started. Again, the discount rate in Eq. (7) is compounding.

2.3. Return rate on capital and internal rate of return

The internal rate of return (IRR) according to Eq. (6) is traditionally computed on cash flow basis [4,7,8]. In this paper, three features are included in the elementary financial treatment. Firstly, the return on capital is discussed as an integrable time rate (RROC), according to Eq. (3). Secondly, the profit rate appearing in Eq. (1) and then substituted to Eq. (3) is discussed on an accrual basis. Thirdly, accrual-basis increments of capitalization are discussed as consequences of a growth process, contributing positively to the profit rate.

An apparently nontrivial question is, how the ratio of the additive expected values in Eq. (3) relates to the compounding discount rate in Eqs. (6) and (7). Accounting is inherently based on simply additive processes, whereas financial computations are often compounding [26,50]. The discount rates in Eqs. (6) and (7) are defined through the compounding equations. The rate of return on capital (RROC) in Eq. (3) is defined through accounting equations, but it is supposed to be used in compounding equations, in the clarification of the change of wealth over time [7,11,17,22–25,47]. This apparently is a remarkable difference between Eq. (3) on the one hand, and (6) and (7) on the other; the former is rigorously derived from accounting, whereas the latter ones are not.

In periodic growth systems, profits often accumulate within a rotation period. On accrual basis, instead of cash basis, the growth of any season adds to profit. In the absence of intermediate divestments, the expected value of the operating profit rate can be written

(8)

where the arbitrary starting point of the integration that appeared in Eq. (1) has been abandoned, and the time origin is placed at the beginning of the rotation cycle. The reason for this arrangement is that divestment often occurring at the end of any rotation cycle does then not become an intermediate divestment. On the other hand, the expected value of the operating profit rate can be given simply by normalizing the total accumulated operating profit by the duration of the rotation period:

(9)

The last form of Eq. (9) shows that the expected value of the profit rate is path-independent: it depends on the time-average value of the spot return rates , rather than the sequence of the return rates. The path-independency does not become violated by eventual negative returns.

On the other hand, in the absence of intermediate divestments, the expected value of the capitalization can be written

(10)

Interestingly, there is no path-independent form of Eq. (10). Correspondingly, the expected value of the capitalization is path dependent. It depends on the time schedule of the spot return rates in the inner integral of Eq. (10). This will naturally render the return rate on capital according to Eq. (3) path dependent. At elevated return rates at the beginning of the rotation period, the expected value of capitalization becomes greater, and the expected value of the return rate on capital is smaller. Reduced return rates at the beginning of the rotation have the opposite effect.

The path-dependence of the return rate on capital appears worthy of investigation. A simple Ansatz for the path-dependency might be to write the spot return rate on capital as a function of time on period, period duration, and the time-average value of the rate of return on capital:

(11)

where is the time-average value of the spot rate of return on capital within a full cycle of the squared periodic function, and a is a modeling parameter. Ansatz here refers to a tentative formulation established for the purpose of illustrating a problem.

The above discussion can be readily compared with an internal rate of return, determined on a cash-flow basis. In the absence of intermediate divestments, Eq. (6) for IRR can be rewritten

(12)

where the time-average value of the rate of return on capital depends on the rotation age τ according to the Ansatz of Eq. (11).

Fig 1 shows the annual spot return rate according to Eq. (11), the expected value of return on capital according to Eq. (3), and the internal rate of return according to Eq. (12) as a function of the phase of the full period cycle of duration π from Eq. (11). In Fig 1, the parameter value a=0.5. As the internal rate of return o is path-independent according to Eqs. (6) and (12), it converges to the full-cycle average at the end of the full cycle.

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Fig 1. Annual spot return rate according to

Eq. (11) with parameter value a=0.5, expected value of the rate of return according to Eq. (3), internal rate of return according to Eq. (12), and the reference case from Eq. (11).

https://doi.org/10.1371/journal.pcsy.0000043.g001

Eq. (12) readily shows that the internal rate of return o corresponds to the time-average of the spot return rates. On the other hand, according to Eq. (3), (8) and (10), the expected value of the return on capital is produced by weighing the spot return rates by current capitalization. In the absence of intermediate divestments, capitalization increases within a growing cycle. In Fig 1, the spot return rate, resulting from Eq. (11), is symmetric with respect to the center point of the growth period, where it displays its maximum. The low-return rate cycle part towards the end of the growth cycle is weighed by the greatest capitalization, rendering the expected rate of return from Eq. (3) at the end of the full growth cycle lower than the time-average .

The maximum value of either IRR (o) or the expected value of the rate of return on capital is not reached at the end of the full period cycle of duration π from Eq. (11), but at an earlier phase. The expected value of the RROC (or ) being path-dependent, and the capitalization being smaller at early stages of the cycle, the maximum value of is greater than that of IRR (o), and reaches its maximum earlier within the cycle.

2.4. Net present value and leverage effect

It is of interest to consider the outcome of a net present value analysis in the present example problem. As a modification of Eqs. (7) and (12), the net present value of all future cash flows is

(13)

where q is a discount rate (as in Eq. (7)), and the denominator corresponds to the contribution of further rotations in the future [1,52,53]. The discount rate is supposed to reflect market interest rates, and it is used to match achievable present value to a utility-indifference curve of different temporal consumption patterns [2–4,11]. Fig 2 shows the effect of the discount rate on the net present value, as a modification of Fig 1. The discount rate is given as multiples of the time-average value of the spot return rate on capital within a full cycle of the squared periodic function. The discount rate affects not only the level of NPV but also the rotation cycle length where the NPV becomes maximized (Fig 2).

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Fig 2. Net present value of future cash events, according to

Eq. (13), normalized by the initial investment K(0). The discount rate q is given in different multiples of (Eq. (11)).

https://doi.org/10.1371/journal.pcsy.0000043.g002

It is found from Fig 2 that the NPV varies very much as a function of the discount rate. In general, utilization of the NPV for consumption requires that any agent can borrow at the market rate of an assumed perfect financial market [2–4,11]. However, there are cases where the discount rate can be derived from the utility function [4]. As the translation along the time axis in terms of discounting is considered financially invariant, and there is no return rate apart from the discount rate, the NPV-approach is apparently unable to discuss any leverage effect.

There is a leverage effect. The leverage ratio L is here defined as

(14)

where K is capitalization, and E is equity. Increasing capitalization increases leverage if equity does not change. The capitalization within the productive system may approach zero, in which case the leverage coefficient reaches negative unity. Leverage may also be changed by increasing or decreasing the amount of equity.

Then, the leverage effect on the return rate on capital can be written as

(15)

where is the expected value of the return rate on equity, and u is a market interest rate [cf. 30]. The rotation cycle length apparently enters Eq. (15) only through (or ). can now be plotted as a function of L and u; in Fig 3, L=1, and the market interest rate is given in terms of multiples of . It is found that whereas the depends on the market interest rate, the rotation cycle length maximizing indeed is independent of the market interest level. For the market interest rates appearing in Fig 3, the leveraging increases the return on equity in the vicinity of the cycle length close to the maximum of . On the other hand, in rotation cycle lengths where is less than u, the leveraging reduces the . It is here worth noting that seminal research has stated the magnitude of productive investments depend on the market interest rates [7].

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Fig 3. Expected value of the rate of return according to

Eq. (3), and leveraged (L=1) rate of return on equity according to Eq. (15) with four different interest rates, expressed as multiples of (Eq. (11)).

https://doi.org/10.1371/journal.pcsy.0000043.g003

As mentioned above, negative leverage is also possible; the range of the leverage is from negative unity to infinity. Negative leverage maximizes the if the market rate is greater than the expected value of return on capital in the periodic growth process (Eq. (3)). Negative leverage corresponds to lending instead of borrowing. The leverage ratio approaching negative unity corresponds to the lending of all available capital, resulting in zero capital allocation to the periodic growth process.

Another question is, how the leverage effect could be applied in the context of the internal rate of return. Firstly, Eq. (15) could be technically used, simply by replacing by the internal rate of return o from Eq. (6) or (12). However, such a solution would not be correct since the IRR does not correspond to the expected value of the rate of return on capital (Fig 2). Alternatively, one could write another IRR-type equation for a leveraged project as

(16)

where Ω is a discount rate forcing the initial value of the leveraged project to zero. This discount rate is not internal, since it depends on the market interest rate u. The leverage effect is readily solvable from Eq. (16) as

(17)

Eq. (16) has been written under the boundary condition that compound loan interest is paid along with the loan itself at the end of the rotation. Another Equation can be written for the case of interest payments during the growth period; such an Equation is resolvable at least numerically.

A few asymptotic values of Eq. (17) are indeed correct. However, there is no proof that it would produce the leverage effect correctly, including the effect of the market interest rate. There rather is proof that it does not: Eq. (15) and Fig 2 show that the market interest rate does not contribute to the rotation cycle length maximizing the return rate on equity (). Eq. (17) conflicts with that result, as the rotation time interacts with the market interest rate in the determination of the leverage effect.

Eq. (16) inspires an eventual application to the leverage effect on NPV. Indeed, it is possible to write a leveraged net present value as

(18)

Equation (18) contains three financial time rates: the time-average return rate on capital , the market interest rate u, and the discount rate d. As the first two first rates can be determined from the production process and from the market, the discount rate remains arbitrary. NPV(L) on the left-hand side being a priori unknown, the Equation is then not solvable. However, it has been postulated that the discount rate must equal a market interest rate [4]. Then, Eq. (18) becomes solvable, and the leveraged NPV in relation to unleveraged NPV becomes

(19)

Eq. (19) contains a remarkable result. Yes, indeed, it does describe the effect of the leverage coefficient reasonably. But the leverage effect does not depend on the market interest rate! The latter finding is in serious conflict with Eq. (15) and indicates that the NPV-approach seriously fails in its description of the leverage effect. It also is worth noting that leverage coefficient of negative unity renders the net present value to zero. In other words, investment in interest-bearing instruments produces no value.

Even if the utilization of the maximized NPV requires borrowing at the discount rate [4], or a more general expansion of Eq. (18) with the help of Eq. (19) might be of interest.

(20)

A few features of Eq. (20) appear logical. The unleveraged NPV is positive if . Then, a positive leverage coefficient makes a positive leverage effect if , and negative leverage effect if . The unleveraged NPV is negative if . Then, a positive leverage coefficient makes a negative leverage effect if , and positive leverage effect if . However, with negative unleveraged NPV, negative leverage effect increases the value of the leveraged NPV. Interestingly, the leverage effect goes to zero if , and to infinity if . The effect of the interest rate vanishes if .

With negative leverage coefficient L, most of the above becomes reversed. For a positive unleveraged NPV, there is a negative leverage effect if , and positive leverage effect if . In other words, the higher the interest, the greater the NPV. This is understandable since negative leverage corresponds to lending, instead of borrowing. For a negative unleveraged NPV, there is a negative leverage effect if , and positive leverage effect if . Again, with negative unleveraged NPV, negative leverage effect increases the value of the leveraged NPV. Again, the effect of the interest rate vanishes if , but if the leverage ratio simultaneously reaches negative unity, the NPV goes to zero. This is apparently controversial since the leverage ratio of negative unity means that all equity is invested in interest-bearing instruments. However, the equality of the interest rate u and the discount rate q means that interest-bearing instruments produce no value.