This chapter presents our simulation model. First, we present the simulation engine that generates the capital investment time series. Second, we present the generation of the cash inflows from the capital investments in terms of alternative contribution distributions. Third, we present the alternative depreciation methods the simulated firm may apply.
2.1 Generation of the Capital Investment Time Series
As discussed in the introduction the firm can be considered a series of cash outflows to capital investments and the cash inflows generated by these capital investments. The earlier discussion of the methods has been based on the implicit assumption of constant, exponential growth of the capital investments that make up the firm. In a previous simulation approach to analyze Kay's method Salmi and Luoma (1981) also assumed capital investments obeying constant, exponential growth. Their engine to generate the capital investments was the standard exponential growth model
where
g 0
= initial level of capital investments,
g t
= capital investments in year t,
k = growth rate.
Assuming a constant growth is a major simplification of the reality of capital investment decisions in business firms. To evaluate the reliability of the IRR estimation methods under observation it is of paramount importance to know whether the methods are sensitive to business cycles, noise and disruptive irregularities in the capital investment activities. To tackle these questions we extend the Salmi-Luoma simulation engine to generate the capital investments with the possibility of business cycles, noise and shocks. We use
For the indexing of the years t in the simulation engine we denote
T = length of the simulation period,
n = length of the observation period (number of years under
observation for the profitability estimation).
In the simulation the index t must run all the way from year 1 to year T. The simulated firm is founded at the beginning of year 1. The transient, initial period from year 1 to year T-n represents the stage needed to reach a going-concern phase. The evaluation of the selected long-run profitability estimation methods is best conducted only after the going-concern phase has been reached. Thus the actual observation period for the evaluation of the profitability estimation methods is from T-n+1 to T. For brevity, the indexing is not presented in the formulas.
The first part in Formula (2) of the simulation engine is equivalent to the constant exponential growth Formula (1) used earlier in the simulation by Salmi and Luoma (1981). We have for the trend component the same g 0, g t and k definitions as in Formula (1).
In our extension we first incorporate a sinusoidal business-cycle component to the engine. For this augmented cyclical fluctuation component we denote
A = amplitude of the cycle,
C = length of the cycle,
f
= technical phase adjustment for the cycle.
In the above, the term f is purely a technical phase adjustment needed in the engine. It is needed to slightly shift the continuous sine curve so that its maximum and minimum values coincide with the discrete observations. For example, for an average length of six years of real-life business cycles the value of f becomes p/6.
Seasonal variations do not arise. This is because our simulation engine is discrete with one-year intervals. In the terms of real-life business practice this is tantamount to using annual financial statements instead of the potential quarterly reports.
Next, we incorporate a random component. We use white noise as the random component and denote
s
= the standard deviation of the random fluctuation in the
capital expenditures,
z = random variable following the (0,1)-normal distribution.
For the shock (disruption) component we have
S = capital investment shock coefficient,
t
= the year of the capital investment shock (
t =
¥
for no shock in the simulation),
d
= Kronecker's delta,
d tt
= 1 when t = t
, and 0 otherwise.
All the new components, which are augmented into Formula (1) to arrive at the generalized capital investments generation engine presented in Formula (2), are multiplicative. In other words, the components are defined relative to the trend-level of the capital investments. This means, for example, that in the terms of statistics the standard deviation of the random fluctuation in the capital expenditures is heteroscedastic in nature. Likewise, the relative amplitude of the business cycles stays constant while the absolute magnitude of the business cycles increases over time.
Compared with the constant-growth approach, the inclusion of the business cycles and noise components make the simulation engine realistic. This is attested by the fact that in simulation the extended engine produces financial time series which resemble the time series profiles observed on actual business firms. See e.g. the sample of the time series drawn in Salmi et al. (1984: 46-48).
2.2 Cash Inflows Produced by the Capital Investments
The capital investments g t induce later, corresponding cash inflows. The relationship between the initial outlay of a capital investment and its cash inflows can be expressed in terms of a contribution distribution. Denote by bi an individual, relative cash-inflow contribution from a capital investment that has been made i years back. This term is called the contribution coefficient. The contribution distribution is naturally made up by the individual contribution coefficients for the life-span of the capital investment. The mathematical formulation below is based on e.g. Ruuhela (1972). The cash flow profiles in Ijiri (1979), Salamon (1982) and Gordon and Hamer (1988) represent the same idea of contributions induced by the capital investments of the firm.
First consider the contributions (the cash inflows) from a single capital investment made at time point t = 0, illustrated by Figure 1.
In a more general denotation, a contribution (i.e. the cash inflow) in year t from a capital investment made in year t-i is given by
where
f ti
= absolute contribution in year t from capital investment i
years back,
bi
= relative contribution from capital investment i years
back,
N = life-span of a capital investment project (the same for
all capital investments).
Under the regular going-concern phase, which is to be used for evaluating the profitability estimation methods, the index i runs from 1 to N. However, during the transient initial period before the year N, i.e. t < N, there can be contributions only from t years back. Hence the term min(N,t).
The total contribution in any year t (i.e. all the cash inflows in that year) is cumulated from the contributions from the capital investments made in the earlier years. Hence we have
which defines
f t = cash inflow in year t.
The accumulation of the contributions (the cash inflows) in year t from all the capital investments in the previous years is illustrated by Figure 2.
A comment on the mathematically discontinuous nature of the simulation model is in order at this stage. As is familiar from capital investment literature, the capital investment model involves a discretization of what basically are partly continuous events. An initial outlay made at time t = 0 is assumed to produce its corresponding contributions at times t = 1,...,N. Likewise, the depreciations for a capital expenditure made at time t = 0 will take place at t = 1,...,N. The same pattern is repeated for all capital investments for the simulation period. Our simulation model considers all the events as discrete as is common in capital investment models.
In line with the standard treatment in literature the contribution distribution bi in our simulation engine is the same for all the capital investments. In other words, the profitability of the capital investments remains the same over the period under observation. Furthermore, constant returns of scale on the capital investments are assumed, as is the custom in growth models. When the firm grows, there are no economics of scale. See e.g. the standard reference Levhari and Shrinivasan (1969: 153).
A contribution distribution bi fixes the internal rate of return. As was noted, the contribution distribution is assumed unchanged for all the consecutive capital investments even though the level of the capital investment outlays varies over the business cycles as defined by Formula (2). Hence the internal rate of return, i.e. the profitability of the simulated firm, is defined by the cash flows of any individual, simulated capital investment. The internal rate of return corresponding to a given contribution distribution is defined by equating the initial outlay with the sum of the future, discounted cash inflows:
which is readily reduced to
The r given by Formula (6) is the true internal rate of return of the simulated firm that is the benchmark in evaluating the various IRR estimation methods. It should be noted that Formula (6) is not suggested to be used as another estimation method for the long-run profitability of the firm from actual business data. Such a direct estimation would not be practical, nor maybe even possible, because the literature does not currently have adequate means readily to identify the contribution distribution of the capital investments making up the firm.
In the simulation evaluation the internal rate of return r which corresponds to a chosen contribution distribution bi can be readily assessed from Formula (6) using the numerical analysis methods such as the bisection method. For the bisection method see any standard text-book of numerical analysis such as Conte (1965: 39-43). The discussion of the specification of the alternative contribution distributions will be postponed till the next section.
It is a well-known fact that under non-conventional cash flows (more than one sign alteration) there can be multiple or no real roots for the internal rate or return r in Equation (6). See e.g. Teichroew, Robichek and Montalbano (1965). This problem does not arise in our simulations. A conventional cash-flow contribution pattern will be used.
Profitability defined as the IRR in our simulation is assessed from the contributions of the capital investments only. The financing issue does not come to the fore. This separation of capital investments from financing is in line with the classic results of Modigliani and Miller. For a discussion on this issue, see for example Yli-Olli (1980). This separation also is in line with the standard usage of IRR in connection with the capital investment decision. In making the decision, the decision maker compares the IRR of the capital investment project prior interest to the cost of capital. Including the interest (i.e. the cost of financing) in the cash estimates for the project's flows would be double accounting as pointed out by any good textbook on capital investments.
The question of financing and its costs do not arise in our simulations as long as it can be safely assumed that the firm remains sufficiently profitable to be able to obtain new capital as the need arises. Hence chronically declining activities (divestments) or infeasible combinations of growth and profitability will not be considered in our research, since in actual business practice this would in the long-run cause restrictions or even a cessation of the availability of capital to the firm. For a discussion of feasible growth / profitability combinations see Suvas (1994).
2.3 Contribution Distribution
As discussed in the previous section the true internal rate of return r determined by Formula (6) is a function of the contribution distribution bi introduced in Formula (4). The true form of the contribution distribution is not generally known for real-life business firms. In order to assess the effect the different, potential contribution patterns of the firm's capital investments we will perform our simulations with three different contribution patterns from the capital investments. Figure 3 illustrates three different types of potential contribution distribution, a neutral, a typical growth-maturity-decline life-cycle pattern and a steadily declining case.
The three distributions we choose are the uniform contribution distribution for the neutral case, the negative binomial contribution distribution for the growth-maturity-decline case and a linearly declining distribution (Anton distribution) for the steady decline.
The uniform contribution distribution is defined by the annuity factor
The uniform contribution distribution for the life-span of the investments is an obviously neutral choice. It produces the same level of contribution each year throughout the entire life-span of the capital investment. In the simulation the numerical values of the contribution coefficients which lead to the preselected true profitability r are given directly by substituting the numerical value r into Formula (7).
The typical life-cycle of a product includes an early growth phase, maturity, and decline. The negative binomial distribution corresponds to this cycle. For our simulation purposes it has the further advantage of being different from the uniform contribution distribution in two important respects. It is not constant and it is not symmetrical.
The general definition for the negative binomial distribution is given by
where q is a shape parameter and m is a location parameter. For our simulation we choose q = 0.15 and m = 2 which leads to a typical life-cycle profile. For the definition and the properties of the negative binomial distribution see Fisz (1967: 167).
For our purposes, two technical adjustments to the generic negative binomial distribution are needed. First, the distribution is cut from the right at the life-span instead of letting it continue to infinity. Second, the distribution is shifted to the left to coincide with the capital investments' life-span. Hence we have as our negative binomial contribution coefficients
where s is a scaling factor inducing the desired level of true profitability. In the simulation the numerical values of the contribution coefficients which lead to the preselected true profitability r are found by finding the value of s which fulfills Formula (6). It is given by
The Anton distribution presented in Anton (1956) is defined as
This is a linearly declining contribution distribution with convenient theoretical properties. It has been shown (see Solomon (1971: 168 footnote and Appendix 2 of the current paper)) that if the contributions from the firm's capital investments follow the Anton distribution the theoretical annuity depreciation (to be discussed in a later section) and the practical straight-line depreciation coincide and hence lead exactly to the same reported profit for the firm. (This is tantamount to the accountant's and the economist's concepts of income agreeing under these special circumstances.)
2.4 Depreciation Methods
To complete the simulation model we need the formulas for alternative depreciation methods in order to have the annual profit and book value figures. First consider the accounting relationships between these concepts.
The profit pt is defined by the cash inflow f t less depreciation d t as
The book value vt of the firm at the end of year t is defined by book value at the beginning of the year plus the capital investments g t less the depreciation d t Hence
In our simulation model the book value of the firm involves only the capital expenditures and the depreciation. For simplicity cash, inventories and other assets are not modeled separately.
Next consider depreciation. The firm's choice of the depreciation method is central to profit measurement and asset valuation both in accounting theory and practice. We build into our simulation model the possibility of three alternative depreciation methods to be employed by the simulated firm in its financial statements. The alternatives are the straight-line depreciation method, the double declining-balance method and the theoretical annuity depreciation method.
An important feature of the current research approach is to be able to evaluate how well the different IRR estimation methods perform under realistic conditions. The first two of the alternative depreciation methods for the simulated firm are prevalent in actual accounting practice. The idea of straight-line depreciation method is that it allocates the costs evenly based on the passage of time over the expected life-span of the asset. Decreasing charge depreciation methods are based on the idea of equipment being more efficient in their early life. We choose double-declining-balance method as a representative of the decreasing charge methods because it is by definition (the doubled rate) related to the corresponding straight-line method. Annuity depreciation is included as one of the alternatives for quite a different reason. It is purely a theoretical concept. It is included to verify whether the simulation and the profitability estimation algorithms produce the correct internal rate of return under annuity depreciation as predicted by theory.
Straight-line depreciation is calculated as
Double-declining-balance depreciation is a decreasing depreciation used in the U.S. practice. See Davidson and Weil (1977). Double- declining-balance depreciation method formula is
The above formula for the double-declining-balance depreciation forms an infinite geometric series. However, in accounting the capital investment expenditure is exhausted at the end of the life-span. We use the historical cost convention. Hence, all the remaining book value of the relevant investment is depreciated in the last year of the life-span.
The well-accepted definition for the annuity depreciation is that the profit (before interest and taxes) pt is assessed as the interest on the initial capital stock vt-1 in year t. Thus
and hence from Formula (12) we get
Annuity depreciation is a theoretical construct. As is evident from Formula (17), a circular reasoning is involved. It is necessary to know in advance the value of r (the internal rate of return) in order to be able to apply the annuity depreciation method. In other words, the profitability information is needed for estimating the profitability. In a simulation model, however, this is possible since the true internal rate r can be fixed in advance.
2.5 Accountant's vs. Economist's Profits and Annuity Depreciation
The construction of our simulation model was concluded in the previous section. However, annuity depreciation has a pivotal role in the relationship between accountant's and the economist's profit and valuation concepts. Hence the discussion started in the section on the problem statement is continued utilizing the notation introduced.
The accountant and the economist have different concepts of income. The accountant's profit and the accountant's rate of return are based on historical data. The accountant needs depreciation in defining annual profits. The accountant's rate of return is given by
The economist's income concept is independent of depreciation. It is based on future cash flows. The well-known economist's valuation of the firm is defined as the present value of the future net cash inflows:
In accordance to the classic results, discussed in the section on the problem statement, IRR and ARR (appropriately weighted, if not constant) agree if the annuity method of depreciation is used for depreciating the book value of the firm's assets. This result is tantamount to proving that if the economist's valuation wt and accountant's valuation vt of the firm's assets agree, then IRR and ARR agree.
A second, relevant classic result is that if the steady state growth of the firm is equal to its internal rate of return, then ARR and IRR agree. See Solomon (1966: 115). For a discussion and a presentation of the proofs see for example Salmi and Luoma (1981).