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2020 | OriginalPaper | Buchkapitel

1. Dynamics. The Law of Motion

verfasst von : Carlo Alberto Magni

Erschienen in: Investment Decisions and the Logic of Valuation

Verlag: Springer International Publishing

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Abstract

We investigate the dynamics of an economic system, explaining how the forces that act upon the system affect the position of the system (i.e., the capital) by means of two sources of increase/decrease: Income and cash flow. The dynamics of an economic system is characterized by the Law of Motion, which may also be framed in terms of income rate, whence the time value-of-money principle is derived. Finally, we define a project as an incremental system and classify capital budgeting projects in terms of expansion, replacement, abandonment.

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Nächstes Kapitel Statics. The Law of Conservation

In Example 1.1, we have

$$ \begin{array}{ll|ll|ll} \hline C_0&{}=\phantom {-}100&{} C_1&{}=80&{} C_2&{}=134\\ I_0&{}= \phantom {100\,\,}0&{} I_1&{}=10 &{} I_2&{}=\phantom {50}4\\ F_0&{}=-100&{} F_1&{}=30&{} F_2&{}=-50\\ \hline \end{array} $$

We will refer to an economic “system”, “unit”, “entity”, or “activity” interchangeably.

For example, suppose a transaction consists of three cash flows available now, in five months, and in one year, respectively. The analyst has two options: (i) choose the year as a unit of time (and, therefore, shift the interim cash flow at the end of the year, so that the transaction is described as a two-date operation), or (ii) choose the month as a unit of time and describe the transaction as an operation with ten equidistant cash flows: Three nonzero cash flows at times 0, 5, and 12, and ten cash flows equal to zero at the remaining dates. The choice depends on the degree of approximation one is willing to work with.

Using the fundamental relation (1.1), one may write

$$\begin{aligned} C_t=\sum _{k=0}^t I_k - \sum _{k=0}^t F_k. \end{aligned}$$

(1.6)

Therefore, the BOP capital can be viewed as the accumulated difference of incomes and cash flows. For $t=n$, Eq. (1.6) becomes

$$\begin{aligned} 0= \sum _{k=0}^n I_k-\sum _{k=0}^n F_k\implies \sum _{k=0}^n I_k= \sum _{k=0}^n F_k. \end{aligned}$$

(1.7)

This means that the total income is equal to the net cash flow. In overall terms, income and cash flow coincide: They are the same overall amount which is distributed differently across the various periods. For example, if the income vector is equal to $\varvec{I}=(10, 30, -20, 50, 70)$, then the overall net cash flow is necessarily equal to 140. This result will turn to be important in developing the notion of internal average rate of return in Chap. 10.

The primary surplus/deficit is the difference between tax revenues and government spending.

For example, ‘revenues’ may be used to denote income or cash flows or even a firm’s sales; ‘return’ may be used to denote an income or, rather, an income rate; ‘income’ itself is sometimes used to denote an inflow payment instead of the remuneration of capital.

The highly specialized jargons of these disciplines may even impede exchanges and communications among scholars of different fields. Paradoxically, some notions, results, and approaches may be well-known in one field long since while they may appear as novel and original in another field (or just known and neglected).

If $I_0\ne 0$, implying $C_0=I_0-F_0\ne -F_0$, then the relation becomes

$$\begin{aligned} C_t=I_0(1+i_1)(1+i_2)\cdot \ldots \cdot (1+i_t) -\sum _{k=0}^t F_k (1+i_{k+1})(1+i_{k+2})\cdot \ldots \cdot (1+i_n). \end{aligned}$$

(1.12)

Considering that $C_n=0$, the above equation implies

$$(I_0-F_0)(1+i_1)(1+i_2)\cdot \ldots \cdot (1+i_n)=\sum _{k=1}^n F_k (1+i_{k+1})(1+i_{k+2})\cdot \ldots \cdot (1+i_t).$$

As $\varDelta C_0=C_0=I_0-F_0$, the latter is equivalent to (1.13).

We may also frame the equation in terms of ending capitals: $C_t=E_{t+1}/(1+i_{t+1})$.

The adjectives retrospective and prospective refer to time t as the reference date. Therefore, in the retrospective relation, cash flows prior to time t are compounded (i.e., ‘ moved’ forward) to time t; in the prospective relation, cash flows after time t are discounted (i.e., ‘moved’ backward) to time t.

If $I_0\ne 0$, then (1.15) becomes

$$I_0-F_0=\sum _{k=1}^n \frac{F_k}{(1+i_{1})(1+i_{2})\cdot \ldots \cdot (1+i_{k})}.$$

As will become apparent later, an outflow from the firm/project represents an inflow for the investor, whereas an inflow into the firm/project represents an outflow for the investor.

In financial markets, the expressions ‘long position’ and ‘short position’ are also often employed to reflect the distinction between investment and financing (see Chap. 5).

The reader is invited to complete the table with other economic transactions.

For example, dividends are inflows for shareholders and outflows for the firm; capital contributions are inflows for the firm and outflows for shareholders; a project’s positive cash flow is an inflow for the firm and an outflow for the project; money injected in a fund by an investor is an inflow for the fund and an outflow for the investor.

In recent years, the government bonds of some countries have been negative for investors. Among others, Germany Switzerland, Sweden, Denmark, and Japan have charged depositors instead of remunerating them. In these (rather extreme) situations, the borrower is remunerated for borrowing and the lender is penalized for lending. In corporate and engineering contexts, negative capitals with negative rates often occur whenever the firm retires from an activity or abandons a given course of action. This may conceptualized by saying that the firm borrows from the project and is remunerated for this borrowing (see Sect. 1.4).

A loan is then a less general concept than savings-and-credit account.

An income/profit/return rate is unambiguously defined as the ratio $i_t=I_t/C_{t\!-\!1}$. The ratio $F_t/C_{t\!-\!1}$ is a rate, but not an income/profit/return rate. However, an income rate may well be expressed in terms of cash flow: From $i_t=I_t/C_{t\!-\!1}$ and considering that $I_t$ is equal to $F_t+\varDelta C_t$, $i_t$ may be written as $i_t=(F_t+\varDelta C_t)/C_{t\!-\!1}=(F_t+ C_t-C_{t\!-\!1})/C_{t\!-\!1}$ (see Eq. (1.9)).

Historically, income has been viewed as a ‘surplus capital’: “income is the surplus net worth (proprietor’s interest) accruing during the period after maintaining opening capital (net worth) intact.” (Whittington 2017, p. 43). Traces of it can still be found in the accounting expression “clean surplus relation” which is the name accountants and accounting scholars give to Eq. (1.1), where $I_t$ is called “comprehensive income” (see, for example, Brief and Peasnell 1996).

This difference must take account of any direct or indirect effect arising as a consequence of the project undertaking: Externalities, cannibalism, spillover effects, etc. (e.g., see Example 9.19 and Rainbow program in Chap. 12).

The incomes and cash flows of the following examples (and, in general, of the first two chapters of the book) may be considered after-tax amounts. Alternatively, one may well assume that they are pre-tax amounts (or that the projects are undertaken in a tax-free world). Taxes will be introduced in Chap. 3.

A disposal of assets is often associated with a gain or (more often) a loss on disposal, whenever the resale price is different from the carrying amount (see treatment in Sects. 3.6.2 and 4.4).

Note that the capital at time $t=-1$, which we assume to be equal to $\$$7,200, is irrelevant: The difference between the cash flows of the alternatives does not depend on it and is, invariably, equal to $\$$6,500.

This terminology might well be extended to expansion projects and abandonment projects.

In principle, one might consider an additional terminal outflow for dismantling the plant.

More precisely, the income rate is not defined for the capital is zero. As the income is positive, one might use the concept of limit to conventionally define it to be equal to $i_t=+\infty $.

See Magni amd Marchioni (2019) for description of a more realistic example of PhV plant.

$I_0=0$ is assumed.

Titel: Dynamics. The Law of Motion
verfasst von: Carlo Alberto Magni
Verlag: Springer International Publishing
Buch: Investment Decisions and the Logic of Valuation
Print ISBN: 978-3-030-26775-9

Electronic ISBN: 978-3-030-27662-1

Copyright-Jahr: 2020
DOI: https://doi.org/10.1007/978-3-030-27662-1_1