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M
A S I
The response of a linear sales
curve imposed to a micro
economical model of a company Hans Jessen Management Simulator M A S I , P.O. Box 171, DK-2630 Taastrup and Department of Mathematical Modelling The Technical
University of Denmark, DK-2800 Lyngby,
Denmark (November 11, 1998) Abstract A model of a company based on common accounting
practice for tactical planning is developed containing physical flow of materials,
manhours and deposits of materials, value flow and deposits of value and
financial flow and deposits as functions of time. In the first place a
graphical model is described naming each part by a mathematical function.
Thereafter the functions of time are determined with respect to accountancy
and their solutions are found imposing a linear sales curve. These solutions
describe fundamental functions in time of basic theory of accountancy with
reference to the flow of resources. E.g. profit and loss account, cash flow,
working capital and main key figures of the Dupont Pyramide are determined as
functions of time. Key words:
Flow of resources, accountancy, cash flow, working
capital, key figures, Dupont Pyramide. 1. INTRODUCTION This
paper is concerned with a model of a company containing common accoun- ting
practice. Such models have been presented by Bela Gold,7 with a keynumber technique, which were based on a
very simple ratio technique. Jay W. Forre- ster,6 developed models based on signal-graph - 2 - techniques,
but these models of system dynamics are difficult to apply in practice
because of the data to be found and to be interpreted. Models more applicable
for management analysis and decisions were developed by Albert Danielsson,2-3 in the form of flow-graphs but containing no
mathematical func- tions for evaluation. Samuel Eilon,4-5 made some mathematical approach to describe the
primary problem of this article, the equations as functions of time between
the working capital and the working system of profit and loss account,
without also considering the derived cash flow. His model as well as others
on this very aggregated data level are not able simultaneously to measure
values from the basic theory of accountancy as functions of time. In the litterature of accountancy
and management e.g. C. J. Malmborg,8
Alfred Rappaport,9 and R.S.
Segal,10 one will find no functions of time describing and
being consistent with accounting practice. Among
all these efforts to describe the processes of products and finance in a
company one will find Dan Ahlmark,1 as a primary source for this study. Dan Ahlmark only
made a general desciption without mathematical modelling of ac- countancy
with functions of time. - 3 – 2. An
analytical graphical business model This
Chapter describes an analytical graphical business model (see Fig. 2.1.).
This model will form the basis of a mathematical analytical descrip- tion of
the business which can be used by the business management in their principal
planning activities. The model will integrate principal elements of
managerial economics and the accounting theory, under the assumption that the
business comprises an activity/ cash flow and related principal assets
(accounts payable, accounts receivable, inventories). It is the management's
task to achieve the best possible composition of this general structure by
using some of the ratios defined in the model. 2.1. Activity parameters 2.1.1. Sales The volume of goods sold by the firm per unit time
is denoted S'u, where S'u = S'u(t). The dot denotes the physial dimension of
“current”. Sales
are here divided into two main components of which one is the reference sales
S'u,kon, which refers to the share of
sales which is paid for in cash. The other component of sales is denoted with
S'u,deb, which refers to the share of sales
which is paid for by the trade accounts receivable the debit time dD after delivery from the firm. Here the following
eguation applies: S'u(t) = S'u,deb(t) + S'u,kon(t) (1) 2.1.2. Purchases The
firm is supplied with a number of labor hours per time unit denoted by a'i and with the volume of goods per time unit denoted
by V'i. The flow of goods consists of two main components
of which one is the reference purchase V'i,kon, and the other one in the goods purchased on credit
V'i,kre, which are
paid for by the firm after the credit time dK. - 4 –
- 5 - The
following equation applies: V'i(t) = V'i,kre(t) + V'i,kon(t) (2) The firm is supplied with the fixed volume of
resources per unit time F'i. This flow of resources may, for example, include
electricity, administration, heating, rent, etc. 2.1.3. Inventories The
volume Q'i of raw materials supplied per unit time is added to the raw materials inventory consisting
of the volume RL. From the
raw materials in- ventory is deduced the raw materials volume Q'u. The following equation appli- es here: t RL = ̣ (Q'i(t) - Q'u(t))dt (3) 0 The volume of finished goods per unit time Z'i is added to the finished goods inventory consisting of the volume FL. From the finished goods invento- ry is deduced the
finished goods volume Z'u. The following equation applies here: t FL = ̣ (Z'i(t) - Z'u(t))dt (4) 0 2.2. Payment parameters, operations 2.2.1. Sales The
total volume of means of payment per time unit from the customers is denoted
with S'i. This payments flow consists of two components. One
component is the payments flow S'i,kon caused by the cash sales flow S'u,kon. The other component S'i,deb is the payment flow caused by the credit sales flow
S'u,deb. Here the
following equation applies: S'i(t) = S'i,kon(t) + S'i,deb(t) (5) - 6 - 2.2.2. Purchases The
total volume of payment per unit time for operations is denoted by U'b. This payment flow is composed of three components,
a'b and V'b and F'b. a'b is the payment flow corresponding to the flow of
labour hours consumed a'i, V'b is the payment flow corresponding to the flow of
raw material purchases V'i, F'b is the payment flow corresponding to the flow of
fixed resources consumed F'i. The following equation applies: U'b(t) = a'b(t) + V'b(t) + F'b(t) (6) The
payments flow V'b is made up of two components. One component is the
pay- ments flow V'b,kon corresponding to the cash purchases of rawmaterials
V'i,kon; the other
component is the payments flow V'b,kre corresponding to the credit purchase of raw
materials V'i,kre. The following equation applies; V'b(t) = V'b,kon(t) + V'b,kre(t) (7) 2.3. Market parameters, sales In
order to depict the fundamental financial effects of the market on the firm
and its effects on earnings, the market is characterized by three basic
components q , p and dD. They also describe the fundamental link between
the firm's sales of goods and the related payment flows. 2.3.1. Cash sales ratio q The
cash sales ratio is defined by the equation: S'u,kon(t) = q S'u(t) (8) where
0 £ q £ 1
In
a manufacturing business q will typically have a value in the interval 0 £ q £ 0.2.
In a supermarket q will typically be in the interval 0.8 £ q £
1. - 7 - 2.3.2. The price p The
price of the firm's product(s) is defined by the eguations S'u,kon,1(t) = p S'u,kon(t) (9) S'i,kon(t) = S'u,kon,1(t) (10) where
S'u,kon,1(t) is the
flow of debts corresponding to the sales flow S'u,kon(t) (i.e. the current invoice flow stating the
amount of debt; see equation (9)). Equation (10) expresses the fact that the
flow of debts S'u,kon,1(t) is equal to the payments flow from the customers
(cash payment). In
practice, it should be noted that there is normally a time lag between
invoicing and sales. However, it has a temporary negative effect on liquidity
and the computation of results. Management will therefore have in view that
the invoicing is done without the mentioned delays. 2.3.3. Debit time dD This
model defines the debit time dD as the time of delivery of the goods from the firm
until the time of payment by the customer for the goods. In practi- ce, dD
is spread over the individual customers but with well defined terms of
payment the mean value can be detemined. The
definition of dD can be expressed by the equations S'u,deb,1(t) = p S'u,deb(t) (11) V'deb,dD(t) = S'u,deb,1(t - dD) (12) S'i,deb(t) = V'deb,dD(t) (13) - 8 - S'u,deb,1 refers here to the invoice flow corresponding to the
credit sales flow S'u,deb cf. equation (11). Equation (12) gives a functional
description of a function V'deb,dD(t), which can be defined as the payments flow (documents)
corresponding to the actual receipt of payments S'i,deb(t) cf. equation (13). In practice, no time lag is
found between the two last mentioned functions. In
pratice, attention should be paid to the fact that there may be a time lag in
the business between invoicing and sales, the result being changes in li-
quidity and the computation of earnings. Management usually aims at applying
equation (11) in practice, i.e. no time lag. 2.4. Market parameters, purchases With
a view to depicting the fundamental financial effects of the purchasing
market on the firm as well as its effects on costs, it is characterized by
four basic components e, q1,
q2 and dK. They describe the fundamental link between the
firm's purchases of resources and the related payment flows. 2.4.1. Cash purchases ratio e The
cash purchases ratio is defined by the equation: V'i,kon(t) = e V'i(t) (14) where
0 £ e £ 1 In,
say, a manufacturing business e will typically have a main value
in the interval 0 £ e £
0.2. This is also a typical
feature in a trading firm. 2.4.2. The price q1 of raw materials The
price of the firms raw materials is defined by the equation: - 9 - V'i,kon,1(t) = q1 V'i,kon(t) (15) V'b,kon(t) = V'i,kon,1(t) (16) where
V'i,kon,1(t) is the
flow of debts corresponding to the raw materials flow V'i,kon(t) (i.e. the current receipt of invoices stating
the amounts of debts); see equation (15). Equation (16) expresses the fact
that the flow of debts V'i,kon,1(t) is equal to the payments flow to suppliers (cash
payment). In
practice, attention should be paid to the fact that the time lag between the
supplier's invoicing and the supplies of raw materials is usually a temporary
feature which has a temporary positive effect on liquidity and the
computation of results. 2.4.3. The price q2 of labor hours The
price of the firm's labor hours is defined by the equations a'i,1(t) = q2 a'i(t) (17) a'b(t) = a'i,1(t) (18) where
a'i,1(t) is the
time ticket flow corresponding to the flow of labor hours used a'i(t) (i.e. the current issuing of time tickets
stating wages earned); see equation (17). Equation (18) expresses the fact
that the time ticket flow a'i,1(t) is equal to the time rate flow a'b(t). In
practice there is a certain time lag between functions on the right hand side
and the left hand side of the equal sign in equation (17). This time lag is
ignored here. There is usually no time lag between the functions of equa-
tion (18), or the time lag is relatively small and of no importance here. - 10 - 2.4.4. Credit time dK This
model defines the credit time dK as the time from the time of delivery of the raw
materials to the firm until the time of payment by the firm for the raw
materials. In practice, dK is spread over the individual suppliers but with
well defined terms of payment the mean value can be used. The definition of dK
can be expressed by the equations: V'i,kre,1(t) = q1 V'i,kre(t) (19) V'kre,dK(t) = V'i,kre,1(t - dK) (20) V'b,kre(t) = V'kre,dK(t) (21) where
V'i,kre,1(t) refers
here to the invoice flow corresponding to the credit purchases flow V'i,kre(t), cf. equation (19). Equation (20) gives a
functional description of a function V'kre,dK(t) which can be defined as the payment order flow
(documents) corresponding to the actual effecting of payments V'b,kre(t), cf. equation (21). In practice, there is no
time lag between the two last mentioned functions. In practice, attention should be
paid to the fact that the time lag between the supplier's invoicing and the
supplies of raw materials is usually a temporary feature which has a
temporary positive affect on liquidity and the computation of results. The
following equations are defined in relation to the fixed resources consumed F'i and the related fixed costs F'b. F'i,1(t) = k F'i(t) (22) F'b(t) = F'i,1(t) (23) - 11 - where
F'i,1(t) in
equation (22) refers to the flow of debts in the form of in- voices (stating
amounts) corresponding to the fixed resources flow F'i(t). k denotes a symbolic operator in the
form of an average price of the fixed re- sources unit. In practice, there is
some time lag between the functions in eguation (23). As, however, the fixed
costs by definition are constant in ti- me, such a time lag is not important
in this context. 3.1 Income statement In
this Chapter an income statement for operations is presented (before depre-
ciation, etc.) using the general main principles of accounting theory. 3.1.1 Sales of goods Sales
of goods are defined on the basis of the following equations: S'u,kon,2(t) = S'u,kon,1(t) (24) S'u,deb,2(t) = S'u,deb,1(t) (25) S'u,1(t) = S'u,kon,2(t) + S'u,deb,2(t) (26) Eguation
(24) expresses the fact that the flow of debts (in the form of in- voices
with statement of amounts) S'u,kon,1(t) gives rise to an equally large information flow
S'u,kon,2(t). This quantity
is identical with the current crediting to the cash sales account. From
equation (25) follows that the flow of debts S'u,deb,1(t) causes an equally large information flow S'u,deb,2(t). This quantity is identical to the current
crediting to the credit sales account. Total
sales in the form of the information flow S'u,1(t) corresponding to the total crediting to the
sales account are then obtained from equation (26). - 12 - 3.1.2 Costs The
costs of the firm in connection with production and sales are defined by the
following equations: V'i,kon,2(t) = V'i,kon,1(t) (27) V'i,kre,2(t) = V'i,kre,1(t) (28) a'i,2(t) = a'i,1(t) (29) F'i,2(t) = F'i,1(t) (30) U'd(t) = V'i,kon,2(t) + V'i,kre,2(t) + a'i,2(t) + F'i,2(t)
(31) Equation
(27) expresses the fact that the invoice flow from the cash purchase V'i,kon,1(t) is currently debited to the cash purchases
account to the extent of the cash flow V'i,kon,2(t). Equation
(28) expresses the fact that the invoice flow from the credit pur- chase V'i,kon,1(t) is currently debited to credit purchases account
to the extent of the cash flow V'i,kre,2(t). Equation
(29) denotes the functional relationship between the time ticket flow a'i,1(t) and the current debiting to the time rate
account of the wage payment flow a'i,2(t). Equation
(30) expresses the functional relationship between the invoice flow F'i,1(t) for fixed costs and the current debiting of the
cash flow F'i,2(t) to the fixed costs account. The
total cost flow is defined by equation (31). - 13 - 3.1.2.1 Inventories, additions (with signs)
By
way of introduction, it is mentioned that the signs relating to additions to
inventories (as a mean time value) are assumed to be the same as those
relating to additions to sales (as a mean time value). Against this
background the ad- ditions to the individual inventories will for principal
planning purposes ha-ve the same signs. The inventories only serve as
"standby stores" in case of emergency events "i.e. in normal
operation state" the materials and products go directly through the
factory. Thus, the following systems of equations apply: The
increase of sales S'u is supplied directly by the production and the
inventories are increased proportionally with S'u. Q'i(t) > 0 Q'u(t) = 0 d S'u ¾¾¾¾ >
0 ̃
(32) dt Z'i(t) > 0 Z'u(t) = 0 Constant
sales S'u is supplied directly by the production and the
inventories remain constant.
Q'i(t) = 0 Q'u(t) = 0 d S'u ¾¾¾¾ =
0 ̃
(33) dt Z'i(t) = 0 Z'u(t) = 0 The
decrease of sales S'u is supplied directly by the production and the flow
from inventories. The inventories are decreased proportionally with S'u. - 14 - Q'i(t) = 0 Q'u(t) > 0 d S'u ¾¾¾¾ < 0 ̃ (34) dt Z'i(t) = 0 Z'u(t) > 0 The
system of equations (32) denotes that inventories rise when sales rise. The
system of equations (33) denotes that inventories are constant when sales
remain unchanqed. The
system of equations (34) denotes that inventories fall when sales fall. Based
on these main principles for the model the following equations can be
developed. Q'i,1(t) = qR Q'i(t) (35) Q'u,1(t) = qR Q'u(t) (36) Z'i,1(t) = qF Q'i(t) (37) Z'u,1(t) = qF Z'u(t) (38) U'tl(t) = Q'i,1(t) + Z'i,l(t) (39) U'al(t) = Q'u,1(t) + Z'u,1(t) (40) where Q'i,1(t) is the flow
of additions to raw materials invento-
ries corresponding to the additions to rawmateri-
als inventory records with statement of amounts. - 15 - Q'u,1(t) is the
flow of deductions to raw materials inven-
tories corresponding to the deductions to raw
materials inventory records with statement of
amounts. Z'i,1(t) is the
flow of additions to finished goods inven-
tories corresponding to the additions to finished
goods inventory records with statement of amounts. Z'u,1(t) is the
flow of deductions to finished goods inven-
tories corresponding to the
deductions to finished
goods inventory records with statement of amounts. qR denotes the calculated rav material price per
unit
of finished goods. qF denotes the calculated direct cost price per
unit
of finished goods. U'tl(t) is
total additions to inventories. U'al(t) is
total deductions from inventories. The
system of equations (32), (33) and (34) can now be given the form: d S'u ¾¾¾¾ > 0 ̃ U'tl(t) > 0
and U'al(t) = 0
(41) dt d S'u ¾¾¾¾ = 0 ̃ U'tl(t) = 0
and U'al(t) = 0
(42) dt d S'u ¾¾¾¾ < 0 ̃ U'tl(t) = 0
and U'al(t) > 0
(43) dt Attention
is drawn to the fact that the physical model based on the FIFO principle can
be desribed mathematically only by - 16 - d S'u sign ( ¾¾¾¾ ) = sign (U'tl(t)) (44) d t given
U'al(t) = 0
(45) and
U'tl(t) is
computed with signs. 3.1.3. Resourceconsumption (incl. F'i,1) Resources
consumed U'd,1,1(t) can be defined by the following equations: d S'u ¾¾¾¾ > 0 ̃ U'd,1,1(t) = U'd(t) - U'tl(t)
(46) dt given U'al(t) = 0 d S'u ¾¾¾¾ = 0 ̃ U'd,1,1(t) = U'd(t) (47) dt d S'u ¾¾¾¾ < 0 ̃ U'd,1,1(t) = U'd(t) + U'al(t)
(48) dt
given U'tl(t) = 0 3.1.4. Operation profit (before interest and
depreciation) The
operating profit (before interest and depreciation etc.) is defined by the
equation: O'(t) = S'u,1(t) - U'd,1,1(t) (49) 3.1.5 Operating profit incl. inventory
depreciation If a tax year of the length T is considered in a
period of time t1
£ t £ t1
+ T where t1 is a time
selected at random, the following functions can be defined: t1+T Vkøb = ̣ q1 V'i(t) dt (50) t1 w
= w(t1) (51) an = an(t) (52) - 17 - In
equation (50) Vkøb represents
the purchases of goods in the period
t1 £ t £ t1 + T. Equation
(51) defines w(t1) as the
total inventory value at time t1.
an(t) in the equation defines the inventory
depreciation rate. Materials
consumed computed for tax purposes is then derived from the follow- ing
equation (53): Vtax = Vkøb + w(t1) - (w(t1)/(1 - an(t1)) t1+T + ̣ (U'tl(t) - U'al(t)) dt) (1 - an(t1 + T)) (53) t1
For
principal planning purposes the mean time value of an(t)
for a given business will be a constant an and limited i.e. 0
< an < 0.3 . Based on this assumption equation (53)
gives
t1+T Vtax = Vkøb - (1 - an) ̣ (U'tl(t) - U'al(t)) dt (54) t1 Materials
consumed for operations is defined by the following equation:
t1+T Vdrift =
Vkøb + w(t1) - (w(t1) + ̣ (U'tl(t) - U'al(t)) dt)
(54a) t1 or t1+T Vdrift =
Vkøb - ̣ (U'tl(t) - U'al(t)) dt) (55) t1 If
equation (55) and equation (54) are combined, the following equations are
developed: t1+T Vtax = Vdrift + an ̣ (U'tl(t) - U'al(t)) dt (56)
t1 - 18 -
t1+T Vtax = Vdrift + ̣ an(U'tl(t) - U'al(t)) dt (57) t1
On
the basis of equation (57) the following functions can be defined: U'tl,1(t) = U'tl(t) (58) U'al,1(t) = U'al(t) (59) In
equation (58) U'tl,1(t) denotes total additions to inventories from a
taxation point of view. U'al,1(t) denotes in equation (59) total deductions from
invento-ries from a taxation point of view. With
the following definition equation: B'ln(t) = an (U'tl,1(t) - U'al,1(t)) (60) equation
(57) can be transformed into
t1+T Vtax = Vdrift + ̣ B'ln(t) dt (61) t1
On
the basis of equation (61) the following equation (62) can be defined: O'DS = O' - B'ln (62) where
O'DS is the
operating profit adjusted for inventory depreciation. 4.1. Change in liquidity (operations) The
cash flow released by operations, the change in liquidity, is defined by the
following equation (63): l'(t) = S'i(t) - U'b(t) (63) -
19 - 5.1. Cash balance The
cash balance of the firm is designated by M, which, in relation to the
present principal planning model, is very small in practice, i.e. M(t) = 0.
The folloving equation can now be developed: i'e = l' + i'K - y'B - y'L - H'S,1 (64) where
i'e is the
self financing flow y'B is the
service of bank loans y'L is the
service of other loans i'K is
current raise of loans for operations H'S,1 is tax
payments 5.2. Bank loans. The
firm is financed currently by trading credits in the form of the cash flow i'B. The equation is defined as follows: i'B,1(t) = i'B(t) (65) where
i'B,1(t) is the
information flow in the form of loan documents with statement of amounts
corresponding to the cash flow i'B(t). The bank charges currently interest r'B(t) on the amount outstanding B
= B(t) where r'B(t) is the document flow with statement of interest.
The following equation applies: n'B(t) = i'B,1 + r'B (66) where
n'B(t) is the
firm's current crediting to the bank account. - 20 - The
current service payments y'B(t) to the bank give rise to a payment order flow
with statement of amounts y'B,1(t). We have: y'B,1(t) = y'B(t) (67) The
payment order flow y'B,1(t) involves a corresponding current debiting to the
bank account in the form of y'B,2(t). The following equation therefore ap- plies: y'B,2(t) = y'B,1(t) (68) 5.3. Loans (long term) The
long term financing of the business is represented by the cash flow i'L. The following equation applies: i'L,1(t) = i'L(t) (69) where
i'L,1(t) is the
information flow in the form of loan documents with statement of amounts
corresponding to the cash flow i'L(t). On the loan L current interest r'L(t) is charged where r'L(t) is the document flow with statement of interest.
The following equation applies: n'L(t) = i'L,1(t) + r'L(t) (70) where
n'L(t) is the
firm's total current crediting to the loan account. The
following equation applies: i'L(t) = i'L,1(t) + i'D(t) (71) where
i'L,D(t) denotes
the long term financing flow to the working capital, and i'L,1(t) is the long term financing flow to the fixed
capital. - 21 - The
folloving equation applies: i'K(t) = i'B(t) + i'L,D(t) (72) The
current service payments y'L(t) to lender give rise to a payment order flow with
statement of amounts y'L,1(t). We have y'L,1(t) = y'B(t) (73) The
payment order flow y'L,1(t) involves a corresponding current debiting to the
loan account in the form of y'L,2(t). The following equation therefore applies: y'L,2(t) = y'L,1(t) (74) 6.1. Investment (in fixed capital) The
firm's current investment in fixed capital is denoted i'(t). The following equation applies: i'(t) = i'L,1(t) + i'e(t) (75) It
is pointed out that, in practice, i'L,D(t) currently converts short term liabilities into
long term liabilities, which means that at a strategic level alone i'L,D = 0. As to i'e(t), there is no unique definition of i'e(t) as it de- pends on the financing and market
situation. Roughly speaking, i'e(t) is the average cash flow which can be withdrawn
from the business without changing the existing product, investnent and
financing structure and the necessary financial reserves set aside for an
appropriate future development of the bu- sinees. - 22 - 7.1. Depreciation (for tax purposes) It
is normal to distinguish between depreciation for tax purposes and depre-
ciation for accounting purposes. Depreciation for accounting purposes is used
with the object of comparing alternative projects on the basis of special
cost principles. These principles are purely OR mathematical models and do
not reflect the physical business situation. Here
we shall only take an overall view of the financial flow of the firm for
which reason depreciation for tax purposes will be used. Such depreciation
will only reflect the actual effects on liquidity (after tax). The
following equations apply: i'1(t) = i'(t) (76) t D(t) = ̣ (i'1(t) - d'1(t))dt (77) 0 where
i'1(t)
represents the current debiting to the tax depreciation account corresponding
to the investment flow i'(t). d'1(t) is the current crediting to the same account
(i.e. current "depreciation"). D(t)
represents the balance of the tax depreciation account. The depreciation
charges d'(t)
are calculated on the basis of this account, and the following expressions
apply: d'1(t) = d'(t) (76a) d'2(t) = d'(t) (76b) where
d'2(t) is the
depreciation flow which is included on the basis of compu- tation of the
taxable income. - 23 - 8.1. Interest (for tax puroses) Interest
is usually computed for two main purposes. One concerns the income statement
for tax purposes, the other concerns internal computation purposes such as
the effect of interest on the income statement as a whole or in con- nection
with special computations. No
distinction will be made here between the two purposes. The interest charges
will be placed in this model with the sole aim of depicting the fundamental
fi-nancial characteristics. The
following equations are defined: r'B,1(t) = r'B(t) (78)
r'L,1(t) = r'L(t) (79) r'BL(t) = r'B,1(t) + r'L,1(t) (80) where r'B,1(t) denotes the current recording of interest
payment to the bank. r'L,1(t) denotes the current recording of interest
payments to other lenders. The recording of total interest payments is
designated r'BL(t). 9.1. Tax payments According
to the principles governing computation of the taxable income the following
equations apply: f'u(t) = d'2(t) + r'BL(t) (81) H'S(t) = s (O'DS(t) - f'u(t)) (82) H'S,1(t) = H'S(t) (82a) - 24 - where
f'u(t) is a
state function for the computation of tax, cf. equation (81), s is the
tax rate, H'S(t) is the computed tax payment and H'S,1(t) is the tax payment flow. 10.1. Principal ratios As
appears from Fig. 2.1, the following principal ratios in the firm are
im-portant to the understanding of the dynamic (tactical) characteristics of
the firm. Operating profit O'(t) Change in liquidity l'(t) Working capital (net) K'(t) Contribution ratio DG(t) Depreciation d'2(t) Interest r'BL(t) These
ratios will be discussed in detail in the following. 10.1.1. Operating profit O'(t)
Using
different assumptions concerning prices and changes in principal assets
(accounts payable, accounts receivable, inventories) it is possible via Fig.
2.1 to assess the effects on the operating profit. A reduction of the raw
materials inventories in a situation with raw materials prices which are
higher than the prices of the raw materials inventories but otherwise
constant will increase the profit temporarily in the period concerned. One
of the things that will be seen is that the profit O'(t) is independent of the volume of trade accounts
payable and the volume of trade accounts recei- vable. - 25 - 10.1.2. Change in liquidity l'(t)
Other
things being equal, the following expression, cf. Fig. 2.1., applies: d S'u ¾¾¾¾ > 0 ̃ l'(t) < O'(t) (83) d t Equation
(83) shows that the profit O'(t) is larger than the change in liqui- dity in the
case of growing sales in the firm, the reason being the funds ti- ed up,
calculated with signs, in principal assets (accounts receivable and
inventories), d S'u ¾¾¾¾ = 0 ̃ l'(t) = O'(t) (84) d t Equation
(84) shows that the change in liquidity is equal to the profit in the case of
constant sales, the reason being an unchanged volume of principal as-sets
(accounts payable, accounts receivable and inventories). d S'u ¾¾¾¾ < 0 ̃ l'(t) > O'(t)
(85) d t From
equation (85) appears that in the case of falling sales the change in
liquidity becomes greater than the operating profit owing to a reduced volume
of principal assets (accounts payable, accounts receivable and inventories). The
above shows how important it is for the business to keep the cash budget
currently up to date as the profit and the financial circumstances of the
business may differ substantially from each other. It should be noted that if
the net principal assets are negative, the inequality signs in (83) and (85)
must be reversed. - 26 - 10.1.3. Working capital K(t) If
the working capital is denoted K(t), the definition eguation for net capi-
tal tied up in the operating system will apply: K(t)
= Vdeb(t) + FL(t) + RL(t) - Vkre(t) (86) The
following definition equation will also apply: d K(t) ¾¾¾¾ + l'(t) = O'(t) (87) d t Equation
(87) shows that the profit is equal to the change in liquidity + the
increment of the net working capital tied up. If
equation (87) is transformed, the following equation is derived: d K(t) ¾¾¾¾ = O'(t) - l'(t) (88) d t |
