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, - 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, In the litterature of accountancy
and management e.g. C. J. Malmborg, Among
all these efforts to describe the processes of products and finance in a
company one will find Dan Ahlmark, - 3 – 2. 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. 2.1.1. The volume of goods sold by the firm per unit time
is denoted S where S Sales
are here divided into two main components of which one is the reference sales
S S 2.1.2. The
firm is supplied with a number of labor hours per time unit denoted by a - 4 – - 5 - The
following equation applies: V The firm is supplied with the fixed volume of
resources per unit time F 2.1.3. The
volume Q t R 0 The volume of finished goods per unit time Z goods inventory consisting of the volume F t F 0 2.2. 2.2.1. The
total volume of means of payment per time unit from the customers is denoted
with S S - 6 - 2.2.2. The
total volume of payment per unit time for operations is denoted by U U The
payments flow V V 2.3. 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 d 2.3.1. The
cash sales ratio is defined by the equation: S 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 of the firm's product(s) is defined by the eguations S S where
S 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. This
model defines the debit time d The
definition of d S V S - 8 - S 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. 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, q 2.4.1. The
cash purchases ratio is defined by the equation: V 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 of the firms raw materials is defined by the equation: - 9 - V V where
V 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 of the firm's labor hours is defined by the equations a a where
a 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. This
model defines the credit time d V V V where
V 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 F F - 11 - where
F 3.1 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 are defined on the basis of the following equations: S S S Eguation
(24) expresses the fact that the flow of debts (in the form of in- voices
with statement of amounts) S From
equation (25) follows that the flow of debts S Total
sales in the form of the information flow S - 12 - 3.1.2 The
costs of the firm in connection with production and sales are defined by the
following equations: V V a F U Equation
(27) expresses the fact that the invoice flow from the cash purchase V Equation
(28) expresses the fact that the invoice flow from the credit pur- chase V Equation
(29) denotes the functional relationship between the time ticket flow a Equation
(30) expresses the functional relationship between the invoice flow F The
total cost flow is defined by equation (31). - 13 - 3.1.2.1 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 Q Q d S ¾¾¾¾ >
0 ̃
(32) dt Z Z Constant
sales S
Q Q d S ¾¾¾¾ =
0 ̃
(33) dt Z Z The
decrease of sales S - 14 - Q Q d S ¾¾¾¾ < 0 ̃ (34) dt Z Z 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 Q Z Z U U where Q
ries corresponding to the additions to rawmateri-
als inventory records with statement of amounts. - 15 - Q
tories corresponding to the deductions to raw
materials inventory records with statement of
amounts. Z
tories corresponding to the additions to finished
goods inventory records with statement of amounts. Z
tories corresponding to the
deductions to finished
goods inventory records with statement of amounts. q
of finished goods. q
of finished goods. U U The
system of equations (32), (33) and (34) can now be given the form: d S ¾¾¾¾ > 0 ̃ U dt d S ¾¾¾¾ = 0 ̃ U dt d S ¾¾¾¾ < 0 ̃ U 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 sign ( ¾¾¾¾ ) = sign (U d t given
U and
U 3.1.3. Resources
consumed U d S ¾¾¾¾ > 0 ̃ U dt given U d S ¾¾¾¾ = 0 ̃ U dt d S ¾¾¾¾ < 0 ̃ U dt
given U 3.1.4. The
operating profit (before interest and depreciation etc.) is defined by the
equation: O 3.1.5 If a tax year of the length T is considered in a
period of time t functions can be defined: t V t w
= w(t a - 17 - In
equation (50) V Equation
(51) defines w(t Materials
consumed computed for tax purposes is then derived from the follow- ing
equation (53): V
t For
principal planning purposes the mean time value of a 0
< a
V t Materials
consumed for operations is defined by the following equation:
t V t or
V t If
equation (55) and equation (54) are combined, the following equations are
developed:
V
t - 18 -
V t On
the basis of equation (57) the following functions can be defined: U U In
equation (58) U With
the following definition equation: B equation
(57) can be transformed into
t V t On
the basis of equation (61) the following equation (62) can be defined: O where
O 4.1. The
cash flow released by operations, the change in liquidity, is defined by the
following equation (63): l -
19 - 5.1. 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 where
i y y i H 5.2. The
firm is financed currently by trading credits in the form of the cash flow i i where
i B
= B(t) where r n where
n - 20 - The
current service payments y y The
payment order flow y y 5.3. The
long term financing of the business is represented by the cash flow i i where
i n where
n The
following equation applies: i where
i - 21 - The
folloving equation applies: i' The
current service payments y y The
payment order flow y y 6.1. The
firm's current investment in fixed capital is denoted i i It
is pointed out that, in practice, i - 22 - 7.1. 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 t D(t) = ̣ (i 0 where
i D(t)
represents the balance of the tax depreciation account. The depreciation
charges d d d where
d - 23 - 8.1. 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 r r where r r 9.1. According
to the principles governing computation of the taxable income the following
equations apply: f H H - 24 - where
f 10.1. 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 Change in liquidity l Working capital (net) K Contribution ratio DG(t) Depreciation d Interest r These
ratios will be discussed in detail in the following. 10.1.1. 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 - 25 - 10.1.2. Other
things being equal, the following expression, cf. Fig. 2.1., applies: d S ¾¾¾¾ > 0 ̃ l d t Equation
(83) shows that the profit O d S ¾¾¾¾ = 0 ̃ l 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 ¾¾¾¾ < 0 ̃ l 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. 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)
= V The
following definition equation will also apply: d K(t) ¾¾¾¾ + l 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 d t Equation
(88) denotes that the difference between the operating profit and the change
in liquidity is equal to the financing requirements for operations in the
period under review. 10.1.4. The
contribution ratio is defined by equation (89): DG(t)
= (O Equation
(89) shows that DG is independent of the amount of sales and defines the
share of sales which will cover fixed costs, etc. The point is stressed here
that a high contribution ratio does not imply that there is "money"
to cover the fixed costs. For further details see section 10.1.2. as the size
of l - 27 - 10.1.5. Depreciation
contributes to influencing the firm's liquidity, cf. equation (82). Assuming
that the investments are made as individual projects at time intervals, it is
shown that depreciation in the periods between investments causes liquidity
to rise owing to the reduction in tax payments. However,
it should be noted that of the cash flow released after tax there must be
funds to cover repayment commitments in connection with loans raised. The
effect of the cash flow released after tax described above is therefore
partial and must be seen in relation to the repayment commitments. Later
there will be shown that for practical reasons the division described here is
desirable for the understanding of the financial components of the cash flow
released. 10.1.6. From
Fig. 2.1 and from equations (81) and (82) is apparent that interest pay-ments
reduce the cash flow released after tax. Thus, the net effect on cash flow released (to be defined
later) stems partly from the computation of income for tax purposes, partly
from the payment of interest on total loans. The
computation of interest on total loans seen in relation to a given level of activity
will be defined later. - 28 - 11. This
Chapter presents a new analytical mathematical model description of the
business. This model has been developed for use in the tactical planning pro-
cess. No reference can be made to a similar model in existing literature. The
theoretical literature which gets nearest is S. Eilon's article discussed in
Chapter A in “Economical Keynumbers”. 11.1. In
the following further definitions of mathematical functions and their rela-
tionships will be established. The sole justification of these definitions is
that they provide the basis of a clear and generally coherent system of
equa-tions between ratios. 11.1.1. A
basic sales volume is defined: S where
S The
development of sales during the time period is defined by equation (91): d S ¾¾¾¾
= a d t where
a - 29 - The
following equation now applies: S where
t ³ 0 11.1.2. Let
a ratio h h for
t ³ 0, hF being a positive constant which is designated
"finished goods inventory time". Another ratio hR is defined so that equation (94) applies: h for
t ³ 0 being a
positive constant which is designated "raw materials inventory
time". From
equation (93) follows: F The
definition equation applies: t F
0 which
substituted into equation (95) gives: |

- 30 - t F 0 or
t ̣ Z 0 If
equation (92) is used in equation (98), the following expression is derived: t ̣ Z 0 For
t = 0 equation (93) gives the following expression: h Using
equation (100) together with equation (99) we have: t ̣ Z 0 The
solution to the integral equation (101) is: Z The
flow of goods Z Z for
a and
- 31 - Z for
a Mathematically
the physical equations (103) and (104) may be described by equation (105) for
all values of a Z for - ¥ < a With
equation (105) the physical inventory system has been converted to a
mathematical model where Z From
equation (94) follows: R The
definition equation applies:
t R
0 which
combined with equation (106) gives: t R 0 or
t ̣ Q 0 If
equation (92) is used in equation (109), the following equation is derived: - 32 - t ̣ Q 0
For
t = 0 equation (94) gives: h Using
equation (110) together with equation (111) we have: t ̣ Q 0 The
solution to the integral equation (112) is: Q The
flow of goods Q Q for
a Q for
a Mathematically
the physically equations (114) and (115) can be described by equation (116)
for all values of a Q for - ¥ < a With
equation (116) the physical inventory system has been converted to a
ma-thematica1 model where Q - 33 - 11.1.3. Total
output T T If
the ratio ba is here
defined as the number of labor hours used per unit of output and the ratio b a V If
equation (117) and equation (119) are combined, the following equation is
obtained: V If
equations (92), (105) and (116) are substituted into equation (120), the
following equation is obtained: V Using
equations (117), (92) and (105), equation (118) gives: a 11.1.4. Using
equations (8), (9) and (10) we obtain payments derived from cash sales: S - 34 - Using
equations (1), (8), (11), (12) and (13) we obtain payments derived from debit
sales: S Equations
(1), (122) and (123) give: S (1 + a or
S 11.1.5. The
outgoing payments flow corresponding to cash purchases of raw materials is
expressed by means of equations (14), (15), (16) and (121) as V Credit
purchases of raw materials cause an outgoing payments flow which by means of
equations (7) (14), (19), (20) and (21) is computed at: V Equation
(127) is transformed by means of equation (121) into: V (1 - e)q Equation
(128) is reduced to: V - 35 - The
total payments flow to purchases of resources is then obtained by using
equations (6), (17), (18) and (129): U (1 - e))) + F By
combining equation (121a) and equation (130) the total outgoing payments flow
is then given by: U (h 11.1.6. The
accounting concept, change in liquidity l l - q
(h 11.2. Depending
on the firm's level of activity capital will be tied up in the ope- rating
system. Capital will be tied up in trade accounts payable, raw materi- als
inventories and finished goods inventories as well as accounts receivable
(the amounts are indicated with signs). 11.2.1. The
volume of trade accounts receivable is defined by the following equation,
equations (1), (8), (11) and (12) being used: d V 0 - 36 - In
this model it is assumed that equation (92) applies. From this equation
combined with (133) follows: d V
0 The
computation of the integral in equation (134) allows equation (134) to be
reduced to: V 11.2.2. The
volume of trade accounts payable is defined by the following equation,
equations (2), (14), (19) and (20) being used: d V 0 Assuming
that sales satisfy equation (92) and that equation (121) applies, equation
(136) develops the following expression: d V 0 By
computing the integral in equation (137) this equation is reduced to: V l1.2.3 The
volume of the raw materials inventory is given by equation (106). The value
of the raw materials inventory R R - 37 - If
equations (106) and (92) are substituted into equation (139), we have: R 10.2.4. The
volume of the finished goods inventory is given by equation (95). The
calculated consumption of materials and labor hours per unit of finished
goods is given by q q The
value of the finished goods inventory F F If
equations (95), (92) and (141) are substituted into equation (142), the
following expression is obtained: F 11.2.5. The
total capital tied up in the operating system, i.e. the working capital K(t),
is through the use of equations (135), (138), (140) and (143) given by: K(t)
= V or
by substituting into the relevant places K(t) = p(1 - q)S - q + q + (b or - 38 - K(t) = S + p(1
- q) d - q 12.1. In
the following, functions are established for the computation of operating
profit based on accounting theory. The
turnover of the firm is obtained by using equations (9), (11), (24), (25) and
(26) and is expressed as: S Using
equation (92) and equation (136) gives: S Raw
materials consumed corresponding to sales S V or
by using equation (92): V The
wages paid, time rates, corresponding to sales S a or by using equation (92) a - 39 - By
using equations (147), (149) and (151) the operating profit O O or
O 12.2. In
this section the operation profit will as an alternative be computed directly
on the basis of Fig. 2.1. The
costs U U + q Computed
with a plus or minus sign (positive for inventory) the following va- lue is
added to the raw materials inventory, cf. equation (35): Q or
equation (116) may be used: Q Here
the definition equation for cost prices of raw materials per unit of finished
goods has been used: q - 40 - The
following value is added to the finished goods inventory, cf. equation (37): Z or
equation (102) may be used: Z The
total value flow to inventories now amounts to, cf. equations (44) and (45): U or
by reduction U The
total operating profit is obtained by using equations (147), (154) and (161)
and is expressed as: O or
by substituting into the right hand side: O - (q + q - S - 41 - or
by reduction: O It
will be seen that equations (153) and (164) are identical, i.e. a systema- tic
use of Fig. 2.1. gives here the same result as the use of a simple
"logi- cal" accounting method. 12.2.1. If
equations (161) and (58) are substituted into equation (60), U B The
operating profit incl. inventory depreciation is given by equation (62). If equations
(164) and (165) are substituted into this equation, the following expression
is derived: O -
a or by reduction: O - a - F - 42 - 12.3.1. This
model takes as its starting point that the net working capital tied up K(t)
can be given the form: K(t)
= K where
K d K ¾¾¾¾¾ = i d t This
means that the increase in the capital tied up in the operating system is
financed by the bank overdraft. If
equation (168) is used together with equation (167), the following equation
will also apply: d K(t) ¾¾¾¾¾ = i d t It
is assumed that: B(0)
= 0 (169) This
means that the overdraft amounts to DKK B(0) = 0 at time t = 0. As
regards the mathematical model it is pointed out that in equation (168) i y - 43 - 12.3.2. It
is assumed that i i for
all t > 0, apart from certain selected times t i with
the condition i In
close connection with the operational financial possibilities of equations
(171) and (172) this model also assumes that equation (173) applies: y 12.3.3. Investments
are defined by i In
this mathematical model equation (75) is changed into: i where
i - 44 - New investments Instalments on
loans Etc. This
change of equation (75) is desirable seen in relation to the possibili- ties
of implementing this mathematical model on a computer. 12.4.1. From
equations (78), (79), (80), (170) and (173) the total interest payment is
derived: y where
r 12.4.2.
Depreciation
to tax computation is obtained from equation (77) and is expressed as: d where a 12.4.3. From
equations (81), (175) and (176) the following equation is derived: f By
using equations (82) and (177) total tax payments are expressed as: H - 45 - 12.4.4. With
the special definition of i i which
together with equation (88) gives: i If
equation (168a) including the related assumption is used here, the follow-
ing equation is obtained: i or
if equations (175) and (178) are used: i - (1 -
s)(r By
using equations (62) and (87), the following equation is derived from
equation (180): i - (1 -
s)(r If
the function O O O Equation
(181) is now transformed into: i - (1 -
s)(r It
appears from equation (183) that it may be appropriate to define the fol-
lowing managerial ratios: - 46 - 12.4.4.1. Interest
relative is defined by equation (183): r r
O r 11.4.4.2.
Depreciation
relative is defined by equation (183): a a
O a - 47 - 13.1. In
this Chapter some traditional ratios will be computed on the basis of the
functional expressions derived in section 12. 13.1.1. The
contribution ratio DG(t) is defined by the following equation (186): S |