Budgeting Models and System Simulation: a dynamic approach

IntroductionSolution

A Budget Model AnalysisA new approach

Conclusion

Budgeting Models and System Simulation: adynamic approach

Luca Gentili, Dario Girardi, Bruno GiacomelloUniversita degli studi di Verona

September 2th, 2010

Luca Gentili, Dario Girardi, Bruno Giacomello Budgeting Models and System Simulation: a dynamic approach



Conclusion

Introduction

The main idea is to create a flexible Budgeting Model which worksin a dynamic framework capable of providing information about thefinancial position, income, assets and liabilities of an enterprise.This new tool should be considered the starting point for a newclass of models able to adapt to the new informative requirementsthat come from the different economic subjects. In order to createa similar model one has to analyze the procedures of double entrybookkeeping and find a mathematical formalization of them.




Conclusion

Introduction

During the last five decades many economists andaccountants have worked to give a mathematical formalizationto bookkeeping double entry and more in general anaxiomatization of accounting procedures;

The attempts of finding a mathematical formalization has notled to a complete defined framework;

Thus in literature we can distinguish many differentframeworks where an accounting axiomatization has beenutilized;




Conclusion

Introduction

The aim of our studies will be to gather previous literature resultsformalizing a model able to provide some information about thefinancial position, income, asset and liabilities of an enterprise in aframework utterly dynamic.




Conclusion

Interesting References

This paper is mainly based on the following previous literatureresults:

1 Mattessich’s articles, “Budgeting Model and System Simulation” (1961)and “Mathematical Models in Business Accounting” (1958). In thesepapers the main idea is to create an a flexible budgeting system, using aset of simultaneous equation, that give to the management theopportunity to measure different management policies or different options.

2 Butterworth’s article “The accounting system as an Information Function”(1972). The accounting model, develop for his researches, could beconsiderate the first accounting model involved in a dynamic framework.

3 Rambaud, Perez (2004), “The accounting system as an algebraicAutomaton”.

4 Melse (2006) achieved interesting results about the temporal perspectiveof accounting recording, modelling the accounting equations with theirtemporal behaviour.




Conclusion

Accounting Items

In this presentation you are going to use a balance sheet as similaras used by Mattessich (1961) with no contingent claims.

1 Ln Cash and Cash equivalence;

2 Cn Trade receivables;

3 Rn Inventories;

4 Kn Property, Plan and Equipment;

5 Dn Trade and other payables;

6 Df n Long term Borrowings.




Conclusion

Assumptions

Constant amortization schedule;

Property, Plant and Equipment in every period will be net ofaccumulation depreciation expense;

Long debts and Mortgages have to pay obligations;

Cash and Cash equivalence are deposited on a bank accountwith a credit interest;

In our model you will consider just unsold stock and theauthors have decided to evaluate them at Weighted AverageCost;

Trade receivables will be net of fund containing bad credits, orcredits we are not sure they will be settled.




Conclusion

A Variable Dynamic Model

Ln = (1 + β) · L(n−1) + ηn · (C1(n−1) + Fn) + γn · K(n−1) − ωn · (D(n−1) + Gn)−dn · Df(n−1) + (n.Fn −m.r.n) + p/mn

Cn = (1 − ηn) · (C(n−1) + Fn)

Rn = VnVn−1

· R(n−1) + Vn · (Y′n − Yn)

Kn = (1 − γn) · K(n−1) − An

Dn = (1 − ωn) · (D(n−1) + Gn)

Dfn = Df(n−1) + (n.Fn −m.r.n)




Conclusion

A Variable Dynamic Model

Some specifications:

initial condition

Ln−1 = ..Cn−1 = ...Rn−1 = ...Kn−1 = ....Dn = ...Dfn = ...

η = η1·Cn−1+η2·Fn

Cn−1+Fnand ω = ω1·Dn−1+ω2·Gn

Dn−1+Gn




Conclusion

Matrix Representation

LnCnRnKnDnDfn

=

1 + βn ηn 0 γn −ωn −dn0 1− ηn 0 0 0 0

0 0 VnVn−1

0 0 0

0 0 0 1− γn 0 00 0 0 0 1− ωn 00 0 0 0 0 1

Ln−1Cn−1Rn−1Kn−1Dn−1Dfn−1

+

0Fn

(∆n)Vn−10Gn0

+

∆fn + ψn

00−An

0∆fn

∆n = Y′n − Yn

∆fn = n.Fn − m.r.n

ψn = p/mn




Conclusion

Variations

It is easy to see that others accounting items can be introduced,as:

Financial instruments (Held for trading/Held to Maturity);

Taxation;

Others;




Conclusion

Example (Held to Maturity)

Held-to-maturity investments are financial assets with fixed ordeterminable payments and fixed maturity that an entity has the positiveintention and ability to hold to maturity. Held-to-maturity investmentsare measured at amortised cost.

αn nominal interest rate;

(1 − λn) movements inside the accounting items;Pn

pn−1Amortised cost;




Conclusion

Example (Held to Maturity)

LnCnRnKfnDnDfn

=

1 + β1 ηn αn + λn Ωn −ωn −dn0 1− ηn 0 0 0 0

0 0 Pnpn−1

(1− λn) 0 0 0

0 0 0 1− γn 0 00 0 0 0 1− ωn 00 0 0 0 0 1

Ln−1Cn−1Kfn−1Kn−1Dn−1Dfn−1

+

0Fn00Gn0

+

∆fn + ψn

00−An

0∆fn

∆fn = n.Fn − m.r.n

ψn = p/mn




Conclusion

No-Costant Model

A general solution for the no constant model is the following:S0 = Mn · (Sn−1 + CEn−1) + Fn

S0 = S

S1 = M1 · (S0 + CE0) + F1

S2 = M2 · (M1S0 + M1CE0 + F1 + CE1) + F2

S2 = M2M1S0 + M2M1CE0 + M2F1 + M2CE1 + F2

S3 = M3 · (M2M1S0 + M2M1CE0 + M2F1 + M2CE1 + F2 + CE2) + F3

S3 = M3M2M1S0 + M3M2M1CE0 + M3M2F1 + M3M2CE1 + M3F2 + M3CE2 + F3

....

Sn =∏1

h=n M(h) · (S0 + CE0) +∑n

j=2[(∏j

h=n M(h))(Fj−1 + CEj−1)]+Fn




Conclusion

No-Costant Model

Hence, given the transition matrices M(1). . . .M(n) =∏j

h=n M(h),the vectors CE1. . .CEn and F1. . .Fn, we can reach a formula thatyields the cash flow and the whole balance sheet at t = n, definingin this way the balance sheet through a sequence no morerecursive. However, looking the matrix you note that its shape is:

m11(h) m12(h) m13(h) m14(h) m15(h) m16(h)

0 m22(h) 0 0 0 00 0 m33(h) 0 0 00 0 0 m44(h) 0 00 0 0 0 m55(h) 00 0 0 0 0 m66(h)




Conclusion

No-Costant Model

It is easy to prove that M(1) ·M(2)... ·M(n) =∏j

h=n M(h) willhave of the same form of M. Thus, if our goal is to write down amathematical expression for each accounting items at time n, itwill be enough to evaluate every matrix’s element at time n.

[∏0

h=1 m11(h)] [m12(0) · m11(1) + m12(1) · m22(0)] ... [m16(0) · m11(1) + m16(1) · m66(0)]

0 [∏0

h=1 m22(h)] 0 00 0 0 00 0 0 00 0 · · · 0

0 0 0 [∏0

h=1 m66(h)]

.




Conclusion

No-Costant model

In order to obtain a close formula, that gives us the balance sheetat t = n, you have to calculate the matrix’s elements, at time n,that compose the part

∏1h=n M(h) and the elements that compose

the part∑n

j=2(∏j

h=n M(h)) of the general solution.




Conclusion

Diagonal and First Row

Using mathematical induction is possible to demonstrate that theevolution for the elements of the first row and the elements thediagonal will be:

diagonal nmii =∏1

h=n mii (h);

first row nm1i =∑n

h=1[m1j(h) · (∏h−1

b=1 mjj(b)) · (∏n

c=h+1 m11(c)]

With the assumptions that:

∏h−1b=1 mjj(b) = 1 if h = 1∏nc=h+1 m11(c) = 1if h = n.




Conclusion

First Row and Diagonal

The next step is to give a mathematical shape to the last part ofthe dynamic solution, namely

∑nj=2(

∏jh=n M(h)). This is enough

simple, because this is nothing else than a summation of productstructures, so both for the element of first row and the diagonal,change only the superior index above the product structure.

With the assumptions that:

∏h−1b=1 mjj(b) = 1 if h = i∏nc=h+1 m11(c) = 1if h < c.




Conclusion

A close formula for each Accounting Item

Ln =∏1

h=n(1 + β(h)) · L0 +∑n

h=1[η(h) · (∏h−1

b=1(1− η(b))) · (

∏nc=h+1(1 + β(c)))] · (C0 + F0)+

+∑n

h=1[γ(h) · (∏h−1

b=1(1− γ(b))) · (

∏nc=h+1(1 + β(c)))] · K0+

+∑n

h=1[ω(h) · (∏h−1

b=1(1− ω(b))) · (

∏nc=h+1(1 + β(c))]·)(D0 + G0)+

+∑n

h=1[d(h) · (∏n

c=h+1(1 + β(c)))] · Df0+

+∑n

j=2 ·[∏j

h=n(1 + β(h))] · (n.Fj−1 − m.rj−1 + p/mj−1)+

+∑n

h=j η(h) · (∏h

b=j (1− η(b))) · (∏n

c=h+1(1 + β(c)) · Fj−1+

+∑n

h=j γ(h) · (∏h

b=j (1− γ(b))) · (∏n

c=h+1(1 + β(c)) · Aj−1+

−∑n

h=j ω(h) · (∏h

b=j (1− ω(b))) · (∏n

c=h+1(1 + β(c)) · Gj−1+

−∑n

h=j d(h) · (∏n

c=h+1(1 + β(c)) · (n.Fj−1 − m.rj−1)] + n.Fn − m.rn




Conclusion

A close formula for each Accounting Item

Cn =∏1

h=n(1− η(h)) · (C0 + F0) +∑n

j=2 ·[∏j

h=n(1− η(h))] · Fj−1

Rn =∏1

h=n(1− VhVh−1

(h)) · (R0 + (Y ′0 − Y0) · V0) +∑n

j=2 ·[∏j

h=n(1− Vh

Vh−1(h)] · (Y ′j−1 − Yj−1) · Vj−1

Kn =∏1

h=n(1− γ(h)) · D0 +∑n

j=2 ·[∏j

h=n(1− γ(h))] · Aj−1 + An

Dn =∏1

h=n(1− ω(h)) · (D0 + G0) +∑n

j=2 ·[∏j

h=n(1− ω(h))] · Gj−1

Dfn = Df0 +∑n

j=2 ·(n.Fj−1 − m.rj−1) + n.Fn − m.rn




Conclusion

A Particular Case the Constant Model

A model with all coefficients constant can be considered aparticular case of the previous model. A similar model it has beenalready presented during the last seminar so we are going justspecified the highlights to introduce our current researches.We are going to specify:

1 the constant model

2 the general close formula;

3 the entire model;

4 the averages;




Conclusion

The Constant Model

Ln = L + η · (C(n−1) + Fn) + γ · K(n−1) − ω · (D(n−1) + G)−d · Df(n−1) + (n.F −m.r + p/m)

Cn = (1 − η) · (C1(n−1) + F)

Rn = V · R(n−1) + V · (Y′− Y)

Kn = (1 − γ) · K(n−1) − A

Dn = (1 − ω) · (D(n−1) + G)

Dfn = Df(n−1) + (n.F −m.r)




Conclusion

The Close Formula

It easy to demonstrate that the close formula is just a particularcase of the no-constant model. In fact if we consider all the matrixequals, then M(1). . . .M(n) =

∏1h=n M(h) = Mn. Thus arranging

the formula we obtain the following:

Sn = Mn · S0 +∑n

l=1 Ml · CE +

∑n−1l=0 M l · F

We have just to point out that we assume that β tent to zero, thismeans that we do no have credit interest.




Conclusion

The Entire Model

Ln = L0 + [1 − (1 − η1)n] · C10 + [1 − (1 − γ)n] · K0 − [(1 − (1 − ω)n] · D10+

dn · Df0 +[n − (1 − η)( 1−(1−η)

η)n]· F −

[n − (1 − ω)( 1−(1−ω)

ω)]· G+

n · (n.F −m.r + p/m) +[n − ( 1−(1−γ)

γ)]· A− d [ n·(n−1)

2] · (n.F −m.r)

Cn =(1 − η)n · C0 + (1 − η)·[ 1−(1−η)n

η] · F

Rn =Cn =V n · R0+V · [ 1−V n

1−V] · (Y ′ − Y )

Kn = (1 − γ)n · K0 + [ 1−(1−γ)n

γ] · A

Dn =(1 − ω)n · D0 + (1 − ω)·[ 1−(1−ω1)n

ω1] · G

Dfn =Df0 + n · (n.F −m.r)




Conclusion

Averages

We have already shown, in the last seminar, the possibility to finda series of averages for every model’s coefficient, endogenous andexogenous, which are able to replicate the result of the model withvariable parameters, after n period.Here we make use the same technique, we just show the underlineidea based on the concept of Chisini’s average.




Conclusion

Average’s idea

The substance of the average concept introduced by Chisini can beformalize in these form:

f (x1, x2, ...., xn) = f (x , x , ..., x) (x for n times)

We are going to use the same concept, to discover a series ofaverages that permit of reproducing the values assumed from themodel after n periods.




Conclusion

Averages

Rewriting the Cash flow formula Ln in this way:

Ln = Wn + Pn + Tn + Zn where

Wn = [1 − (1 − η1)n] · C0 +[n − (1 − η)( 1−(1−η)

η )n]· F

Pn = [1 − (1 − γ)n] · K0 +[n − ( 1−(1−γ)

γ )]· A

Tn = [(1 − (1 − ω)n] · D10 +[n − (1 − ω)( 1−(1−ω)

ω )]· G

Zn +∑

(p/m) =

dn · Df0 + n · (n.F −m.r + p/m) − d [n·(n−1)2 ] · (n.F −m.r)




Conclusion

Averages

Analyzing the Cash flow formula Ln, it is of primary importance tonote that not everything is a function of the whole set of variables,in fact we notice, for example, that just Wn depend on η and Fand looking the other equations is possible to realize that just Cn isfunction of the same variables. Thus we can write down anequation system in order to find an average for coefficient η and F .




Conclusion

η and FWn = [1 − (1 − η1)n] · C10 +

[n − (1 − η)( 1−(1−η)

η )n]· F

Cn = (1 − η)n · C0 + (1 − η) · [ 1−(1−η)n

η ] · F

Wn + Cn = C0 +∑n

i=1 Fi All the revenues plus the initial creditshave to be allocated between Lnand Cn;

Substituting the previous relation, we obtain that the average of F

is equal to F =∑n

i=1 Fi

n

and if y = (1 − η)

Cn = yn · C0 + y · [ 1−yn

1−y ] ·∑n

i=1 Fi

n =

yn+1 · (C0 +∑n

i=1 Fi

n )−yn · C0 + yn · (C0 +∑n

i=1 Fi

n ) + Cn




Conclusion

Averages

Using the same technique we can define the average for:

γ, A;

ω, G ;

(n.F −m.r), p/m, d ;

V , (Y ′ − Y );




Conclusion

XXX

XXX is a small mutual company which core business is based onmicrocredit and others mutual functions. XXX is a stockholder ofYYY and utilizing the latter to finance both others Mutual AidSocieties and retail customers. Thus we can define its businesssimilar to a normal consumer bank, in fact it receives funds fromdeposits and it invests in microcredit operations, acquiring sharesfrom YYY.




Conclusion

XXX Model

Ln = L(n−1) + ηn · (C1(n−1) + Fn) + γn · K(n−1) + Ω · Kf(n−1) − ωn · (D(n−1) + Gn)

+∆fn

Cn = (1− ηn) · (C(n−1) + Fn)

Kn = (1− γn) · K(n−1) − An

Kfn = (1− Ωn) · Kfn−1

Dn = (1− ωn) · (D(n−1) + Gn)

Dfn = Df(n−1) + n.Fn

∆fn = n.Fn + m.rn + p/mn − dn+αn




Conclusion

Financial Statement XXX

From the Financial Statements available on-line we have:

Matrix t1 t2 t3 t4 t5

η 0, 35 0, 59 0, 52 0, 46 0, 61γ 0 −0, 62 −22, 7 −0, 032 −0, 56Ω 0, 0068 0, 21 0, 71 0, 048 −0, 001ω −0, 75 −0, 70 −0, 75 −0, 83 −0, 81

∣∣∣∣∣∣∣∣∣

Vectors t1 t2 t3 t4 t5

F 99.977 183.079 211.585 206.335 221.449G 122.281 190.617 211.187 205.062 207.256∆f 244.658 275.819 105.524 −25.988 152.509A −1.740 −2.145 −4.828 −7.482 −9.545n.F 226.169 254.969 87.424 −75.113 113.909

∣∣∣∣∣∣∣∣∣∣∣

S03

L0 365.697C0 66.506K0 8.266Kf0 1.501.080D0 18.152Df0 1.265.434




Conclusion

XXX Averages

Matrixη 0, 534γ −1, 31Ω −0, 054ω −0, 795

∣∣∣∣∣∣∣∣∣∣

VectorsF 184.485G 187.280

∆f 150.785A −5.148n.F 121.471

∣∣∣∣∣∣∣∣∣∣∣∣

S03

L0 365.697C0 66.506K0 8.266Kf0 1.501.080D0 18.152Df0 1.265.434




Conclusion

Model in Average

Using this close formula

Sn = Mn · S0 +∑n

l=1 Ml · CE +

∑n−1l=0 M l · F we obtain:

S08

L08 341.072C08 158.391K08 295.875Kf08 1.952.615D08 48.135Df08 1.872.792




Conclusion

Financial Ratio Analysis

Applying the typical financial ratio to XXX we can say that itsFinancial situation does not seem very steady. In fact we have:

t1 t2 t3 t4 t5

Quick ratio 0, 242 0, 72 0, 205 0, 242 0, 260Net worth to FR 0, 435 0, 565 0, 311 0, 348 0, 35Current ratio 0, 242 0, 72 0, 205 0, 242 0, 260

Ratios are not that good. The problems are the amount offinancial debts and their temporal outlook, they are consideredpayable “within the 12 months”.




Conclusion

A new approach to Analysis

We can start by noticing that in our constant model there is ahuge gap between the value of n.F given by our Chisini’s averages,replicating the Balance sheet in the period 2003 - 2008, which isn.F = 121.471, and the same Chisini’s averages value of n.F thatwill bring our firm to a zero cash flow value in the year 2008 whichis n.F = 49.541 Euro. This could suggest that maybe the situationof our firm XXX is not that bad after all. But let’s get a little bitdeeper into this subjet through the next example.




Conclusion

Budgeting Model 2014

We assume this is the bugeting model that has been planned bythe management for the period 2010 - 2014

S10 S11 S12 S13 S14

L 385.653 422.013 449.535 467.753 526.563C 159.484 159.993 160.230 160.340 74.580K 290.727 285.579 280.431 275.283 270.135Kf 2.058.065 2.169.210 2.286.358 2.409.832 2.539.974D 48.143 48.145 48.145 48.145 9.846Df 1.994.263 2.115.735 2.237.206 2.358.678 2.480.150

In this case ratios would keep on giving us a bad image on thehipothetical life of firm XXX for the period 2010 - 2014.




Conclusion

Constant Model

Our CM (constant model) describes the year 2014 with thefollowing matrix and vectors of parameters in Chisini averages:

1 0, 534 0 −0, 024 −0, 795 00 0, 466 0 0 0 00 0 1 0 0 00 0 0 1, 054 0 00 0 0 0 0, 204 00 0 0 0 0 1

*

*

341.072158.391295.875

1.952.61548.135

1.872.792

+

0

184.48500

187.2800

+

121.471 + 28.714

0−5.148

00

121.471

where we remind that the Chisini average for the case of n.F issimply the aritmetic mean n.F = 1

h

∑i=hi=1 n.Fi (= 121.471 in our

case).Luca Gentili, Dario Girardi, Bruno Giacomello Budgeting Models and System Simulation: a dynamic approach



Conclusion

Cash Flow equals zero in C.M.

Now we look for the value of n.F that makes our cash flowequal to zero in the year 2014, this is 16.159, 1;

We would like to underline that there is a very important linkbetween C.M. and V.M. (Variable Model). Given theparameters coming from our Budgeting Model year by year forV.M. (except for n.Fh ) the C.M. replicates perfectly V.M.and viceversa for every set of parameters n.F1....n.Fh such

that∑h

i n.Fi

h = n.F ;

This is the reason why we can use only n.F in order to analysethe behaviour of our Cash flow. Always remembering that weare in the case where the other parameters are all givenaccording to the supposed Budgeting Model.




Conclusion

A probabilistic Approach

Now we suppose that n.Fh is a Random Variable for everystep h;

We have seen that our C.M. uses the aritmetic mean of n.Fh ,thus the Random Variable we are concerned with will be∑h

i n.Fi

h = n.F ;

Because of the Central Limit Theorems family we will assumethat our R.V. will have a Gaussian distribution;




Conclusion

A probabilistic Approach

The density function that we assume will be set on asignificant number of values coming from the historicalFinancial Statements of our firm and others from similar firms;

The average and the St. deviation of the statistical set weused are µ = 180.218, 5 σ = 55.955, 32;

Given the aforementioned distribution the probability that n.Fis less than 16.159, 1 (so that the Cash flow is less than zeroin the year 2014 ) will be 0, 1684%;

According to our analysis the probability of a companydistress appears to be really low, a conclusion that would notdefenitly be drawn looking at ratios.




Conclusion

Conclusions

We hope that the results shown above, could became the startingpoint of a new routes of research on this environment and wewould like to think they can be also a starting point for a new viewin the Management literature.


Documents

Budgeting Models and System Simulation: a dynamic approach