Application of mathematics models in short - term investment decisions

VIETNAM NATIONAL UNIVERSITY, HANOI
SCHOOL OF BUSINESS







DO THI QUYNH AN





APPLICATION OF MATHEMATICS MODELS IN
SHORT – TERM INVESTMENT DECISIONS





MASTER OF BUSINESS ADMINISTRATION THESIS

Hanoi - 2007
VIETNAM NATIONAL UNIVERSITY, HANOI
SCHOOL OF BUSINESS





DO THI QUYNH AN





APPLICATION OF MATHEMATICS MODELS IN
SHORT – TERM INVESTMENT DECISIONS

Major: Business Administration
Code: 60 34 05



MASTER OF BUSINESS ADMINISTRATION THESIS









Supervisors:
1. Dr. Chu Thanh
2. Mrs. Tran Phuong Lan, MBA








Hanoi – 2007

iv

TABLE OF CONTENTS

ACKNOWLEDGEMENTS i
ABSTRACT ii
TÓM TẮT iii
TABLE OF CONTENTS iv
LIST OF TABLES viii
LIST OF FIGURES x

INTRODUCTION 1
1. NECESSITY OF THE THESIS 1
2. OBJECTIVE OF THE RESEARCH 1
3. KEY RESEARCH AREA 1
4. METHODOLOGY 2
5. CONTRIBUTIONS OF THE THESIS 2
6. THESIS STRUCTURE 2
CHAPTER 1: LITERATURE REVIEW 3
1.1.Overview of financial decision making 3
1.1.1.Investment decision 3
1.1.2.Financing decision 4
1.1.3.Dividend decision 5
1.1.4.Other decisions 5
1.2.Overview about the model building 5
1.2.1.Definition of Models 5
1.2.2.Classification of Models 6
1.2.3.Basic Modeling Concepts 7
1.2.4.The method to set up a model and apply in the financial decision making 9
1.3.Models using in investment decisions in current assets 9
v
1.3.1.Determining the target cash balance 9

1.3.1.1.The BAT model 11
1.3.1.2.The Miller – Orr Model: A more general Approach. 16
1.3.1.3.Implications of the BAT and Miller-Orr Models 18
1.3.1.4.Other Factors Influencing the Target Cash Balance 19
1.3.2. Inventory management decisions 20
1.3.3. Accounts receivable management 26
1.3.3.1. Credit Standards 27
1.3.3.2. Credit terms 32
1.3.3.3. Collection Effort 40
CHAPTER 2: ANALYZING BUSINESS ACTIVITIES OF SONADEZI
LONG THANH SHAREHOLDING COMPANY AND TUONG AN
VEGETABLE OIL JOINT STOCK COMPANY 42
2.1. Sonadezi Longthanh Shareholding Company 42
2.1.1. The introduction of Sonadezi Longthanh Shareholding Company 42
2.1.2. The Operating results of Sonadezi Long Thanh Shareholding Company. 47
2.1.2.1. The operating results in 2005, 2006 and 9 months 2007. 47
2.1.2.2. Some ratios assess the financial stability and business activities results 48
2.1.3. Characteristic of cash in Sonadezi Long Thanh Shareholding Company
and relating decisions 49
2.1.3.1. Characteristic of cash in Sonadezi Long Thanh Shareholding Company 49
2.1.3.2. Decision making relate to cash in Sonadezi Long Thanh 51
2.2. Tuong An Vegetable Oil Joint Stock Company (TAC) 52
2.2.1. The introduction of Tuong An Vegetable Oil Joint Stock Company 52
2.2.2. The Operating result of Tuong An Vegetable Oil Joint Stock Company. 56
2.2.2.1. The operating result in 2005, 2006 and 9 months 2007. 56
2.2.2.2. Some ratios show the financial situation and business activities results. 58
2.2.3. Characteristic of current assets in Tuong An Vegetable Oil Joint Stock
Company and relating decisions 60
vi
2.2.3.1. Characteristic of Inventory in Tuong An Vegetable Oil Joint Stock

Company 60
2.2.3.2. Decision making relate to inventory in Tuong An Vegetable Oil Joint Stock
Company 62
2.2.4. Characteristic of Account Receivables in Tuong An Vegetable Oil Joint
Stock Company and relating decisions 63
2.2.4.1. Characteristic of Account Receivables in Tuong An Vegetable Oil Joint
Stock Company 63
2.2.4.2. Decision making relate to Account Receivables in Tuong An Vegetable Oil
Joint Stock Company 65
2.3. Conclusion 65
CHAPTER 3: APPLICATION OF MATHEMATICS MODELS IN SHORT-
TERM INVESTMENT DECISIONS IN SONADEZI LONGTHANH
SHAREHOLDING COMPANY AND TUONG AN VEGETABLE OIL JOINT
STOCK COMPANY. 66
3.1. Apply the cash management model in decision making in Sonadezi LongThanh
Shareholding Company 66
3.1.1. The BAT (Baumol) model 66
3.1.1.1. The guideline to apply the BAT (Baumol) model 66
3.1.1.2. Apply the BAT model in determining the target cash balance 67
3.1.2. The Miller – Orr Model 70
3.1.2.1. The guideline to apply the Miller – Orr model 70
3.1.2.2. Apply the Miller – Orr model in determining the target cash balance 71
3.2. Apply the Inventory Management Model and Credit and Receivable
Management Model in decision making in Tuong An Vegetable Oil Joint Stock
Company 73
3.2.1. The Economic Order Quantity Model 73
3.2.1.1. The guideline to apply the Economic Order Quantity model 73
vii
3.2.1.2. Apply the Economic Order Quantity model in determining the optimal size
of inventory orders 73

3.2.2. Credit and receivables management models 75
3.2.2.1. The guideline to apply the Credit and Receivables Management Models 75
3.2.2.2. Apply the Credit and receivables management model 80
REFERENCES 86
APPENDICES 88
Appendix A: The development of Vietnamese businesses 88
Appendix B: The fact of using mathematics models in short – term investment
decisions 95
Appendix C: Number of acting enterprises as of Annual 31 Dec. by type of
enterprise 106
Appendix D: Number of employees in enterprises as of Annual 31 Dec. by type of
enterprise 107
Appendix E: Annual average capital of enterprises by type of enterprise 108
Appendix F: Net turnover of enterprises by type of enterprise 109


viii
LIST OF TABLES

Table 1.1: Classification of models 7
Table 1.2: Credit Evaluation Data Compiled by Bassett Furniture Industries 28
Table 1.3: Bassett Furniture Industry’s Analysis of the Decision to Relax Credit
Standards by Extending Full Credit to Customers in Credit Risk Group 4. 31
Table 1.4: Nike’s Analysis of the Decision to Change Its Credit Terms from “Net
30” to “Net 60” 34
Table 1.5: CBS Record Company’s Analysis of the Decision to Offer a 1 Percent
Cash Discount 38
Table 2.1: Capital structure of Sonadezi Long Thanh 45
Table 2.2: Operating results in 2005, 2006 and 9 months 2007. 47
Table 2.3: Financial stability and business activities results of SZL 48

Table 2.4: Cash account of Sonadezi Long thanh 50
Table 2.5: Liquidity ratios of Sonadezi Long thanh 50
Table 2.6: Capital structure of Tuong An Vegetable Oil Joint Stock Company 55
Table 2.7: Operating result in 2005, 2006 and 9 months 2007 of TAC 57
Table 2.8: Structure of sales and expenses in 2006 57
Table 2.9: Financial stability and business activities results of TAC 58
Table 2.10: Liabilities structure of TAC 59
Table 2.11: Inventory structure of TAC 60
Table 2.12: Inventory turnover of TAC 61
Table 2.13: Comparison inventory turnover of Tuong An oils and Marvella oils 61
Table 2.14: Comparison inventory turnover between Tuong An oils and Marvella
oils 63
Table 2.15: Account receivable turnover of TAC 64
Table 2.16: Comparison account receivable turnover between Tuong An Oils and
Marvella Oils 64
ix
Table 3.1: The optimal cash balance of Sonadezi Longthanh from January to June,
2007. 68
Table 3.2: Total cost of the optimal cash balance from January to June in 2007. 68
Table 3.3: The total cost in optimal cash balance compare with others cash balance
in January 69
Table 3.4: Comparison of the total cost of holding cash in the case of using Baumol
model and in the case of basing on experiences 70
Table 3.5: The optimal cash balance of Sonadezi Longthanh from January to June,
2007. 72
Table 3.6: The average cash balance of Sonadezi Longthanh from January to June,
2007. 72
Table 3.7: The economic order quantity (EOQ) of Tuong An Vegetable Oil Joint
Stock Company per year form 2005 - 2007. 74
Table 3.8: The total cost of Tuong An Vegetable Oil Joint Stock Company per year

form 2005 - 2007. 74
Table 3.9: Comparison of the total cost of holding inventory in the case of using
EOQ model and in the case of basing on experiences 75
Table 3.10: Tuong An Vegetable Oil Joint Stock Company’s Analysis of the
Decision to relax Credit Standard by Extending Full Credit to customers. 81
Table 3.11: Tuong An Vegetable Oil Joint Stock Company’s Analysis of the
Decision to change its credit term from “net 30” to ” net 60”. 82
Table 3.12: Tuong An Vegetable Oil Joint Stock Company’s Analysis of the
Decision to offer a 1 percent cash discount 84


x
LIST OF FIGURES

Figure 1.1: The various categories of variables are related. 8
Figure 1.2: Cost of holding cash 10
Figure 1.3: Cash Balance for the Company A 11
Figure 1.4: The Miller – Orr Mode 16
Figure 1.5: Costs of holding inventory 23
Figure 1.7: Liberal credit policy model 32
Figure 1.8: Illiberal credit policy model 32
Figure 1.9: Lengthen the credit period model 36
Figure 1.10: Shorten the credit period model 36
Figure 1.11: Increase Cash discount policy model 39
Figure 1.12: Decrease Cash discount policy model 40
Figure 1.13: Credit and receivables management model 41
Figure 2.1: Sonadezi Corporation Structure 44
Figure 2.2: Capital structure of Sonadezi Long Thanh 46
Figure 2.3: Company Organizing Structure of Sonadezi Long Thanh 46
Figure 2.4: Operating results in 2005, 2006 and 9 months 2007. 48

Figure 2.5: Tuong An Vegetable Oil Joint Stock Company Ownership 55
Figure 2.6: Company Organizing Structure 56
Figure 2.7: Operating result in 2005, 2006 and 9 months 2007 of TAC 58
Figure 3.1: Liberal credit policy model 76
Figure 3.2: Illiberal credit policy model 77
Figure 3.3: Lengthen the credit period model 78
Figure 3.4: Shorten the credit period model 78
Figure 3.5: Increase Cash discount policy model 79
Figure 3.6: Decrease Cash discount policy model 80

1
INTRODUCTION

1. NECESSITY OF THE THESIS
In fact, Board of Director and Chief Financial Officer often face with decisions
making relate to financial issue, such as how to choose the target cash management,
how to manage the inventory, how to management the accounts receivable,…These
decisions play an important role, sometime it impact directly on company ‗s success
or failure.
From observation the financial management method in some companies, talking
with some directors, studying the management experiences in some countries and
through the time I work in Sonadezi Longthanh. I think we can apply some
mathematical models in financial decision making. Its gives managers with the
analyzing and making decision tool base on scientific and quantitative.
So I decide to choose the topic: ―Application of mathematics models in short - term
investment decisions‖

2. OBJECTIVE OF THE RESEARCH
The focus of this thesis will be on the researching some mathematical models and
how those models can apply in making decision of Director or Chief Financial

Officer. This thesis has two aims. The first aim is to research the way to apply
financial models to resolve the issues and find out the best solution for each
decision. The second aim is to guide to manager apply the models in making
decision.

3. KEY RESEARCH AREA
Thesis only concentrates on how to use the model in making decision in investment
decision in current assets such as: the target cash balance, inventory management,
and accounts receivable management. These models will be applied in Sonadezi
Longthanh Shareholding Company and in Tuong An Vegetable Oil Joint Stock
Company.

2
4. METHODOLOGY
Thesis is used methodology of researching the secondary data, primary data, logic
reason combination materialistic history, methods in raising the issues,
interpretation, analysis and giving the conclusion.
Thesis is also used statistic, formula illustration, interpreting the issues means and
quantitative method.

5. CONTRIBUTIONS OF THE THESIS
The research of this thesis has the important meaning both the science and reality.
About the science, this thesis chose and improves some theory models appropriate
to conditions and management level of Vietnam.
About the reality, this thesis provide for managers the effective tools in analyzing
and making decision base on quantitative method and apply mathematical model.

6. THESIS STRUCTURE
Topic: ―Application of mathematics models in short - term investment decisions‖
PREFACE

INTRODUCTION
CHAPTER 1: LITERATURE REVIEW
CHAPTER 2: ANALYZING BUSINESS ACTIVITIES OF SONADEZI LONG
THANH SHAREHOLDING COMPANY AND TUONG AN VEGETABLE OIL
JOINT STOCK COMPANY.
CHAPTER 3: APPLICATION OF MATHEMATICS MODELS IN SHORT –
TERM INVESTMENT DECISIONS IN SONADEZI LONGTHANH
SHAREHOLDING COMPANY AND TUONG AN VEGETABLE OIL JOINT
STOCK COMPANY.
REFERENCES
APPENDIXES

3
CHAPTER 1: LITERATURE REVIEW

1.1. Overview of financial decision making
Financial decision making is talked so much in corporate financial management.
Van Horne and Wachowics (2001) state that financial management interested in
buying and selling, financing and asset management follow the general objective.
The studies by McMahon encompass this thesis that financial management
interested in finding the capital to buy the asset and operating the company,
analyzing the limited capital for different purposes, guaranteeing the capital is used
effectively to get the target.
Other researchers such as Brealey and Myers (2003), Ross and other authors (2003)
are believed that financial management interesting in investment, financing, asset
management to get the target. Through the definitions above, we can see the
financial decision making have 3 kinds: investment, financing and dividend
decisions. Besides, there are a lot of decisions relate to company operation but in
the field of research, the thesis just study some decisions can quantitative analysis
and use the model to make decisions. There are some main financial decisions

making.

1.1.1. Investment decision
Investment decisions are decisions relate to (1) total asset values and the values of
each assets (current assets and fixed assets) and (2) the balance between these
assets. We are used to with the balance sheet in accounting. Investment decisions
are related to the left hand sight of the balance sheet. Concretely as follows:
- Investment decisions in current assets, include:
o Cash management decisions
o Inventory management decisions
o Credit decisions
- Investment decisions in fix assets, include:

4
o Financing new fixed assets decisions
o Replacing old fixed assets decisions
o Investing in project decisions
o Long – term financial decisions
- The relationship between investing in current assets and investing in fixed
assets decisions, include:
o Using operating leverage decisions
o Break – even point decisions
Investment decision is considered the most important decision in financial decision
making because it creates the value of firm (Hawawini & Vialiet, 2002). The right
investment decision will contribute to increase the value of firm; and the
shareholder wealth will increase too. Vice versa, the wrong investment decision will
decrease the value of firm, so the shareholder wealth decreases.

1.1.2. Financing decision
If the investment decisions relate to the left hand sight of the balance sheet then

financing decisions relate to the right hand sight of the balance sheet. Its relate to
chose which source of capital finance to purchase the assets, use owner‘s equity or
debt, short term or long term capital. In addition, financing decision also consider
the relationship between retained earnings and payout earnings by dividend. After
choosing one kind in those, the second step the manager should make decision how
can mobilize that source of capital. Concretely as follows:
- Short term financial decisions, include:
o Short term debt or trade credit decisions
o Short term borrowing or commercial paper decisions
- Long term financial decisions, include:
o Long term borrowing: bank loans or bond decisions
o Common Equity or long term debt decisions
o Common equity or preferred equity decisions

5
- The ratio of total debt to total assets decisions (Financial Leverage)
- Borrowing to buy the assets or leasing decisions.
Those above relate to financing decisions in operating of company. If manager lack
the knowledge of analysis tools before making decision, in order to make the right
decision is a big challenge.

1.1.3. Dividend decision
The third decision in financial decision making is the disposition of profits or
dividend policy. In this decision the Chief financial officer (CFO) must be choose
between retained earnings or payout earnings by dividend. In addition, CFO must
be deciding to use what dividend policy and what effect of dividend policy on the
value of firm or stock‘s value on the market.

1.1.4. Other decisions
Besides 3 kinds of decisions in financial decision making, there is a lot of other

decisions relate to the operations of the business. But focus of this thesis will be on
the decisions that the model can be applied to make decisions.

1.2. Overview about the model building
1

1.2.1. Definition of Models
A model is a simplified representation of an empirical situation. Ideally, it strips a
natural phenomenon of its bewildering complexity and duplicated the essential
behavior of the natural phenomenon with a few variables that are simply related.
The simpler the model, the better for the decision maker, provided the model serves
as a reasonably reliable counterpart of the empirical problem. The advantages of a
simple model are:
- It is economical of time and thought.


1
Bonini, Hausman, Bierman,(1997), Quantitative Analysis for Management (9
th
Edition), McGraw
– Hill/Irwin.

6
- It can be understood readily by the decision maker.
- If necessary, the model can be modified quickly and effectively.
The object of the decision maker is not to construct a model that is close as
possible to reality in every respect. Such a model would require an excessive
length of time to construct, and then it might be beyond human comprehension.
Rather, the decision maker wants the simplest model that predicts outcomes
reasonably well and is consistent with effective action.

1.2.2. Classification of Models
There are several types of decision models. To understand and build the models,
first we need to know how to classify the model base on different criteria.
- The nature of models
o Physical model
o Notion model
o Mathematical model
- The level of complication
o Simple Problems
 A case or scenario model
 Decision analysis models
o Complex Problems
 Linear and integer programming models
 Simulation
o Dynamic Problems
 Inventory models
 PERT or Critical path models (CPM)
 Queuing models
- The dynamic nature‘s models
o Certain models
o Uncertain models

7
Table 1.1: Classification of models
Decision problem
is:
Major variables in a decision problem are

Certain

Uncertain
Simple
Case models
Decision analysis
(decision trees)
Complex
Case models
Linear and integer
programming
Simulation

Dynamic
Inventory models
PERT or Critical path models
Simulation
Inventory models
Queuing models
( Sources: Bonini, Hausman, Bierman. 1997. Quantitative analysis for management. 9
th

Edition. New York: McGraw – Hill/ Irwin)

1.2.3. Basic Modeling Concepts
A model is a simplification of a business decision problem. The simplification is
accomplished by including only important elements and omitting the nonessential
consideration. Because it is simplified, it is highly useful. The factors or variables
that the decision maker considers important, includes:
a. Decision Variables: the decision variables are those under the control of the
decision maker. They represent alternative choices for managers. These are
the major choices, these are the decision variables.

b. Exogenous variables: exogenous or external variables are those that are
important to the decision problem but are controlled by factors outside the
purview of the decision maker. Generally, economic conditions, actions of
competitors, prices of raw materials and similar factors are exogenous
variables.

8
c. Policies and constraints: A decision maker often operates within constraints
imposed by company policy, legal restraints or physical limitations. For
example, there may be limited capacity available in the plant, and this may
restrict the sales that can be made.
d. Performance measures: In making a decision, managers have goals or
objectives that they are trying to achieve. Criteria or performance measures
are quantitative expression of these objectives.
e. Intermediate variables: A number of other variables are usually needed to
include all the important factors in the decision problem. Often these are
accounting variables that relate to cost or revenue factors. They are used to
relate the decision variables and exogenous variables to the performance
measures. They are thus intermediate variables in the sense that they are
between the other variables.
Figure 1.1 show how the various categories of variables are related. Decision
variables, exogenous variablea and policies and constraints are input to the model,
and performance measures are outputs. The model itself represents the set of all
relationships among the variables.



Figure 1.1: The various categories of variables are related.

Model set of

relationships
Exogenous
variables
Decision variables
Policies and
constraints
Performance
measures

9

1.2.4. The method to set up a model and apply in the financial decision making
Depend on the simple or complex problems and the aim of decision maker, the
model can be a simple models, complex models or very complex models. These
steps to set up a model:
Step 1: define the aim or the nature of decision.
Step 2: define the variables affect decision
Step 3: define the relationship among the variables and the aim of decision (Set up
the model)
Step 4: input the data of variables into the model, check the result.
Step 5: change the data of variables and check again effect on the result.

1.3. Models using in investment decisions in current assets
In the field of research, the thesis just study some model using in investment
decisions in current assets, include:
- Cash management decisions
- Inventory management decisions
- Credit decisions

1.3.1. Determining the target cash balance

2

Target cash balance: involves a trade-off between the opportunity costs of holding
too much cash (the carrying costs) and the costs of holding too little (the shortage
costs, also called adjustment costs). The nature of these costs depends on the firm‘s
working capital policy.
If the firm has a flexible working capital policy, then it will probably maintain a
marketable securities portfolio. In this case the adjustment, or shortage, costs will be
the trading costs associated with buying and selling securities. If the firm has a
restrictive working capital policy, it will probably borrow in the short term to meet


2
Ross and et al, (2003), Fundamentals of Corporate Finance,
6
th

Edition, McGraw-Hill/Irwin.


10
cash shortages. The costs in this case will be the interest and other expenses
associated with arranging a loan.
Figure 1.2 presents the cash management problem for our flexible firm. If a firm
tries to keep its cash holdings too low, it will find itself running out of cash more
often than is desirable, and thus selling marketable securities (and perhaps later
buying marketable securities to replace these sold) more frequently than would be
the cash balance were higher. Thus, trading costs will be high when the cash
balance is small. These costs will fall as the cash balance becomes larger.

Figure 1.2: Cost of holding cash

In contrast, the opportunity costs of holdings cash are very low if the firm holds
very little cash. These costs increase as the cash holdings rise because the firm is
giving up more and more interest that could have been earned.
Figure 1.2, the sum of the costs is given by the total cost curve. As show, the
minimum total cost occurs where the two individual cost curves cross at point C*.
At this point, the opportunity costs and the trading costs are equal. This point
represents the target cash balance, and it is the point the firm should try to find.

Cost of holding cash
($)
Total costs of holding
cash
Opportunity costs
Trading costs
Size of cash
balance (C)
C*

11
1.3.1.1. The BAT model
The Baumol – Allais – Tobin (BAT) model is a classic means of analyzing our cash
management problem. This model can be used to actually build the target cash
balance. It is a straightforward model and very useful for illustrating the factors in
cash management and, more generally, current asset management.
To develop the BAT model, suppose Company A starts off at week 0 with a cash
balance of C = $1.2 million. Each week, outflows exceed inflows by $600,000. As

a result, the cash balance will drop to zero at the end of week 2. The average cash
balance will be the beginning balance ($1.2 million) plus the ending balance ($0)
divided by 2, or ($1.2 million + 0)/2 = $600,000, over the two – week period. At the
end of week 2, Company A replenishes its cash by depositing another $1.2 million.
As we have described, the cash management strategy for Company A is very simple
and boils down to depositing $1.2 million every two weeks. This policy is shown in
Figure 1.3 Notice how the cash balance declines by $600,000 per week. Because the
company brings the account up to $1.2 million, the balance hits zero every two
weeks. This result in the saw tooth pattern displayed in Figure 1.3.


Figure 1.3: Cash Balance for the Company A

Implicitly, we assume that the net cash outflow is the same every day and that it is
known with certainty. These two assumptions make the model easy to handle. We
will indicate what happens when they do not hold.
Starting cash
C = $1,200,000
$600,000 = C /2
Ending cash: 0
Average cash

Weeks

12
If C were set higher, say, at $2.4 million, cash would last four weeks before the firm
would have to sell marketable securities, but the firm‘s average cash balance would
in- crease to $1.2 million (from $600,000). If C were set at $600,000, cash would
run out in one week, and the firm would have to replenish cash more frequently, but
the average cash balance would fall from $600,000 to $300,000.

Because transactions costs (for example, the brokerage costs of selling marketable
securities) must be incurred whenever cash is replenished, establishing large initial
balances will lower the trading costs connected with cash management. However,
the larger the average cash balance, the greater is the opportunity cost (the return that
could have been earned on marketable securities).
To determine the optimal strategy, Company A needs to know the following three
things:
F

The fixed cost of making a securities trade to replenish cash.
T

The total amount of new cash needed for transactions purposes over the
relevant planning period, say, one year.
R

The opportunity cost of holding cash. This is the interest rate on marketable
securities
a. The Opportunity Costs
To determine the opportunity costs of holding cash, we have to find out how much
interest is forgone. Company A has, on average, C/2 in cash. This amount could be
earning interest at rate R. So the total dollar opportunity costs of cash balances are
equal to the average cash balance multiplied by the interest rate:
Opportunity costs = (C/2) x R
For example, the opportunity costs of various alternatives are given here assuming
that the interest rate is 10 percent:

13
Initial Cash Balance
Average Cash Balance
Opportunity Cost
(R=0.1)
C
C/2
(C/2) x R
$4,800,000
$2,400,000
$240,000
2,400,000
1,200,000
120,000
1,200,000
600,000
60,000
600,000
300,000
30,000
300,000
150,000
15,000

In our original case, in which the initial cash balance is $1.2 million, the average
balance is $600,000. The Interest Company A could have earned on this (at 10
percent) is $60,000, so this is what the firm gives up with this strategy. Notice that
the opportunity costs increase as the initial (and average) cash balance rises.
b. The Trading Costs

To determine the total trading costs for the year, we need to know how many times
Company A will have to sell marketable securities during the year. First of all, the
total amount of cash disbursed during the year is $600,000 per week, so T =
$600,000 x 52 weeks = $31.2 million. If the initial cash balance is set at C = $1.2
million, then Company A will sell $1.2 million in marketable securities T/C =
$31.2 million/1.2 million = 26 times per year. It costs F dollars each time, so
trading costs are given by:
million $1.2
million $31.2
x F = 26 x F
In general, the total trading costs will be given by:
Trading costs = (T/C) x F
In this example, if F were $1,000 (an unrealistically large amount), then the trading
costs would be $26,000.
We can calculate the trading costs associated with some different strategies
as follows:

14
Total Amount of
Disbursements
during Relevant Period
Initial Cash
Balance
Trading Costs
(F = $1,000)
T
C
(T/C) x F
$31,200,000
$4,800,000

$6,500
31,200,000
2,400,000
13,000
31,200,000
1,200,000
26,000
31,200,000
600,000
52,000
31,200,000
300,000
104,000

c. The Total Cost
Now that we have the opportunity costs and the trading costs, we can calculate the
total cost by adding them together:
Total cost = Opportunity costs + Trading costs
(C/2) x R + (T/C) x F
Using the numbers generated earlier, we have:
Cash
Balance
Opportunity Costs
+
Trading Costs
=
Total Cost
$4,800,000
$240,000

$6,500

$246,500
2,400,000
120,000

13,000

133,000
1,200,000
60,000

26,000

86,000
600,000
30,000

52,000

82,000
300,000
15,000

104,000

119,000

Notice how the total cost starts out at almost $250,000 and declines to about
$82,000 before starting to rise again.

d. The Solution
We can see from the preceding schedule that a $600,000 cash balance results in the
lowest total cost of the possibilities presented $82,000. But what about $700,000 or

15
$500,000 or other possibilities? It appears that the optimum balance is somewhere
between $300,000 and $1.2 million. With this in mind, we could easily proceed by
trial and error to find the optimum balance. It is not difficult to find it directly,
however, so we do this next.
Take a look back at Figure 1.2 As the figure is drawn, the optimal size of the cash
balance, C*, occurs right where the two lines cross. At this point, the opportunity
costs and the trading costs are exactly equal. So, at C*, we must have that:
Opportunity costs = Trading costs
(C*/2) x R = (T/C*) x F
With a little algebra, we can write:
C*
2
= (2T x F)/R
To solve for C*, we take the square root of both sides to get:
C* =
F)/R x (2T

This is the optimum initial cash balance.
For Company A, we have T = $31.2 million, F = $1,000, and R = 10%.
We can now find the optimum cash balance:
C* =
1,000)/.10 x 0$31,200,00 x (2
=
$624
billion = $789,937

We can verify this answer by calculating the various costs at this balance, as well
as a little above and a little below:
Cash Balance
Opportunity
Costs
+
Trading Costs
=
Total Cost
$850,000
$42,500

$36,706

$79,206
800,000
40,000

39,000

79,000
789,937
39,497

39,497

78,994
750,000
37,500

41,600

79,100
700,000
35,000

44,571

79,571

The total cost at the optimum cash level is $78,994, and it does appear to increase
as we move in either direction.

16
e. Conclusion
The BAT model is possibly the simplest and most stripped-down sensible model
for determining the optimal cash position. Its chief weakness is that it assumes
steady, certain cash outflows.

1.3.1.2. The Miller – Orr Model: A more general Approach.
Different with Baumol, Merton Miller and Daniel Orr develop the target cash
balance model with cash inflows and outflows that fluctuate randomly from day to
day. With this model, we again concentrate on the cash balance, but, in contrast to
the situation with the BAT model, we assume that this balance fluctuates up and
down randomly and that the average change is zero.


Figure 1.4: The Miller – Orr Mode

U* is the upper control limit. L is the lower control limit. The target cash balance

is C*. As long as cash is between L and U*, no transaction is made.


Cash
Cash
L
C*

U*
X
Y
Cash balance