**CERTIFIED INVESTMENT AND FINANCIAL ANALYSTS**

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**PART ONE**

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**SECTION ONE**

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**STUDY TEXT**

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** ****FINANCIAL MATHEMATICS**

**Table of contents**

CHAPTER ONE……………………………………………………………………. 3

Introduction to financial mathematics

CHAPTER TWO…………………………………………………………………….. 8

Financial algebra

CHAPTER THREE………………………………………. 25

Descriptive Statistics

CHAPTER FOUR…………………………………….. 53

Time value of money

CHAPTER FIVE……………………………………… 85

Financial forecasting

CHAPTER SIX……………………………………… 105

Financial calculus

CHAPTER SEVEN……………………………… 115

Probability theory

CHAPTER EIGHT………………………………. 132

Index numbers

# TOPIC ONE

**INTRODUCTION TO FINANCIAL MATHEMATICS**

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** **Nature of financial decision

** **Financial decisions are those made by financial managers of a firm. It‟s broadly classified into two.

- Managerial decision
- Routine decision

##### Managerial decisions

These are Decisions that require technical skills, planning and expertise of a financial manager. It‟s classified into four:

- Financing decision

It involves looking for finance to acquire assets of the firm and may include:

- Issue of ordinary shares
- Long term loan
- Preference shares
- Investment decision

** **It‟s the responsibility of a financial manager to determine whether acquired funds should be invested in order to generate revenue. Financial manager must do a proper appraisal of any investment that may be undertaken.

- Dividend decision

** **Dividends are part of the earnings distributed to ordinary shareholders for their investment in the company. Financial manager has to consider the following:

- How much to pay
- When to pay
- How to pay e. cash or bonus issue
- Why to pay

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# TOPIC TWO

# FINANCIAL ALGEBRA

**Functions**

It’s the relationship between independent variable and the dependent variable. It consists of a constant and a variable.

A constant – This is a quantity whose value remains unchanged throughout a particular analysis

e.g. fixed cost, rent, and salary.

A variable – This is a quantity which takes various values in a particular problem

Illustration

Suppose an item is sold at Sh 11 per unit. Let S represent sales rate revenue in shillings and let Q represents quantity sold.

Then the function representing these two variables is given as: S = 11Q

S and Q are variables whereas the price – Sh 11 – is a constant.

Types of variables

** **Independent variable – this is a variable which determines the quantity or the value of some other variable referred to as the dependent variable. In Illustration 1.1, Q is the independent variable while S is the dependent variable.

An independent variable is also called a predictor variable while the dependent variable is also known as the response variable i.e. Q predicts S and S responds to Q.

A function – This is a relationship in which values of a dependent variable are determined by the values of one or more independent In illustration 1.1 sales is a function of quantity, written as S = f(Q)

Demand = f( price, prices of substitutes and complements, income levels,….) Savings = f(investment, interest rates, income levels,….)

**Note **that the dependent variable is always one while the independent variable can be more than one.

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**TOPIC THREE **

**DESCRIPTIVE STATISTICS**

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#### Introduction

**Statistics **is the art and science of getting information from data or numbers to help in decision making.

As a science, statistics follows **a systematic procedure **to reach objective decisions or solutions to problems.

As an art statistics utilizes **personal judgment and intuition **to reach a solution. It depends on experience of the individual involved. It is more subjective.

Statistics provides us with tools that aid decision making. For example, using statistics we can estimate that expected returns and associated risks of a given investment opportunity.

Statistics involves collection of data, analysis, presentation and interpretation of data.

There are various types of summary measures including averages and measures of dispersion. An average is a figure which represents the whole data. It removes all unnecessary details and gives a clear picture of the data under investigation.

__Qualities of a good average__

- Should be based on all values or observation
- Should be easily understood and calculated
- Should be capable of further statistical investigation/treatment
- Should be least affected by fluctuations of sampling

##### Definitions of key terms

**Measures of central tendency **are single numbers that are used to summarize a larger set of data in a distribution of scores. The three measures of central tendency are ** mean, median, and mode**. They are also called

*types of averages***Measures of dispersion – **These are important for describing the spread of the data, or its variation around a central value. Such measures of dispersion include: standard deviation, inter- quartile range, range, mean difference, median absolute deviation, average absolute deviation (or simply average deviation)

**Variance **is the sum of squared deviations divided by the number of observations. It is the average of the squares of the deviation of the individual values from their means.

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**TOPIC FOUR**

**TIME VALUE OF MONEY**

Objectives

At the end of this chapter you should be able to:

- Explain meaning of time value of money and its role in
- Explain the concept of future value and perform compounding
- Explain the concept of present value and perform discounting
- Apply the mathematics of finance to accumulate a future sum, preparing loan amortization schedules, and determining interest or growth

Introduction

A shilling today is worth more than a shilling tomorrow. An individual would thus prefer to receive money now rather than that same amount later. A shilling in ones possession today is more valuable than a shilling to be received in future because, first, the shilling in hand can be put to immediate productive use, and, secondly, a shilling in hand is free from the uncertainties of future expectations (It is a *sure shilling)*.

Financial values and decisions can be assessed by using either future value (FV) or present value (PV) techniques. These techniques result in the same decisions, but adopt different approaches to the decision.

Future value techniques

Measure cash flow at the some future point in time – typically at the end of a projects life. The **Future Value (FV), **or terminal value, is the value at some time in future of a present sum of money, or a series of payments or receipts. In other words the FV refers to the amount of money an investment will grow to over some period of time at some given interest rate. FV techniques use **compounding **to find the future value of each cash flow at the given future date and the sums those values to find the value of cash flows.

Present value techniques

Measure each cash flows at the start of a projects life (time zero).The **Present Value (PV) **is the current value of a future amount of money, or a series of future payments or receipts. Present value is just like cash in hand today. PV techniques use **discounting **to find the PV of each cash flow at time zero and then sum these values to find the total value of the cash flows.

Although FV and PV techniques result in the same decisions, since financial managers make decisions in the present, they tend to rely primarily on PV techniques.

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**TOPIC FIVE **

**FINANCIAL FORECASTING**

** **It involves determining the future financial requirements of the firm. This requires financial planning using budgets.

#### Importance of financial forecasting

- Facilitates financial planning e. determination of cash surplus or deficit that are likely to occur in future.
- Facilitates control of expenditure so as to minimize wastage of financial
- Forecasting using targets and budgets acts as a motivation to employees who aim at achieving targets set

### Strategic plan:

It‟s a blue print road map that indicates what the firm intends to do and how to do it. It consists of:

- The mission/purpose of existence
- Scope: these are the lines of business
- Objectives: specific goals in quantitative
- Strategies: instruments to be used in achieving firms

### Role of financial manager in strategic planning

- Educate strategic planning team on financial implication on various options
- Ensure strategic plan is viable financially
- Translate strategic plan into long range financial plans

### Elements of financial planning

- Assumption: that they should be clearly
- sales/revenue forecast: it‟s the starting point of financing planning since most of other variables it relates to sales
- pro-forma financial statement: balance sheet, cash flow, income statement
- assets/investment requirement: this will reveal investment required to achieve forecasted/budgets sales in short-term and long term
- Financial plan: spells out proposed means of financing investment.
- Cash budget: it indicates the cash inflow and cash outflow

### Importance of financial forecasting/need

- Forces management to plan in advance and allocate resources efficiently
- Forces managers to avoid surprises which may occur in the course of operations e.g if there is a cash deficit the managers will decide how to finance the
- Used for control purposes e. it enables the company to control expenses and to avoid wastage due to operations of the firm.
- Used for motivation purpose e.g. since managers and employees are aware of what is

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# TOPIC SIX

# FINANCIAL CALCULUS

It explains how the value of the variable varies as the other variable changes.

Calculus is concerned with the mathematical analysis of change or movement. There are two basic operations in calculus:

- Differentiation
- Integration

These two basic operations are reverse of one another in the same way as addition and subtraction or multiplication and division

##### Differentiation

It is concerned with rates of change e.g. profit with respect to output

- Revenue with respect to output
- Change of sales with respect to level of advertisement Savings with respect to income, interest rates

Rates of changes and slope (gradients)

It estimates a slope (gradient of graph) of a particular point. The derivative of a function

𝚫𝒚 gives the exact change of a point. It‟s the process of finding the derivative of a function.

𝚫𝒙

##### TOPIC SEVEN PROBABILITY THEORY

**Introduction**

Probability is a measure of likelihood, the possibility or chance that an event will happen in future.

It can be considered as a quantification of uncertainty.

Uncertainty may also be expressed as likelihood, chance or risk theory. It is a branch of mathematics concerned with the concept and measurement of uncertainty.

Much in life is characterized by uncertainty in actual decision making.

Probability can only assume a value between 0 and 1 inclusive. The closer a probability is to zero the more improbable that something will happen. The closer the probability is to one the more likely it will happen.

##### Definitions of key terms

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**Random experiment **results in one of a number of possible outcomes e.g. tossing a coin

**Outcome **is the result of an experiment e.g. head up, gain, loss, etc. Specific outcomes are known as events.

**Trial**– Each repetition of an experiment can be thought of as a trial which has an observable outcome e.g. in tossing a coin, a single toss is a trial which has an outcome as either head or tail

**Sample space **is the set of all possible outcomes in an experiment e.g. a single toss of a coin, S=(H,T). The sample space can be finite or infinite. A finite sample space has a finite number of possible outcomes e.g. in tossing a coin only 2 outcomes are possible.

An infinite sample space has an infinite number of possible outcomes e.g. time between arrival of telephone calls and telephone exchange.

**An Event **of an experiment is a subset of a sample space e.g. in tossing a coin twice S= (HH, HT,

TH, TT) HH is a subset of a sample space.

**Mutually exclusive event **– A set of events is said to be mutually exclusive if the occurrence of any one of the events precludes the occurrence of other events i.e. the occurrence of any one event means none of the others can occur at the same time e.g. the events head and tail are mutually exclusive

**Collectively exclusive event **– A set of events is said to be collectively exclusive if their union accounts for all possible outcomes i.e. one of their events must occur when an experiment is conducted.

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# TOPIC EIGHT

# INDEX NUMBERS

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** **It’s a number which indicates the level of a certain phenomena at any given date in comparison with the level of the same phenomena at the same standard date.

It‟s a series of number by which changes in magnitude of phenomena or measures from time to time or from place to place. It provides an opportunity for measuring the relative change of a valuable where measure of its actual change is inconvenient or impossible. An index number is constructed by selecting a base year as a starting point.

##### Price index numbers

They are important because they show that the value of money is fluctuating i.e. appreciating or depreciating.

A rise in the index numbers will signify that there is deterioration in value of money and vice versa.

__Factors to be considered when constructing price index numbers__

- Purpose of index numbers

** **The purpose must be determined before their construction otherwise the results will be useless. When purpose is known correctly, true results can be obtained.

- Selection of commodities

** **When purpose is known, the selection of commodities for that purpose becomes easy and accurate to a responsible when selecting commodities.

- Price quotation

** **It is impossible to collect the prices of all selected commodities from all places in country where they are marketed. A sample of the market needs to be selected and it should be from those places where given commodities are marketed in large numbers.

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