# Percentages and Ratios, Simple and Compound Interest, Discounted Cash Flow

#### RATIOS

##### Introduction

A ratio is another form of fraction just like a percentage, decimal or common fraction. It is a relationship between two numbers or two like values. Consider the following situation concerning a small town:

##### To Reduce the Ratio to its Lowest Terms

Put the first figure over the second figure and cancel the resulting fraction. Then re-express as a ratio in the form numerator : denominator

##### Divide a Quantity According to a Given Ratio

Add the terms of the ratio to find the total number of parts. Find what fraction each term of the ratio is to the whole. Divide the total quantity into parts according to the fraction

#### SIMPLE INTEREST

Interest (I) is a charge for the use of money for a specific time. This charge is usually expressed as a percentage called the rate per cent per annum. Three factors determine the amount of interest:

• The sum of money on which the interest is payable; this is known as the principal

(P).

• The rate (R).
• The length of time (Y) for which the money is borrowed.

When the interest due is added to the principal, the sum is called the amount (A), which is the amount to be repaid.

Simple interest is interest reckoned on a fixed principal. Simple interest is, therefore, the same for each year, and the total is found by multiplying the interest for one year by the number of years.

#### INTRODUCTION TO DISCOUNTED CASH FLOW PROBLEMS

If a business is to continue earning profit, its management should always be alive to the need to replace or augment fixed assets. This usually involves investing money (capital expenditure) for long periods. The longer the period the greater is the uncertainty and, therefore, the risk involved. With the advent of automation, machinery, equipment and other fixed assets have tended to become more complex and costly. Careful selection of projects has never been so important. One method of selecting the most profitable investments follows.

These techniques do not replace judgement and the other qualities required for making decisions. However, it is true to say that the more information available, the better able a manager is to understand a problem and reach a rational decision.

##### Classification of Investment Problems

Capital investment problems may be classified into the following types, and each is amenable to discounted cash flow analysis.

• The replacement of, or improvement in, existing assets by more efficient plant and equipment (often measured by the estimated cost savings).
• The expansion of business facilities to produce and market new products (measured by the forecast of additional profitability against the proposed capital investment).
• Decisions regarding the choice between alternatives where there is more than one way of achieving the desired result.
• Decisions whether to purchase or lease assets.
##### Basis of the Method

The method is based on the criterion that the total present value of all increments of income from a project should, when calculated at a suitable rate of return on capital, be at least sufficient to cover the total capital cost. It takes account of the fact that the earlier the return the more valuable it is, for it can be invested to earn further income meanwhile.

By deciding on a satisfactory rate of return for a business, this can then be applied to several projects over their total life to see which gives the best present cash value.

For any capital investment to be worthwhile, it must give a return sufficient to cover the initial cost and also a fair income on the investment. The rate which will be regarded as “fair income” will vary with different types of business, but as a general rule it should certainly be higher than could be obtained by an equivalent investment in shares.

##### Information Required

To make use of DCF we must have accurate information on a number of points. The method can only be as accurate as the information which is supplied.

The following are necessary as a basis for calculation:

• Estimated cash expenditure on the capital project.
• Estimated cash expenditure over each year.
• Estimated receipts each year, including scrap or sale value, if any, at the end of the asset’s life.
• The life of the asset.
• The rate of return expected (in some cases you will be given a figure for “cost of capital” and you can easily use this rate in the same way to see whether the investment is justified).

The cash flow each year is the actual amount of cash which the business receives or pays each year in respect of the particular project or asset (a net figure is used). This represents the difference between (c) and (b).

Clearly the receipts and expenditures may occur at irregular intervals throughout the year, but calculations on this basis would be excessively complicated for problems such as may arise in your examination. So, unless you are told otherwise, you can assume that the net receipt or expenditure for the year occurs at the end of the relevant year

##### Importance of “Present Value”

Before we proceed to a detailed examination of the method used by DCF there is one important concept which you need to understand – the idea of present value.

#### .   TWO BASIC DCF METHODS

You have now seen a simple example of how DCF is used, and you already have a basic knowledge of the principles which the technique employs. There are two different ways of using DCF – the yield (or rate of return) method, and the net present value method, which was used in the above example.

The important point to remember is that both these methods give identical results. The difference between them is simply the way they are used in practice, as each provides an easier way of solving its own particular type of problem.

As you will shortly see, the yield method involves a certain amount of trial-and-error calculation. Questions on either type are possible, and you must be able to distinguish between the methods and to decide which is called for in a particular set of circumstances.

In both types of calculation there is the same need for accurate information as to cash flow, which includes the initial cost of a project, its net income or outgoings for each year of its life, and the final scrap value of any machinery.

##### Yield (Internal Rate of Return) Method

This method is used to find the yield, or rate of return, on a particular investment. By “yield” we mean the percentage of profit per year of its life in relation to the capital employed. In other words, we must allow for repayment of capital before we consider income as being profit for this purpose. The profit may vary over the years of the life of a project, and so may the capital employed, so an average figure needs to be produced.

DCF, by its very nature, takes all these factors into account.

The primary use of the method is to evaluate a particular investment possibility against a guideline for yield which has been laid down by the company concerned. For example, a company may rule that investment may only be undertaken if a 10% yield is obtainable. We then have to see whether the yield on the desired investment measures up to this criterion. In another case, a company may simply wish to know what rate of return is obtainable from a particular investment; thus, if a rate of 9% is obtainable, and the company’s cost of capital is estimated at 7%, it is worth its while to undertake the investment.

What we are trying to find in assessing the figures for a project is the yield which its profits give in relation to its cost. We want to find the exact rate at which it would be breaking even,

i.e. the rate at which discounted future cash flow will exactly equal the present cost, giving an NPV of 0. Thus if the rate of return is found to be 8%, this is the rate at which it is equally profitable to undertake the investment or not to undertake it; the NPV is 0. Having found this rate, we know that if the cost of capital is above 8%, the investment will be unprofitable, whereas if it is less than 8%, the investment will show a profit. We thus reach the important conclusion that once we have assembled all the information about a project, the yield, or rate of return, will be the rate which, when used to discount future increments of income, will give an NPV of 0. We shall then know that we have found the correct yield.

You should ensure that you know exactly how and when to use the method, as practical questions are very much more likely than theoretical ones in the examination.

When to Use the Yield Method

This is not a difficult problem, because you will use the method whenever you require to know the rate of return, or yield, which certain increments of income represent on capital employed. You must judge carefully from any DCF question whether this is what you need to know.

How to Use the Yield Method

The calculation is largely dependent on trial and error. When you use this method, you know already that you are trying to find the rate which, when used to discount the various increments of income, will give an NPV of 0. You can do this only by trying out a number of different rates until you hit on the correct result. A positive NPV means that the rate being tried is lower than the real rate; conversely, a negative NPV means that too high a rate is being used. So you need to work the problem out as many times as is necessary to hit on the appropriate rate for obtaining the NPV of 0. If this process is done sensibly, for simple problems such as those which we are going to encounter, it should not take many steps to hit upon the right result. Watch out for any instructions concerning “rounding” of yields – for example, “to the nearest ½ %”.

##### Net Present Value (NPV) Method

The NPV method is probably more widely used than the yield method, and its particular value is in comparing two or more possible investments between which a choice must be made. If a company insists on a minimum yield from investments of, say, 10%, we could check each potential project by the yield method to find out whether it measures up to this. But if there are several projects each of which yields above this figure, we still have to find some way of choosing between them if we cannot afford to undertake all of them.

At first sight the obvious choice would be that which offered the highest yield. Unfortunately this would not necessarily be the best choice, because a project with a lower yield might have a much longer life, and so might give a greater profit.

However, we can solve the problem in practice by comparing the net present values of projects instead of their yields. The higher the NPV of a project or group of projects, the greater is its value and the profits it will bring.

We must remember that in some instances the cost of capital will be higher for one project than for another. For example, a company which manufactures goods may well be able to borrow more cheaply for its normal trade than it could if it decided to take part in some more speculative process. So each project may need to be assessed at a different rate in accordance with its cost of capital. This does not present any particular problems for DCF.

##### NPV Method and Yield Method Contrasted

You should now be able to see the important difference between the NPV method and the yield method. In the yield method we were trying to find the yield of a project by discovering the rate at which future income must be discounted to obtain a fixed NPV of 0. In the NPV method we already know the discounting rate for each project (it will be the same as the cost of capital) and the factor which we are now trying to find for each project is its NPV. The project with the highest NPV will be the most profitable in the long run, even though its yield may be lower than other projects.

So you can see that comparison of projects by NPV may give a different result from comparison by yields. You must decide for each particular problem which method is appropriate for it.

##### How to Use the NPV Method

We must first assemble the cash flow figures for each project. Then, carry out the discounting process on each annual net figure at the appropriate rate for that project, and calculate and compare the NPVs of the projects. As we have seen, that with the highest NPV will be the most profitable.

##### Allowance for Risk and Uncertainty

All investments are subject to risk. In general terms, we mean normal business risk, i.e. not that the investment plans will collapse completely as a total write-off, but that unforeseen factors will emerge, such as new legislation, changes in fashion, etc. which make the original estimates of costs and sales, etc. no longer valid. There are two accepted methods for incorporating risk into a capital investment appraisal:

Inclusion of a Risk Premium in the Discount Rate

The inclusion of a risk premium in the discount rate means that if the normal discount rate to be used were, say, 12%, then an additional amount, say 4%, might be allowed to cover for risk, making a 16% discount rate in total. The premium to be added is largely arrived at by subjective rather than objective measurement, and is correspondingly weak. As we have seen also, higher discount rates “bite” more savagely at the more distant cash flows, so that two projects, one short and one long in life-span, would be treated differently for risk by this method.

Attaching Probabilities to Cash Flows

With the first method we effectively looked at the project “normally” – with our usual discount rate and in a “least favourable” position, by requiring the project to provide a higher return to cover risk. We can, in fact, refine this method further by attaching individual probabilities to each cash inflow and outflow, rather than a once-off blanket cover by upping the discount rate.

### INTRODUCTION TO FINANCIAL MATHS

The following are the areas which  we cover in this section.

Simple and compound interest, annual percentage rate (APR), depreciation (straight line and reducing balance), discounting, present value and investment appraisal, Annuities, mortgages, amortization, sinking funds.

In this area the letter i and r both stand for the interest rate. The interest rate is often referred to in financial maths as: the discount rate, the cost of capital, the rate of return.

##### Annual percentage rate (APR)

In the above example, we assumed interest was added or compounded annually, however sometimes interest may accrue ever six months (twice a year) or over 3 months (4 times per year). (This would of course be better for the customer).

If interest is at 10% per annum we call this the nominal rate, however if it is compounded every six months then the actual return is greater than 10%. We call the rate you are actually getting on your investment the effective rate or actual percentage rate (APR).

Depreciation: Straight line and reducing balance.

Depreciation is an allowance made in estimates, valuations or balance sheets, normally for “wear and tear”. There are two techniques for calculating depreciation:

− Straight line or equal instalment depreciation &

− Reducing balance depreciation.

Annuities, Mortgages, Amortization, Sinking funds.

This topic deals with various techniques associated with fixed payments (or receipts) over time, otherwise known as annuities.

An annuity is a sequence of fixed equal payments (or receipts) made over uniform time intervals. Some examples are monthly salaries, insurance premiums, mortgage repayments, hire-purchase agreements.

Annuities are used in all areas of business and commerce. Loans are normally repaid with an annuity, investment funds are made up to meet fixed future commitments for example asset replacement, by the payment of an annuity. Perpetual annuities can be purchased with a single lump-sum payment to enhance pensions. Annuities may be paid

− At the end of payment intervals(an ordinary annuity) or

− At the beginning of a payment interval (a due annuity)

Amortization of a debt.

If an amount of money is borrowed over a period of time, one way of repaying the debt is by paying an amortization annuity. This consists of a regular annuity in which each payment accounts for both repayment of capital and interest. The debt is said to be amortized if this method is used. Many of the loans issued for houses are like this. This is known as a repayment mortgage.

The standard question is: given the amount borrowed P, with interest of r%, what must the annual payments be A, in order to pay off (amortize) the debt in a certain number of years.

The easiest way to do this is with an “Amortization Schedule”.

An amortization schedule is a specification, period by period (normally year by year) of the state of the debt. It is usual to show for each year:

• Amount of debt outstanding at the beginning of the year.
• Interest paid
• Annual payment
• Amount of principle repaid.

Sinking fund

Sinking funds are commonly used for the following purposes:

• Repayment of debt
• To provide funds to purchase a new asset when the existing asset is fully depreciated.

Dept repayment using a sinking fund:

Here, a debt is incurred over a fixed period of time, subject to a given interest rate. A sinking fund must be set up to mature to the outstanding amount of the debt.

##### Break-even Analysis

For any business there is a certain level of sales at which there is neither a profit nor a loss, i.e. the total income and the total costs are equal.  This point is known as the break-even point.  It is very easy to calculate, and it can also be found by drawing a graph called a breakeven chart.

##### Break-even Chart

Information Required

Sales Revenue

When we are drawing a break-even chart for a single product, it is a simple matter to calculate the total sales revenue which would be received at various outputs.

Fixed Costs

We must establish which elements of cost are fixed in nature.  The fixed element of any semi-variable costs must also be taken into account.

Plotting the Graph

The graph is drawn with level of output (or sales value) represented along the horizontal axis and costs/revenues up the vertical axis.  The following are the stages in the construction of the graph:

• Plot the sales line from the above figures.
• Plot the fixed expenses line. This line will be parallel to the horizontal axis.

Break-even Chart for More Than One Product

Because we were looking at one product only in the above example, we were able to plot “volume of output” and straight lines were obtained for both sales revenue and costs.  If we wish to take into account more than one product, it is necessary to plot “level of activity” instead of volume of output.  This would be expressed as a percentage of the normal level of activity, and would take into account the mix of products at different levels of activity.

Even so, the break-even chart is not a very satisfactory form of presentation when we are concerned with more than one product:  a better graph, the profit-volume graph, is discussed in the next study unit.  The problem with the break-even chart is that we should find that, because of the different mixes of products at the different activity levels, the points plotted for sales revenue and variable costs would not lie on a straight line.

##### Fixed, Variable and Marginal Costs

Introduction

Costs can be divided either into direct and indirect costs, or variable and fixed costs.

Direct costs are variable, that is the total cost varies in direct proportion to output.  If, for instance, it requires RWF10 worth of material to make one item it will require RWF20 worth to make two items and RWF100 worth to make ten items and so on.

Overhead costs, however, may be either fixed, variable or semi-variable.

##### Fixed Cost

A fixed cost is one which can vary with the passage of time but, within limits, tends to remain fixed irrespective of the variations in the level of output.  All fixed costs are overhead.  Examples of fixed overhead are: executive salaries, rent, rates and depreciation.

##### Variable Cost

This is a cost which tends to follow (in the short term) the level of activity in a business.

As already stated, direct costs are by their nature variable.  Examples of variable overhead are: repairs and maintenance of machinery; electric power used in the factory; consumable stores used in the factory.

##### Semi-Variable (or Semi-Fixed) Cost

This is a cost containing both fixed and variable elements, and which is thus partly affected by fluctuations in the level of activity.

For examination purposes, semi-variable costs usually have to be separated into their fixed and variable components.  This can be done if data is given for two different levels of output.

A Step Cost

Many items of cost are a fixed cost in nature within certain levels of activity.

Semi-Variable Costs

This is a cost containing both fixed and variable components and which is thus partly affected by fluctuations in the level of activity (CIMA official DFN).

##### Scattergraphs

Information about two variables that are considered to be related in some way can be plotted on a scattergraph. This is simply a graph on which historical data can be plotted. For cost behaviour analysis, the scattergraph would be used to record cost against output level for a large number of recorded “pairs” of data.

Then by plotting cost level against activity level on a scattergraph, the shape of the resulting figure might indicate whether or not a relationship exists.

In such a scattergraph, the y axis represents cost and the x axis represents the output or activity level.

One advantage of the scattergraph is that it is possible to see quite easily if the points indicate that a relationship exists between the variables, i.e. to see if any correlation exists between them.

Positive correlation exists where the values of the variables increase together (for example, when the volume of output increases, total costs increase).

##### Regression Analysis

This is a technically superior way to identify the “slope” of the line. It is also known as “Least Squares Regression”. This statistical method is used to predict a linear relationship between two variables. It uses all past data (not just the high and low points) to calculate the line of best fit.

##### The Linear Assumption of Cost Behaviour
• Cost are assumed to be either fixed, variable or semi-variable within a normal range of output.
• Fixed and variable costs can be estimated with degrees of probable accuracy. Certain methods maybe used to access this (High-Low method).
• Costs will rise in a straight line/linear fashion as the activity increases.
##### Factors Affecting the Activity Level
• The economic environment.
• The individual firm – its staff, their motivation and industrial relations.
• The ability and talent of management.
• The workforce (unskilled, semi-skilled and highly skilled).
• The capacity of machines.
• The availability of raw material.
##### Cost Behaviour and Decision Making

Factors to Consider:

• Future plans for the company.
• Current competition to the company.
• Should the selling price of a single unit be reduced in order to attract more customers.
• Should sale staff be on a fixed salary or on a basic wage with bonus/commission. 5) Is a new machine required for current year.

Will the company make the product internally or buy it.

For all of the above factors, management must estimate costs at all levels and evaluate different courses of action.  Management must take all eventualities into account when making decisions for the company.

Example of things management would need to know is fixed costs do not generally change as a result of a decision unless the company have to rent an additional building for a new job etc.

##### Cost Variability and Inflation

Care must be taken in interpreting cost data over a period of time if there is inflation.  It may appear that costs have risen relative to output, but this may be purely because of inflation rather than because the amount of resources used has increased.

If a cost index, such as the Retail Price Index, is available the effects of inflation can be eliminated and the true cost behaviour pattern revealed.

It is essential for the index selected to be relevant to the company; if one of the many Central Statistical Office indices is not appropriate, it may be possible for the company to construct one from its own data.

##### Calculus

Introduction

Calculus is the study of change.  It constitutes a major part of modern mathematics education.  It has two major branches:-

• Differential Calculus
• Integral Calculus

Differential calculus is the study of the definition, properties, and applications of the derivative of a function. The process of finding the derivative is called differentiation. Given a function and a point in the domain, the derivative at that point is a way of encoding the small-scale behavior of the function near that point. By finding the derivative of a function at every point in its domain, it is possible to produce a new function, called the derivative function or just the derivative of the original function. In mathematical terms the derivative is a linear operator which inputs a function and outputs a second function. This is more abstract than many of the processes studied in elementary algebra, where functions usually input a number and output another number. For example, if the doubling function is given the input three, then it outputs six, and if the squaring function is given the input three, then it outputs nine. The derivative, however, can take the squaring function as an input. This means that the derivative takes all the information of the squaring function—such as that two is sent to four, three is sent to nine, four is sent to sixteen, and so on—and uses this information to produce another function. (The function it produces turns out to be the doubling function.)

###### Integral calculus

Integral calculus is the study of the definitions, properties, and applications of two related concepts, the indefinite integral and the definite integral. The process of finding the value of an integral is called integration. Integral calculus studies two related linear operators.

The indefinite integral is the antiderivative, the inverse operation to the derivative. F is an indefinite integral of f when f is a derivative of F. (This use of lower- and upper-case letters for a function and its indefinite integral is common in calculus.)

The definite integral inputs a function and outputs a number, which gives the algebraic sum of areas between the graph of the input and the x-axis. The technical definition of the definite integral is the limit of a sum of areas of rectangles, called a Riemann sum.

A motivating example is the distances travelled in a given time.

Distance = Speed x Time

If the speed is constant, only multiplication is needed, but if the speed changes, then we need a more powerful method of finding the distance. One such method is to approximate the distance travelled by breaking up the time into many short intervals of time, then multiplying the time elapsed in each interval by one of the speeds in that interval, and then taking the sum (a Riemann sum) of the approximate distance travelled in each interval. The basic idea is that if only a short time elapses, then the speed will stay more or less the same. However, a Riemann sum only gives an approximation of the distance traveled. We must take the limit of all such Riemann sums to find the exact distance traveled.

###### Fundamental theorem

The fundamental theorem of calculus states that differentiation and integration are inverse operations. More precisely, it relates the values of antiderivatives to definite integrals. Because it is usually easier to compute an antiderivative than to apply the definition of a definite integral, the Fundamental Theorem of Calculus provides a practical way of computing definite integrals. It can also be interpreted as a precise statement of the fact that differentiation is the inverse of integration

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