Quantitative Methods for Business chapter 8 potx

Counting the cost – summarizing money variables over time8 Chapter objectives This chapter will help you to: ■ employ simple and aggregate index numbers to measure pricechanges over time

Counting the cost – summarizing money variables over time

8

Chapter objectives

This chapter will help you to:

■ employ simple and aggregate index numbers to measure pricechanges over time

■ work out weighted aggregate price indices: Laspeyre andPaasche indices

■ adjust figures for the effects of inflation using price indices

■ apply methods of investment appraisal: accounting rate ofreturn, payback period, net present value, and internal rate ofreturn

■ use the technology: investment appraisal methods in EXCEL

■ become acquainted with business uses of investment appraisal

In the last two chapters we have looked at ways of summarizing data InChapter 6 we concentrated on measuring the location and spread in uni-variate (single variable) data, in Chapter 7 we focused on measuringthe strength and direction in bivariate data In both chapters the dataconcerned were cross-sectional data, data relating to the same point orperiod of time In this chapter and the next we will consider ways of sum-marizing data relating to different periods of time

Time-based data consist of numerical observations that can be ured and summarized using the techniques you met in the previous twochapters We could, for instance, collect the price of gold at various points

meas-in time and calculate the mean price of gold over the period, or usecorrelation analysis to measure the association between the price of goldand the price of silver at various points in time However, often the mostimportant aspect of time-based data is the time factor and the tech-niques in the previous two chapters would not allow us to bring that out

of the data

In this chapter we will look at techniques to summarize money variablesthat relate to different time periods We will start by exploring indexnumbers and how they can be used to summarize the general movements

in prices over time Then we will look at how such price indices can beused to adjust money amounts for the effects of inflation Later in thechapter we will consider summarizing amounts of interest accumulatedover time and how this approach is used to assess the worth of investmentprojects

8.1 Index numbers

Data collected over time are very important for the successful ance of organizations For instance, such data can reveal trends in con-sumer expenditure and taste that companies need to follow

perform-Businesses use information based on data collected by other agenciesover time to help them understand and evaluate the environment inwhich they operate Perhaps the most important and widespread example

of this is the use of index numbers to monitor general trends in prices

and costs For instance, the Retail Price Index is used as a benchmarkfigure in the context of wage bargaining, and Share Price Indices arereference points in financial decisions companies face

Most businesses attach a great deal of importance to changes in thecosts of things they buy and the prices of things they sell During periods

of high inflation these changes are more dramatic; in periods of lowinflation they are modest Over recent decades, when the level of infla-tion has fluctuated, companies have had to track general price and costmovements carefully To help them do this they use index numbers.Index numbers can be used to represent movements over time in aseries of single figures A simple index number is the value of something

at one point in time, maybe the current value, in relation to its value atanother point in time, the base period, multiplied by 100 to give a per-centage (although the percent sign, %, is not usually written alongside it)

where pcrepresents the price in the current year and p0represent theprice in the base year (i.e period zero).

At this point you may find it useful to try Review Question 8.1 at the

end of the chapter

Since businesses usually buy and sell more than a single item, a simple

price index is of limited use Of much greater importance are aggregate

indices that summarize price movements of many items in a single figure

We can calculate a simple aggregate price index for a combination

of goods by taking the sum of the prices for the goods in the currentperiod and dividing it by the sum of the prices of the same goods in thebase period That is

Simple aggregate price index  Σ c * 100

Σ

p

p0

Simple price index current price

base period price * 100 * 100

c 0

p

Example 8.1

Full exhaust systems cost the Remont Repairs garage £156 each in 2003 They cost £125

in 2000 Calculate a simple price index to represent the change in price over theperiod

This tells us that the price of an exhaust system has increased by 24.8% over this period

Simple price index current price

base period price * 100 * 100156

125 * 100 124.8 to 1 decimal place

c 0

p p

At this point you may find it useful to try Review Questions 8.2 and 8.3

at the end of the chapter

The result we obtained in Example 8.2 may well be more usefulbecause it is an overall figure that includes all the commodities However,

it does not differentiate between prices of items that may be purchased

in greater quantity than other items, which implies that their prices are

of much greater significance than prices of less important items

In a simple aggregate price index each price is given equal inence, you can see that it appears once in the expression Its numerical

prom-‘clout’ depends simply on whether it is a large or small price In Example8.2, the result, 125.3, is close to the value of the simple price index ofthe exhaust system calculated in Example 8.1, 124.8 This is becausethe exhaust system happens to have the largest price in the set

In practice, the importance of the price of an item is a reflection of thequantity that is bought as well as the price itself To measure changes in

movements of prices in a more realistic way we need to weight each

price in proportion to the quantity purchased and calculate a weightedaggregate price index

There are two ways we can do this The first is to use the quantity figure

from the base year, represented by the symbol q0, to weight the price ofeach item This type of index is known as the Laspeyre price index Tocalculate it we need to work out the total cost of the base period quan-tities at current prices, divide that by the total cost of the base periodquantities at base period prices, and multiply the result by 100:

Laspeyre price index  Σ 0 c * 100

Σ

q p

q p0 0

Simple aggregate price index:

This result indicates that prices paid by the garage increased by 25.3% from 2000 to2003

The Laspeyre technique uses quantities that are historical The advantage

of this is that such figures are usually readily available The disadvantage

is that they may not accurately reflect the quantities used in the currentperiod

The alternative approach, which is more useful when quantities usedhave changed considerably, is to use quantity figures from the current

period, qc This type of index is known as the Paasche price index Tocalculate it you work out the total cost of the current period quantities

at current prices, divide that by the total cost of the current periodquantities at base period prices, and multiply the result by 100:

Paasche price index c c * 100

c

 ΣΣ

This suggests that the prices have increased by 21.6% between 2000 and 2003.

The result is lower than the figure obtained in Example 8.2, 125.3, because the exhaustsystem price has the lowest weighting and tyres, which have the lowest price change,have the highest weighting

44250 * 100 121.6 to 1 decimal place

ΣΣ

q p

q p

c c c

* 100 (50 * 156) (600 * 35) (750 * 32)

* 125 (600 * 25) (750 * 28) * 10052800

The advantage of using a Paasche price index is that the quantityfigures used are more up-to-date and therefore realistic But it is notalways possible to get current period quantity figures, particularly whenthere is a wide range of items and a large number of organizations orconsumers that buy them.

The other disadvantage of using the Paasche price index is that newquantity figures must be available for each period we want to comparewith the base period If the garage proprietor wants a Paashce price indexfor prices in 2004 compared to 2000 you could not provide one untilyou know both the quantities and the prices used in 2004 By contrast,

to calculate a Laspeyre price index for 2004 you only need to know theprices in 2004 because you would use quantities from 2000

If you look carefully at Example 8.3 and 8.4 you will see that whicheverindex is used the same quantity figures weight the prices from the dif-

ferent years This is an important point; they are price indices and they

are used to compare prices across the time period, not quantities

At this point you may find it useful to try Review Questions 8.4 to 8.7

at the end of the chapter

Organizations tend to use index numbers that have already beencompiled rather than construct their own Probably the most commonuse of index numbers that you will meet is in the adjustment of finan-cial amounts to take into account changes in price levels

A sum of money in one period is not necessarily the same as the sameamount in another period because its purchasing power changes Thismeans that if we want to compare an amount from one period with anamount from another period we have to make some adjustment forprice changes The most common way of doing this is to use the RetailPrice Index (RPI), an index the Government Statistical Service calcu-lates to monitor price changes, changes in the cost of living

This result suggests that the prices have increased by 25.0% between 2000 and 2003.The figure is higher than the result in Example 8.3 because there is a greater weighting

on the battery price, which has changed most, and a lower weighting on the tyre price,which has changed least

Example 8.5

The annual salary of the manager of the Zdorovy sports goods shop has changed in thefollowing way between 2000 and 2003 Use the RPI figures for those years to seewhether the increases in her salary have kept up with the cost of living

At this point you may find it useful to try Review Questions 8.8 to 8.11

at the end of the chapter

is purchased, yet the income which it is intended to help generatearises in the future, perhaps over many years

In this section we will look at techniques that enable managers toappraise, or weigh up, investment in long-lasting assets by relating theinitial outlay to the future revenue These techniques are used by busi-nesses both to assess specific investments and to decide between alter-native investments Companies take these decisions very seriously becausethey involve large amounts of resources and once made they cannot bereversed

We can ‘deflate’ the figures for 2001, 2002 and 2003 so that they are expressed in ‘2000pounds’ by multiplying each of them by the ratio between the RPI for 2000 and the RPIfor the year concerned

These results suggest that her salary has increased more than the cost of living out the period

through-Adjusted 2001 salary 29 * 170.3

173.3 28.498 i.e £28,498Adjusted 2002 salary 30 * 170.3

176.2 28.995 i.e £28,995Adjusted 2003 salary 33 * 170.3

We will begin with the accounting rate of return method then we willconsider the payback period approach, and finally the more sophisti-cated discounting techniques Despite the differences between themthey all involve the determination of single figures that summarize thefinancial appeal of an investment project.

8.2.1 The accounting rate of return

Generally, a rate of return expresses the return or profit resulting from

the use of assets such as machinery or equipment in terms of theexpenditure involved in purchasing them, usually in percentage terms.You will find that accountants make extensive use of these types of sum-mary measure; look at a business newspaper or a company report andyou will probably find reference to measures like the ROCE (Return

on Capital Employed) These measures are used by companies to cate how effectively they have managed the assets under their control.The accounting rate of return, often abbreviated to ARR, is the use

indi-of this approach to weigh up the attraction indi-of an investment proposal

To apply it we need to establish the average (mean) profit per year anddivide that by the average level of investment per year

To calculate the average profit per year we add up the annual profitsand divide by the number of years over which the investment will helpgenerate these revenues Having said that, the profit figures we use must

be profits after allowing for depreciation Depreciation is the spreading of

the cost of an asset over its useful life The simplest way of doing this is to

subtract the residual value of the asset, which is the amount that the

com-pany expects to get from the sale of the asset when it is no longer of use,from the purchase cost of the asset and divide by the number of years of

useful life the asset is expected to have This approach is known as

straight-line depreciation and it assumes that the usefulness of the asset, in terms of

helping to generate profits, is reasonably consistent over its useful life

To work out the average level of investment, we need to know the cost

of the asset and the residual value of the asset The average investmentvalue is the difference between the initial cost and the residual valuedivided by two, in other words we split the difference between the high-est and lowest values of the asset while it is in use After dividing theaverage return by the average investment we multiply by 100 so that wehave a percentage result The procedure can be represented as:

accounting rate of return average annual return

average annual investment * 100



What is the accounting rate of return for this investment?

The average annual profit before depreciation is:

From this amount we must subtract the annual cost of depreciation, which is:

The annual average profit after depreciation is: 27000 16000  £11000

The average annual investment is:

The accounting rate of return is:

of return for alternative investments that it could make with the same

money, or perhaps they have a company minimum rate that any projecthas to exceed to be approved.

The accounting rate of return is widely used to evaluate investmentprojects It produces a percentage figure which managers can easily com-pare to interest rates and it is essentially the same approach to futureinvestment as accountants take when working out the ROCE (Return

on Capital Employed) to evaluate a company’s past performance.The critical weakness in using the accounting rate of return toappraise investments is that it is completely blind to the timing of the

initial expenditure and future income It ignores what is called the time

value of money The value that an individual or business puts on a sum

of money is related to when the money is received; for example ifyou were offered the choice of a gift of £1000 now or £1000 in two year’s time you would most likely prefer the cash now This may bebecause you need cash now rather than then, but even if you have suf-ficient funds now you would still be better off having the money nowbecause you could invest the money in a savings account and receiveinterest on it

The other investment appraisal techniques we shall examine havethe advantage of bringing the time element into consideration The otherdifference between them and the accounting rate of return approach

is that they are based on net cash flows into the company, which areessentially net profits before depreciation

8.2.2 Payback period

The payback period approach to investment appraisal does take thetiming of cash flows into account and is based on a straightforwardconcept – the time it will take for the net profits earned using the asset

to cover the purchase of the asset We need only accumulate the tive (expenditure) and positive (net profits before depreciation) cashflows relating to the investment over time and ascertain when thecumulative cash flow reaches zero At this point the initial outlay onthe asset will have been paid back

nega-Example 8.7

Work out the payback period for the investment proposal being considered by theBudisha Bus Company in Example 8.6

Note that in the net cash flow column of the table in Example 8.7the initial outlay for the coach has a negative sign to indicate that it

is a flow of cash out of the business You will find that accountants use round brackets to indicate an outflow of cash, so where we havewritten120000 for the outgoing cash to buy the coach an accountantwould represent it as (120000)

The payback period we found in Example 8.7 might be comparedwith a minimum payback period the company required for any invest-ment or with alternative investments that could be made with the sameresources

At this point you may find it useful to try Review Questions 8.12 at

the end of the chapter

The payback period is a simple concept for managers to apply and it

is particularly appropriate when firms are very sensitive to risk because

it indicates the time during which they are exposed to the risk of notrecouping their initial outlay A cautious manager would probably becomfortable with the idea of preferring investment opportunities thathave shorter payback periods

The weakness of the payback approach is that it ignores cash flowsthat arise in periods beyond the payback period Where there are twoalternative projects it may not suggest the one that performs betteroverall

The net cash flows associated with the acquisition of the luxury coach can be marized as follows:

sum-Payback is achieved in year 5 We can be more precise by adding the extra cash flowrequired after the end of year four to reach zero cumulative cash flow (£5000) divided

by the net cash flow received by the end of the fifth year (£20,000):

In Example 8.8 the payback period for the Smeshnoy machine is fouryears and for the Pazorna machine three years Applying the paybackperiod criterion we should choose the Pazorna machine, but in doing

so we would be passing up the opportunity of achieving the rather higherreturns from investing in the Smeshnoy machine

A better approach would be to base our assessment of investments

on all of the cash flows involved rather than just the earlier ones, and

to bring into our calculations the time value of money Techniques that

allow us to do this adjust or discount cash flows to compensate for the time

that passes before they arrive The first of these techniques that weshall consider is the net present value

Example 8.8

Gravura Print specialize in precision graphics for the art poster market To expandtheir business they want to purchase a flying-arm stamper There are two manufacturersthat produce such machines: Smeshnoy and Pazorna The cash flows arising from thetwo ventures are expected to be as follows:

Smeshnoy machine

Pazorna machine

End of year Cost/receipt Net cash flow (£) Cumulative cash flow (£)

End of year Cost/receipt Net cash flow (£) Cumulative cash flow (£)

8.2.3 Net present value

The net present value (NPV) of an investment is a single figure thatsummarizes all the cash flows arising from an investment, bothexpenditure and receipts, each of which have been adjusted so that

whenever they arise in the future it is their current or present value that

is used in the calculation Adjusting, or discounting them to get their

present value means working out how much money would have to beinvested now in order to generate that specific amount at that time inthe future

To do this we use the same approach as we would to calculate theamount of money accumulating in a savings account We need to knowthe rate of interest and the amount of money initially deposited Theamount in the account at the end of one year is the original amount

deposited plus the rate of interest, r, applied to the original amount:

Amount at the end of the year Deposit  (Deposit * r)

We can express this as:

Amount at the end of the year Deposit * (1  r)

If the money stays in the account for a second year:

Amount at the end of the second year Deposit * (1  r) * (1  r)

 Deposit * (1  r)2

Example 8.9

If you invested £1000 in a savings account paying 5% interest per annum, how muchmoney would you have in the account after two years?

Amount at the end of the first year 1000 * (1  0.05)  £1050

If we invested £1050 for a year at 5%, at the end of one year it would be worth:

1050 * (1 0.05)  £1102.5

We can combine these calculations:

Amount at the end of the second year 1000 * (1  0.05)2

 1000 * 1.052 1000 * 1.1025  £1102.5

In general if we deposit an amount in an account paying an annual

interest rate r for n years, the amount accumulated in the account at

the end of the period will be:

Deposit * (1 r) n

The deposit is, of course, the sum of money we start with, it is the

pre-sent value (PV) of our investment, so we can express this procedure as:

Amount at the end of year n  PV * (1  r) n

This expression enables us to work out the future value of a knownpresent value, like the amount we deposit in an account When we assessinvestment projects we want to know how much a known (or at leastexpected) amount to be received in the future is worth now Instead ofknowing the present value and wanting to work out the future, we need

to reverse the process and determine the present value of a knownfuture amount To obtain this we can rearrange the expression we used

to work out the amount accumulated at the end of a period:

Present value (PV) Amount at the end of year

The present value of £1000 received in two years’ time is £907.03, to the nearest penny

In other words, if you invested £907.03 at 5% now in two years’ time the amount would

be worth

Amount at the end of year two 907.03 * (1  0.05)2 907.03 * 1.1025

 £1000.00 to the nearest penny

Present value 1000

(1 0.05)

10001.05

10001.1025 907.029

of interest to use In Example 8.10 we used 5% as it was a viable

alterna-tive that in effect reflected the opportunity cost of not receiving the money

for two years, that is, the amount you have had to forego by having to wait

The interest, or discount, rate a company uses is likely to reflect the

opportunity cost, which may be the interest it could earn by investingthe money in a bank It may also reflect the prevailing rate of inflationand the risk of the investment project not working out as planned

The net present value of the project in Example 8.11 is £8936 The tial outlay of £120,000 in effect purchases future returns that are worth

ini-£128,936 Because the discount rate used is in effect a threshold ofacceptable returns from a project, any opportunity that results in a pos-itive NPV such as in Example 8.11 should be approved and any oppor-tunity producing a negative NPV should be declined

The calculation of present values of a series of cash flows is an

ardu-ous process, so it is easier to use discount tables, tables that give values of

the discount factor, 1/(1 r) n for different values of r and n You can

find discount tables in Table 1 on page 617

Example 8.11

What is the net present value of the proposed investment in a luxury coach by theBudisha Bus Company in Example 8.6? Use a 10% interest rate

The cash flows involved in the project were:

End of year Cash flow (£) Calculation for PV PV (to the nearest £)

Purchase the Smeshnoy machine

End of year Cash flow (£) Discount factor PV (Cash flow * discount factor)