Basic Mathematics for Economists - Rosser - Chapter 7 pps

7 Financial mathematicsSeries, time and investment Learning objectives After completing this chapter students should be able to: • Calculate the final sum, the initial sum, the time perio

7 Financial mathematics

Series, time and investment

Learning objectives

After completing this chapter students should be able to:

• Calculate the final sum, the initial sum, the time period and the interest rate for aninvestment

• Calculate the Annual Equivalent Rate for part year investments and compare thiswith the nominal annual rate of return

• Calculate the Net Present Value and Internal Rate of Return on an investment,constructing relevant spreadsheets when required

• Use the appropriate investment appraisal method to decide if an investment project

is worthwhile

• Find the sum of finite and infinite geometric series

• Calculate the value of an annuity

• Calculate monthly repayments and the APR for a loan

• Apply appropriate mathematical methods to solve problems involving the growthand decline over discrete time periods of other economic variables, including thedepletion of natural resources

7.1 Discrete and continuous growth

In economics we come across many variables that grow, or decline, over time A sum of moneyinvested in a deposit account will grow as interest accumulates on it The amount of oil left

in an oilfield will decline as production continues over the years This chapter explains howmathematics can help answer certain problems concerned with these variables that changeover time The main area of application is finance, including methods of appraising differentforms of investment Other applications include the management of natural resources, wherethe implications of different depletion rates are analysed

The interest earned on money invested in a deposit account is normally paid at set regularintervals Calculations of the return are therefore usually made with respect to specific timeintervals For example,Figure 7.1(a)shows the amount of money in a deposit account at anygiven moment in time assuming an initial deposit of £1,000 and interest credited at the end ofeach year at a rate of 10% There is not a continuous relationship between time and the totalsum in the deposit account Instead there is a ‘jump’ at the end of each year when the interest

(a) Deposit account balance

A discrete function can therefore be defined as one where the value of the dependent

variable is known for specific values of the independent variable but does not continuouslychange between these values Hence one gets a series of values rather than a continuum.For example, teachers’ salaries are based on scales with series of increments A hypotheticalscale linking completed years of service to salary might be:

0 yrs= £20,000, 1 yr = £21,800, 2 yrs = £23,600, 3 yrs = £25,400

The relationship between salary and years of service is a discrete function At any moment

in time one knows what a teacher’s salary will be but there is not a continuous relationshipbetween time and salary level

An example of a continuous function is illustrated in Figure 7.1(b) This shows the lative total amount of oil extracted from an oilfield when there is a steady 5 million barrels peryear extraction rate There is a continuous smooth function showing the relationship betweenthe amount of oil extracted and the time elapsed

cumu-In this chapter we analyse a number of discrete-variable problems Algebraic formulaeare developed to solve some applications of discrete functions and methods of solution using

spreadsheets are explained where appropriate, for investment appraisal analysis in particular.The analysis of continuous growth requires the use of the exponential function and will beexplained inChapter 14.

7.2 Interest

Time is money If you borrow money you have to pay interest on it If you invest money in

a deposit account you expect to earn interest on it From an investor’s viewpoint the interestrate can be looked on as the ‘opportunity cost of capital’ If a sum of money is tied up in

a project for a year then the investor loses the interest that could have been earned by investingthe money elsewhere, perhaps by putting it in a deposit account

Simple interest is the interest that accrues on a given sum in a set time period It is not

reinvested along with the original capital The amount of interest earned on a given investmenteach time period will be the same (if interest rates do not change) as the total amount of capitalinvested remains unaltered

Example 7.1

An investor puts £20,000 into a deposit account and has the annual interest paid directly into

a separate current account and then spends it The deposit account pays 8.5% interest Howmuch interest is earned in the fifth year?

If you can remember that 1%= 0.01 then you should be able to transform any interest ratespecified in percentage terms into a decimal fraction in your head Try to do this for thefollowing interest rates:

(i) 1.5% (ii) 30% (iii) 0.075% (iv) 1.02% (v) 0.6%

Now check your answers with a calculator If you got any wrong you really ought to go backand revise Section 2.5 before proceeding Converting decimal fractions back to percentageinterest rates is, of course, simply a matter of multiplying by 100;

e.g 0.02= 2%, 0.4 = 40%, 1.25 = 125%, 0.008 = 0.8%

Compound interest is interest which is added to the original investment every time it

accrues The interest added in one time period will itself earn interest in the following timeperiod The total value of an investment will therefore grow over time

Interest at end of year 1= 0.08 × 600 48.00

Total sum invested for year 2 648.00

Interest at end of year 2= 0.08 × 648 51.84

Total sum invested for year 3 699.84

Interest at end of year 3= 0.08 × 699.84 55.99

Final value of investment 755.83

The above examples only involved the calculation of interest for a few years and did nottake too long to solve from first principles To work out the final sum of an investment afterlonger time periods one could construct a spreadsheet, but an even quicker method is to usethe formula explained below.

Calculating the final value of an investment

Consider an investment at compound interest where:

A is the initial sum invested,

F is the final value of the investment,

i is the interest rate per time period (as a decimal fraction) and

n is the number of time periods.

The value of the investment at the end of each year will be 1+ i times the sum invested

at the start of the year For instance, the £648 at the start of year 2 is 1.08 times the initialinvestment of £600 in Example 7.3 above The value of the investment at the start of year 3

is 1.08 times the value at the start of year 2, and so on Thus, for any investment

Value after 1 year= A(1 + i)

Value after 2 years= A(1 + i)(1 + i) = A(1 + i)2

Value after 3 years= A(1 + i)2

(Refer back toChapter 2, Section 8 if you cannot remember how to use the [y x] function key

on your calculator to work out large powers of numbers.)

Sometimes a compound interest problem may be specified in a rather different format, butthe method of solution is still the same

Example 7.6

You estimate that you will need £8,000 in 3 years’ time to buy a new car, assuming

a reasonable trade-in price for your old car You have £7,000 which you can put into

a fixed interest building society account earning 4.5% Will you have enough to buythe car?

Solution

You need to work out the final value of your savings to see whether it will be greater than

£8,000 Using the usual notation,

A= £7,000 n= 3 i= 0.045

F = A(1 + i) n= 7,000(1.045)3= 7,000(1.141166) = £7,988.16

So the answer is ‘almost’ You will have to find another £12 to get to £8,000, but perhapsyou can get the dealer to knock this off the price.

Changes in interest rates

What if interest rates are expected to change before the end of the investment period? Thefinal sum can be calculated by slightly adjusting the usual formula

Example 7.7

Interest rates are expected to be 14% for the next 2 years and then fall to 10% for the following

3 years How much will £2,000 be worth if it is invested for 5 years?

Test Yourself, Exercise 7.1

1 If £4,000 is invested at 5% interest for 3 years what will the final sum be?

2 How much will £200 invested at 12% be worth at the end of 4 years?

3 A parent invests £6,000 for a 7-year-old child in a fixed interest scheme whichguarantees 8% interest How much will the child have at the age of 21?

4 If £525 is invested in a deposit account that pays 6% interest for 6 years, what willthe final sum be?

5 What will £24,000 invested at 11% be worth at the end of 5 years?

6 Interest rates are expected to be 10% for the next 3 years and then to fall to 8%for the following 3 years How much will an investment of £3,000 be worth at theend of 6 years?

7.3 Part year investment and the annual equivalent rate

If the duration of an investment is less than a year the usual final sum formula does notalways apply It is usually the custom to specify interest rates on an annual basis for part yearinvestments, but two different types of annual interest rates can be used:

(a) the nominal annual interest rate, and

(b) the Annual Equivalent Rate (AER)

The ways that these annual interest rates relate to part year investments differ They are alsoused in different circumstances

Nominal annual interest rates

For large institutional investors on the money markets, and for some forms of individualsavings accounts, a nominal annual interest rate is quoted for part year investments To findthe interest that will actually be paid, this nominal annual rate is multiplied by the fraction

of the year that it is quoted for

Example 7.9

What interest is payable on a £100,000 investment for 6 months at a nominal annual interestrate of 6%?

Solution

6 months is 0.5 of one year and so the interest rate that applies is

0.5× (nominal annual rate) = 0.5 × 6% = 3%

Therefore interest earned is

3% of £100,000= £3,000

and the final sum is

F= (1.03)100,000 = £103,000

If this nominal annual interest rate of 6% applied to a 3-month investment then the actualinterest payable would be a quarter of 6% which is 1.5% If it applied to an investment forone month then the interest payable would be 6% divided by 12 which gives 0.5%

The calculation of part year interest payments on this basis can actually give investors

a total annual return that is greater than the nominal interest rate if they can keep reinvestingthrough the year at the same part year interest rate The total final value of the investment

can be calculated with reference to these new time periods using the F = A(1 + i) nformula

as long as the interest rate i and the number of time periods n refer to the same time periods.

For example, if £100,000 can be invested for four successive three month periods at

a nominal annual interest rate of 6% then, letting i represent the effective quarterly interest rate and n represent the number of three month periods, we get

A= £100,000 n= 4 i= 0.25 × 6% = 1.5% = 0.015

F = A(1 + i) n= 100,000(1.015)4= £106,136.35

This final sum gives a 6.13635% return on the initial £100,000 sum invested (Although inpractice interest rates are usually only specified to 2 dp.)

The more frequently that interest based on the nominal annual rate is paid the greater will

be the total annual return when all the interest is compounded For example, if a nominalannual interest rate of 6% is paid monthly at 0.5% a month and £100,000 is invested for

The Annual Equivalent Rate (AER) and Annual Percentage Rate (APR)

Although some part year investments on money markets may earn a return which is notequivalent to the nominal annual interest rate, individual investors are usually quoted anannual equivalent rate (AER) which is an accurate reflection of the interest that they earn

on investments For example, interest on the money you may have in a building society willnormally be worked out on a daily basis although you will only be told the AER and theinterest on your account may only be credited once a year For loan repayments the annualequivalent rate is usually referred to as the annual percentage rate (APR) If you take out

a bank loan you will usually be quoted an APR even though you will be asked to makemonthly repayments

The examples above have already demonstrated that the AER is not simply 12 times themonthly interest rate To determine the relationship between part year interest rates and theirtrue AER, consider another example

Example 7.10

If interest is credited monthly at a monthly rate of 0.9% how much will £100 invested for

12 months accumulate to?

Because (1+ im)12gives the ratio of the final sum F to the initial amount A the−1 has to be

added to the formula in order to get the proportional increase in F over A The APR on loans

is the same thing as the annual equivalent rate and so the same formula applies

inter-LloydsTSB Trustcard had an interest rate per month of 1.527% and quoted the APR as 19.9%.

We can check this using the formula

is much more useful to savers to help them make comparisons between different possible

investment opportunities The relationship between the daily interest rate id on a depositaccount and the AER can be formulated as

AER= (1 + id)365− 1

For example, if a building society tells you that it will pay you an AER of 6% on a savingsaccount, what it actually will do is credit interest at a rate of 0.015954% a day We can checkthis out using the formula

A building society account pays interest on a daily basis at an AER of 4.5% If you deposited

£2,750 in such an account on 1st October how much would you get back if you closed theaccount 254 days later?

Interest rates on Treasury Bills

A government Treasury Bill, like certain other forms of bond, guarantees the owner a fixedsome of money payable at a fixed date in the future So, for example, a 3-month Treasury Billfor £100,000 is effectively a promise from the government that it will pay £100,000 to theowner on a date 3 months from when it was issued The prices that the institutional investorswho trade in these bills will pay for them will reflect the returns that can be made on othersimilar investments

Suppose that investors are currently willing to pay £95,000 for 12-month Treasury Billswhen they are issued This would mean that they consider an annual return of £5,000 on their

Although the above example considered a 12-month Treasury Bill so that the equivalentannual rate of return could be easily compared, in practice UK government Treasury Billsare normally issued for shorter periods Also, the nominal annual rates are quoted usingfractions, such as 4165%, rather than in decimal format

Example 7.13

If an annual discount rate of 478% is quoted for 3-month Treasury Bills, what would it cost

to buy a tranch of these bills with redemption value of £100,000? What would be the annualequivalent rate of return on the sum paid for them?

and the amount of the discount is £1,218.75.

Therefore, the rate of return on the sum of £98,781.25 invested for 3 months is

Test Yourself, Exercise 7.2

1 If £40,000 is invested at a monthly rate of 1% what will it be worth after 9 months?What is the corresponding AER?

2 A sum of £450,000 is invested at a monthly interest rate of 0.6% What will thefinal sum be after 18 months? What is the corresponding AER?

3 Which is the better investment for someone wishing to invest a sum of money fortwo years:

(a) an account which pays 0.9% monthly, or

(b) an account which pays 11% annually?

4 If £1,600 is invested at a quarterly rate of interest of 4.5% what will the final sum

be after 18 months? What is the corresponding AER?

5 How much interest is earned on £50,000 invested for three months at a nominalannual interest rate of 5%? If money can be reinvested each quarter at the samerate, what is the AER?

6 If a credit card company charges 1.48% a month on any outstanding balance, whatAPR is it charging?

7 A building society pays an AER of 5.5% on an investment account, calculated on

a daily basis What daily rate of interest will it pay?

8 If 3-month government Treasury Bills are offered at an annual discount rate of

4167%, what would it cost to buy bills with redemption value of £500,000? Whatwould the AER be for this investment?

7.4 Time periods, initial amounts and interest rates

The formula for the final sum of an investment contains the four variables F, A, i and n.

So far we have only calculated F for given values of A, i and n However, if the values

of any three of the variables in this equation are given then one can usually calculate thefourth

Initial amount

A formula to calculate A, when values for F, i and n are given, can be derived as follows.

Since the final sum formula is

How much money needs to be invested now in order to accumulate a final sum of £12,000

in 4 years’ time at an annual rate of interest of 10%?

What we have actually done in the above example is find the sum of money that is equivalent

to £12,000 in 4 years’ time if interest rates are 10% An investor would therefore be indifferentbetween (a) £8,196.16 now and (b) £12,000 in 4 years’ time The £8,196.16 is therefore known

as the ‘present value’ (PV) of the £12,000 in 4 years’ time We shall come back to this concept

in the next few sections when methods of appraising different types of investment project areexplained

Time period

Calculating the time period is rather more tricky than the calculation of the initial amount.From the final sum formula

F = A(1 + i) n

F

A = (1 + i) n

If the values of F, A and i are given and one is trying to find n this means that one has to

work out to what power (1+ i) has to be raised to equal F/A One way of doing this is via

since to find the nth power of a number its logarithm must be multiplied by n Finding logs,

this means that (2) becomes

An alternative approach is to use the iterative method and plot different values on

a spreadsheet To find the value of n for which

1.6= (1.1)n

this entails setting up a formula to calculate the function y = (1.1)n and then computing

it for different values of n until the answer 1.6 is reached Although some students who

find it difficult to use logarithms will prefer to use a spreadsheet, logarithms are used in theother examples in this section Logarithms are needed to analyse other concepts related toinvestment and so you really need to understand how to use them

If £4,000 invested for 10 years is projected to accumulate to £6,000, what interest rate is used

to derive this forecast?

A general formula for calculating the interest rate can be derived Starting with the familiarfinal sum formula

Substituting these values into the interest rate formula gives

Test Yourself, Exercise 7.3

1 How much needs to be invested now in order to accumulate £10,000 in 6 years’time if the interest rate is 8%?

2 What sum invested now will be worth £500 in 3 years’ time if it earns interest at12%?

3 Do you need to invest more than £10,000 now if you wish to have £65,000 in

15 years’ time and you have a deposit account which guarantees 14%?

4 You need to have £7,500 on 1 January next year How much do you need to invest

at 1.3% per month if your investment is made on 1 June?

5 How much do you need to invest now in order to earn £25,000 in 10 years’ time

if the interest rate is

10 How long will £70,000 take to accumulate to £100,000 if it is invested at 11%?

11 If £6,000 is to accumulate to £10,000 after being invested for 5 years, what ratemust it earn interest at?

12 What interest rate will turn £50,000 into £60,000 after 2 years?

13 At what interest rate will £3,000 accumulate to £4,000 after 4 years?

14 What monthly rate of interest must be paid on a sum of £2,800 if it is to accumulate

to £3,000 after 8 months?

15 What rate of interest would turn £3,000 into £8,000 in 10 years?

16 At what rate of interest will £600 accumulate to £900 in 5 years?

17 Would you prefer (a) £5,000 now or (b) £8,000 in 4 years’ time if money can beborrowed or lent at 11%?

7.5Investment appraisal: net present value

Assume that you have £10,000 to invest and that someone offers you the following proposal:pay £10,000 now and get £11,000 back in 12 months’ time Assume that the returns on thisinvestment are guaranteed and there are no other costs involved What would you do? Perhaps

you would compare this return of 10% with the rate of interest your money could earn in

a deposit account, say 4% In a simple example like this the comparison of rates of return,known as the internal rate of return (IRR) method, is perhaps the most intuitively obviousmethod of judging the proposal

This is not the preferred method for investment appraisal, however The net present value(NPV) method has several advantages over the IRR method of comparing the project rate

of return with the market interest rate These advantages are explained more fully in thefollowing section, but first it is necessary to understand what the NPV method involves

We have already come across the concept of present value (PV) in Section 7.4 If a certainsum of money will be paid to you at some given time in the future its PV is the amount ofmoney that would accumulate to this sum if it was invested now at the ruling rate of interest

An investor would be indifferent between £1,333.49 now and £1,500 in 3 years’ time Thus

£1,333.49 is the PV of £1,500 in 3 years’ time at 4% interest

In all the examples in this chapter it is assumed that future returns are assured with 100%certainty Of course, in reality some people may place greater importance on earlier returnsjust because the future is thought to be more risky If some form of measure of the degree

of risk can be estimated then more advanced mathematical methods exist which can be used

to adjust the investment appraisal methods explained in this chapter However, here we justassume that estimated future returns and costs, are correct An investor has to try to make themost rational decision based on whatever information is available

The net present value (NPV) of an investment project is defined as the PV of the future

returns minus the cost of setting up the project

Example 7.22

An investment project involves an initial outlay of £600 now and a return of £1,000 in 5 years’time Money can be invested at 9% What is the NPV?

Another way of looking at the situation is to consider what alternative sum could be earned

by the investor’s £600 If £649.93 was invested for 5 years at 9% it would accumulate to

£1,000 Therefore the lesser sum of £600 must obviously accumulate to a smaller sum Usingthe final sum investment formula this can be calculated as

F = A(1 + i) n= 600(1.09)5= 600(1.538624) = £923.17

The investor thus has the choice of

(a) putting £600 into this investment project and securing £1,000 in 5 years’ time, or(b) investing £600 at 9%, accumulating £923.17 in 5 years

Option (a) is clearly the winner

If the outlay is less than the PV of the future return an investment must be a profitable

ven-ture The basic criterion for deciding whether or not an investment project is worthwhile

is therefore

NPV > 0

As well as deciding whether specific projects are profitable or not, an investor may have to

decide how to allocate limited capital resources to competing investment projects The rule for choosing between projects is that they should be ranked according to their NPV If only

one out of a set of possible projects can be undertaken then the one with the largest NPVshould be chosen, as long as its NPV is positive

Example 7.23

An investor can put money into any one of the following three ventures:

Project A costs £2,000 now and pays back £3,000 in 4 years

Project B costs £2,000 now and pays back £4,000 in 6 years

Project C costs £3,000 now and pays back £4,800 in 5 years

The current interest rate is 10% Which project should be chosen?

NPV of project A= 3,000(1.1)−4− 2,000

= 2,049.04 − 2,000 = £49.04NPV of project B= 4,000(1.1)−6− 2,000

= 2,257.90 − 2,000 = £257.90NPV of project C= 4,800(1.1)−5− 3,000

= 2,980.42 − 3,000 = −£19.58Project B has the largest NPV and is therefore the best investment Project C has a negativeNPV and so would not be worthwhile even if there was no competition

The investment examples considered so far have only involved a single return payment atsome given time in the future However, most real investment projects involve a stream ofreturns occurring over several time periods The same principle for calculating NPV is used

to assess these projects, the initial outlay being subtracted from the sum of the PVs of thedifferent future returns

Example 7.24

An investment proposal involves an initial payment now of £40,000 and then returns of

£10,000, £30,000 and £20,000 respectively in 1, 2 and 3 years’ time If money can beinvested at 10% is this a worthwhile investment?

This NPV is greater than zero and so the project is worthwhile At an interest rate of 10% onewould need to invest a total of £48,910.60 to get back the projected returns and so £40,000

is clearly a bargain price

The further into the future the expected return occurs the greater will be the discountingfactor This is made obvious in Example 7.25 below, where the returns are the same eachtime period The PV of each successive year’s return is smaller than that of the previous yearbecause it is multiplied by (1+ i)−1.

Example 7.25

An investment project requires an initial outlay of £7,500 and will pay back £2,000

at the end of the next 5 years Is it worthwhile if capital can be invested elsewhere

The NPV < 0 and so this is not a worthwhile investment.

Investment appraisal using a spreadsheet

From the above examples one can see that the mathematics involved in calculating the NPV

of a project can be quite time-consuming For this type of problem a spreadsheet programcan be a great help Although Excel has a built in NPV formula, this does not take the initialoutlay into account and so care has to be taken when using it We shall therefore construct

a spreadsheet to calculate NPV from first principles

To derive an algebraic formula for calculating NPV assume that Rj is the net return in

year j, i is the given rate of interest, n is the number of time periods in which returns occur and C is the initial cost of the project Then

NPV= R1

1+ i+

R2(1+ i)2 + · · · + R n

(1+ i) n − C Using the  notation this becomes

If the initial outlay C is considered as a negative return at time 0 (i.e R0= −C) the formula

can be more neatly stated as

since (1+ i)0= 1 (Remember that x0= 1 whatever the value of x.)

The following example shows how an Excel spreadsheet program based on this formulacan be used to work out the NPV of a project The answer obtained is then compared withthe solution using the Excel built in NPV function

Example 7.26

An investment project requires an initial outlay of £25,000 with the following expectedreturns:

£5,000 at the end of year 1

£6,000 at the end of year 2

£10,000 at the end of year 3

£10,000 at the end of year 4

£10,000 at the end of year 5

Is this a viable investment if money can be invested elsewhere at 15%?

Solution

Follow the instructions for creating an Excel spreadsheet set out inTable 7.1, which shouldgive you the spreadsheet inTable 7.2 This calculates the PVs of the returns in each year sep-arately, including the outlay in year 0 It then sums the PVs, giving a total NPV of £1,149.15which is positive and hence means that the project is a viable investment opportunity.This can be compared with the answer obtained using the Excel built-in NPV formula.Because this formula always treats the number in the first cell of the range as the return at theend of year 1, the computed answer of £26,149.15 is the total PV of the returns in years 1 to

5 only To get the overall NPV of the project one has to subtract the initial outlay (The outlayamount was entered as a negative quantity and so this is actually added in the formula.) Thisadjusted Excel NPV figure should be the same as the NPV calculated from first principles,which it is Having an answer computed by two separate methods is a useful check If yousave this spreadsheet and adapt it for other problems then, if you do not get the same answerfrom both methods, you will know that a mistake has been made somewhere

The spreadsheet created for the above example can be used to work out the NPV for otherprojects The initial cost and returns need to be entered in cells B4 to B9 and the new interestrate goes in cell D2 Obviously if there are more (or less) years when returns occur then rowswill need to be added (or deleted or left blank)

As investment appraisal involves the comparison of different projects, as well as the ment of the financial viability of individual projects, a spreadsheet can be adapted to work

assess-Table 7.1

A1 Ex.7.26 Label to remind you what example this is

C1 Interest rate = Label to tell you interest rate goes in next cell.D1 15% Value of interest rate (NB Excel automatically

treats this % format as 0.15 in any calculations.)A4 to A9 Enter numbers 0 to 5 These are the time periods

B4 -25000 Initial outlay (negative because it is a cost)

Returns at end of years 1 to 5

C4 =B4/(1+$D$1)^A4 Formula calculates PV corresponding to return

in cell B4, time period in cell A4 and interestrate in cell D1 Note the $ to anchor cell D1 C5 to C9 Copy cell C4 formula

down column C

Calculates PV for return in each time period.Format to 2 d.p as monetary valuesB11 NPV = Label to tell you NPV goes in next cell.C11 =SUM(C4:C9) Calculates NPV of project by summing PVs for

each year in cells C4-C9, which includes the negative return of the initial outlay

B13 Excel NPV Label tells you Excel NPV goes in next cell.B14 less cost = Label tells you what goes in next cell

C13 =NPV(D1,B5:B9) The Excel NPV formula will calculate NPV

based only on the interest rate in D1 and the 5 years of future returns in cells B5 to B9 C14 =C13+B4 Adjusts the Excel computed NPV in C13 by

subtracting initial outlay in B4 (This was entered as a negative number so it is added.)

out the NPV for more than one project The following example shows how the spreadsheetcreated for Example 7.26 can be extended so that two projects can be compared.

Example 7.27

An investor has to choose between two projects A and B whose outlay and returns are set out

in Table 7.3 Which is the better investment if the going rate of interest is 10%?

Table 7.3

(All values in £) Project AProject B

Return in 1 year’s time 6,000 8,000

Return in 2 years’ time 10,000 8,000

Return in 3 years’ time 10,000 8,000

Return in 4 years’ time 10,000 8,000

Return in 5 years’ time 8,000 8,000

Table 7.4

B3 PROJECT A Changed column heading label

D3 PROJECT B New column heading label for project B returns

E3 PV B New column heading label for project B PVs

Project A returns at end of years 1 to 5

Project B returns at end of years 1 to 5

E4 =D4/(1+$D$1)^A4 Formula calculates PV for project B

corresponding to return in cell D4

E5 to E9 Copy cell E4 formula

E14 =E13+D4 Adjusts the Excel NPV for project B

If you do not have access to a spreadsheet program then you can still work out the NPV ofdifferent projects from first principles However, there are now available financial calculatorswith an NPV function which may be a cheaper alternative than a computer To assist studentswithout a spreadsheet program or a financial calculator, a set of discounting factors is repro-duced inTable 7.6 Although the actual monetary returns will differ from project to projectthe discounting factor will be the same for a given time period and a given rate of interest.

For example, the PV of a sum of money £x payable in 8 years’ time when the interest rate is

7% will be

£x

(1.07)8 or £x(1.07)−8

The value of (1.07)−8can be read off from Table 7.6 by looking at the column headed 7%

and the row corresponding to year 8, giving a figure of 0.582009 If £x was £525 then the

Table 7.6 Discounting factors for Net Present Value

Test Yourself, Exercise 7.4

1 The following investment projects all involve an outlay now and a single return

at some point in the future Calculate the NPV and say whether or not each is aworthwhile investment:

(a) £1,100 outlay, £1,500 return after 3 years, interest rate 8%

(b) £750 outlay, £1,000 return after 5 years, interest rate 9%

(c) £10,000 outlay, £12,000 return after 3 years, interest rate 8%

(d) £50,000 outlay, £75,000 return after 3 years, interest rate 14%

(e) £50,000 outlay, £100,000 return after 5 years, interest rate 14%

(f) £5,000 outlay, £7,000 return after 3 years, interest rate 6%

(g) £5,000 outlay, £7,750 return after 5 years, interest rate 6%

(h) £5,000 outlay, £8,500 return after 6 years, interest rate 6%

2 An investor has to choose between the following three projects:

Project A requires an outlay of £35,000 and returns £60,000 after 4 yearsProject B requires an outlay of £40,000 and returns £75,000 after 5 yearsProject C requires an outlay of £25,000 and returns £50,000 after 6 years

Which project would you advise this investor to put money into if the cost ofcapital is 10%?

3 A firm has a choice between three investment projects, all of which involve aninitial outlay of £36,000 The returns at the end of the next 4 years are given inTable 7.7 If the interest rate is 15%, say (a) whether each project is viable or not,and (b) which is the best investment

All values are given in £.

4 If money can be invested elsewhere at 6%, is the following project worthwhile?Initial outlay £100,000

Return at end of year 1 £10,000

Return at end of year 2 £12,000

Return at end of year 3 £15,000

Return at end of year 4 £18,000

Return at end of year 5 £20,000

Return at end of year 6 £20,000