chapter 10: comparing monetary returns over time

Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE

ISBN 1-86152-991-0 © Thomson Learning 2004


ISBN 1-86152-991-0 © Cengage

Chapter 10: Comparing Monetary Returns

Over Time




ISBN 1-86152-991-0 © Cengage

Objectives

• Find the payback time for a project

• Understand the concept of time value of money

• Calculate a net present value for a project

• Criticise the process

• Discuss the selection of discount factors




ISBN 1-86152-991-0 © Cengage

Costs & Benefits

• Costs usually come at the beginning of a project

• Their level is often known with some certainty

• Benefits come over some future time period

• They are open to considerable variation

• They are also uncertain




ISBN 1-86152-991-0 © Cengage

PaybackThis method only takes into account how long it will take to pay back the initial investment in nominal termsIf we invest £1,000

and get back £500 in year 1 and £500 in year 2

Then it takes 2 years to payback

If the money back were £750 in year 1And £750 in year 2

Then it would take one and a third years to pay back




ISBN 1-86152-991-0 © Cengage

Interest Calculations

Interest is earned on a sum of money invested over a period of timeThe amount of interest depends upon the interest rate and the time period,But also on the method of interest accumulation

SIMPLE INTEREST: Here the same amount is earned each yearSo if you invest £100 at 10% you get

£10 interest in year 1£10 interest in year 2

and so on ………..So the total interest is(the amount)x(interest rate)x(number of years)




ISBN 1-86152-991-0 © Cengage

Compound InterestWith compound interest, the money earned is left invested from year to yearAnd hence you get interest on interest

If you invest £100 at 10%, you get£10 interest at the end of year 1

In year 2, you get £10 interest on your £100plus £1 interest on the £10

A formula has been developed to help work out the total amount:

A0(1+r)n

Where A0 is the initial amount, r is the decimal interest rate and n is the number of years




ISBN 1-86152-991-0 © Cengage

Time-value of Money

Money which we get in the future will not buy as much as the same amount received now.

One reason is inflation.

To work out the present value of a sum of money, we need to assume a rate of interest.We can then use the formula:

tt

r

AA

10

A0 - start yearAt - in t years time

This is just a manipulation of the compound interest formula




ISBN 1-86152-991-0 © Cengage

TablesYou could work out the values of the present value formula:

by hand, or you could use a spreadsheet,

or you could use tables

To find the figure for4%

And 6 years

You get




ISBN 1-86152-991-0 © Cengage

Present ValueFor example:

How much would you need to invest now at 10% interest, to have £242 in two years time?

At = £242 r =0.1

201.01

242

A = 242 * 0.826446

= £200

So invest £200 to get £242 two years from now




ISBN 1-86152-991-0 © Cengage

Choosing between Opportunities

You are offered a choice between two dealsThe first gives you £700 in 4 years time

The second gives £850 in 6 years timeThe rate of interest is set at 8%

Option 1:

Option 2:

4008.1

700A = 700 * 0.735

= £514.50

6008.1

850A =850 * 0.6302

=£535.67CHOICE?




ISBN 1-86152-991-0 © Cengage

Investment AppraisalBusinesses often have several competing uses for their fundsThey need to find a way of objectively comparing themThis needs to take account of the time value of money

Net Present Value calculations meet these criteria

Method: For each project or use of funds we need to determine

1. Initial cost2. Income in each year3. Costs in each year4. An interest rate to be used5. The projected life of the project or asset




ISBN 1-86152-991-0 © Cengage

Example - 1A company needs to invest in new manufacturing capacity and can buy either two Xenion Producers at £50,000 each or one Yeoming Producer at £120,000

The Xenion Producer will need to be upgraded in year two at a cost of £20,000 per machine

There are no expected future costs with the Yeoming Producer during its lifetime

All Producers are expected to last for 6 years and have zero scrap value

Expected revenues are given in the table

A 8% interest rate is used




ISBN 1-86152-991-0 © Cengage

Example - 2

Yr X Y

0 0 0

1 25000 20000

2 30000 25000

3 30000 40000

4 35000 40000

5 25000 30000

6 15000 20000

Expected Revenues Expected Costs

X Y

-100000 -120000

-40000

Net Revenues

X Y

-100000 -120000

25000 20000

-10000 25000

30000 40000

35000 40000

25000 30000

15000 20000

R - C




ISBN 1-86152-991-0 © Cengage

Tables 2

To help answer this problem we need six years of present value factors




ISBN 1-86152-991-0 © Cengage

Example - 3

X Y-100000 -120000

25000 20000-10000 2500030000 4000035000 4000025000 3000015000 20000

Net Revenues

0.92590.85730.79380.735

0.68060.6302

Present Value Factor PV1 PV2

-100000 -12000023147.5 18518

-8573 21432.523814 3175225725 2940017015 204189453 12604

CHOICE

-£9,418.15 £14,124.50

Net Revenue times Present Value Factor

Total Net Present Value




ISBN 1-86152-991-0 © Cengage

Choosing r

No-one publishes a specific value of r to useThere are a range of alternatives:

• The rate of inflation

• The rate used in the past

• The rate of return on capital (from the accounts)

• The rate available on the stock market

• The rate currently paid on the bond market

• A rate to reflect the riskiness of the project




ISBN 1-86152-991-0 © Cengage

Ranges of Benefits

We already know that the future is uncertain

But the future expected income may possibly be labelled

By the likelihood of it happening

And then we could assign probabilities to the sets of

outcomes

The next example considers this situation




ISBN 1-86152-991-0 © Cengage

Ranges 2A company is assessing a project and has 3 sets of projections

of contribution. These are shown in the table below.

Year

Pessimistic General Optimistic

Cost

£100,000 £100,000 £100,000

Expected Contribution, Year 1

£10,000 £12,000 £20,000


£20,000 £25,000 £40,000


£40,000 £50,000 £70,000


£25,000 £40,000 £60,000


£10,000 £20,000 £30,000

The company uses a discount rate of 12% and you have determined the probabilities of the three scenarios as 0.2, 0.7 and 0.1 respectively.




ISBN 1-86152-991-0 © Cengage

Ranges 3

£100,000£100,0

00 £100,000

-£100,00

0

-£100,00

0

-£100,00

0

£10,000£12,00

0 £20,0000.89285

7 £8,929 £10,714 £17,857

£20,000£25,00

0 £40,0000.79719

4 £15,944 £19,930 £31,888

£40,000£50,00

0 £70,000 0.71178 £28,471 £35,589 £49,825

£25,000£40,00

0 £60,0000.63551

8 £15,888 £25,421 £38,131

£10,000£20,00

0 £30,0000.56742

7

NPV -£25,094 £3,002 £54,723

Year Pessimistic

General

Optimistic PV1 PV2 PV3

Cost





Expected Contribution, Year 5 £5,674 £11,349 £17,023

The first step is to find NPV’s in the normal way




ISBN 1-86152-991-0 © Cengage

Expected NPV

(-£25,094 x 0.2) + (£3,002 x 0.7) + (£54,723 x 0.1)= £2,555.20

You then take each NPV and multiply it by the appropriate probability

Where there are several projects competing for the same funds, this method suggests that you choose

the one with the highest expected NPV




ISBN 1-86152-991-0 © Cengage

ConclusionsNet Present Value takes account of the time value of moneyOther methods are available:

Discounted Cash Flow Looks for the rate of return on the investment which gives zero NPV

Internal Rate of Return Accounting ratio

Payback period

Ignores time value of money

Just counts up income until total equals the cost

chapter 10: comparing monetary returns over time

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