Chapter 10: Comparing Monetary Returns Over Time

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Chapter 10: Comparing Monetary Returns Over Time. 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. Costs &amp; Benefits.

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### Chapter 10: Comparing Monetary Returns Over Time

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
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
Payback

This method only takes into account how long it will take to pay back the initial investment in nominal terms

If 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 1

And £750 in year 2

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

Interest Calculations

Interest is earned on a sum of money invested over a period of time

The 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 year

So 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)

Compound Interest

With compound interest, the money earned is left invested from year to year

And 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 £100

plus £1 interest on the £10

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

A0(1+r)n

Where A0is the initial amount,ris the decimal interest rate andnis the number of years

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 thepresent valueof a sum of money, we need to assume a rate of interest.

We can then use the formula:

A0 - start year

At - in t years time

This is just a manipulation of the compound interest formula

Tables

You 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 for

4%

And 6 years

You get

Present Value

For example:

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

At = £242

r =0.1

= 242 * 0.826446

= £200

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

Choosing between Opportunities

You are offered a choice between two deals

The first gives you £700 in 4 years time

The second gives £850 in 6 years time

The rate of interest is set at 8%

Option 1:

= 700 * 0.735

= £514.50

Option 2:

=850 * 0.6302

=£535.67

CHOICE?

Investment Appraisal

Businesses often have several competing uses for their funds

They need to find a way of objectively comparing them

This needs to take account of the time value of money

Net Present Valuecalculations meet these criteria

Method:

For each project or use of funds we need to determine

• Initial cost
• Income in each year
• Costs in each year
• An interest rate to be used
• The projected life of the project or asset
Example - 1

A 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

R - C

Example - 2

Expected Revenues

Expected Costs

Net Revenues

Tables 2

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

Net Revenue times Present Value Factor

Example - 3

-£9,418.15

£14,124.50

CHOICE

Total Net Present Value

Choosing r

No-one publishes a specific value of r to use

There 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
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

Year

Pessimistic

General

Optimistic

Cost

£100,000

£100,000

£100,000

Expected Contribution, Year 1

£10,000

£12,000

£20,000

Expected Contribution, Year 2

£20,000

£25,000

£40,000

Expected Contribution, Year 3

£40,000

£50,000

£70,000

Expected Contribution, Year 4

£25,000

£40,000

£60,000

Expected Contribution, Year 5

£10,000

£20,000

£30,000

Ranges 2

A company is assessing a project and has 3 sets of projections of contribution. These are shown in the table below.

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.

Year

Pessimistic

General

Optimistic

PV1

PV2

PV3

Cost

£100,000

£100,000

£100,000

-£100,000

-£100,000

-£100,000

Expected Contribution, Year 1

£10,000

£12,000

£20,000

0.892857

£8,929

£10,714

£17,857

Expected Contribution, Year 2

£20,000

£25,000

£40,000

0.797194

£15,944

£19,930

£31,888

Expected Contribution, Year 3

£40,000

£50,000

£70,000

0.71178

£28,471

£35,589

£49,825

Expected Contribution, Year 4

£25,000

£40,000

£60,000

0.635518

£15,888

£25,421

£38,131

Expected Contribution, Year 5

£10,000

£20,000

£30,000

0.567427

£5,674

£11,349

£17,023

NPV

-£25,094

£3,002

£54,723

Ranges 3

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

Expected NPV

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

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

= £2,555.20

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

the one with the highest expected NPV

Conclusions

Net Present Value takes account of the time value of money

Other 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

Just counts up income until total equals the cost

Ignores time value of money