ACCA F9 Financial Management - 2010 - Study text

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Paper F9

Financial management

CHAPTER

Discounted cash flow

Contents

1 Thetimevalueofmoney:compoundingand

discounting

2 Netpresentvalue(NPV)methodofinvestment

appraisal

3 Internalrateofreturn(IRR)method

4 Annuitiesandperpetuities

5 LayoutofNPVcalculations

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The time value of money: compounding and discounting

 The time value of money

 Compounding

 Discounting

 Discount factors

 Introduction to discounted cash flow (DCF) analysis

1 The time value of money: compounding and

discounting

Discounted cash flow analysis (DCF) is a method of evaluating proposed capital

investments, that:

 evaluates the expected cash flows of the investment, not accounting profits, and

 recognises the relevance of the time value of money.

1.1 The time value of money

Companies make investments in order to earn a return. They want to recover their

investment, and in addition make a profit. However investments and investment

returns should be measured by their cash flows, not their accounting profits.

Companies that invest are no different in this respect from individuals who invest.

Money has a time value, because an investor expects a return that allows for the

length of time that the money is invested. Larger cash returns should be required for

investing for a longer term.

Example

If $1,000 is invested at 10% annual interest, the investor will want a return of:

$1,100 (= $1,000 × 1.10

) if it is a one-year investment, but

$1,210 (= $1,000 × 1.10

) if it is a two-year investment and

$1,331 (= $1,000 × 1.10

) if it is a three-year investment.

Investment returns can be measured by compounding or discounting.

1.2 Compounding

Compounding is used to calculate the future value of an investment, where the

investment earns a compound rate of interest. If an investment is made ‘now’ and is

expected to earn interest at r% in each time period, for example each year, the future

value of the investment can be calculated as follows.

Future return = Initial investment × (1 + r)

Chapter 8: Discounted cash flow

The term ‘future value’ or ‘FV’means the value of an investment or cash flow at a

future date. ‘Present value’ or ‘PV’ refers to value now. The above compound

interest formula can therefore be stated as:

FV = PV × (1 + r)

Notes

r = the return on the investment each time period (year). This might be an actual

return or a required return.

n = the number of time periods (years) covered by the investment.

r is expressed as a proportion. For example:

 if the return is 12%, r = 0.12

 if the return is 7%, r = 0.07

 if the return is 8.5%, r = 0.085.

Example

A company is investing $200,000 to earn an annual return of 6% over three years. If

there are no cash returns before the end of Year 3, what will be the return from the

investment after three years?

Answer

$200,000 × 1.06

= $238,203.

1.3 Discounting

Discounting is the reverse of compounding. Future cash flows from an investment

can be converted to an equivalent present value amount.

Present value of future return = Future value of return × [1/(1 + r)

]

PV = FV × [1/(1 + r)

The present value of a future cash flow from an investment is the amount that would

have to be invested now, at the investment cost of capital, to earn that future cash flow.

Example

How much would an investor need to invest now in order to have $1,000 after 12

months, if the compound interest on the investment is 0.5% each month?

Answer

The future value of the investment return is $1,000 after 12 months. The investment

‘now’ would have to be the present value of $1,000 after 12 months, discounted at

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0.5% per month.

Present value = $1,000 × [1/(1.005)

]= $1,000 × 0.942 = $942.

Example

An investor wants to make a return on his investments of at least 7% per year. He

has been offered the chance to invest in a bond that will cost $200,000 and will pay

$270,000 at the end of four years. If there is no investment risk with the bond,

should he undertake the investment?

Answer

If the investor could invest his money at 7% per year, then in order to earn $270,000

after four years the amount of his investment now would need to be:

PV = $270,000 × 1/(1.07)

= $270,000 × 0.763

= $206,010.

He is required to invest only $200,000 to earn $270,000 after 4 years. This indicates

that the bond will provide a return in excess of 7% per year. It can also be suggested

that the bond would give the investor an immediate increase in wealth of $6,010

because an investment costing $200,000 is actually ‘worth’ $206,010 to him.

Exercise 1

An investor has just invested a sum of money at an interest rate of 8% per year. The

investment will be worth $125,000 in five years’ time.

How much has he invested?

1.4 Discount factors

A discount factor is 1/(1 + r)

. Future cash flows or investment values are

multiplied by the appropriate discount factor to convert them into a present value.

Present value = Future cash flow × Discount factor

The discount factor is smaller for higher values of r and higher values of n.

1.5 Introduction to discounted cash flow (DCF) analysis

Discounted cash flow is a technique for evaluating proposed investments, to decide

whether they are financially worthwhile.

The expected future cash flows from the investment (cash payments and cash

receipts) are all converted to a present value by discounting them at the cost of

capital r. The present value of investment costs and the present value of the

investment returns (cash benefits or returns) can be compared.

Chapter 8: Discounted cash flow

There are two methods of DCF:

 Net present value (NPV) method: the cost of capital r is the return required by

the investor or company

 Internal rate of return (IRR) method: the cost of capital r is the actual return

expected from the investment.

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Net present value (NPV) method of investment appraisal

 Calculating the NPV of an investment project

 Assumptions about the timing of cash flows

 Discount tables

 Advantages and disadvantages of the NPV method

2 Net present value (NPV) method of investment

appraisal

With the NPV method of investment appraisal, all the future cash flows from an

investment are converted into a present value by discounting each future cash flow

at the investment cost of capital. This cost of capital is the return required from the

investment.

The present value of a future cash inflow from a capital project is the amount that

would have to be invested now at the cost of capital to obtain that cash flow in the

future. For example suppose that a project is expected to provide a cash return of

$40,000 after two years and a further $50,000 after three years, and the company

needs to make a return of 10% per year. The NPV approach to investment appraisal

is to convert these expected future cash inflows into their present value equivalent.

 The present value of these future cash flows would be the amount that the

company would need to invest now at 10% per year to obtain a return of $40,000

after two years and another $50,000 after three years.

 The present value of the expected cash flows is therefore the value to the

company, in terms of ‘today’s value’ of those cash flows in the future.

2.1 Calculating the NPV of an investment project

In NPV analysis, all future cash flows from a project are converted into a present

value, so that the value of all the annual cash outflows and cash inflows can be

expressed in terms of ‘today’s value’.

The net present value (NPV) of a project is the net difference between the present

value of all the costs incurred and the present value of all the cash flow benefits

(savings or revenues).

 If the present value of benefits exceeds the present value of costs, the NPV is

positive.

 If the present value of benefits is less than the present value of costs, the NPV is

negative.

 The NPV is 0 when the PV of benefits and the PV of costs are equal.

The decision rule is that, ignoring other factors such as risk and uncertainty, and

non-financial considerations, a project is worthwhile financially if the NPV is

positive or zero. It is not worthwhile if the NPV is negative.

Chapter 8: Discounted cash flow

The net present value of an investment project is also a measure of the value of the

investment. For example, if a company invests in a project that has a NPV of $2

million, the value of the company should increase by $2 million.

2.2 Assumptions about the timing of cash flows

In DCF analysis, the following assumptions are made about the timing of cash flows

during each year:

 All cash flows for the investment are assumed to occur at the end of the year

 If a cash flow will occur early during a particular year, it is assumed that it will

occur at the end of the previous year. Therefore cash expenditure early in Year 1,

for example, is assumed to occur in Year 0.

Year 0 cash flows

Cash flows at the beginning of the investment, in Year 0, are already stated at their

present value.

The discount factor for a cash flow in Year 0 is 1/(1 + r)

Any value to the power of 0 is always = 1. Therefore the discount factor for Year 0 is

always = 1.000, for any cost of capital.

This means that the present value of $1 in year 0 is always $1, for any cost of capital.

Example

A company estimates that its cost of capital is 10%. It is considering whether to

invest in a project with the following cash flows and will make the decision on the

basis of the net present value of the project:

Year

0 (10,000)

1 6,000

2 8,000

Should the project be undertaken?

Answer

The cash flow in each year must be converted into a present value, using a discount

rate of 10%. Negative cash flows have a negative present value and positive cash

flows have a positive present value. The sum of the present values in each year is

the net present value of the investment.

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Year

Cash

flow

Discount

factor at

10%

Present

value

$ $

0 (10,000) 1 (10,000)

1 6,000 1/(1.10) 5,456

2 8,000 1/(1.10)

6,612

NPV 2,068

The project should be accepted, ignoring other factors such as risk and uncertainty,

because it has a positive NPV of $2,068.

(Note: The figures here are rounded to the nearest $1. It might be more appropriate

to round to the nearest $100 or even the nearest $1,000, depending on the degree of

accuracy that is required or that is appropriate, given the inevitable uncertainties in

the cash flow estimates).

The positive NPV shows that in order to earn a cash return of $6,000 after one year

and $8,000 after two years, for an investment return of 10%, the company would

need to invest $12,068 now. Instead, by investing just $10,000 in the project, it will

obtain the same returns so which is better: investing $10,000 or investing $12,068 in

order to obtain exactly the same returns? Clearly, the answer is that it is better to

invest $10,000. The project will provide a return in excess of 10% per year.

Example

A company is considering whether to undertake an investment. The cost of capital is

10%. The initial cost of the investment would be $50,000 and the expected annual

cash flows from the project would be:

Year Revenue Costs

Netcash

flow

 $ $ $

1 40,000 30,000 10,000

2 55,000 35,000 20,000

3 82,000 40,000 42,000

Required

(a) Use compounding arithmetic to calculate what the investment should be

worth at the end of Year 3.

(b) Using discounting, calculate the NPV of the project.

with the NPV.

Chapter 8: Discounted cash flow

Answer

Compounding $

Investment in Year 0 (50,000)

Interest required (10%), Year 1 (5,000)

Return required, end of Year 1 (55,000)

Net cash flow, Year 1 10,000

(45,000)

Interest required (10%), Year 2 (4,500)

Return required, end of Year 2 (49,500)

Net cash flow, Year 2 20,000

(29,500)

Interest required (10%), Year 3 (2,950)

Return required, end of Year 3 (32,450)

Net cash flow, Year 3 42,000

Future value, end of Year 3 9,550

Discounting: NPV method

Year Cashflow

Discountfactor

at10%

Presentvalue

 $$

0 (50,000) 1.0 (50,000)

1 10,000 1/(1.10)

 9,091

2 20,000 1/(1.10)

 16,529

3 42,000 1/(1.10)

 31,555



–

–––––––

Netpresentvalue +7,175



–

–––––––

Reconciliation of present value and future value

NPV × (1 + r)

= Future value: $7,175 × (1.10)

= $9,550

This example shows a simple capital project with an initial capital outlay in Year 0

and cash inflows for three years. The same technique can be applied to much bigger

and longer capital projects, and projects with negative cash flows in years other than

Year 0.

2.3 Discount tables

Discount tables are available. They take away the need to calculate the value of

discount factors [1/(1 + r)

]. Discount tables are included in the formula and tables

sheets near the end of this study text.

Discount tables are provided in your examination, and the discount factors in the

tables are rounded to three decimal places.

Paper F9: Financial management

An extract from discount tables is shown below.

Discount rates (r)

Periods

(n) 1% 2% 3% 4% 5% 6% 7% 8% 9% 10%

1 0.990 0.980 0.971 0.962 0.952 0.943 0.935 0.926 0.917 0.909

2 0.980 0.961 0.943 0.925 0.907 0.890 0.873 0.857 0.842 0.826

3 0.971 0.942 0.915 0.889 0.864 0.840 0.816 0.794 0.772 0.751

4 0.961 0.924 0.888 0.855 0.823 0.792 0.763 0.735 0.708 0.683

5 0.951 0.906 0.863 0.822 0.784 0.747 0.713 0.681 0.650 0.621

For example, suppose that you need to calculate the present value of $60,000 in Year

4 if the cost of capital (discount rate) is 7%.

You could use your calculator to calculate: $60,000 × 1/(1.07)

= $45,774.

Alternatively, you could use discount tables to calculate: $60,000 × 0.763 = $45,780.

The rounding difference is insignificant.

Exercise 2

The following exercise tests your basic understanding of the NPV method and the

use of discount tables. It should also demonstrate that when the cost of capital is

lower, the NPV is higher.

A company is considering an investment in equipment costing $70,000. Working

capital of $5,000 will also be required early in Year 1. The equipment will have a

resale value of $7,000 at the end of Year 5. The operating profits from the

investment, in cash flows, will be:

Year Cashflow

 $

1 25,000

2 20,000

3 30,000

4 20,000

5 3,000

Required

Using discount tables for the discount factors, calculate the NPV of the project if the

cost of capital is:

(a) 12% and

(b) 8%

2.4 Advantages and disadvantages of the NPV method

The advantages of the NPV method of investment appraisal are that:

 NPV takes account of the timing of the cash flows by calculating the present

value for each cash flow at the investor’s cost of capital.

ACCA F9 Financial Management - 2010 - Study text - Emile Woolf Publishing

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