ACCA F9 Financial Management - 2010 - Study text - Emile Woolf Publishing
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Chapter 8: Discounted cash flow
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DCF is based on cash flows, not accounting profits. It is therefore much more
suitable than the ARR method for investment appraisal.
It evaluates all cash flows from the project, unlike the payback method which
considers only those cash flows in the payback period.
It gives a single figure, the NPV, which can be used to assess the value of the
investment project. The NPV of a project is the amount by which the project
should add to the value of the company, in terms of ‘today’s value’.
The NPV method provides a decision rule which is consistent with objective of
maximisation of shareholders’ wealth. In theory, a company ought to increase in
value by the NPV of an investment project (assuming that the NPV is positive).
The main disadvantages of the NPV method are:
The time value of money and present value are concepts that are not easily
understood
There might be some uncertainty about what the appropriate cost of capital or
discount rate should be for applying to any project. Cost of capital is considered
in more detail in a later chapter.
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Internal rate of return (IRR) method
The investment decision rule with IRR
Calculating the IRR of an investment project
Advantages and disadvantages of the IRR method
Summary: comparison of the four investment appraisal methods
3 Internal rate of return (IRR) method
The internal rate of return method (IRR method) is another method of investment
appraisal using DCF.
The internal rate of return of a project is the discounted rate of return on the
investment.
It is the average annual investment return from the project
Discounted at the IRR, the NPV of the project cash flows must come to 0.
The internal rate of return is therefore the discount rate that will give a net present
value = $0.
3.1 The investment decision rule with IRR
A company might establish the minimum rate of return that it wants to earn on an
investment. If other factors such as non-financial considerations and risk and
uncertainty are ignored:
If a project IRR is equal to or higher than the minimum acceptable rate of return,
it should be undertaken
It the IRR is lower than the minimum required return, it should be rejected.
Since NPV and IRR are both methods of DCF analysis, the same investment decision
should normally be reached using either method.
The internal rate of return is illustrated in the diagram below:
Chapter 8: Discounted cash flow
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3.2 Calculating the IRR of an investment project
The IRR of a project can be calculated by inputting the project cash flows into a
financial calculator. In you examination, you might be required to calculate an IRR
without a financial calculator. An approximate IRR can be calculated using
interpolation.
To calculate the IRR, you should begin by calculating the NPV of the project at two
different discount rates.
One of the NPVs should be positive, and the other NPV should be negative.
(This is not essential. Both NPVs might be positive or both might be negative,
but the estimate of the IRR will then be less reliable.)
Ideally, the NPVs should both be close to zero, for better accuracy in the estimate
of the IRR.
When the NPV for one discount rate is positive NPV and the NPV for another
discount rate is negative, the IRR must be somewhere between these two discount
rates.
Although in reality the graph of NPVs at various discount rates is a curved line, as
shown in the diagram above. Using the interpolation method we assume that the
graph is a straight line between the two NPVs that we have calculated. We can then
use linear interpolation to estimate the IRR, to a reasonable level of accuracy.
The interpolation formula
If the NPV at A% is positive, + $P
and if the NPV at B% is negative, - $N
()
⎥
⎦
⎤
⎢
⎣
⎡
−×
+
+= %A B
N P
P
A% IRR
Ignore the minus sign for the negative NPV. For example, if P = + 75 and N = - 30,
then P + N = 105.
Example
A business requires a minimum expected rate of return of 12% on its investments. A
proposed capital investment has the following expected cash flows.
Year
$
0 (80,000)
1 20,000
2 36,000
3 30,000
4 17,000
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Required
Calculate the NPV at a cost of capital of 10% and a cost of capital of 15%. Use these
NPV figures to estimate the IRR
Answer
Year
Cash
flow
Discount
factorat10%
Presentvalue
at10%
Discount
factorat15%
Present
valueat15%
$ $ $
0 (80,000) 1.000 (80,000) 1,000 (80,000)
1 20,000 0.909 18,180 0.870 17,400
2 36,000 0.826 29,736 0.756 27,216
3 30,000 0.751 22,530 0.658 19,740
4 17,000 0.683 11,611 0.572 9,724
–
–––––––
––––––––
NPV +2,057 (5,920)
–
–––––––
––––––––
The IRR is above 10% but below 15%.
Using the interpolation method:
The NPV is + 2,057 at 10%.
The NPV is – 5,920 at 15%.
The NPV therefore falls by 7,977 between 10% and 15%.
The estimated IRR is:
()
()
⎥
⎦
⎤
⎢
⎣
⎡
−×
+
+= %10 15
5,920 2,057
2,057
10% IRR
= 10% + 1.3%
= 11.3%
Recommendation
The project is expected to earn a DCF return below the target rate of 12%, and on
financial grounds it is not a worthwhile investment.
3.3 Advantages and disadvantages of the IRR method
The main advantages of the IRR method of investment appraisal are:
As a DCF appraisal method, it is based on cash flows, not accounting profits.
Like the NPV method, it recognises the time value of money.
It is easier to understand an investment return as a percentage return on
investment than as a money value NPV in $.
For accept/reject decisions on individual projects, the IRR method will reach the
same decision as the NPV method.
Chapter 8: Discounted cash flow
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The disadvantages of the IRR method are:
It is a relative measure (% on investment) not absolute measure in $. Because it is
a relative measure, it ignores the absolute size of the investment. For example,
which is the better investment if the cost of capital is 10%:
− an investment with an IRR of 15% or
− an investment with an IRR of 20%?
If the investments are mutually exclusive, and only one of them can be
undertaken the correct answer is that it depends on the size of each of the
investments. This means that the IRR method of appraisal can give an incorrect
decision if it is used to make a choice between mutually exclusive projects
Unlike the NPV method, the IRR method does not indicate by how much an
investment project should add to the value of the company.
Example
There are two mutually exclusive projects.
Year Project 1
Project 2
$ $
(1,000) (10,000)
1,200 4,600
- 4,600
- 4,600
IRR 20%
18%
NPV at 15% + $43 + $503
Which is better?
Answer
Project 2 is better, because it has the higher NPV. Project 2 will add to value by $503
but Project 1 will add value of just $43.
3.4 Summary: comparison of the four investment appraisal methods
A comparison of the four investment appraisal methods is given in the table below.
The key points to note are that:
DCF is superior to the ARR method and payback method of investment appraisal
It is often equally as good to use NPV or IRR
However, NPV has two advantages over IRR
− The NPV method indicates the value that the investment should add (if the
NPV is positive) or the value that it will destroy (if the NPV is negative).
− When there are two or more mutually exclusive projects, the NPV will
always identify the project that should be selected. This is the project that
will provide the highest value (NPV).
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The IRR method has the advantage of being more easily understood by non-
accountants
Another disadvantage of the IRR method is that a project might have two or
more different IRRs, when some annual cash flows during the life of the project
are negative. (The mathematics that demonstrate this point are not shown here.)
ARR Payback Discounted cash flow
Disadvantages Advantages Advantages
Financial
profits, not
cash flows
Balance sheet
values, not
cash
investment
cost.
Cash flows, not
accounting
values
Focus on
recovering the
cost of the
investment
Based on investment cash flows, not
accounting profit
Recognises the time value of money
Recognises all cash flows, over the full life of
the project
By far the best investment appraisal method
Required return is based on the organisation’s
cost of finance.
Disadvantages NPV IRR
Choice of
maximum
payback period
is arbitrary
Indicates the increase in
the value of the
company that should
be expected if it were to
undertake the
investment.
More easily
understood than
NPV by a non-
accountant.
Ignores cash
from the project
after payback
If a choice has to be made between two (or
more) mutually exclusive projects, the NPV
method is more reliable than IRR.
Ignores the time
value of money.
Chapter 8: Discounted cash flow
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Annuities and perpetuities
Definition of an annuity
Calculating the PV of an annuity
Annuity discount tables
Using annuities and annuity factors for investment appraisal
Definition of a perpetuity
Present value of a perpetuity
4 Annuities and perpetuities
4.1 Definition of an annuity
An annuity is a constant cash flow for a given number of time periods. A capital
project might include estimated annual cash flows that are an annuity.
Examples of annuities are:
$30,000 each year for years 1 – 5
$20,000 each year for years 3 – 10
$500 each month for months 1 – 24.
The present value of an annuity can be calculated using annuity factors, rather than
using discount factors to calculate the present value of the cash flow for each
individual year.
4.2 Calculating the PV of an annuity
If you need to calculate the present value of $50,000 per year for years 1 – 3 at a
discount rate of 9%, you could calculate this as follows:
Year
Cash flow
Discount factor
at 9%
Present
value
$
$
1 50,000 1/(1.09) = 0.917
45,850
2 50,000 1/(1.09)
2
= 0 842
42,100
3 50,000 1/(1.09)
3
= 0.772
38,600
NPV
126,550
There is a formula for calculating the present value of an annuity. The formula is:
PV =
A
r
× 1 −
1
1 + r
()
n
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
Where:
A = the constant annual cash flow (the annuity)
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r = discount rate, as a proportion
n = number of time periods
The present value of $50,000 per year for three years at a discount rate of 9% can
therefore be calculated as:
$50,000
0.09
× 1 −
1
1.09
()
3
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
= [$50,000/0.09] × (1 – 0.0.77218) = $50,000 × 0.22782
= $126,567.
4.3 Annuity discount tables
Another way of calculating the PV of an annuity is to multiply the annuity by the
sum of the discount factors for the years in which the cash flows occur. In the
example above, the PV of $50,000 for years 1 – 3 at a discount rate of 9% could be
calculated as:
$50,000 × (0.917 + 0.842 + 0.772) = $50,000 × 2.531 = $126,550.
The discount factors for annuities are simply the sum of the annual discount factors
for each year of the annuity. Discount tables for annuities are included in the
formula and tables sheets near the end of this text. These tables will be provided in
your examination. An extract is shown below.
Extract from annuity tables
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 1.970 1.942 1.913 1.886 1.859 1.833 1.808 1.783 1.759 1.736
3 2.941 2.884 2.829 2.775 2.723 2.673 2.624 2.577 2.531 2.487
4 3.902 3.808 3.717 3.630 3.546 3.465 3.387 3.312 3.240 3.170
5 4.853 4.713 4.580 4.452 4.329 4.212 4.100 3.993 3.890 3.791
The annuity factors are for periods starting in period 1 (year 1). For example, the
annuity factor for years 1 – 3 at 9%, from the table, is 2.531.
Examples
The annuity factor for years 1 – 2 at a cost of capital of 8% = 1.783 (n = 2, discount
factor 8%. This is the sum of the discount factors at 8% for years 1, and 2 (0.926 +
0.857).
The annuity factor for years 1 – 5 at a cost of capital of 10% = 3.791 (n = 5, discount
factor = 10%). This is the sum of the discount factors at 10% for years 1, 2, 3, 4 and 5
(0.909 + 0.826 + 0.751 + 0.683 + 0.621).
Chapter 8: Discounted cash flow
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4.4 Using annuities and annuity factors for investment appraisal
Annuity discount factors can be used in DCF investment analysis, mainly to make
the calculations easier and quicker.
Example
What is the present value of the cash flows for a project, if the cash flows are $60,000
each year for years 1 – 7, and the cost of capital is 15%?
Answer
$60,000 × 4.160 (annuity factor at 15%, n = 7) = $249,600.
Example
What is the present value of the following cash flows, when the cost of capital is
12%?
Year
Annualcash
flow
Discount
factorat12%
Present
value
$ $
0 (100,000)
1 10,000
2 15,000
3–15 20,000
NPV
Answer
Annuity factor at 12%, years 1 – 15 = 6.811
Annuity factor at 12%, years 1 – 2 = 1.690
Therefore annuity factor at 12%, years 3 – 15 = 6.811 – 1.690 = 5.121
Year
Annualcash
flow
Discount
factorat12%
Present
value
$ $
0 (100,000) 1.000 (100,000)
1 10,000 0.893 8,930
2 15,000 0.797 11,955
3–15 20,000 5.121 102,420
NPV +23,305
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Example
A company is considering an investment of $70,000 in a project. The project life
would be five years.
What must be the minimum annual cash returns from the project to earn a return of
at least 9% per annum?
Answer
Investment = $70,000
Annuity factor at 9%, years 1 – 5 = 3.890
Minimum annuity required = $17,995 (= $70,000/3.890)
Exercise 3
A company is considering an investment of $70,000 in a project. The project life
would be ten years. The cash flows would be $15,000 each year for years 1 – 5 and
$10,000 each year for years 6 – 10. The cost of capital is 8%.
What is the NPV of the project?
4.5 Definition of a perpetuity
A perpetuity is a constant annual cash flow ‘forever’, or into the long-term future.
In investment appraisal, an annuity might be assumed when a constant annual cash
flow is expected for a long time into the future.
4.6 Present value of a perpetuity
The present value of a perpetuity is C/r, where:
C is the constant annual cash flow in perpetuity
r is the cost of capital, for example 0.08, 0.10 etcetera.
Examples
The present value of $2,000 in perpetuity, starting in Year 1, given a cost of capital of
8%, is: $2,000/0.08 = $25,000.
The present value of $5,500 in perpetuity, starting in Year 4, given a cost of capital of
11%, is calculated as follows:
The perpetuity starts in Year 4, therefore the ‘present value’ as at the end of Year
3 = $5,500/0.11 = $50,000.