Discounted measures

DISCOUNTED MEASURES
OF PROJECT WORTH

Net Present Value (NPV)
 The cashflows estimated for the project

are in the future; they are not yet
realised
 The future is not here yet, but decisions
would have to be taken in the present
time

 The question then is, what is the value of

these future estimated cashflows in the
present or current period, or better still
today?
 future estimated cashflows would have
to be ‘brought’ to the current or present
period


Bt - Ct
NPV = ∑
t
(1+r)
t=1
T


Bt
Ct
NPV = ∑
-∑
t
t
(1+r) t=1 (1+r)
t=1
T

T


Where :
Bt

is periodic benefit

Ct

is periodic cos t

∑

is the summation sign

 Decision Rule:




NPV > 0; project is viable, accept.
NPV < 0; project is not viable, reject.
NPV = 0; project is neither viable nor not
viable

 The value of NPV suggests how much a

project is adding in value terms to an
existing entity or how much value the
project is creating.
 A positive NPV means that the project is
expected to add value to the firm and will
therefore increase the wealth of the
owners.

 Since the goal of projects is to add value

or increase owner’s wealth, NPV is a
direct measure of how well this project
will meet the goal.
 NPV has units of currency such as Rs or
US dollars (US$).

Year

0

1

2

3

4

5

6

7

A

-100

30

30

40

20

10

0

0

B

-100

30

30

30

30

30

10

10

Cashflow Analysis for Project A and B
Cashflow

Discount Factor

Year

A

B

(1+0.30)^-t

0

-100

-100

1

30

2

Discounted Cashflow
A

B

1.0000

-100.00

-100.00

30

0.7692

23.08

23.08

30

30

0.5917

17.75

17.75

3

40

30

0.4552

18.21

13.65

4

20

30

0.3501

7.00

10.50

5

10

30

0.2693

2.69

8.08

6

0

10

0.2072

0.00

2.07

7

0

10

0.1594

0.00

1.59

-31.2691

-23.2675

Net Present Value

 Advantages
 Takes opportunity cost of money into

account.
 A single measure, which takes the
amount and timing of cashflows into
account.
 With NPV one can consider different
scenarios.

Results are expressed in value terms
units of currency. So one is able to
know the impact the value that the
project would create.
 It is based on cashflows, which are
less subjective than profits.


 Disadvantages
 Complex to calculate and communicate.
 Meaning of the result is often misunderstood.
 Only comparable between projects if the initial

investment is the same.

 It can be difficult to identify an

appropriate discount rate.
 Cashflows are usually assumed to occur
at the end of a year, but in practice this
is over simplistic.

Net Benefit Investment Ratio
 Investments are required for project

benefits to be realised.
 These investments in the project
cashflow can be identified as negatives.

 The procedure:







discount all the positive cashflows
separately
discount all the negative cashflows
separately.
Sum each of them
The sum of positive discounted cashflows
is divided by sum of negative discounted
cashflows.

T

∑B
t =1

NBIR =

t

(1 + i)

t

T

∑K
t =1

t

(1 + i)
where K is sum of negative net benefit or investment
t

 The decision rule:



NBIR > 1 accept;
NBIR < 1 reject.

Cashflow Analysis for Project A and B
Cashflow

Discount Factor

Year

A

B

(1+0.30)^-t

0
1

-100
30

-100
30

2

30

3

Discounted Cashflow
A

B

1.0000

-100.00

-100.00

0.7692

23.08

23.08

30

0.5917

17.75

17.75

40

30

0.4552

18.21

13.65

4

20

30

0.3501

7.00

10.50

5

10

30

0.2693

2.69

8.08

6

0

10

0.2072

0.00

2.07

7

0

10

0.1594

0.00

1.59

Sum of +ves

68.7309

76.7325

Sum of -ves

100.00

100.00

0.687309

0.767325

NBIR

 NBIR is also referred to as Profitability

Index by the accounting profession.
 It is often used for ranking projects
especially if rationing is in place.

Benefit – Cost Ratio (BCR)
 A variant of the formula for NPV uses the

subtraction of discounted cash outflow
from discounted cash inflow.
 In the case of BCR, the discounted cash
inflow is expressed in terms of the
discounted cash outflow.

T

Bt
∑ +r )t
(1
t
BCR = T
Ct
∑ +r )t
(1
t

 This can be viewed as:


how many times the discounted cash
inflow covers the discounted cash outflow
over the project horizon.

 Decision criteria




For a single project, a B/C ratio which is
greater than 1 indicates acceptability
For multiple (competing) projects, the
project(s) with the highest B/C ratios
(greater than 1) should receive highest
priority

 NPV measures totals, indicates the

amount by which benefits exceed (or do
not exceed) costs.
 B/C measures the ratio (or rate) by
which benefits do or do not exceed
costs.
 They are clearly similar, but not identical.
 With multiple projects, some may do
better under NPV analysis, others under
B/C.

Internal Rate of Return (IRR)
 IRR is the rate of return or discount rate

that makes the NPV = 0.
 Decision Rule:


Accept the project if the IRR is greater than
the required return

 This is the most important alternative to NPV.
 It is often used in practice and is intuitively

appealing.
 It is based entirely on the estimated cashflows and
is independent of interest rates found elsewhere.
 Without a financial calculator, this becomes a trial
and error process.

 A critical thing to note is that there

should be at least one change of sign in
order to realise IRR.
 there should be a negative net cashflow
among positive net cashflows or a
positive cashflow among negative
cashflows.
 The change in sign is crucial.

 Using a spreadsheet;
 Start with the cashflows.
 You first enter your range of cashflows,

beginning with the initial cash outlay
(negative).

 Call the IRR function







Choose insert on the menu bar
Select function
Choose IRR from among the list
Select the range of cashflows
Enter a guess rate, but it is not necessary; Excel
will start at 10% as a default
The default format is a whole percent – you will
normally want to increase the decimal places to at
least two to get the most accurate output.

 NPV and IRR will generally give us the

same decision.
 There are however some exceptions.


Non-conventional cashflows
 cashflow



signs change more than once

Mutually exclusive projects
 Initial

investments are substantially different
 Timing of cashflows is substantially different

 When the cashflows change sign more than

once, there is more than one IRR.
 When we solve for IRR it would be noticed that
we are solving for the root of an equation and
when we cross the x-axis more than once, there
will be more than one return that solves the
equation.
 Therefore, IRR may be unreliable if we have
any negative cashflows after our original
investment.

 Suppose an investment will cost ¢90,000

initially and will generate the following
cashflows:




Year 1: 132,000
Year 2: 100,000
Year 3: -150,000

 The required return is 15%.
 Should we accept or reject the project?

Year 0

-90,000

Year 1

132,000

Year 2

100,000

Year 3

-150,000

IRR

NPV fx 15%
Less inv.
NPV at 15%

10.11%

reject

91,770
-90,000
1,770

IRR says to reject, but NPV says to accept. Go with NPV.

accept

 Mutually exclusive projects




If you choose one, you can’t choose the
other
Example: You can choose to attend
graduate school next year at either Legon
or Central, but not both

 Intuitively you would use the following

decision rules:




NPV – choose the project with the higher
NPV
IRR – choose the project with the higher
IRR

Period

Project A

Project B

0

-500

-400

1

325

325

2

325

200

IRR

19.43%

22.17%

NPV

64.05

60.74

 The required return for both projects is

10%.
 Which project should you accept and
why?
 (Accept Project A because of NPV)

 Conflicts between NPV and IRR








NPV directly measures the increase in value to the
firm.
Whenever there is a conflict between NPV and
another decision rule, you should always use NPV.
IRR is unreliable in the following situations
Non-conventional cashflows
Mutually exclusive projects


 Advantages of IRR
 It takes into account the time value of money,

which is a good basis for decision-making.
 Results are expressed as a simple percentage,
and are more easily understood than some other
methods.
 It indicates how sensitive decisions are to a
change in interest rates.

 Advantages of IRR
 It is a simple way to communicate

the value of a project to someone
who doesn’t know all the estimation
details.
 If the IRR is high enough, you may
not need to estimate a required
return, which is often a difficult task.

 Disadvantages
 For mutually exclusive projects: timing and scale

differences. This may lead to incorrect decisions
in comparisons of mutually exclusive
investments.
 Assumes funds are re-invested at a rate
equivalent to the IRR itself, which may be
unrealistically high.

 IRR will produce more than one

mathematically correct rate for each year
in which inflows are followed by outflows
and vice versa. This is common with
projects with unconventional cashflows.
This can create some confusion to the
user.

Choice of Discount Rate
 Cost of capital - weighted average and marginal

(financing rate)
 ‘Opportunity cost’ of capital - what could they
earn if that money was elsewhere
 Current capital position and expected capital
position over next few years
 The rates of return for alternative investments.
 Market sentiments.

Sources of discount rate
 Banks
 Long term government papers
 Ministry of Finance
 Sponsors

Suggestions
 For industrial projects use market rate or

cost of borrowing funds.
 For public sector projects use social time
preference rate.
 For public projects to be funded from
international loans use the cost of
borrowing.

Suggestions
 Generally, in financial analysis, the market rate is

used, whilst the social time preference rate is
used for public sector projects.
 When funding comes from various sources or
from the same source but at different rates, then,
compute and use the weighted average.

Choosing Year 0 or Year 1
 World Bank




World Bank believes that since investment
is made and some returns may accrue
from the first year, then discounting should
start from 0 to first year.
In this case, the initial year is Year 1.

Choosing Year 0 or Year 1
 Others





Other international originations use Year 0.
Their argument is that investment must
take place before benefits accrue.
Thus, discounting should start from the
second year.

 Choose any convention but be

consistent.

Deciding on a Project
 We should consider several investment

criteria when making decisions.
 NPV and IRR are the most commonly
used primary investment criteria.
 Payback is a commonly used secondary
investment criteria, but only because of
its ease of use.

Deciding on a Project
 For a single project, a positive NPV

indicates acceptability.
 For multiple (competing) projects, the
project(s) with the highest NPVs should
receive highest priority.

Discounted measures

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