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 futu...
Net Present Value (NPV)
 The question then is, what is the value of

these future estimated cashflows in the
present or c...
Net Present Value (NPV)

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

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

T
Net Present Value (NPV)

Where :
Bt

is periodic benefit

Ct

is periodic cos t

∑

is the summation sign
Net Present Value (NPV)
 Decision Rule:




NPV > 0; project is viable, accept.
NPV < 0; project is not viable, reject...
Net Present Value (NPV)
 The value of NPV suggests how much a

project is adding in value terms to an
existing entity or ...
Net Present Value (NPV)
 Since the goal of projects is to add value

or increase owner’s wealth, NPV is a
direct measure ...
Net Present Value (NPV)
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

...
Net Present Value (NPV)
Cashflow Analysis for Project A and B
Cashflow

Discount Factor

Year

A

B

(1+0.30)^-t

0

-100
...
Net Present Value (NPV)
 Advantages
 Takes opportunity cost of money into

account.
 A single measure, which takes the
...
Net Present Value (NPV)
Results are expressed in value terms
units of currency. So one is able to
know the impact the valu...
Net Present Value (NPV)
 Disadvantages
 Complex to calculate and communicate.
 Meaning of the result is often misunders...
Net Present Value (NPV)
 It can be difficult to identify an

appropriate discount rate.
 Cashflows are usually assumed t...
Net Benefit Investment Ratio
 Investments are required for project

benefits to be realised.
 These investments in the p...
Net Benefit Investment Ratio
 The procedure:







discount all the positive cashflows
separately
discount all the n...
Net Benefit Investment Ratio
T

∑B
t =1

NBIR =

t

(1 + i)

t

T

∑K
t =1

t

(1 + i)
where K is sum of negative net bene...
Net Benefit Investment Ratio
 The decision rule:



NBIR > 1 accept;
NBIR < 1 reject.
Net Benefit Investment Ratio
Cashflow Analysis for Project A and B
Cashflow

Discount Factor

Year

A

B

(1+0.30)^-t

0
1...
Net Benefit Investment Ratio
 NBIR is also referred to as Profitability

Index by the accounting profession.
 It is ofte...
Benefit – Cost Ratio (BCR)
 A variant of the formula for NPV uses the

subtraction of discounted cash outflow
from discou...
Benefit – Cost Ratio (BCR)
T

Bt
∑ +r )t
(1
t
BCR = T
Ct
∑ +r )t
(1
t
Benefit – Cost Ratio (BCR)
 This can be viewed as:


how many times the discounted cash
inflow covers the discounted cas...
Benefit – Cost Ratio (BCR)
 Decision criteria




For a single project, a B/C ratio which is
greater than 1 indicates a...
Benefit – Cost Ratio (BCR)
 NPV measures totals, indicates the

amount by which benefits exceed (or do
not exceed) costs....
Internal Rate of Return (IRR)
 IRR is the rate of return or discount rate

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

Ac...
Internal Rate of Return (IRR)
 This is the most important alternative to NPV.
 It is often used in practice and is intui...
Internal Rate of Return (IRR)
 A critical thing to note is that there

should be at least one change of sign in
order to ...
Internal Rate of Return (IRR)
 Using a spreadsheet;
 Start with the cashflows.
 You first enter your range of cashflows...
Internal Rate of Return (IRR)
 Call the IRR function







Choose insert on the menu bar
Select function
Choose IR...
Internal Rate of Return (IRR)
 NPV and IRR will generally give us the

same decision.
 There are however some exceptions...
Internal Rate of Return (IRR)
 When the cashflows change sign more than

once, there is more than one IRR.
 When we solv...
Internal Rate of Return (IRR)
 Suppose an investment will cost ¢90,000

initially and will generate the following
cashflo...
Internal Rate of Return (IRR)
Year 0

-90,000

Year 1

132,000

Year 2

100,000

Year 3

-150,000

IRR

NPV fx 15%
Less in...
Internal Rate of Return (IRR)
 Mutually exclusive projects




If you choose one, you can’t choose the
other
Example: Y...
Internal Rate of Return (IRR)
 Intuitively you would use the following

decision rules:




NPV – choose the project wi...
Internal Rate of Return (IRR)
Period

Project A

Project B

0

-500

-400

1

325

325

2

325

200

IRR

19.43%

22.17%

...
Internal Rate of Return (IRR)
 The required return for both projects is

10%.
 Which project should you accept and
why?
...
Internal Rate of Return (IRR)
 Conflicts between NPV and IRR








NPV directly measures the increase in value to ...
Internal Rate of Return (IRR)

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

which is a good basis...
Internal Rate of Return (IRR)
 Advantages of IRR
 It is a simple way to communicate

the value of a project to someone
w...
Internal Rate of Return (IRR)
 Disadvantages
 For mutually exclusive projects: timing and scale

differences. This may l...
Internal Rate of Return (IRR)
 IRR will produce more than one

mathematically correct rate for each year
in which inflows...
Choice of Discount Rate
 Cost of capital - weighted average and marginal

(financing rate)
 ‘Opportunity cost’ of capita...
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...
Suggestions
 Generally, in financial analysis, the market rate is

used, whilst the social time preference rate is
used f...
Choosing Year 0 or Year 1
 World Bank




World Bank believes that since investment
is made and some returns may accrue...
Choosing Year 0 or Year 1
 Others





Other international originations use Year 0.
Their argument is that investment ...
Deciding on a Project
 We should consider several investment

criteria when making decisions.
 NPV and IRR are the most ...
Deciding on a Project
 For a single project, a positive NPV

indicates acceptability.
 For multiple (competing) projects...
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Discounted measures

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Discounted measures

  1. 1. DISCOUNTED MEASURES OF PROJECT WORTH
  2. 2. 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
  3. 3. Net Present Value (NPV)  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
  4. 4. Net Present Value (NPV) Bt - Ct NPV = ∑ t (1+r) t=1 T
  5. 5. Net Present Value (NPV) Bt Ct NPV = ∑ -∑ t t (1+r) t=1 (1+r) t=1 T T
  6. 6. Net Present Value (NPV) Where : Bt is periodic benefit Ct is periodic cos t ∑ is the summation sign
  7. 7. Net Present Value (NPV)  Decision Rule:    NPV > 0; project is viable, accept. NPV < 0; project is not viable, reject. NPV = 0; project is neither viable nor not viable
  8. 8. Net Present Value (NPV)  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.
  9. 9. Net Present Value (NPV)  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$).
  10. 10. Net Present Value (NPV) 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
  11. 11. Net Present Value (NPV) 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
  12. 12. Net Present Value (NPV)  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.
  13. 13. Net Present Value (NPV) 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. 
  14. 14. Net Present Value (NPV)  Disadvantages  Complex to calculate and communicate.  Meaning of the result is often misunderstood.  Only comparable between projects if the initial investment is the same.
  15. 15. Net Present Value (NPV)  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.
  16. 16. Net Benefit Investment Ratio  Investments are required for project benefits to be realised.  These investments in the project cashflow can be identified as negatives.
  17. 17. Net Benefit Investment Ratio  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.
  18. 18. Net Benefit Investment Ratio 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
  19. 19. Net Benefit Investment Ratio  The decision rule:   NBIR > 1 accept; NBIR < 1 reject.
  20. 20. Net Benefit Investment Ratio 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
  21. 21. Net Benefit Investment Ratio  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.
  22. 22. 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.
  23. 23. Benefit – Cost Ratio (BCR) T Bt ∑ +r )t (1 t BCR = T Ct ∑ +r )t (1 t
  24. 24. Benefit – Cost Ratio (BCR)  This can be viewed as:  how many times the discounted cash inflow covers the discounted cash outflow over the project horizon.
  25. 25. Benefit – Cost Ratio (BCR)  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
  26. 26. Benefit – Cost Ratio (BCR)  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.
  27. 27. 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
  28. 28. Internal Rate of Return (IRR)  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.
  29. 29. Internal Rate of Return (IRR)  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.
  30. 30. Internal Rate of Return (IRR)  Using a spreadsheet;  Start with the cashflows.  You first enter your range of cashflows, beginning with the initial cash outlay (negative).
  31. 31. Internal Rate of Return (IRR)  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.
  32. 32. Internal Rate of Return (IRR)  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
  33. 33. Internal Rate of Return (IRR)  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.
  34. 34. Internal Rate of Return (IRR)  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?
  35. 35. Internal Rate of Return (IRR) 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
  36. 36. Internal Rate of Return (IRR)  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
  37. 37. Internal Rate of Return (IRR)  Intuitively you would use the following decision rules:   NPV – choose the project with the higher NPV IRR – choose the project with the higher IRR
  38. 38. Internal Rate of Return (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
  39. 39. Internal Rate of Return (IRR)  The required return for both projects is 10%.  Which project should you accept and why?  (Accept Project A because of NPV)
  40. 40. Internal Rate of Return (IRR)  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
  41. 41. Internal Rate of Return (IRR)  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.
  42. 42. Internal Rate of Return (IRR)  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.
  43. 43. Internal Rate of Return (IRR)  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.
  44. 44. Internal Rate of Return (IRR)  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.
  45. 45. 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.
  46. 46. Sources of discount rate  Banks  Long term government papers  Ministry of Finance  Sponsors
  47. 47. 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.
  48. 48. 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.
  49. 49. 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.
  50. 50. 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.
  51. 51. 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.
  52. 52. 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.
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