1. Topics Covered
 Competitors to NPV
 The Payback Period
 The Book Rate of Return
 Internal Rate of Return
ALSO Consider
 Capital Rationing

2. Payback
 The payback period of a project is the number
of years it takes before the cumulative
forecasted cash flow equals the initial outlay.
 The payback rule says only accept projects
that “payback” in the desired time frame.
 This method is very flawed, primarily because
it ignores later year cash flows and the the
present value of future cash flows.

3. Payback
Example
Examine the three projects and note the mistake we
would make if we insisted on only taking projects
with a payback period of 2 years or less.
Payback
Project C0 C1 C2 C3 NPV@ 10%
Period
A - 2000 500 500 5000
B - 2000 500 1800 0
C - 2000 1800 500 0

4. Payback
Example
Examine the three projects and note the mistake we
would make if we insisted on only taking projects
with a payback period of 2 years or less.
Payback
Project C0 C1 C2 C3 NPV@ 10%
Period
A - 2000 500 500 5000 3 + 2,624
B - 2000 500 1800 0 2 - 58
C - 2000 1800 500 0 2 + 50

5. Book Rate of Return
Book Rate of Return - Average income divided by
average book value over project life. Also called
accounting rate of return.
book income
Book rate of return =
book assets

6. The book rate of return
 It does recognise that capital is required to
earn income.
 However it has shortcomings as an
investment appraisal method
 It does not have a hurdle rate
 It does not recognise the time value of money
 Ambiguity regarding its definition.

7. The Internal Rate of Return
 Like the NPV this is a discounted cash flow
criterion. In the single period case where we
are merely interested in accepting or rejecting
a project it is exactly equivalent to the NPV
rule. However in all other situations
congruence of the IRR and NPV rules cannot
be guaranteed. When a conflict does occur
the NPV gives the correct answers.

8. Definition of IRR
 The IRR is the discount rate that when used to
discount a projects cash flow gives an NPV of
zero.

9. IRR in Single Period Case
C1
− 1 = IRR = ρ
C0

10. Proof
C1
NPV = - C 0 +
1+ i
IF NPV = 0
C1
C0 =
1+ ρ
C 0 (1 + ρ) = C 1
C1 - C0
IRR = ρ=
C0

11. Example
 Cost of project 1000 at time 0
 Payoff in time 1 is 1400
 IRR = (1400/1000) – 1 = 40%
1400
− 1000 = 0
1 + IRR

12. IRR Decision Rule
 Accept all projects with a rate of return that is
greater than the cost of capital.

13. IRR versus NPV
For accept/reject decisions in a single period
case the NPV and IRR decision rules give
exactly the same answer.
For ranking decisions conflicts may occur

14. Structure of Analysis
 Single Period Case
 Accept / Reject Decisions
 Ranking Decisions
 Multi-period Case
 Accept / Reject Decisions
 Ranking Decisions

15. Single Period: Mutually Exclusive
Projects – Differing Scales
Project B
Project A Cash Flows
Cash Flows
t0 (10) (100)
t1 20 130
IRR 100% 30%
NPV @ 10% (10) + 20/1.1 = -100 + 130/1.1 =
8.2 18.2

16. Internal Rate of Return
 IRR ignores the magnitude of the project.
 The following two projects illustrate that problem.
Project C0 Ct IRR NPV @ 10%
E −10,000 + 20,000 100 + 8,182
F − 20,000 + 35,000 75 +11,818

17. IRR in Multi-period Case
 The IRR generally gives the same answer in
accept/reject decisions.
 However, there may be some technical
difficulties with the IRR.
 First we will examine how to compute the
IRR in the multiperiod case.

18. Internal Rate of Return
Example
You can purchase a turbo powered machine tool
gadget for $4,000. The investment will generate
$2,000 and $4,000 in cash flows for two years,
respectively. What is the IRR on this investment?

19. Internal Rate of Return
Example
You can purchase a turbo powered machine tool gadget for $4,000. The
investment will generate $2,000 and $4,000 in cash flows for two years,
respectively. What is the IRR on this investment?
2,000 4,000
NPV = −4,000 + + =0
(1 + IRR ) (1 + IRR )
1 2

20. Internal Rate of Return
Example
You can purchase a turbo powered machine tool gadget for $4,000. The
investment will generate $2,000 and $4,000 in cash flows for two years,
respectively. What is the IRR on this investment?
2,000 4,000
NPV = −4,000 + + =0
(1 + IRR ) (1 + IRR )
1 2
IRR = 28.08%

21. Internal Rate of Return
2500
2000
IRR=28%
1500
1000
NPV (,000s)
500
0
-500
20
60
10
30
40
50
70
80
90
0
10
-1000
-1500
-2000
Discount rate (%)

22. IRR computation with Excel ®
 See IRRDR.xls

23. Multi-period Computation of IRR
without a computer
 1. Calculate the NPV at a high discount rate so that it is < 0.
 2. Calculate the NPV at a low discount rate so that it is > 0.
(e.g. use 0%)
 3. Divide the difference between the positive NPV and the
negative NPV by the change in the discount rate to get the
approximate change in NPV for a one- percent change in the
discount rate.
 4. Using the information in 3 compute the required increase
(decrease) in the rate to reduce (increase) the positive
(negative) NPV to zero.
 5. Use the rate computed in 4 to discount the cash flows. If
the NPV is not equal to zero then alter your estimate of the
IRR accordingly.

24. Calculation of IRR in multi-period
case
TIME 0 1 2 3
CASH -350 110 121 200
FLOW
The NPV @ 0% = 81
The NPV @ 20% = (59)
Thus there is a spread of 140(7) in the NPV for a spread of
20%(1%) in the discount rate.

25. IRR Calculation Continued
 On the basis of this information we should
reduce the discount rate by (59/7)% = 8.43%
from 20%.
 If we try a rate of 12% the NPV = (13).
Therefore we have to reduce by a further 2%
or so to get NPV = 0.
 Discounting at 10% gives an NPV of 0.
Hence the IRR is 10%.

26. Internal Rate of Return
Accept / Reject Decisions: Technical Problem -
Multiple Rates of Return
 Certain cash flows can generate NPV=0 at two different
discount rates.
 The following cash flow generates NPV=0 at both (-50%)
and 15.2%.
C0 C1 C2 C3 C4 C5 C6
− 1,000 + 800 + 150 + 150 + 150 + 150 − 150

27. Internal Rate of Return
Pitfall 2 - Multiple Rates of Return
 Certain cash flows can generate NPV=0 at two different discount rates.
 NPV
The following cash flow generates NPV=0 at both (-50%) and 15.2%.
1000
IRR=15.2%
500
0 Discount
Rate
-500 IRR=-50%
-1000

28. Multiple IRRs
 A project has as many IRRs as it has
changes in the sign of its cash flows.
Number 0 1 2 3
of IRRS
1 (100) 200 300 400
3 (100) 200 (100) 500

29. Internal Rate of Return
Mutually Exclusive Projects

30. Internal Rate of Return
Reinvestment rate assumption
 There is an implicit assumption that all intermediate cash
inflows are reinvested at the IRR
 It makes far more sense to assume that the intermediate cash
flows are reinvested at the opportunity cost of capital.
 We assume that discount rates are stable during the term of
the project.

31. Profitability Index
 Profitability Index is PV/C0
 The rule is to accept all projects which have a
PI > 1.
 One gets the same answers as the NPV for
accept/reject decisions.
 For ranking decisions conflicts arise in a
manner similar to the IRR case e.g.
differences in the scale of the project.

32. The Profitability Index and Capital
Rationing
 There is one case where the Profitability
Index is superior to the NPV. This is where
the firm faces a limit on the amount it can
invest in a single year and projects are
divisible and there is no postponement. In
this special case one should maximise the
NPV per £1 invested.

33. Profitability Index
 When resources are limited, the profitability
index (PI) provides a tool for selecting among
various project combinations and alternatives
 A set of limited resources and projects can
yield various combinations.

34. Project appraisal: capital rationing,
•Coping with investment appraisal in an environment of
capital rationing, taxation and inflation
•More specifically:
– Explain why capital rationing exists and be able to use
the profitability ratio in one-period rationing situations

35. Capital rationing
•Capital rationing occurs when funds are not
available to finance all wealth-enhancing projects
•Soft rationing
•Hard rationing
•One-period capital rationing
– 1 Divisible projects
– 2 Indivisible projects

36. One-period capital rationing
with divisible projects
•Capital at time zero has been rationed to £4.5m

37. Bigtasks plc (continued)
Ranking according to absolute NPV
Present value
Profitability index =
–––––––––––––––– Initial
outlay
Net present value
Benefit–cost ratio =
–––––––––––––– Initial
outlay

38. Bigtasks plc: Profitability indices
and benefit–cost ratios

39. Bigtasks plc: Ranking according
to the highest profitability index

40. Indivisible projects

41. NPV – the pros
 NPV is the theoretically correct criteria for
making investment decisions when
maximisation of shareholder wealth is the
objective
 It recognises the time value of money
 It forces managers to consider their
projections carefully when estimating future
cash flows

42. NPV – the pros continued
 It is generally easy to use
 It has a clear decision rule
 It can deal with multiple discount rates –
unlike IRR
 It is not affected by differences in scale –
unlike IRR and Profitability Index

43. NPV – the cons
 It does not help find value creating projects
 It does not lend itself to ex-post evaluation of
managers in a straightforward manner. But
residual income can help here.

44. NPV vs. IRR
Single- Multi-Period
Period Case
Case
Accept/Reject Always Can be
Consistent technical
Problems with
IRR
Non Independent Conflicts Conflicts
Projects (e.g.
Ranking)

45. IRR – the pros
 It is theoretically correct for accept/reject
decsions
 It recognises the time value of money

46. IRR – the cons
 It cannot be relied upon to signal the correct decision
for non-independent projects – e.g. mutually
exclusive projects of differing scales.
 Can have multiple IRRs
 Some projects have no IRR
 The re-investment assumptions of the rule do not
make economic sense

47. IRR – the cons continued
 It cannot cope with multiple discount rates
 In addition it has all the other the drawbacks
of the NPV criterion.