1. Paper F9 Financial Management
Cai Ji-fu
Accounting School

2. Investment appraisal using DCF methods
Chapter 8
Investment appraisal using DCF methods

3. Topic list Discounted cash flow The net present value method
The internal rate of return method
NPV and IRR compared
Assessment of DCF methods of project appraisal

4. Exam guide Applying the various investment appraisal techniques
Discussing theirs relative merits

5. 1 Discounted cash flows Fast forward
There are two methods of using DCF to evaluate capital investments, the NPV method and the IRR method.
DCF is an investment appraisal technique that takes into account the timing of cash flows and also total profitability over a project’s life.

6. 1 Discounted cash flows Two important points about DCF are as follows:
DCF analysis is based on future cash flows, not accounting profits or losses.
The timing of cash flows is taken into account by discounting them to a ‘present value’.

7. 1.1 Compounding
Key terms
A sum money invested or borrowed is known as principal.
When money is invested it earns interest, similarly when money is borrowed, interest is payable.
Interest on an investment can be calculated as either simple interest or compound interest.

8. 1.1 Compounding Simple interest
Simple interest is interest that is paid (earned) on only the original amount, or principal, borrowed (lent).
With simple interest, the interest is payable or recoverable each year but it is not added to the principle.
The amount of simple interest is an function of three variables: the original amount borrowed (lent), principal, the interest rate per time period; and the number of time period for which the principal is borrowed (lent).

9. 1.1 Compounding The formula for calculating simple interest is:
where SI = simple interest in pound
P = principal, or original amount borrowed (lent) at time period 0
i = interest rate per time period
n = number of time period

10. 1.1 Compounding
Key terms
Discounting computes the present value of future cash flows.
Compounding computes the future value of cash flows from the past.
The interest rate used in discounting or compounding is an opportunity cost of what the money invested could earn in its best alternative investment in the same risk class.

11. 1.1 Compounding Future value (terminal value) of simple interest
The value at some future time of a present amount of money, or a series of payments, evaluated at a given interest rate.
Present value
The current value of a future amount of money, or a series of payments, evaluated at a given interest rate.

12. 1.1 Compounding
Mrs. Peal has $1000 invested in a Certificate of Deposit (CD) earning 8% over the next year.
1000
1080
i=8%
n=1
compounding
discounting

13. 1.1 Compounding Compound interest
Interest is paid (earned) any previous interest earned, as well as on the principal borrowed (lent).
With compound interest, the interest is added each year to the principal. The next year the interest is calculated on the combined sum.

14. 1.1 Compounding Future value of compound interest
The formula for calculating future value of compound interest at the end of n periods is:
where is the future value interest factor at i% for n periods.

15. 1.1 Compounding Example (future value)
$1,000 is deposited in a savings account on January 1, 20X1. Interest of 12% is compounded annually. What is the balance in the account at the end of the fifth year (December 31,20X5)?
1000 ?
i=12% n=5

16. 1.1 Compounding
Solution

17. Future Value of $100 Initially Deposited and Compounded at 0,5,10 Percent 50
250
200
150
100
300
year
Future Value (dollars)

18. 1.2 Discounting Present values of compound interest
Present value is the cash equivalent now of a sum of money receivable or payable at a stated future date, discounted at a specified rate of return.
A present value of an investment can be described as the amount of money (a lump sum) that you would have to invest now for n periods, earning interest at r per time period, to build up the value of your investment to pound F at the end of that time.

19. 1.2 Discounting Present values of compound interest
Finding the present value is simply the reverse of compounding.
where is the present value interest factor at i% for n periods.

20. 1.2 Discounting
Example
What is the present value of a single payment of $10,000, which is to be received three years from now using a discount rate of 10%?
10000 ?
i=10% n=3

21. Present Value of $100 to be received at a Future date and Discounted back to the Present at 0,5,10 Percent 80 60 40 20
100
10 percent
5 percent
0 percent
year
Present Value (dollars)

22. 1.3 Annuities
Key terms
An annuity is a regular payment of the same amount each year.
An annuity is a series of identical payments, at identical periods or intervals, over a specified term, at a constant interest rate
The term ‘annuity’ is also used more generally to mean a constant amount invested every time period, not necessarily every year
The most common example of an annuity is a series of mortgage payments. Automobile leasing, which requires regular fixed monthly payments

23. 1.3 Annuities Types of annuities: ordinary annuity annuity due
In an ordinary annuity, payments or receipts occur at the end of each period
annuity due
In an annuity due, payments or receipts occur at the beginning of each period
deferred annuity
perpetual annuity

24. 1.3 Annuities
Calculations for annuities involve the following four items:
payment amount per period (pmt→A)
interest rate per period (rate→i)
number of interest periods (nper→n)
future value or present value of the annuity (PV,FV)

25. 1.3 Annuities
The future value of an annuity A at the end of year n can be calculated using the following formula
FVAn is simple equal to the periodic receipt (A) times the “sum of the future value interest factors at i percent for time periods 0 to n-1”.
Where FVIFAi,n stands for the future value interest factor of an annuity at i% for n periods.

26. Future value of an ordinary annuity (A, i, n →FV)
$100.00
$110.00
$121.00
$133.10
$146.41
FV=$610.51
i=10%

27. Future value of an ordinary annuity (A, i, n →FV)

28. Future value of an ordinary annuity (FA, i, n →A)
How much must we deposit in an 10 percent savings account at the end of each year to accumulate $1000 at the end of four years?
1000
? ? ? ?
i=10%

29. Future value of an ordinary annuity (FV, i, n →A)

30. 1.3 Annuities Present value and annuities
The present value of an annuity A for n periods can be calculated using the following formula
PVA0 is simple equal to the periodic receipt (A) times the “sum of the present value interest factors at i percent for time periods 1 to n”.
Where PVIFAi,n stands for the present value interest factor of an ordinary annuity at i% for n periods.

31. Present value of an ordinary annuity (A, i, n →PV)
$109.09 $ $ $ $
PV=$454.89
i=10%

32. Present value of an ordinary annuity (A, i, n →PV)

33. Present value of an ordinary annuity (PV, i, n →A)
A 1000 loan carries an interest rate of 12 percent and calls for equal annual repayments over ten years. What is the amount of each repayment?
1000
i=12% A=? ……

34. Present value of an ordinary annuity (PV, i, n →A)

35. 1.4 Perpetuity perpetual annuity
An ordinary annuity whose payments or receipts continue forever.

36. 1.5 The cost of capital The cost of capital has two aspects to it:
It is the cost of funds that a company raises and uses.
The required return that investors expect to be paid for putting funds into the company. It is therefore the minimum return that a company should make from its own investments, to earn the cash flows out of which investors can be paid their return.

37. 1.5 The cost of capital
The cost of capital is the minimum after-tax rate of return the firm must earn on new investment (in its own risk class) to just compensate the firm’s investors with their required rates of return.
The cost of capital can therefore be measured by studying the returns required by investors, and used to derive a discount rate for DCF analysis and investment appraisal.

38. 2 The NPV method Fast forward
The NPV method of investment appraisal is to accept projects with a positive NPV.
Ensure that you are aware of the three conventions concerning the timings of cash flows.
An annuity is a constant cash flow for a number of years.
A perpetuity is constant cash flow forever.

39. 2 The NPV method
Key term
The NPV is the value obtained by discounting all cash outflows and inflows of a capital investment project by a chosen target rate of return or cost of capital.
Cash outflows are negative and inflows are positive value, the sum of the present value of all the cash flows from the project is the ‘net’ present value amount.

40. 2 The NPV method
NPV is the sum of all discounted cash flows generated by a project. It represents the economic gain from the project and thus it measures the increase in value to the firm.
The amount of the net present value indicates the increase in shareholder wealth that accrues as a consequence of the investment

41. Net Present Value (NPV)
The NPV is the sum of the present value of all the cash inflows from a project minus the PV of all the cash outflows.

42. The NPV Rule
If NPV ＞0, it means that the cash inflows from a capital investment will yield a return in excess of the cost of capital, The project is therefore seems financially attractive.
If NPV=0, it means that the cash inflows from a capital investment will yield a return exactly equal to the cost of capital. The project is therefore just about financially attractive.
If NPV <0 it means that the cash inflows from a capital investment will yield a return below the cost of capital. The project is therefore financially unattractive. then reject the project

43. example
$10000
$ $ $6000
k=10%

44. Net present value method
Assumptions in DCF about the timing of cash flows
A cash outlay to be incurred at the beginning of an investment project (‘now’) occurs in year 0.
A cash flow (outlay, savings and inflows) that occurs during the course of a time period is assumed to occur all at once at the end of the time period.
If a cash flow occurs at the beginning of a time period, it is assumed that the cash flow happens at the end of the previous year.

45. 3 Internal rate of return method (IRR)
Fast forward
The IRR method of investment appraisal is to accept projects whose IRR exceeds a target rate of return. The IRR is calculating using interpolation.
The internal rate of return (IRR) method of DCF analysis is to calculate the exact DCF rate of return that the project is expected to achieve.
The internal rate of return (IRR) is that discount rate which gives a net present value of zero.

46. The Internal Rate of Return (IRR)
IRR is another discount cash flow method that calculates the break-even rate of return on the project such that the NPV = 0.
IRR is the discount rate at which NPV equals zero

47. Internal rate of return method (IRR)
Acceptance criterion
If the expected rate of return (IRR) is higher than a target rate of return (the cost of capital), the project is financially worth undertaking.

48. Internal rate of return method (IRR)
Interpolation
For a project with uneven cash flows, the IRR may be approximated by assuming a linear relationship between NPV and discount rate.

49. Internal rate of return method (IRR)
NPV($) k1
IRR
discount rate(%)
NPV(k1) k2
Net present-value profile showing net present value as a function of the discount rate
NPV(k2)
Graph of NPV v discount rate

50. internal rate of return method (IRR)
Formula for calculating IRR
Where
If the NPV at H% (the higher rate used) is NH
If the NPV at L% (the lower rate used) is NL

51. internal rate of return method (IRR)
IRR of even annual cash flows
Find the present value interest factor of an ordinary annuity
Find the life of project, n
Look along the n year row of the present value interest factor of an ordinary annuity till the closest value is found
The column in which this figure appears is IRR.

52. internal rate of return method (IRR)
IRR of perpetuities
IRR of perpetuities

53. 4 NPV and IRR compared Fast forward
there are advantages and disadvantages to each appraisal method, make sure that you can discuss them

54. 4 NPV and IRR compared
For deciding whether a independent project should be undertaken or not, both methods usually gives the same decision, to invest or not invest.
However, they can give conflicting guidance when a choice has to be made between mutually exclusive projects, in this case, the NPV method should be used.
Independent project – A project whose acceptance (or rejection) does not prevent the acceptance of other projects under consideration.

55. 4 NPV and IRR compared Mutually exclusive investments
Mutually exclusive projects: choice of one project, preclude the other choices. Alternatives are analyzed as a group and only the best one is chosen. Ex: Purchase of land in one location precludes purchasing other lots

56. 4 NPV and IRR compared Mutually exclusive investments
The golden rule for deciding between mutually exclusive projects is to accept the project with the higher NPV.
To maximize shareholder wealth, i.e. the market value of their share, we wish to maximize absolute return, i.e. NPV

57. Comparing NPV & IRR NPV($)k1
IRR
discount rate(%)
NPV(k1) k2
Net present-value profile showing net present value as a function of the discount rate
NPV(k2)

58. Comparing NPV & IRR Conditions that lead to conflict
When project size (or scale) differences exist, meaning that the investment of one project is larger than that of the other.
When timing differences exist, meaning that the timing of cash flows from the two projects differs such that most of the cash flows from one project come in the early years and most of the cash flows from the other project come in the later years.

59. Comparing NPV & IRR Cash flow of mutually exclusive project1 2 3 4 5 X Y
X-Y
-50,000
-32,000
-18,000
17,000
12,000
5,000
NPV(K=8%) IRR PI
Alternative X $ 17, %
Alternative Y , %
X minus Y , %

60. Comparing NPV & IRR B+(A-B)=A (A-B):incremental investment 1 2 3 NPV1 2 3
NPV
(12%)
IRR A B
A-B
$10000
10000 $
- 8000
$ 4000
3000
1000
$12000
9000
$3518
3457 61
27%
38%
12.5%
B+(A-B)=A (A-B):incremental investment

61. Comparing NPV & IRR NPV($) NPV IRR If K<12.5% A B 8000
6000
K=12% rA=27% rB= 38%
Project A
Project B
discount rate(%)
Crossover rate=12.5%
NPV IRR
If K<12.5% A B
If K>12.5% ? ?
NPV($)

62. Comparing NPV & IRR Causes of conflict: reinvestment rate assumptions
The net present-value method implies that cash flows released from any project are reinvestment at the discount rate that was used in calculating its net present value (k=12%)
reliance on the internal rate of return approach implies that funds released from any project can be reinvested at that particular project’s internal rate of return
(IRRA=27% IRRB=38%)

63. Comparing NPV & IRR
CF0=-10, CF1=2,000 CF2=4,000 CF3=12,000

64. Modified IRR (MIRR)
PV costs=PV terminal value

65. Modified IRR (MIRR) 0 1 2 3 CF ($10000) 2000 4000 12000 K=12% 4480 CF
($10000)
K=12%
4480
K=12%
2509
Terminal Value(TV) $ 18989
MIRR=23.83%
PV of TV $10000
NPV $
=RATE(3,,-10000,18989)→ %
=PV( %,3,,-18989)→$10000

66. Modified IRR (MIRR) NPV(K=12%) IRR MIRR
Alternative A $ 3, % %
Alternative B , % %

67. Comparing NPV & IRR Non-conventional cash flows
Conventional cash flows-an initial cash outflow followed by a series of inflows, otherwise is called non-conventional cash flows.
Certain non-conventional cash-flow streams may have more than one IRR.

68. Multiple IRRs CF0 CF1 CF2 NPV(13%) IRR
% or 400%

69. Multiple IRRs
discount rate(%)
IRR1=25%
IRR2=400%

70. Comparing NPV & IRR Summary of NPV and IRR comparison
When cash flow patterns are conventional both two methods gives the same accept or reject decision.
The IRR method is more easily understand.
NPV is technically superior to IRR and simpler to calculate
IRR and accounting ROCE can be confused.
IRR ignores the relative sizes of investments.

71. Comparing NPV & IRR Summary of NPV and IRR comparison
Where cash flow patterns are non-conventional, there may be several IRRs which decision makers must be aware of to avoid making the wrong decision.
The NPV method is superior for ranking mutually exclusive projects in order of attractiveness.

72. Comparing NPV & IRR Summary of NPV and IRR comparison
The reinvestment assumption underlying the IRR method can’t be substantiated.
When discount rates are expected to differ over the life of the project, such variations can be incorporated easily into NPV calculations, but not into IRR calculations
Despite the advantages of the NPV method over the IRR method, the IRR method is widely used in practice.

73. 5 Assessment of DCF methods of project appraisal
Fast forward
DCF methods of appraisal have a number of advantages over other appraisal methods
The time value of money is taken into account.
The method takes account of all of a project’s cash flows
It allows for the timing of cash flows.
There are universally accepted methods of calculating the NPV and IRR.

74. Assessment of DCF methods of project appraisal
Promblems with DCF methods
DCF methods use future cash flows that may be difficult to forecast. Although other methods use these as well, arguably the problem is greater with DCF methods that take cash flows into the longer-term.
The basic decision rule, accept all projects with a positive NPV, will not apply when the capital available for investment is rationed

75. Assessment of DCF methods of project appraisal
Promblems with DCF methods
The cost of capital used in DCF calculations may be difficult to estimate
The cost of capital may change over the life of the investment