BNFN 521
INVESTMENT APPRAISAL
Lecture Notes
Lecture Three
DISCOUNTING
AND
ALTERNATIVE INVESTMENT
CRITERIA
1
Project Cash Flow Profile
Benefits Less Costs
(+)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Year of Project Life
(-)
Initial Investment
Period
Operating Stage
Project Life
Liquidation
2
Discounting and Alternative Investment Criteria
Basic Concepts:
A.
Discounting
•
Recognizes time value of money
a. Funds when invested yield a return
b. Future consumption worth less than present consumption
o
o
PVB = (B /(1+r) +(B 1/(1+r) 1+.…….+(Bn /(1+r) n
r
o
o
o
PVC = (C /(1+r) +(C 1/(1+r) 1+.…….+(Cn /(1+r)
r
o
n
o
NPV = (B o-C o)/(1+r) o+(B 1-C 1)/(1+r) 1+.…….+(B n-C n)/(1+r) n
r
3
Discounting and Alternative Investment Criteria
(Cont’d)
B. Cumulative Values
• The calendar year to which all projects are discounted to is important
• All mutually exclusive projects need to be compared as of same
calendar year
1
If NPV r= (B o-C o)(1+r) 1+(B 1-C 1) +..+..+(B n-C n)/(1+r) n-1 and
NPV 3= (B o-Co)(1+r) 3+(B 1-C 1)(1+r) 2+(B 2-C 2)(1+r)+(B 3-C 3)+...(B n-C n)/(1+r) n-3
r
3
1
Then NPV r = (1+r) 2 NPV r
4
Example of Discounting (10% Discount Rate)
Year
Net Cash Flow
PV00.1  1000 
PV01.1
0
-1000
1
200
2
300
3
350
4
1440
200 300
350
1440



 676.25
2
3
4
1.1 (1.1)
(1.1)
(1.1)
300
350
1440
 1000(1.1)  200 


 743.88
2
3
1.1 (1.1)
(1.1)
PV02.1  1000(1.1) 2  200(1.1)  300 
350
1
(1.1)
1440

(1.1)
2
 818.26
Note: All of the transactions are done at the beginning of the year.
5
C. Variable Discount Rates
• Adjustment of Cost of Funds Through Time
r0
r1
r2
r3
r4
r5
If funds currently are
abnormally scarce
Normal or historical
average cost of funds
r *4
r *3
r *2
r *1
r *0
If funds currently are
abnormally abundant
0
1
2
3
4
5
Years from
present period
•For variable discount rates r1, r2, & r3 in years 1, 2, and 3, the discount factors
are, respectively, as follows:
1/(1+r1), 1/[(1+r1)(1+r2)] & 1/[(1+r1)(1+r2)(1+r3)]
6
Example of Discounting (multiple rates)
Year
Net Cash Flow
r
NPV 0  1000
0
-1000
18%
1
200
16%
2
300
14%
3
350
12%
4
1440
10%
200
300
350
1440



 436.91
1.18 (1.18)(1.16) (1.18)(1.16)(1.14) (1.18)(1.16)(1.14)(1.12)
300
350
1440
NPV  1000(1.18)  200


 515.55
1.16 (1.16)(1.14) (1.16)(1.14)(1.12)
1
NPV 2  1000(1.18)(1.16)  200(1.16)  300
350
1440

 598.04
(1.14) (1.14)(1.12)
Note: All of the transactions are done at the beginning of the year.
7
ALTERNATIVE INVESTMENT
CRITERIA
1.
2.
3.
4.
Net Present Value (NPV)
Benefit-Cost Ratio (BCR)
Pay-out or Pay-back Period
Internal Rate of Return (IRR)
8
Net Present Value (NPV)
1. The NPV is the algebraic sum of the discounted values of the
incremental expected positive and negative net cashflows over a
project’s anticipated lifetime.
2. What does net present value mean?
– Measures the change in wealth created by the project.
– If this sum is equal to zero, then investors can expect to recover
their incremental investment and to earn a rate of return on their
capital equal to the private cost of funds used to compute the
present values.
– Investors would be no further ahead with a zero-NPV project than
they would have been if they had left the funds in the capital
market.
– In this case there is no change in wealth.
9
Alternative Investment Criteria
First Criterion: Net Present Value (NPV)
• Use as a decision criterion to answer
following:
a. When to reject projects?
b. Select project(s) under a budget constraint?
c. Compare mutually exclusive projects?
d. How to choose between highly profitable
mutually exclusive projects with different
lengths of life?
10
Net Present Value Criterion
a. When to Reject Projects?
Rule: “Do not accept any project unless it generates a positive net
present value when discounted by the opportunity cost of funds”
Examples:
Project A: Present Value Costs $1 million, NPV + $70,000
Project B: Present Value Costs $5 million, NPV - $50,000
Project C: Present Value Costs $2 million, NPV + $100,000
Project D: Present Value Costs $3 million, NPV - $25,000
Result:
Only projects A and C are acceptable. The country is made worse
off if projects B and D are undertaken.
11
Net Present Value Criterion (Cont’d)
b. When You Have a Budget Constraint?
Rule: “Within the limit of a fixed budget, choose that subset of the
available projects which maximizes the net present value”
Example:
If budget constraint is $4 million and 4 projects with positive NPV:
Project E:
Costs $1 million, NPV + $60,000
Project F:
Costs $3 million, NPV + $400,000
Project G:
Costs $2 million, NPV + $150,000
Project H:
Costs $2 million, NPV + $225,000
Result:
Combinations FG and FH are impossible, as they cost too much. EG
and EH are within the budget, but are dominated by the combination
EF, which has a total NPV of $460,000. GH is also possible, but its
NPV of $375,000 is not as high as EF.
12
Net Present Value Criterion (Cont’d)
c. When You Need to Compare Mutually Exclusive Projects?
Rule: “In a situation where there is no budget constraint but a project
must be chosen from mutually exclusive alternatives, we should always
choose the alternative that generates the largest net present value”
Example:
Assume that we must make a choice between the following three
mutually exclusive projects:
Project I: PV costs $1.0 million, NPV $300,000
Project J: PV costs $4.0 million, NPV $700,000
Projects K: PV costs $1.5 million, NPV $600,000
Result:
Projects J should be chosen because it has the largest NPV.
13
Alternative Investment Criteria
Benefit-Cost Ratio (R)
• As its name indicates, the benefit-cost ratio (R), or what is
sometimes referred to as the profitability index, is the ratio of
the PV of the net cash inflows (or economic benefits) to the
PV of the net cash outflows (or economic costs):
R
PV of Cash Inflows (or Econom icBenefits)
PV of Cash Outflows (or Econom icCosts)
14
Benefit-Cost Ratio (Cont’d)
Basic rule:
If benefit-cost ratio (R) >1, then the project should be undertaken.
Problems?
Sometimes it is not possible to rank projects with the Benefit-Cost Ratio
• Mutually exclusive projects of different sizes
• Mutually exclusive projects and recurrent costs subtracted out of
benefits or benefits reported gross of operating costs
• Not necessarily true that RA>RB
that project “A” is better
15
Benefit-Cost Ratio (Cont’d)
First Problem:The Benefit-Cost Ratio Does Not Adjust for Mutually Exclusive Projects
of Different Sizes. For example:
Project A: 
PV0of Costs = $5.0 M,
PV0 of Benefits = $7.0 M
NPV0A = $2.0 M
RA = 7/5 = 1.4
Project B: 
PV0 of Costs = $20.0 M,
PV0 of Benefits = $24.0 M
NPV0B = $4.0 M
RB = 24/20 = 1.2
According to the Benefit-Cost Ratio criterion, project A should be chosen over project B
because RA>RB, but the NPV of project B is greater than the NPV of project A. So,
project B should be chosen
Second Problem: The Benefit-Cost Ratio Does Not Adjust for Mutually Exclusive
Projects and Recurrent Costs Subtracted Out of Benefits or Benefits Reported As Gross of
Operating Costs. For example:
Project A: PV0 Total Costs = $5.0 M
PV0 Recurrent Costs = $1.0 M
(i.e. Fixed Costs = $4.0 M)
PV0 of Gross Benefits= $7.0 M
RA = (7-1)/(5-1) = 6/4 = 1.5
Project B: Total Costs
= $20.0 M
Recurrent Costs
= $18.0 M
(i.e. Fixed Costs = $2.0 M)
PV0 of Gross Benefits= $24.0 M
RB = (24-18)/(20-18) = 6/2 =3
Hence, project B should be chosen over project A under Benefit-Cost Criterion.
Conclusion: The Benefit-Cost Ratio should not be used to rank projects
16
Alternative Investment Criteria
Pay-out or Pay-back period
• The pay-out period measures the number of years it will
take for the undiscounted net benefits (positive net
cashflows) to repay the investment.
• A more sophisticated version of this rule compares the
discounted benefits over a given number of years from the
beginning of the project with the discounted investment
costs.
• An arbitrary limit is set on the maximum number of years
allowed and only those investments having enough
benefits to offset all investment costs within this period
will be acceptable.
17
• Project with shortest payback period is preferred by this criteria
Comparison of Two Projects With Differing Lives Using Pay-Out Period
Bt - Ct
Ba
Bb
ta
0
tb
Ca
=
Cb
Payout period for
project a
Time
Payout period for
project b
18
Pay-Out or Pay-Back Period
• Assumes all benefits that are produced by in longer
life project have an expected value of zero after the
pay-out period.
• The criteria may be useful when the project is
subject to high level of political risk.
19
Alternative Investment Criteria
Internal Rate of Return (IRR)
• IRR is
the discount rate (K) at which the present
value of benefits are just equal to the present
value of costs for the particular project
t

i=0
Bt - Ct
=0
(1 + K)t
Note: the IRR is a mathematical concept, not an
economic or financial criterion
20
Common uses of IRR:
(a)
If the IRR is larger than the cost of funds then the
project should be undertaken
(b)
Often the IRR is used to rank mutually exclusive
projects. The highest IRR project should be chosen
•
An advantage of the IRR is that it only uses
information from the project
21
Difficulties With the Internal Rate of Return Criterion
First Difficulty: Multiple rates of return for project
Bt - Ct
+300
Time
-100
-200
Solution 1:
K = 100%; NPV= -100 + 300/(1+1) + -200/(1+1)2 = 0
Solution 2:
K = 0%;
NPV= -100+300/(1+0)+-200/(1+0)2 = 0
22
Difficulties With The Internal Rate of Return Criterion (Cont’d)
Second difficulty: Projects of different sizes and also mutually exclusive
Year
1
2
3
...
...

+600
+4,000
+600
+4,000
+600
+4,000
+600
+4,000
+600
+4,000
+600
+4,000
0
Project A -2,000
Project B -20,000
NPV and IRR provide different Conclusions:
Opportunity cost of funds = 10%
0
NPV A : 600/0.10 - 2,000 = 6,000 - 2,000 = 4,000
NPV 0B : 4,000/0.10 - 20,000 = 40,000 - 20,000 = 20,000
Hence, NPV 0B> NPV
0
A
IRR A : 600/K A - 2,000 = 0 or K A = 0.30
IRR B : 4,000/K B - 20,000 = 0 or K B = 0.20
Hence, K A>K B
23
Difficulties With The Internal Rate of Return Criterion (Cont’d)
Third difficulty: Projects of different lengths of life and mutually exclusive
Opportunity cost of funds = 8%
Project A: Investment costs = 1,000 in year 0
Benefits = 3,200 in year 5
Project B: Investment costs = 1,000 in year 0
Benefits = 5,200 in year 10
NPV 0A : -1,000 + 3,200/(1.08) 5 = 1,177.86
NPV 0B : -1,000 + 5,200/(1.08)10= 1,408.60
Hence, NPVB0 > NPVA0
IRRA : -1,000 + 3,200/(1+KA)5 = 0 which implies that KA = 0.262
IRRB : -1,000 + 5,200/(1+KB)10 = 0 which implies that KB = 0.179
Hence, KA>KB
24
Difficulties With The Internal Rate of Return Criterion (Cont’d)
Fourth difficulty: Same project but started at different times
Project A: Investment costs = 1,000 in year 0
Benefits = 1,500 in year 1
Project B: Investment costs = 1,000 in year 5
Benefits = 1,600 in year 6
NPV 0A : -1,000 + 1,500/(1.08) = 388.88
NPV 0B : -1,000/(1.08) 5 + 1,600/(1.08) 6 = 327.68
0
Hence, NPV A> NPV
0
B
IRR A : -1,000 + 1,500/(1+K A) = 0 which implies that K A = 0.5
IRR B : -1,000/(1+K B) 5+ 1,600/(1+K B) 6= 0 which implies that K B = 0.6
Hence, K B >KA
25
IRR FOR IRREGULAR CASHFLOWS
For Example: Look at a Private BOT Project from the perspective of the
Government
Year 
Project A
IRR A
0
1
2
3
4
1000
1200
800
3600
-8000
3600
-6400
10%
Compares Project A and Project B ?
Project B
IRR B
1000
1200
800
-2%
Project B is obviously better than A, yet IRR A > IRR B
Project C
IRR C
1000
1200
800
3600
-4800
-16%
Project C is obviously better than B, yet IRR B > IRR C
Project D
IRR D
-1000
1200
800
3600
-4800
4%
Project D is worse than C, yet IRR D > IRR C
Project E
IRR E
-1325
1200
800
3600
-4800
20%
Project E is worse than D, yet IRR E > IRR D
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