chapter 5
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Chapter 5
THE IMPORTANCE OF SCALE
AND
TIMING IN PROJECT APPRAISAL
Why is scale important?
Too large or too small can destroy a good project One of the most important decision that a project
analyst is to make is the "scale" of the investment. This is mostly thought as a technical issue but it has a financial and economic dimension as well.
Right scale should be chosen to maximize NPV. In evaluating a project to determine its best scale,
the most important principle is to treat each incremental change in its size as a project in itself
Why is scale important? (Cont’d)
By comparing the present value of the incremental benefits with the present value of the incremental costs, scale is increased until NPV of the incremental net benefits is negative. (incremental NPV is called Marginal Net Present Value (MNPV)
We must first make sure that the NPV of the overall project is positive. Secondly, the net present value of the last addition must also be greater than or equal to zero.
Choice of Scale Rule: Optimal scale is when NPV = 0 for the last addition
to scale and NPV > 0 for the whole project Net benefit profiles for alternative scales of a facility
C1
C2
C3
B1
B2B3
Bt - Ct
Time0NPV (B1 – C1) 0 ?
NPV (B2 – C2) 0 ?
NPV (B3 – C3) 0 ?
Determination of Scale of Project
Relationship between net present value and scale
NPV
A
Scale of Project0
B C ED F G H I J LK M N
(+)
(-)
NPV of Project
Internal Rate of Return (IRR) Criterion
The optimal scale of a project can also be determined
by the use of the IRR. Here it is assumed that each
successive increment of investment has a unique
IRR.
Incremental investment is made as long as the
MIRR is above or equal to the discount rate.
Table 5-1Determination of Optimum Scale of Irrigation Dam (Cont’d)
S0
S1
S2
S3
S4
S5
S6
-3000
-4000
-5000
-6000
-7000
-8000
-9000
50
125
400
800
1000
1101
1150
-2500
-2750
-1000
2000
3000
3010
2500
0.017
0.031
0.080
0.133
0.143
0.138
0.128
50
125
400
800
1000
1101
1150
50
125
400
800
1000
1101
1150
50
125
400
800
1000
1101
1150
50
125
400
800
1000
1101
1150
0 1 2 3 4 5 - Costs Benefits
Year
Scale NPV 10% IRR
Opportunity cost of funds (discount rate) = 10%
Table 5-1Determination of Optimum Scale of Irrigation Dam
S0
S1
S2
S3
S4
S5
S6
-3000
-4000
-5000
-6000
-7000
-8000
-9000
50
125
400
800
1000
1101
1150
-2500
-2750
-1000
2000
3000
3010
2500
0.017
0.031
0.080
0.133
0.143
0.138
0.128
50
125
400
800
1000
1101
1150
50
125
400
800
1000
1101
1150
50
125
400
800
1000
1101
1150
50
125
400
800
1000
1101
1150
0 1 2 3 4 5 - Costs Benefits
Year
Scale NPV 10% IRR
Opportunity cost of funds (discount rate) = 10%
Note:
1. NPV of last increment to scale 0 at scale S5. i.e. NPV of scale 5 = 10.
2. NPV of project is maximized at scale of 5, i.e. NPV1-5 = 3010.
3. IRR is maximized at scale 4.4. When the IRR on the last increment
to scale (MIRR) is equal to discount rate the NPV of project is maximized.
1. at Scale 3: Maximum point of MIRR (0.40)between Scale 3 and Scale 4: MIRR is greater than IRR; MIRR and IRR are greater than r
2. at Scale 4: Maximum point of IRR (0.143) and MIRR intersects with IRR between Scale 4 and Scale 5: MIRR is smaller than IRR; MIRR and IRR are greater than r
3. at Scale 5: MIRR is equal to Discount Ratebetween Scale 5 and Scale N: MIRR is smaller than IRR; MIRR is smaller than r; IRR is greater than r
4. at some Scale N: IRR is equal to Discount Rate
Figure 5-3Relationship between MIRR, IRR and DR
Scale
IRR>r
Sn
MIRR<r
MaximumIRR (0.14)
Discount Rate (r) Opp. Cost
of Funds (0.10)
MIRR>r
S3S4 S5
Percent
MaximumMIRR
Figure 5-4Relationship between MNPV and NPV
1. at Scale 3: Maximum point of MNPV ($3000) at 0.10 Discount rate2. at Scale 4: Maximum point of NPV (zero) at 0.14 Discount Rate
between Scale 0 and Scale 5: NPV is positive and NPV it increases3. at Scale 5: Maximum point of NPV and MNPV is equal to zero
between Scale 5 and Scale N: NPV is positive and it decreases4. at some Scale N: NPV is equal to zero
after Scale N: NPV is negative and it decreases
$3010$3000
Scale
NPV (+)
NPV(0.10)
NPV(0.14)
NPV(0.10) 0
S4
S5
Sn 0
NPV (-)MNPV (0.10)
Per
cent
S3
Maximum NPV Maximum MNPV
Figure 5-5Relationship between MIRR, IRR, MNPV and NPV
Maximum IRR
Maximum MIRR
Scale
NPV (+)
NPV (-)
Maximum NPV
Maximum MNPV
MNPV (0.10)
IRR
Discount Rate (r)Opp. Cost of Funds
(0.10)
MIRR
Percent
NPV (0.10)
S4 S5Sn 0S3
0
Maximum IRR
Maximum MIRR
Scale
NPV (+)
NPV (-)
Maximum NPV
Maximum MNPV
MNPV (0.10)
IRR
Discount Rate (r)Opp. Cost of Funds
(0.10)
MIRR
Percent
NPV (0.10)
S4 S5Sn 0S3
0
Relationship between MIRR, IRR, MNPV and NPV
When MNPV is positive – NPV is increasing When MNPV is zero – NPV is at the maximum and MIRR
is equal to Discount Rate When NPV is zero – IRR is equal to Discount Rate
When MIRR is greater than IRR – IRR is increasing When MIRR is equal to IRR – IRR is at the maximum When MIRR is smaller than IRR – IRR is decreasing
IRR is greater than Discount Rate as long as NPV is positive
MIRR is greater than Discount Rate as long as NPV is increasing
Relationship between MIRR, IRR and NPV (cont’d.)
Figure 5.5 gives the relationship between MIRR, IRR and NPV.
MIRR cuts IRR from above at its maximum point.
Scale of the project must be increased until MIRR is just equal to the discount rate. This is the optimal scale (S5).
At the optimum scale NPV is maximum and MIRR is equal to the discount rate (10%).
When NPV is equal to zero, IRR is equal to the discount rate (10%).
To illustrate the procedure, construction of an irrigation dam which could be built at different heights is given as an example in Table 5.1.
Timing of Investments
Key Questions:1.What is right time to start a project?2.What is right time to end a project?
Four Illustrative Cases of Project TimingCase 1. Benefits (net of operating costs) increasing
continuously with calendar time. Investments costs are independent of calendar time
Case 2. Benefits (net of operating costs) increasing with calendar time. Investment costs function of calendar time
Case 3. Benefits (net of operating costs) rise and fall with calendar time. Investment costs are independent of calendar time
Case 4. Costs and benefits do not change systematically with calendar time
Case 1: Timing of Projects:When Potential Benefits Are a Continuously Rising Function of Calendar Time but Are Independent of
Time of Starting Project
rKI D E
Timet0 t2
A C
K
B (t)
rKt Bt+1
rKt > Bt+1 Postpone
rKt < Bt+1 Start
<>
t1
K
B1
Benefits and Costs
5.52
4.09
3.25
2.19
1.981.80
1.651.52
1.411.31
1.22 1.14 1.07 1.03 1.03
0.00
1.00
2.00
3.00
4.00
5.00
6.00
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Timing for Start of Operation of Roojport Dam, South Africa of Marginal Economic Unit Water Cost
Numbers of Years Postponed
Eco
no
mic
Wat
er C
ost
Ran
d/m
3
Case 2: Timing of Projects: When Both Potential Benefits and Investments Are
A Function of Calendar Time
rKt < Bt+1 + (Kt+1-Kt) Start
rKt >Bt+1+ (Kt+1-Kt) Postpone
rK0
D E
Time
A C
B (t)
B1
t2 t3
K1
K0
K1
F
I H
t1
K0
G
B2
0
Benefits and Costs
Case 3: Timing of Projects: When Potential Benefits Rise and Decline
According to Calendar Time
Time
rK
A
C
K0
B
K
Start if: rKt* < Bt*+1
B (t)
0
K1 K2
I
rSV
t0 t1 t* tn tn+1
SV
Benefits and Costs
Stop if: rSVt - B(tn+1) - ΔSVt > 0 ; SVt = SVt - SVt n n+1 n+1 n+1 n
Do project if: NPV = ∑i=t*+1
tn
> 0- Kt* +SVt n
(1+r)t - t*n
t*r (1+r)i-t*
Bi
Do not do project if: NPV = ∑t*r <0
i=t*+1
tn
(1+r)i-t*
Bi - Kt* +SVt n
(1+r)t - t*n
The Decision Rule
If (rSVt - Bt - ΔSVt ) > 0 Stop
(ΔSVt = SVt - SVt ) < 0 Continue
This rule has 5 special cases:1. SV > 0 and ΔSV < 0, e.g. Machinery2. SV > 0 but ΔSV > 0, e.g. Land3. SV < 0, but ΔSV = 0, e.g. A nuclear plant4. SV < 0, but ΔSV > 0, e.g. Severance pay for workers5. SV < 0 and ΔSV < 0 e.g. Clean-up costs
Timing of Projects:When The Patterns of Both Potential Benefits and
CostsDepend on Time of Starting Project
t0 t1 t2
A
C
K0
B
K0
Benefits From K1
K1
0 tn tn+1
K1
D
Benefits From K0
Benefits and Costs
NPV FOR THE BASE SCENARIO WITH DIFFERENT STARTING YEARS
(thousands of 1998 US$)
Beginning Construction Year (Operation of Bridge)
Financial Economic
Argentina
Economic
Uruguay
1999 (2003) 190,925 610,730 218,044
2000 (2004) 189,296 571,933 203,859
2001 (2005) 185,499 536,248 190,791
2002 (2006) 180,160 502,502 178,650
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