# Ch05 Scale Timing n Int by O8F7bJ

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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
Bt - Ct

B3
B2

B1

0                                                                    Time
C1
NPV (B1 – C1)    0?
C2
NPV (B2 – C2)    0?
C3
NPV (B3 – C3)     0?
Determination of Scale of
Project
   Relationship between net present value and scale

NPV
(+)
NPV of Project

Scale of
0     A   B   C   D E   F   G   H   I   J   K   L M N   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 5-1
Determination of Optimum Scale of Irrigation Dam (Cont’d)

Year        0        1      2        3       4      5-
Scale      Costs                  Benefits                 NPV 10% IRR
S0          -3000    50      50       50       50     50     -2500   0.017
S1          -4000    125    125       125     125    125     -2750   0.031
S2          -5000    400    400       400     400    400     -1000   0.080
S3          -6000    800    800       800     800    800     2000    0.133
S4          -7000   1000    1000     1000     1000   1000    3000    0.143
S5          -8000   1101    1101     1101     1101   1101    3010    0.138
S6          -9000   1150    1150     1150     1150   1150    2500    0.128

Opportunity cost of funds (discount rate) = 10%
Table 5-1
Determination of Optimum Scale of Irrigation Dam

Year     0       1      2        3       4       5-
Scale   Costs                 Benefits                  NPV 10% IRR
S0       -3000    50     50       50       50      50     -2500   0.017
S1       -4000   125    125       125     125     125     -2750   0.031
S2       -5000   400    400       400     400     400     -1000   0.080
S3       -6000   800    800       800     800     800     2000    0.133
S4       -7000   1000   1000     1000     1000    1000    3000    0.143
S5       -8000   1101   1101     1101     1101    1101    3010    0.138
S6       -9000   1150   1150     1150     1150    1150    2500    0.128

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.
Figure 5-3
Relationship between MIRR, IRR and DR
1.        at Scale 3: Maximum point of
MIRR (0.40)
Percent                                            between Scale 3 and Scale 4:
MIRR is greater than IRR; MIRR
Maximum                                                    and IRR are greater than r
MIRR                                          2.        at Scale 4: Maximum point of
IRR (0.143) and MIRR intersects
Maximum                                                     with IRR
IRR (0.14)                                                  between Scale 4 and Scale 5:
IRR>r
MIRR is smaller than IRR; MIRR
and IRR are greater than r
3.        at Scale 5: MIRR is equal to
Discount Rate
between Scale 5 and Scale
MIRR>r                       N: MIRR is smaller than IRR;
MIRR is smaller than r; IRR is
greater than r
at some Scale N: IRR is equal

4.

Discount Rate                                                to Discount Rate
(r) Opp. Cost                            MIRR<r
of Funds (0.10)
S3   S4     S5                     Sn        Scale
Figure 5-4
Relationship between MNPV and NPV
1.   at Scale 3: Maximum point of MNPV (\$3000) at 0.10 Discount rate
2.   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 increases
3.   at Scale 5: Maximum point of NPV and MNPV is equal to zero
between Scale 5 and Scale N: NPV is positive and it decreases
4.   at some Scale N: NPV is equal to zero                             NPV (+)
after Scale N: NPV is negative and it decreases

                                        \$3010 Maximum NPV
\$3000 Maximum MNPV
NPV(0.10)
Percent


S5
Scale
S3     S4                             Sn                   0
NPV(0.14)                                                                  NPV (-)
MNPV (0.10)        NPV(0.10)  0
Figure 5-5
Relationship between MIRR, IRR, MNPV and NPV

Percent
NPV (+)


Maximum MIRR
Maximum NPV
Maximum MNPV
Maximum IRR

Discount Rate (r)                  

Opp. Cost of Funds
(0.10)
Scale
0   S3   S4   S                Sn              0
5
MIRR                  IRR
NPV (-)
NPV (0.10)
MNPV (0.10)
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 Timing
Case 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
Benefits and Costs
B (t)
I        D   E
rK
A            C

B1

t0   t1       t2                                      Time

rKt   <   Bt+1
K                          >

rKt > Bt+1       Postpone
rKt < Bt+1       Start
K
Timing for Start of Operation of Roojport Dam, South
Africa of Marginal Economic Unit Water Cost
6.00

5.52

5.00

4.00
4.09
Economic Water Cost Rand/m3

3.25
3.00

2.00                           2.19
1.98
1.80
1.65
1.52
1.41
1.31
1.00                                                                              1.22
1.14   1.07   1.03   1.03

0.00
0    1      2     3          4    5      6      7      8      9     10     11     12     13     14      15

Numbers of Years Postponed
Case 2: Timing of Projects:
When Both Potential Benefits and Investments Are
A Function of Calendar Time

Benefits and Costs

B (t)
D         E
rK0
A            C

B1 B2

0            t1    t2      t3                                      Time

K0        K1
rKt >Bt+1+ (Kt+1-Kt)             Postpone
rKt < Bt+1 + (Kt+1-Kt)           Start
K0
F            G
K1
I         H
Case 3: Timing of Projects:
When Potential Benefits Rise and Decline
According to Calendar Time
Benefits and Costs

SV

rK             B    C
I

A                                                                             rSV

B (t)

0   t0    t1      t*                                                  tn       tn+1      Time
Start if: rKt* < Bt*+1

Stop if: rSVt n - B(tn+1) - ΔSVt n+1 > 0 ;           SVt n+1= SVt n+1- SVt       n
tn
Bi                     SVtn
Do project if: NPV = ∑ (1+r)i-t*
t*
r             - Kt* +
(1+r)t n- t* > 0
K0       K1   K2                          i=t*+1
tn
K                                                                         Bi                     SVtn
Do not do project if: NPV r = ∑
t*
- Kt* +
(1+r)t n- t*       <0
i=t*+1 (1+r)
i-t*
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. Machinery
2. SV > 0 but      ΔSV > 0, e.g. Land
3. SV < 0, but     ΔSV = 0, e.g. A nuclear plant
4. SV < 0, but     ΔSV > 0, e.g. Severance pay for workers
5. SV < 0 and      ΔSV < 0 e.g. Clean-up costs
Timing of Projects:
When The Patterns of Both Potential Benefits and Costs
Depend on Time of Starting Project

Benefits and Costs

Benefits From K1     D
C
B

A

Benefits From K0

0    t0        t1         t2                 tn           tn+1

K0        K1

K0

K1
NPV FOR THE BASE SCENARIO
WITH DIFFERENT STARTING YEARS
(thousands of 1998 US\$)

Beginning Construction Year   Financial   Economic    Economic
(Operation of Bridge)                  Argentina   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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