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