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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.

B3

B2

B1

0

Time

C1

C2

C3

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

NPV (B1 – C1) 0 ?

NPV (B2 – C2) 0 ?

NPV (B3 – C3) 0 ?

(+)

NPV of Project

0

Scale of Project

A

B

C

D

E

F

G

H

I

J

K

L

M

N

(-)

Determination of Scale of Project- Relationship between net present value and scale

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.

Determination of Optimum Scale of Irrigation Dam (Cont’d)

Year

Scale

0 1 2 3 4 5 -

Costs Benefits

NPV 10% IRR

S0

S1

S2

S3

S4

S5

S6

-3000

-4000

-5000

-6000

-7000

-8000

-9000

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

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

Opportunity cost of funds (discount rate) = 10%

Determination of Optimum Scale of Irrigation Dam

Year

Scale

0 1 2 3 4 5 -

Costs Benefits

NPV 10% IRR

S0

S1

S2

S3

S4

S5

S6

-3000

-4000

-5000

-6000

-7000

-8000

-9000

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

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

Opportunity cost of funds (discount rate) = 10%

Note:

- NPV of last increment to scale 0 at scale S5. i.e. NPV of scale 5 = 10.
- NPV of project is maximized at scale of 5, i.e. NPV1-5 = 3010.
- IRR is maximized at scale 4.
- When the IRR on the last increment to scale (MIRR) is equal to discount rate the NPV of project is maximized.

Maximum

MIRR

Maximum

IRR (0.14)

IRR>r

MIRR>r

Discount Rate (r) Opp. Cost of Funds (0.10)

MIRR<r

Sn

S4

Scale

S3

S5

Figure 5-3

Relationship between MIRR, IRR and DR

- 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

- 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

- at Scale 5: MIRR is equal to Discount Rate

between Scale 5 and Scale N: MIRR is smaller than IRR; MIRR is smaller than r; IRR is greater than r

- at some Scale N: IRR is equal to Discount Rate

Relationship between MNPV and NPV

- at Scale 3: Maximum point of MNPV ($3000) at 0.10 Discount rate
- 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

- 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

- at some Scale N: NPV is equal to zero
- after Scale N: NPV is negative and it decreases

NPV (+)

Maximum NPV

$3010

$3000

Maximum MNPV

NPV(0.10)

Percent

S5

Scale

Sn

S3

S4

0

NPV(0.14)

NPV (-)

MNPV (0.10)

NPV(0.10) 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 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

B (t)

I

D

E

rK

A

C

B1

Time

t0

t1

t2

rKt Bt+1

<>

K

rKt > Bt+1 Postpone

rKt < Bt+1 Start

K

Case 1: Timing of Projects:When Potential Benefits Are a Continuously Rising Function of Calendar Time but Are Independent of Time of Starting Project

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

3.25

3.00

2.00

2.19

Economic Water Cost Rand/m3

1.98

1.80

1.65

1.52

1.41

1.31

1.22

1.00

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

B (t)

D

E

rK0

A

C

B1

B2

0

Time

t1

t2

t3

K0

K1

rKt >Bt+1+ (Kt+1-Kt) Postpone

rKt < Bt+1 + (Kt+1-Kt) Start

K0

F

G

K1

I

H

Case 2: Timing of Projects: When Both Potential Benefits and Investments AreA Function of Calendar Time

SV

B

C

rK

I

A

rSV

B (t)

0

Time

t0

t1

tn

tn+1

t*

Start if: rKt* < Bt*+1

Stop if: rSVt - B(tn+1) - ΔSVt > 0 ; SVt = SVt -SVt

n

n

n+1

n+1

n+1

SVt

SVt

tn

tn

Bi

Bi

> 0

n

n

t*r

Do project if: NPV = ∑

- Kt* +

(1+r)t - t*

(1+r)t - t*

(1+r)i-t*

(1+r)i-t*

n

n

K0

K1

K2

i=t*+1

i=t*+1

K

<0

t*r

Do not do project if: NPV = ∑

- Kt* +

Case 3: Timing of Projects: When Potential Benefits Rise and Decline According to Calendar Time

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

D

Benefits From K1

C

B

A

Benefits From K0

0

t0

t1

t2

tn

tn+1

K0

K1

K0

K1

Timing of Projects:When The Patterns of Both Potential Benefits and CostsDepend on Time of Starting Project

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