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Chapter 5 THE IMPORTANCE OF SCALE AND TIMING IN PROJECT APPRAISAL

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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

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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.

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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 B2 B3 B t - C t Time 0 NPV (B 1 – C 1 ) 0 ? NPV (B2 – C2) 0 ? NPV (B 3 – C 3 ) 0 ?

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Determination of Scale of Project Relationship between net present value and scale NPV A Scale of Project 0 BCEDFGHIJLKMN (+) (-) NPV of Project

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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.

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Table 5-1 Determination 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 ScaleNPV 10%IRR Opportunity cost of funds (discount rate) = 10%

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Table 5-1 Determination 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 ScaleNPV 10%IRR Opportunity cost of funds (discount rate) = 10%

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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. NPV 1-5 = 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.

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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 Rate between 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 5-3 Relationship between MIRR, IRR and DR Scale IRR>r SnSn MIRR

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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 after Scale N: NPV is negative and it decreases $3010 $3000 Scale NPV (+) NPV(0.10) NPV(0.14) NPV(0.10) 0 S4S4 S5S5 SnSn 0 NPV (-) MNPV (0.10) Percent S3S3 Maximum NPV Maximum MNPV

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Figure 5-5 Relationship between MIRR, IRR, MNPV and NPV

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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

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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.

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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

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Case 1: Timing of Projects: When Potential Benefits Are a Continuously Rising Function of Calendar Time but Are Independent of Time of Starting Project rK IDE Time t0t0 t2t2 AC K B (t) rK t B t+1 rK t > B t+1 Postpone rK t < B t+1 Start <><> t1t1 K B1B1 Benefits and Costs

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5.52 4.09 3.25 2.19 1.98 1.80 1.65 1.52 1.41 1.31 1.22 1.14 1.07 1.03 0.00 1.00 2.00 3.00 4.00 5.00 6.00 0123456789101112131415 Timing for Start of Operation of Roojport Dam, South Africa of Marginal Economic Unit Water Cost Numbers of Years Postponed Economic Water Cost Rand/m 3

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Case 2: Timing of Projects: When Both Potential Benefits and Investments Are A Function of Calendar Time rK t < B t+1 + (K t+1 -K t ) Start rK t >B t+1 + (K t+1 -K t ) Postpone rK 0 DE Time AC B (t) B1B1 t2t2 t3t3 K1K1 K0K0 K1K1 F IH t1t1 K0K0 G B2B2 0 Benefits and Costs

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Case 3: Timing of Projects: When Potential Benefits Rise and Decline According to Calendar Time Time rK A C K0K0 B K Start if: rK t* < B t*+1 B (t) 0 K1K1 K2K2 I rSV t0t0 t1t1 t* tntn t n+1 SV Benefits and Costs Stop if: rSV t - B(t n+1 ) - ΔSV t > 0 ; SV t = SV t - SV t n n+1 n Do project if: NPV = ∑ i=t*+1 tntn > 0 - K t* + SV t n (1+r) t - t* n t* r (1+r) i-t* BiBi Do not do project if: NPV = ∑ t* r <0 i=t*+1 tntn (1+r) i-t* BiBi - K t* + SV t n (1+r) t - t* n

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The Decision Rule If (rSV t - B t - ΔSV t ) > 0 Stop (ΔSV t = SV t - SV t ) < 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, e.g. Severance pay for workers 5. SV < 0 and ΔSV < 0 e.g. Clean-up costs

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Timing of Projects: When The Patterns of Both Potential Benefits and Costs Depend on Time of Starting Project t0t0 t1t1 t2t2 A C K0K0 B K0K0 Benefits From K 1 K1K1 0 tntn t n+1 K1K1 D Benefits From K 0 Benefits and Costs

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