![C. Variable Discount Rates
• Adjustment of Cost of Funds Through Time
9
•For variable discount rates r0, r1, r2, & r3 in years 0, 1, 2, and 3, the discount
factors are, respectively, as follows:
1 , 1/(1+r0), 1/[(1+r0)(1+r1)] & 1/[(1+r0)(1+r1)(1+r2)]
0 1 2 3 4 5
r0
r1
r2
r3
r4
r5
r *
4
r *
3
r *
2
r *
1
r *
0
If funds currently are
abnormally scarce
Normal or historical
average cost of funds
If funds currently are
abnormally abundant
Years from
present period](https://image.slidesharecdn.com/fina521lecture3investmentcriteria2013-10-09-1-141224153333-conversion-gate01/85/Fina521-lecture-3_investment_criteria_2013-10-09-1-9-320.jpg)
Fina521 lecture 3_investment_criteria_2013-10-09-1
- 2. Discounting and Alternative Investment Criteria 2
- 3. How do we know this cash flow profile is good project? We need an evaluation criteria. 3 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 BenefitsLessCosts (-) (+) Year of Project Life Initial Investment Period Operating Stage Residual Value Project Life
- 4. ALTERNATIVE INVESTMENT CRITERIA 1. Net Present Value (NPV) 2. Benefit-Cost Ratio (BCR) 3. Pay-out or Pay-back Period 4. Internal Rate of Return (IRR) 5. Debt Service Coverage Ratios -ADSCR -LLCR 4
- 5. 1. The NPV is the algebraic sum of the discounted values of the incremental expected positive and negative net cashflows over a project’s anticipated lifetime. 2. What does net present value mean? – Measures the change in wealth created by the project. – If this sum is equal to zero, then investors can expect to recover their incremental investment and to earn a rate of return on their capital equal to the private cost of funds used to compute the present values. In this case there is no change in wealth. – Investors would be no further ahead with a zero-NPV project than they would have been if they had left the funds in the capital market. 5 Alternative Investment Criteria 1. Net Present Value (NPV)
- 6. Discounting and Net Present Value Criterion Basic Concepts: A. Discounting • Recognizes time value of money a. Funds when invested yield a return b. Future consumption worth less than present consumption 6 PVB = (B o /(1+r) o +(B 1/(1+r)1+.…….+(Bn /(1+r) n PVC = (C o /(1+r) +(C 1/(1+r)1+.…….+(Cn /(1+r) o o r r NPV = (B o-Co)/(1+r) o+(B1-C1)/(1+r) 1+.…….+(B n-Cn)/(1+r) n o n o r
- 7. Discounting and Net Present Value Criteria (Cont’d) B. Cumulative Values • The calendar year to which all projects are discounted to is important • All mutually exclusive projects need to be compared as of same calendar year 7 If NPV = (Bo-Co)(1+r)1 +(B1-C1) +..+..+(B n-Cn)/(1+r)n-1 and NPV = (Bo-Co)(1+r)3 +(B1-C1)(1+r)2 +(B2-C2)(1+r)+(B 3-C3)+...(Bn-Cn)/(1+r) n-3 Then NPV = (1+r) 2 NPV 1 r 3 r 3 r 1 r
- 8. 8 Year 0 1 2 3 4 Net Cash Flow -1000 200 300 350 1440 Example of Discounting (10% Discount Rate) 25.676 )1.1( 1440 )1.1( 350 )1.1( 300 1.1 200 1000PV 432 0 1.0 =++++−= 88.743 )1.1( 1440 )1.1( 350 1.1 300 200)1.1(1000PV 32 1 1.0 =++++−= 26.818 )1.1( 1440 )1.1( 350 300)1.1(200)1.1(1000PV 21 22 1.0 =++++−= Note: All of the transactions are done at the beginning of the year.
- 9. C. Variable Discount Rates • Adjustment of Cost of Funds Through Time 9 •For variable discount rates r0, r1, r2, & r3 in years 0, 1, 2, and 3, the discount factors are, respectively, as follows: 1 , 1/(1+r0), 1/[(1+r0)(1+r1)] & 1/[(1+r0)(1+r1)(1+r2)] 0 1 2 3 4 5 r0 r1 r2 r3 r4 r5 r * 4 r * 3 r * 2 r * 1 r * 0 If funds currently are abnormally scarce Normal or historical average cost of funds If funds currently are abnormally abundant Years from present period
- 10. 10 Year 0 1 2 3 4 Net Cash Flow -1000 200 300 350 1440 r 18% 16% 14% 12% 10% Example of Discounting (multiple rates) 55.515 )12.1)(14.1)(16.1( 1440 )14.1)(16.1( 350 16.1 300 200)18.1(10001 =++++−=NPV 04.598 )12.1)(14.1( 1440 )14.1( 350 300)16.1(200)16.1)(18.1(10002 =++++−=NPV 91.436 )12.1)(14.1)(16.1)(18.1( 1440 )14.1)(16.1)(18.1( 350 )16.1)(18.1( 300 18.1 200 10000 =++++−=NPV Note: All of the transactions are done at the beginning of the year.
- 11. Net Present Value (NPV) • Used as a decision criterion to answer following: a. When to reject projects? b. Select project(s) under a budget constraint? c. Compare mutually exclusive projects? d. How to choose between highly profitable mutually exclusive projects with different lengths of life? 11 Alternative Investment Criteria
- 12. Net Present Value Criterion a. When to Reject Projects? Rule: “Do not accept any project unless it generates a positive net present value when discounted by the opportunity cost of funds” Examples: Project A: Present Value Costs $1 million, NPV + $70,000 Project B: Present Value Costs $5 million, NPV - $50,000 Project C: Present Value Costs $2 million, NPV + $100,000 Project D: Present Value Costs $3 million, NPV - $25,000 Result: Only projects A and C are acceptable. The country is made worse off if projects B and D are undertaken. 12
- 13. Net Present Value Criterion (Cont’d) b. When You Have a Budget Constraint? Rule: “Within the limit of a fixed budget, choose that subset of the available projects which maximizes the net present value” Example: If budget constraint is $4 million and 4 projects with positive NPV: Project E: Costs $1 million, NPV + $60,000 Project F: Costs $3 million, NPV + $400,000 Project G: Costs $2 million, NPV + $150,000 Project H: Costs $2 million, NPV + $225,000 Result: Combinations FG and FH are impossible, as they cost too much. EG and EH are within the budget, but are dominated by the combination EF, which has a total NPV of $460,000. GH is also possible, but its NPV of $375,000 is not as high as EF. 13
- 14. Net Present Value Criterion (Cont’d) c. When You Need to Compare Mutually Exclusive Projects? Rule: “In a situation where there is no budget constraint but a project must be chosen from mutually exclusive alternatives, we should always choose the alternative that generates the largest net present value” Example: Assume that we must make a choice between the following three mutually exclusive projects: Project I: PV costs $1.0 million, NPV $300,000 Project J: PV costs $4.0 million, NPV $700,000 Projects K: PV costs $1.5 million, NPV $600,000 Result: Projects J should be chosen because it has the largest NPV. 14
- 15. 2. Benefit-Cost Ratio (R) • As its name indicates, the benefit-cost ratio (R), or what is sometimes referred to as the profitability index, is the ratio of the PV of the net cash inflows (or economic benefits) to the PV of the net cash outflows (or economic costs): 15 )( )( CostsEconomicorOutflowsCashofPV BenefitsEconomicorInflowsCashofPV R = Alternative Investment Criteria
- 16. Benefit-Cost Ratio (Cont’d) Basic rule: If benefit-cost ratio (R) >1, then the project should be undertaken. Problems? Sometimes it is not possible to rank projects with the Benefit-Cost Ratio • Mutually exclusive projects of different sizes • Mutually exclusive projects and recurrent costs subtracted out of benefits or benefits reported gross of operating costs • Not necessarily true that if RA>RB then project “A” is better 16
- 17. 17 Benefit-Cost Ratio (Cont’d) First Problem: The Benefit-Cost Ratio Does Not Adjust for Mutually Exclusive Projects of Different Sizes. For example: Project A: PV0 of Costs = $5.0 M, PV0 of Benefits = $7.0 M NPV0 A = $2.0 M RA = 7/5 = 1.4 Project B: PV0 of Costs = $20.0 M, PV0 of Benefits = $24.0 M NPV0 B = $4.0 M RB = 24/20 = 1.2 According to the Benefit-Cost Ratio criterion, project A should be chosen over project B because RA>RB, but the NPV of project B is greater than the NPV of project A. So, project B should be chosen. 17
- 18. 18 Second Problem: The Benefit-Cost Ratio Does Not Adjust for Mutually Exclusive Projects and Recurrent Costs Subtracted Out of Benefits or Benefits Reported As Gross of Operating Costs. For example: Project A: PV0 Total Costs= $5.0 M PV0 Recurrent Costs = $1.0 M (i.e. Fixed Costs= $4.0 M) PV0 of Gross Benefits= $7.0 M RA = (7-1)/(5-1) = 6/4 = 1.5 When not subtracting the recurrent costs: RA = 7/5 = 1.4 Project B: Total Costs= $20.0 M Recurrent Costs= $18.0 M (i.e. Fixed Costs= $2.0 M) PV0 of Gross Benefits= $24.0 M RB = (24-18)/(20-18) = 6/2 =3 When not subtracting the recurrent costs: RB = 24/20 = 1.2 Hence, project B should be chosen over project A under Benefit-Cost Criterion. Conclusion: The Benefit-Cost Ratio should not be used to rank projects
- 19. 3. Pay-out or Pay-back period • The pay-out period measures the number of years it will take for the undiscounted net benefits (positive net cashflows) to repay the investment. • A more sophisticated version of this rule compares the discounted benefits over a given number of years from the beginning of the project with the discounted investment costs. • An arbitrary limit is set on the maximum number of years allowed and only those investments having enough benefits to offset all investment costs within this period will be acceptable. 19 Alternative Investment Criteria
- 20. • Project with shortest payback period is preferred by this criteria Comparison of Two Projects With Differing Lives Using Pay-Out Period 20 Bt - Ct Ba Bb ta tb Ca = Cb Payout period for project a Payout period for project b 0 Time
- 21. Pay-Out or Pay-Back Period • Assumes all benefits that are produced by in longer life project have an expected value of zero after the pay-out period. • The criteria may be useful when the project is subject to high level of political risk. 21
- 22. 4. Internal Rate of Return (IRR) • IRR is the discount rate (K) at which the present value of benefits are just equal to the present value of costs for the particular project: Bt - Ct (1 + K)t Note: the IRR is a mathematical concept, not an economic or financial criterion 22 = 0Σ n t=0 Alternative Investment Criteria
- 23. Common uses of IRR: (a) If the IRR is larger than the cost of funds then the project should be undertaken (b) Often the IRR is used to rank mutually exclusive projects. The highest IRR project should be chosen • An advantage of the IRR is that it only uses information from the project 23
- 24. Difficulties with the Internal Rate of Return Criterion First Difficulty: Multiple rates of return for project Solution 1: K = 100%; NPV= -100 + 300/(1+1) + (-200)/(1+1)2 = 0 Solution 2: K = 0%; NPV= -100+300/(1+0)+(-200)/(1+0)2 = 0 24 +300 Bt - Ct -200 -100 Time
- 25. 25 Second difficulty: Projects of different sizes and also mutually exclusive Year 0 1 2 3 ... ... Ґ Project A -2,000 +600 +600 +600 +600 +600 +600 Project B -20,000 +4,000 +4,000 +4,000 +4,000 +4,000 +4,000 NPV and IRR provide different Conclusions: Opportunity cost of funds = 10% NPV : 600/0.10 - 2,000 = 6,000 - 2,000 = 4,000 NPV : 4,000/0.10 - 20,000 = 40,000 - 20,000 = 20,000 Hence, NPV > NPV IRRA : 600/KA - 2,000 = 0 or K A = 0.30 IRRB : 4,000/K B - 20,000 = 0 or K B = 0.20 Hence, K A>KB 0 B 0 A 0 B 0 A Difficulties with the Internal Rate of Return Criterion (Cont’d)
- 26. 26 Third difficulty: Projects of different lengths of life and mutually exclusive Opportunity cost of funds = 8% Project A: Investment costs = 1,000 in year 0 Benefits = 3,200 in year 5 Project B: Investment costs = 1,000 in year 0 Benefits = 5,200 in year 10 NPV : -1,000 + 3,200/(1.08)5 = 1,177.86 NPV : -1,000 + 5,200/(1.08)10 = 1,408.60 Hence, NPV > NPV IRRA : -1,000 + 3,200/(1+KA)5 = 0 which implies that KA = 0.262 IRRB : -1,000 + 5,200/(1+KB)10 = 0 which implies that KB = 0.179 Hence, KA>KB 0 B 0 A 0 B 0 A Difficulties with the Internal Rate of Return Criterion (Cont’d)
- 27. 27 Fourth difficulty: Same project but started at different times Project A: Investment costs = 1,000 in year 0 Benefits = 1,500 in year 1 Project B: Investment costs = 1,000 in year 5 Benefits = 1,600 in year 6 NPVA : -1,000 + 1,500/(1.08) = 388.88 NPVB : -1,000/(1.08) 5 + 1,600/(1.08) 6 = 327.68 Hence, NPV > NPV IRRA : -1,000 + 1,500/(1+K A) = 0 which implies that KA= 0.5 IRRB : -1,000/(1+K B)5+ 1,600/(1+K B)6= 0 which implies that KB = 0.6 Hence, K B >KA 0 B 0 A Difficulties with the Internal Rate of Return Criterion (Cont’d) 0 0
- 28. Year 0 1 2 3 4 Project A 1000 1200 800 3600 -8000 IRR A 10% Compares Project A and Project B ? Project B 1000 1200 800 3600 -6400 IRR B -2% Project B is obviously better than A, yet IRR A > IRR B Project C 1000 1200 800 3600 -4800 IRR C -16% Project C is obviously better than B, yet IRR B > IRR C Project D -1000 1200 800 3600 -4800 IRR D 4% Project D is worse than C, yet IRR D > IRR C Project E -1325 1200 800 3600 -4800 IRR E 20% Project E is worse than D, yet IRR E > IRR D 28 IRR FOR IRREGULAR CASHFLOWS For Example: Look at a Private BOT Project from the perspective of the Government