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FINE 3010-01 Financial Management

FINE 3010-01 Financial Management. Instructor: Rogério Mazali Lecture 05: 09/21/2011. Agenda. Multiple Cash Flows FV of Multiple Cash Flows PV of Multiple Cash Flows Perpetuities How to Value Perpetuities How to Value Annuities Effective Annual Interest Rates

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FINE 3010-01 Financial Management

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  1. FINE 3010-01Financial Management

    Instructor: RogérioMazali Lecture 05: 09/21/2011
  2. Agenda Multiple Cash Flows FV of Multiple Cash Flows PV of Multiple Cash Flows Perpetuities How to Value Perpetuities How to Value Annuities Effective Annual Interest Rates Real vs. Nominal Cash Flows Inflation and Interest Rates Valuing Real Cash Payments Real or Nominal?
  3. Multiple Cash Flows Consider the following problem: Receive stream of cash flows as follows Assume r = 10% How to calculate the Present Value of such CF stream? A: Just calculate the PV of each CF and then sum them up. Year 0 Year 1 Year 2 Year 3 Year 4 PV0 = ??? $200 $350 $150 $100
  4. Multiple Cash Flows Year 0 Year 1 Year 2 Year 3 Year 4 PV0 = ??? $200 $350 $150 $100 $200/(1.10) $350/(1.10)2 $150/(1.10)3 $100/(1.10)4
  5. Multiple Cash Flows Year 0 Year 1 Year 2 Year 3 Year 4 PV0 = ??? $200 $350 $150 $100 $ 181.82 $289.26 $ 112.70 $ 68.30 $652.08
  6. Multiple Cash Flows More generally, if you have T periods: Year 0 Year 1 Year 2 … Year T PV0 = ??? CF1 CF2 … CFT
  7. Multiple Cash Flows Consider the following problem: Receive stream of cash flows as follows Assume r = 10% How to calculate the Future Value of such CF stream? A: Just calculate the FV of each CF and then sum them up. Year 0 Year 1 Year 2 Year 3 Year 4 $100 $200 $350 $150 FV4 = ???
  8. Multiple Cash Flows Year 0 Year 1 Year 2 Year 3 Year 4 $100 $200 $350 $150 FV4 = ??? $150 × (1.10) $350 × (1.10)2 $200 × (1.10)3 $100 × (1.10)4
  9. Multiple Cash Flows Year 0 Year 1 Year 2 Year 3 Year 4 $100 $200 $350 $150 FV4 = ??? $ 165.00 $423.50 $266.20 $146.41 $1,001.11
  10. Multiple Cash Flows More generally, if you have T periods: Year 0 Year 1 Year 2 … Year T CF0 CF1 CF2 … FVT
  11. Example You are planning to climb Mt. Everest next summer. You Need $65,000 to make the trip. How much money should I put in my bank account at time 2 so to be able to afford the 65,000 at time 2, if the discount rate is 2.5%? Cash Flows: Year 0 Year 1 Year 2 Year 3 Year 4 $20,000 $10,000 ????? FV3 = $65,000
  12. Example This is a typical Future Value problem. In algebraic terms,
  13. Multiple Discount Rates Consider the following problem: Issue: The discount rates change over time Year 0 Year 1 Year 2 Year 3 PV0 = ??? $10 $20 $30 r = 10% r = 15% r = 20%
  14. Compounding Periods The interest rate has been quoted per year. However, there is no reason to think that this should be always the case. Cash flows are usually compounded over periods other than annually Example: You borrow for one year at a semiannual compounded rate of 5%. Question: How much should you give at the end of the year? Is 10% the actual interest rate you are paying?
  15. Compounding Periods You borrow $100 for one year compounded semiannually at the semiannual percentage rate of 5%. FV1-year = $100 * (1 + 0.05)2 = $110.25 Therefore, the Effective Annual Rate you are paying is ($110.25/$100) – 1 = 10.25% Another way to write it: (1 + rAnnual) = (1 + rSemi-Annual)2 rAnnual = (1 + 0.05)2 -1 = 10.25%
  16. APR – Annual Percentage Rate Example: You borrow $100 for one year from your credit card at an APR of 12%. How much money should you give back? What is the effective annual rate? APR is Different from Effective Annual Rate. Effective annual interest rate: yearly rate that will give you the same amount of money as you obtain with the monthly rate after 1 year (1 + effective annual rate) = (1 + monthly rate)12 Annual Percentage Rates (APRs): approximatedyearly rate: APR = 12 × monthly rate
  17. APR – Annual Percentage Rate Step 1: Find the Monthly Interest Rate rmonthly = APR/12 = 12%/12 = 1% Step 2: Convert it to Effective Annual Rate rannual = (1 + 0.01)12-1 = 12.62% Step 3: Find the Future Value FV1-year= $100 × (1+ rannual) = $100 × (1.1262) = $112.62
  18. Compounding (r=10%) Annual: |-----------------------------------------------| 100 110.00 Semi-annual: |-----------------------|-----------------------| 100 105 110.25 Quarterly: |----------|------------|------------|------------| 100 102.5 105.06 107.69 110.38
  19. Effective Annual Interest Rates
  20. Compounding Periods The relationship between present and future value when interest is compounded M times per year: FV1 = PV × ( 1+ r / M)M The relationship between present and future value for N years when interest is compounded M times per year: FVN = PV × ( 1+ r /M)M × N
  21. Example Find the PV of $500 received in the future under the following conditions: 12% nominal rate, semiannual compounding, 5 years 12% nominal rate, quarterly compounding, 5 years
  22. Continuous Compounding As M approaches infinity, (1+ r/M)M approaches er=(2.718)r So (1+ r/M)MN approaches erN=(2.718)rN Example: The future value of $100 continuously compounded at 10% for one year is : FV1-year = 100*e0.1 = 110.517
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