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# Time value of money

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Time value of money

Time value of money

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• 1. <ul><li>Future value </li></ul><ul><li>Present value </li></ul><ul><li>Rates of return </li></ul>CHAPTER 8 Time Value of Money
• 2. <ul><li>Time lines show timing of cash flows. </li></ul>CF 0 CF 1 CF 3 CF 2 0 1 2 3 i% Tick marks at ends of periods, so Time 0 is today; Time 1 is the end of Period 1; or the beginning of Period 2.
• 3. Time line for a \$100 lump sum due at the end of Year 2. 100 0 1 2 Year i%
• 4. Time line for an ordinary annuity of \$100 for 3 years. 100 100 100 0 1 2 3 i%
• 5. Time line for uneven CFs: -\$50 at t = 0 and \$100, \$75, and \$50 at the end of Years 1 through 3. 100 50 75 0 1 2 3 i% -50
• 6. What’s the FV of an initial \$100 after 3 years if i = 10%? FV = ? 0 1 2 3 10% Finding FVs (moving to the right on a time line) is called compounding. 100
• 7. After 1 year: FV 1 = PV + INT 1 = PV + PV (i) = PV(1 + i) = \$100(1.10) = \$110.00. After 2 years: FV 2 = PV(1 + i) 2 = \$100(1.10) 2 = \$121.00.
• 8. After 3 years: FV 3 = PV(1 + i) 3 = \$100(1.10) 3 = \$133.10. In general, FV n = PV(1 + i) n .
• 9. What’s the PV of \$100 due in 3 years if i = 10%? 10% Finding PVs is discounting, and it’s the reverse of compounding. 100 0 1 2 3 PV = ?
• 10. Solve FV n = PV(1 + i ) n for PV:   PV = \$100 1 1.10 = \$100 0.7513 = \$75.13.       3
• 11. Ordinary Annuity PMT PMT PMT 0 1 2 3 i% PMT PMT 0 1 2 3 i% PMT Annuity Due What’s the difference between an ordinary annuity and an annuity due ? PV FV
• 12. What’s the FV of a 3-year ordinary annuity of \$100 at 10%? 100 100 100 0 1 2 3 10% 110 121 FV = 331
• 13. What’s the PV of this ordinary annuity? 100 100 100 0 1 2 3 10% 90.91 82.64 75.13 248.69 = PV
• 14. Find the FV and PV if the annuity were an annuity due. 100 100 0 1 2 3 10% 100
• 15. What is the PV of this uneven cash flow stream? 0 100 1 300 2 300 3 10% -50 4 90.91 247.93 225.39 -34.15 530.08 = PV
• 16. 0 1 2 3 10% 0 1 2 3 5% 4 5 6 134.01 100 133.10 1 2 3 0 100 Annually: FV 3 = \$100(1.10) 3 = \$133.10. Semiannually: FV 6 = \$100(1.05) 6 = \$134.01.
• 17. We will deal with 3 different rates: i Nom = nominal, or stated, or quoted, rate per year. i Per = periodic rate. EAR = EFF% = . effective annual rate
• 18. <ul><li>i Nom is stated in contracts. Periods per year (m) must also be given. </li></ul><ul><li>Examples: </li></ul><ul><ul><li>8%; Quarterly </li></ul></ul><ul><ul><li>8%, Daily interest (365 days) </li></ul></ul>
• 19. <ul><li>Periodic rate = i Per = i Nom /m, where m is number of compounding periods per year. m = 4 for quarterly, 12 for monthly, and 360 or 365 for daily compounding. </li></ul><ul><li>Examples: </li></ul><ul><ul><li>8% quarterly: i Per = 8%/4 = 2%. </li></ul></ul><ul><ul><li>8% daily (365): i Per = 8%/365 = 0.021918%. </li></ul></ul>
• 20. <ul><li>Effective Annual Rate (EAR = EFF%): </li></ul><ul><li>The annual rate which causes PV to grow to the same FV as under multi-period compounding. </li></ul><ul><li>Example: EFF% for 10%, semiannual: </li></ul><ul><ul><li>FV = (1 + i Nom /m) m </li></ul></ul><ul><li>= (1.05) 2 = 1.1025. </li></ul><ul><li> EFF% = 10.25% because </li></ul><ul><ul><li> (1.1025) 1 = 1.1025. </li></ul></ul><ul><li>Any PV would grow to same FV at 10.25% annually or 10% semiannually. </li></ul>
• 21. How do we find EFF% for a nominal rate of 10%, compounded semiannually? Or use a financial calculator. = - 1.0 ( 1 + ) 0.10 2 2 = (1.05) 2 - 1.0 = 0.1025 = 10.25%. EFF% = - 1 ( 1 + ) i Nom m m
• 22. EAR = EFF% of 10% EAR Annual = 10%. EAR Q = (1 + 0.10/4) 4 - 1 = 10.38%. EAR M = (1 + 0.10/12) 12 - 1 = 10.47%. EAR D(360) = (1 + 0.10/360) 360 - 1 = 10.52%.
• 23. FV of \$100 after 3 years under 10% semiannual compounding? Quarterly? = \$100(1.05) 6 = \$134.01. FV 3Q = \$100(1.025) 12 = \$134.49. FV = PV 1 . + i m n Nom mn       FV = \$100 1 + 0.10 2 3S 2x3      
• 24. What’s the value at the end of Year 3 of the following CF stream if the quoted interest rate is 10%, compounded semiannually? 0 1 100 2 3 5% 4 5 6 6-mos. periods 100 100
• 25. Could you find the FV with a financial calculator? Yes, by following these steps: a. Find the EAR for the quoted rate: 2nd Method: Treat as an Annuity EAR = ( 1 + ) - 1 = 10.25%. 0.10 2 2
• 26. What’s the PV of this stream? 0 100 1 5% 2 3 100 100 90.70 82.27 74.62 247.59