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[object Object],[object Object],[object Object],CHAPTER 8 Time Value of Money
[object Object],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.
Time line for a $100 lump sum due at the end of Year 2. 100 0 1 2  Year i%
Time line for an ordinary annuity of $100 for 3 years. 100 100 100 0 1 2 3 i%
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
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
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.
After 3 years: FV 3 = PV(1 + i) 3 = $100(1.10) 3 =  $133.10. In general, FV n = PV(1 + i) n .
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 = ?
Solve FV n  = PV(1 + i ) n  for PV:   PV  = $100 1 1.10 = $100 0.7513 = $75.13.       3
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
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
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
Find the FV and PV if the annuity were an annuity due. 100 100 0 1 2 3 10% 100
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
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.
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
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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
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%.
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      
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
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
What’s the PV of this stream? 0 100 1 5% 2 3 100 100 90.70 82.27 74.62 247.59

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Time value of money

  • 1.
  • 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.
  • 19.
  • 20.
  • 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