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The document discusses the time value of money concepts of future value and present value for single sums and annuities. It provides examples of calculating future value and present value for single cash flows using formulas and a financial calculator. It also discusses the differences between ordinary annuities, where cash flows occur at the end of periods, and annuity dues, where cash flows occur at the beginning of periods.

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The Time Value of Money References: Keown, 2005, Financial Management: Principles and Applications, 10th ed., Prentice Hall Ross, Westerfield, and Jordan, 2006, Fundamentals of Corporate Finance, 7th ed., McGraw-Hill Brigham & Houston, Fundamentals of Financial Management, 8th ed. Prepared By: Liem Pei Fun, S.E., MCom.
About me  1998-2002 Bachelor of Economics (Majoring in Finance), Petra Christian University  2004-2005 Postgraduate Diploma in Finance, The University of Melbourne  2005-2006 Master of Commerce (Finance), The University of Melbourne  Pass CFA Exam Level I  Bloomberg Global Product Certification – Equity Level One  Head of Finance Program, Faculty of Economics, Petra Christian University
OUTLINE  Future Value Single Sum  Present Value Single Sum  Future Value Annuities  Present Value Annuities  Ordinary Annuity vs Annuity Due  Perpetuities  Practice Problems
The Time Value of Money  2002, Prentice Hall, Inc.
Compounding and Discounting Single Sums
We know that receiving $1 today is worth more than $1 in the future. This is due to opportunity costs. The opportunity cost of receiving $1 in the future is the interest we could have earned if we had received the $1 sooner. Today Future
If we can measure this opportunity cost, we can:  Translate $1 today into its equivalent in the future (compounding).  Translate $1 in the future into its equivalent today (discounting). ? Today Future Today ? Future
Future Value
Future Value - single sums If you deposit $100 in an account earning 6%, how much would you have in the account after 1 year? Mathematical Solution: (Arithmetic Method) FV = PV (1 + i)n FV = 100 (1.06)1 = $106 0 1 PV = -100 FV = 106
Calculator Keys  Texas Instruments BA-II Plus  FV = future value  PV = present value  I/Y = period interest rate  P/Y must equal 1 for the I/Y to be the period rate  Interest is entered as a percent, not a decimal  N = number of periods  Remember to clear the registers (CLR TVM) after each problem  Other calculators are similar in format
Future Value - single sums If you deposit $100 in an account earning 6%, how much would you have in the account after 1 year? Calculator Solution: P/Y = 1 I/Y = 6 N = 1 PV = -100 FV = $106 0 1 PV = -100 FV = 106
Future Value - single sums If you deposit $100 in an account earning 6%, how much would you have in the account after 5 years? Mathematical Solution: (Arithmetic Method) FV = PV (1 + i)n FV = 100 (1.06)5 = $133.82 0 5 PV = -100 FV = 133.82
Future Value - single sums If you deposit $100 in an account earning 6%, how much would you have in the account after 5 years? Calculator Solution: P/Y = 1 I/Y = 6 N = 5 PV = -100 FV = $133.82 0 5 PV = -100 FV = 133.82
Mathematical Solution: (Arithmetic Method) FV = PV (1 + i/m) m x n FV = 100 (1.015)20 = $134.68 0 20 PV = -100 FV = 134.68 Future Value - single sums If you deposit $100 in an account earning 6% with quarterly compounding, how much would you have in the account after 5 years?
Calculator Solution: P/Y = 4 I/Y = 6 N = 20 PV = -100 FV = $134.68 0 20 PV = -100 FV = 134.68 Future Value - single sums If you deposit $100 in an account earning 6% with quarterly compounding, how much would you have in the account after 5 years?
Mathematical Solution: (Arithmetic Method) FV = PV (1 + i/m) m x n FV = 100 (1.005)60 = $134.89 0 60 PV = -100 FV = 134.89 Future Value - single sums If you deposit $100 in an account earning 6% with monthly compounding, how much would you have in the account after 5 years?
Calculator Solution: P/Y = 12 I/Y = 6 N = 60 PV = -100 FV = $134.89 0 60 PV = -100 FV = 134.89 Future Value - single sums If you deposit $100 in an account earning 6% with monthly compounding, how much would you have in the account after 5 years?
Mathematical Solution: (Arithmetic Method) FV = PV (e in) FV = 1000 (e .08x100) = 1000 (e 8) FV = $2,980,957.99 Future Value - continuous compounding What is the FV of $1,000 earning 8% with continuous compounding, after 100 years? 0 100 PV = -1000 FV = $2.98m
Present Value
Mathematical Solution: (Arithmetic Method) PV = FV / (1 + i)n PV = 100 / (1.06)1 = $94.34 PV = -94.34 FV = 100 0 1 Present Value - single sums If you receive $100 one year from now, what is the PV of that $100 if your opportunity cost is 6%?
Calculator Solution: P/Y = 1 I/Y = 6 N = 1 FV = 100 PV = -94.34 PV = -94.34 FV = 100 0 1 Present Value - single sums If you receive $100 one year from now, what is the PV of that $100 if your opportunity cost is 6%?
Mathematical Solution: (Arithmetic Method) PV = FV / (1 + i)n PV = 100 / (1.06)5 = $74.73 Present Value - single sums If you receive $100 five years from now, what is the PV of that $100 if your opportunity cost is 6%? 0 5 PV = -74.73 FV = 100
Calculator Solution: P/Y = 1 I/Y = 6 N = 5 FV = 100 PV = -74.73 Present Value - single sums If you receive $100 five years from now, what is the PV of that $100 if your opportunity cost is 6%? 0 5 PV = -74.73 FV = 100
Mathematical Solution: PV = FV / (1 + i)n 5,000 = 11,933 / (1+ i)5 .419 = ((1/ (1+i)5) 2.3866 = (1+i)5 (2.3866)1/5 = (1+i) i = .19 Present Value - single sums If you sold land for $11,933 that you bought 5 years ago for $5,000, what is your annual rate of return?
Calculator Solution: P/Y = 1 N = 5 PV = -5,000 FV = 11,933 I/Y = 19% 0 5 PV = -5000 FV = 11,933 Present Value - single sums If you sold land for $11,933 that you bought 5 years ago for $5,000, what is your annual rate of return?
Present Value - single sums Suppose you placed $100 in an account that pays 9.6% interest, compounded monthly. How long will it take for your account to grow to $500? Mathematical Solution: PV = FV / (1 + i)N 100 = 500 / (1+ .008)N 5 = (1.008)N ln 5 = ln (1.008)N ln 5 = N ln (1.008) 1.60944 = .007968 N N = 202 months
Calculator Solution:  P/Y = 12 FV = 500  I/Y = 9.6 PV = -100  N = 202 months Present Value - single sums Suppose you placed $100 in an account that pays 9.6% interest, compounded monthly. How long will it take for your account to grow to $500? 0 ? PV = -100 FV = 500
Compounding and Discounting Cash Flow Streams 0 1 2 3 4
What is the difference between an ordinary annuity and an annuity due? Ordinary Annuity PMT PMT PMT 0 1 2 3 i% PMT PMT 0 1 2 3 i% PMT Annuity Due
 a sequence of equal cash flows, occurring at the end of each period. 0 1 2 3 4 Ordinary Annuity
 If you buy a bond, you will receive equal semi-annual coupon interest payments over the life of the bond.  If you borrow money to buy a house or a car, you will pay a stream of equal payments. Examples of Annuities:
0 1 2 3 Future Value - annuity If you invest $1,000 each year at 8%, how much would you have after 3 years?
Future Value - annuity If you invest $1,000 each year at 8%, how much would you have after 3 years? Mathematical Solution: = $3,246.40                  08 . 0 1 ) 08 . 1 ( 000 , 1 1 ) 1 ( 3 FV i i PMT FV n
Calculator Solution: P/Y = 1 I/Y = 8 PMT = -1,000 N = 3 FV = $3,246.40 Future Value - annuity If you invest $1,000 each year at 8%, how much would you have after 3 years? 0 1 2 3 1000 1000 1000
0 1 2 3 Present Value - annuity What is the PV of $1,000 at the end of each of the next 3 years, if the opportunity cost is 8%?
Present Value - annuity What is the PV of $1,000 at the end of each of the next 3 years, if the opportunity cost is 8%? Mathematical Solution: 10 . 577 , 2 $ 08 . 0 ) 08 . 1 ( 1 1 000 , 1 ) 1 ( 1 1 3                               PV PV i i PMT PV n
Calculator Solution: P/Y = 1 I/Y = 8 PMT = -1,000 N = 3 PV = $2,577.10 0 1 2 3 1000 1000 1000 Present Value - annuity What is the PV of $1,000 at the end of each of the next 3 years, if the opportunity cost is 8%?
Ordinary Annuity vs. Annuity Due $1000 $1000 $1000 4 5 6 7 8
Begin Mode vs. End Mode 1000 1000 1000 4 5 6 7 8
Begin Mode vs. End Mode 1000 1000 1000 4 5 6 7 8 year year year 5 6 7
Begin Mode vs. End Mode 1000 1000 1000 4 5 6 7 8 year year year 5 6 7 PV in END Mode
Begin Mode vs. End Mode 1000 1000 1000 4 5 6 7 8 year year year 5 6 7 PV in END Mode FV in END Mode
Begin Mode vs. End Mode 1000 1000 1000 4 5 6 7 8 year year year 6 7 8
Begin Mode vs. End Mode 1000 1000 1000 4 5 6 7 8 year year year 6 7 8 PV in BEGIN Mode
Begin Mode vs. End Mode 1000 1000 1000 4 5 6 7 8 year year year 6 7 8 PV in BEGIN Mode FV in BEGIN Mode
Earlier, we examined this “ordinary” annuity: Using an interest rate of 8%, we find that:  The Future Value (at 3) is $3,246.40.  The Present Value (at 0) is $2,577.10. 0 1 2 3 1000 1000 1000
What about this annuity?  Same 3-year time line,  Same 3 $1000 cash flows, but  The cash flows occur at the beginning of each year, rather than at the end of each year.  This is an “annuity due.” 0 1 2 3 1000 1000 1000
0 1 2 3 Future Value - annuity due If you invest $1,000 at the beginning of each of the next 3 years at 8%, how much would you have at the end of year 3?
Future Value - annuity due If you invest $1,000 at the beginning of each of the next 3 years at 8%, how much would you have at the end of year 3? Mathematical Solution: Simply compound the FV of the ordinary annuity one more period: FV = PMT (FVIFA i, n ) (1 + i) FV = 1,000 (FVIFA .08, 3 ) (1.08) use FVIFA table or = $3,506.11 ) 08 . 1 ( 08 . 0 1 ) 08 . 1 ( 000 , 1 ) 1 ( 1 ) 1 ( 3                   FV i i i PMT FV n
Present Value - annuity due What is the PV of $1,000 at the beginning of each of the next 3 years, if your opportunity cost is 8%? 0 1 2 3
Present Value - annuity due Mathematical Solution: Simply compound the FV of the ordinary annuity one more period: PV = PMT (PVIFA i, n ) (1 + i) PV = 1,000 (PVIFA .08, 3 ) (1.08) use PVIFA table or = $2,783.26 ) 08 . 1 ( 08 . 0 ) 08 . 1 ( 1 1 000 , 1 ) 1 ( ) 1 ( 1 1 3                               PV i i i PMT PV n
Other Cash Flow Patterns 0 1 2 3
Perpetuities  Suppose you will receive a fixed payment every period (month, year, etc.) forever. This is an example of a perpetuity.  You can think of a perpetuity as an annuity that goes on forever.
Present Value of a Perpetuity  When we find the PV of an annuity, we think of the following relationship: PV = PMT (PVIFA i, n )
Mathematically, (PVIFA i, n ) = We said that a perpetuity is an annuity where n = infinity. What happens to this formula when n gets very, very large? 1 - 1 (1 + i)n i
When n gets very large, this becomes zero. So we’re left with PVIFA = 1 - 1 (1 + i)n i 1 i
PMT i PV =  So, the PV of a perpetuity is very simple to find: Present Value of a Perpetuity
What should you be willing to pay in order to receive $10,000 annually forever, if you require 8% per year on the investment? PMT $10,000 i .08 PV = = = $125,000
 Is this an annuity?  How do we find the PV of a cash flow stream when all of the cash flows are different? (Use a 10% discount rate). Uneven Cash Flows 0 1 2 3 4 -10,000 2,000 4,000 6,000 7,000
 Sorry! There’s no quickie for this one. We have to discount each cash flow back separately. 0 1 2 3 4 -10,000 2,000 4,000 6,000 7,000 Uneven Cash Flows
 Sorry! There’s no quickie for this one. We have to discount each cash flow back separately. 0 1 2 3 4 -10,000 2,000 4,000 6,000 7,000 Uneven Cash Flows
 Sorry! There’s no quickie for this one. We have to discount each cash flow back separately. 0 1 2 3 4 -10,000 2,000 4,000 6,000 7,000 Uneven Cash Flows
 Sorry! There’s no quickie for this one. We have to discount each cash flow back separately. 0 1 2 3 4 -10,000 2,000 4,000 6,000 7,000 Uneven Cash Flows
 Sorry! There’s no quickie for this one. We have to discount each cash flow back separately. 0 1 2 3 4 -10,000 2,000 4,000 6,000 7,000 Uneven Cash Flows
period CF PV (CF) 0 -10,000 -10,000.00 1 2,000 1,818.18 2 4,000 3,305.79 3 6,000 4,507.89 4 7,000 4,781.09 PV of Cash Flow Stream: $ 4,412.95 0 1 2 3 4 -10,000 2,000 4,000 6,000 7,000
Annual Percentage Yield (APY) Which is the better loan:  8% compounded annually, or  7.85% compounded quarterly?  We can’t compare these nominal (quoted) interest rates, because they don’t include the same number of compounding periods per year! We need to calculate the APY.
Annual Percentage Yield (APY)
Annual Percentage Yield (APY) APY = ( 1 + ) m - 1 quoted rate m
Annual Percentage Yield (APY)  Find the APY for the quarterly loan: APY = ( 1 + ) m - 1 quoted rate m
Annual Percentage Yield (APY)  Find the APY for the quarterly loan: APY = ( 1 + ) m - 1 quoted rate m APY = ( 1 + ) 4 - 1 .0785 4
Annual Percentage Yield (APY)  Find the APY for the quarterly loan: APY = ( 1 + ) m - 1 quoted rate m APY = ( 1 + ) 4 - 1 APY = .0808, or 8.08% .0785 4
Annual Percentage Yield (APY)  Find the APY for the quarterly loan:  The quarterly loan is more expensive than the 8% loan with annual compounding! APY = ( 1 + ) m - 1 quoted rate m APY = ( 1 + ) 4 - 1 APY = .0808, or 8.08% .0785 4
Spreadsheet Example  Use the following formulas for TVM calculations  FV(rate,nper,pmt,pv)  PV(rate,nper,pmt,fv)  RATE(nper,pmt,pv,fv)  NPER(rate,pmt,pv,fv)  The formula icon is very useful when you can’t remember the exact formula  Click on the Excel icon to open a spreadsheet containing four different examples.
Practice Problems  2002, Prentice Hall, Inc.
Example  Cash flows from an investment are expected to be $40,000 per year at the end of years 4, 5, 6, 7, and 8. If you require a 20% rate of return, what is the PV of these cash flows?
Example 0 1 2 3 4 5 6 7 8 $0 0 0 0 40 40 40 40 40  Cash flows from an investment are expected to be $40,000 per year at the end of years 4, 5, 6, 7, and 8. If you require a 20% rate of return, what is the PV of these cash flows?
Retirement Example  If you invest $400 at the end of each month for the next 30 years at 12%, how much would you have at the end of year 30?
House Payment Example If you borrow $100,000 at 7% fixed interest for 30 years in order to buy a house, what will be your monthly house payment?
Example  Baim ingin menabung untuk biaya wisata keluar negeri. Bajuri ingin keluar negeri pada akhir tahun 2012. Ia ingin memulai tabungan ini pada awal tahun 2008. Biaya yang diperlukan untuk keluar negeri adalah Rp. 150.000.000,-.  Berapakah yang harus ditabung oleh Baim bila ia ingin menabung sekali saja pada awal tahun 2008 bila suku bunga adalah 15 % ?  Berapakah yang harus ditabung Baim bila ia ingin menabung setiap awal tahun mulai tahun 2008 hingga tahun 2012, asumsi suku bunga 15 % ?

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TimeValueOfMoney.ppt

  • 1. The Time Value of Money References: Keown, 2005, Financial Management: Principles and Applications, 10th ed., Prentice Hall Ross, Westerfield, and Jordan, 2006, Fundamentals of Corporate Finance, 7th ed., McGraw-Hill Brigham & Houston, Fundamentals of Financial Management, 8th ed. Prepared By: Liem Pei Fun, S.E., MCom.
  • 2. About me  1998-2002 Bachelor of Economics (Majoring in Finance), Petra Christian University  2004-2005 Postgraduate Diploma in Finance, The University of Melbourne  2005-2006 Master of Commerce (Finance), The University of Melbourne  Pass CFA Exam Level I  Bloomberg Global Product Certification – Equity Level One  Head of Finance Program, Faculty of Economics, Petra Christian University
  • 3. OUTLINE  Future Value Single Sum  Present Value Single Sum  Future Value Annuities  Present Value Annuities  Ordinary Annuity vs Annuity Due  Perpetuities  Practice Problems
  • 4. The Time Value of Money  2002, Prentice Hall, Inc.
  • 6. We know that receiving $1 today is worth more than $1 in the future. This is due to opportunity costs. The opportunity cost of receiving $1 in the future is the interest we could have earned if we had received the $1 sooner. Today Future
  • 7. If we can measure this opportunity cost, we can:  Translate $1 today into its equivalent in the future (compounding).  Translate $1 in the future into its equivalent today (discounting). ? Today Future Today ? Future
  • 9. Future Value - single sums If you deposit $100 in an account earning 6%, how much would you have in the account after 1 year? Mathematical Solution: (Arithmetic Method) FV = PV (1 + i)n FV = 100 (1.06)1 = $106 0 1 PV = -100 FV = 106
  • 10. Calculator Keys  Texas Instruments BA-II Plus  FV = future value  PV = present value  I/Y = period interest rate  P/Y must equal 1 for the I/Y to be the period rate  Interest is entered as a percent, not a decimal  N = number of periods  Remember to clear the registers (CLR TVM) after each problem  Other calculators are similar in format
  • 11. Future Value - single sums If you deposit $100 in an account earning 6%, how much would you have in the account after 1 year? Calculator Solution: P/Y = 1 I/Y = 6 N = 1 PV = -100 FV = $106 0 1 PV = -100 FV = 106
  • 12. Future Value - single sums If you deposit $100 in an account earning 6%, how much would you have in the account after 5 years? Mathematical Solution: (Arithmetic Method) FV = PV (1 + i)n FV = 100 (1.06)5 = $133.82 0 5 PV = -100 FV = 133.82
  • 13. Future Value - single sums If you deposit $100 in an account earning 6%, how much would you have in the account after 5 years? Calculator Solution: P/Y = 1 I/Y = 6 N = 5 PV = -100 FV = $133.82 0 5 PV = -100 FV = 133.82
  • 14. Mathematical Solution: (Arithmetic Method) FV = PV (1 + i/m) m x n FV = 100 (1.015)20 = $134.68 0 20 PV = -100 FV = 134.68 Future Value - single sums If you deposit $100 in an account earning 6% with quarterly compounding, how much would you have in the account after 5 years?
  • 15. Calculator Solution: P/Y = 4 I/Y = 6 N = 20 PV = -100 FV = $134.68 0 20 PV = -100 FV = 134.68 Future Value - single sums If you deposit $100 in an account earning 6% with quarterly compounding, how much would you have in the account after 5 years?
  • 16. Mathematical Solution: (Arithmetic Method) FV = PV (1 + i/m) m x n FV = 100 (1.005)60 = $134.89 0 60 PV = -100 FV = 134.89 Future Value - single sums If you deposit $100 in an account earning 6% with monthly compounding, how much would you have in the account after 5 years?
  • 17. Calculator Solution: P/Y = 12 I/Y = 6 N = 60 PV = -100 FV = $134.89 0 60 PV = -100 FV = 134.89 Future Value - single sums If you deposit $100 in an account earning 6% with monthly compounding, how much would you have in the account after 5 years?
  • 18. Mathematical Solution: (Arithmetic Method) FV = PV (e in) FV = 1000 (e .08x100) = 1000 (e 8) FV = $2,980,957.99 Future Value - continuous compounding What is the FV of $1,000 earning 8% with continuous compounding, after 100 years? 0 100 PV = -1000 FV = $2.98m
  • 20. Mathematical Solution: (Arithmetic Method) PV = FV / (1 + i)n PV = 100 / (1.06)1 = $94.34 PV = -94.34 FV = 100 0 1 Present Value - single sums If you receive $100 one year from now, what is the PV of that $100 if your opportunity cost is 6%?
  • 21. Calculator Solution: P/Y = 1 I/Y = 6 N = 1 FV = 100 PV = -94.34 PV = -94.34 FV = 100 0 1 Present Value - single sums If you receive $100 one year from now, what is the PV of that $100 if your opportunity cost is 6%?
  • 22. Mathematical Solution: (Arithmetic Method) PV = FV / (1 + i)n PV = 100 / (1.06)5 = $74.73 Present Value - single sums If you receive $100 five years from now, what is the PV of that $100 if your opportunity cost is 6%? 0 5 PV = -74.73 FV = 100
  • 23. Calculator Solution: P/Y = 1 I/Y = 6 N = 5 FV = 100 PV = -74.73 Present Value - single sums If you receive $100 five years from now, what is the PV of that $100 if your opportunity cost is 6%? 0 5 PV = -74.73 FV = 100
  • 24. Mathematical Solution: PV = FV / (1 + i)n 5,000 = 11,933 / (1+ i)5 .419 = ((1/ (1+i)5) 2.3866 = (1+i)5 (2.3866)1/5 = (1+i) i = .19 Present Value - single sums If you sold land for $11,933 that you bought 5 years ago for $5,000, what is your annual rate of return?
  • 25. Calculator Solution: P/Y = 1 N = 5 PV = -5,000 FV = 11,933 I/Y = 19% 0 5 PV = -5000 FV = 11,933 Present Value - single sums If you sold land for $11,933 that you bought 5 years ago for $5,000, what is your annual rate of return?
  • 26. Present Value - single sums Suppose you placed $100 in an account that pays 9.6% interest, compounded monthly. How long will it take for your account to grow to $500? Mathematical Solution: PV = FV / (1 + i)N 100 = 500 / (1+ .008)N 5 = (1.008)N ln 5 = ln (1.008)N ln 5 = N ln (1.008) 1.60944 = .007968 N N = 202 months
  • 27. Calculator Solution:  P/Y = 12 FV = 500  I/Y = 9.6 PV = -100  N = 202 months Present Value - single sums Suppose you placed $100 in an account that pays 9.6% interest, compounded monthly. How long will it take for your account to grow to $500? 0 ? PV = -100 FV = 500
  • 29. What is the difference between an ordinary annuity and an annuity due? Ordinary Annuity PMT PMT PMT 0 1 2 3 i% PMT PMT 0 1 2 3 i% PMT Annuity Due
  • 30.  a sequence of equal cash flows, occurring at the end of each period. 0 1 2 3 4 Ordinary Annuity
  • 31.  If you buy a bond, you will receive equal semi-annual coupon interest payments over the life of the bond.  If you borrow money to buy a house or a car, you will pay a stream of equal payments. Examples of Annuities:
  • 32. 0 1 2 3 Future Value - annuity If you invest $1,000 each year at 8%, how much would you have after 3 years?
  • 33. Future Value - annuity If you invest $1,000 each year at 8%, how much would you have after 3 years? Mathematical Solution: = $3,246.40                  08 . 0 1 ) 08 . 1 ( 000 , 1 1 ) 1 ( 3 FV i i PMT FV n
  • 34. Calculator Solution: P/Y = 1 I/Y = 8 PMT = -1,000 N = 3 FV = $3,246.40 Future Value - annuity If you invest $1,000 each year at 8%, how much would you have after 3 years? 0 1 2 3 1000 1000 1000
  • 35. 0 1 2 3 Present Value - annuity What is the PV of $1,000 at the end of each of the next 3 years, if the opportunity cost is 8%?
  • 36. Present Value - annuity What is the PV of $1,000 at the end of each of the next 3 years, if the opportunity cost is 8%? Mathematical Solution: 10 . 577 , 2 $ 08 . 0 ) 08 . 1 ( 1 1 000 , 1 ) 1 ( 1 1 3                               PV PV i i PMT PV n
  • 37. Calculator Solution: P/Y = 1 I/Y = 8 PMT = -1,000 N = 3 PV = $2,577.10 0 1 2 3 1000 1000 1000 Present Value - annuity What is the PV of $1,000 at the end of each of the next 3 years, if the opportunity cost is 8%?
  • 38. Ordinary Annuity vs. Annuity Due $1000 $1000 $1000 4 5 6 7 8
  • 39. Begin Mode vs. End Mode 1000 1000 1000 4 5 6 7 8
  • 40. Begin Mode vs. End Mode 1000 1000 1000 4 5 6 7 8 year year year 5 6 7
  • 41. Begin Mode vs. End Mode 1000 1000 1000 4 5 6 7 8 year year year 5 6 7 PV in END Mode
  • 42. Begin Mode vs. End Mode 1000 1000 1000 4 5 6 7 8 year year year 5 6 7 PV in END Mode FV in END Mode
  • 43. Begin Mode vs. End Mode 1000 1000 1000 4 5 6 7 8 year year year 6 7 8
  • 44. Begin Mode vs. End Mode 1000 1000 1000 4 5 6 7 8 year year year 6 7 8 PV in BEGIN Mode
  • 45. Begin Mode vs. End Mode 1000 1000 1000 4 5 6 7 8 year year year 6 7 8 PV in BEGIN Mode FV in BEGIN Mode
  • 46. Earlier, we examined this “ordinary” annuity: Using an interest rate of 8%, we find that:  The Future Value (at 3) is $3,246.40.  The Present Value (at 0) is $2,577.10. 0 1 2 3 1000 1000 1000
  • 47. What about this annuity?  Same 3-year time line,  Same 3 $1000 cash flows, but  The cash flows occur at the beginning of each year, rather than at the end of each year.  This is an “annuity due.” 0 1 2 3 1000 1000 1000
  • 48. 0 1 2 3 Future Value - annuity due If you invest $1,000 at the beginning of each of the next 3 years at 8%, how much would you have at the end of year 3?
  • 49. Future Value - annuity due If you invest $1,000 at the beginning of each of the next 3 years at 8%, how much would you have at the end of year 3? Mathematical Solution: Simply compound the FV of the ordinary annuity one more period: FV = PMT (FVIFA i, n ) (1 + i) FV = 1,000 (FVIFA .08, 3 ) (1.08) use FVIFA table or = $3,506.11 ) 08 . 1 ( 08 . 0 1 ) 08 . 1 ( 000 , 1 ) 1 ( 1 ) 1 ( 3                   FV i i i PMT FV n
  • 50. Present Value - annuity due What is the PV of $1,000 at the beginning of each of the next 3 years, if your opportunity cost is 8%? 0 1 2 3
  • 51. Present Value - annuity due Mathematical Solution: Simply compound the FV of the ordinary annuity one more period: PV = PMT (PVIFA i, n ) (1 + i) PV = 1,000 (PVIFA .08, 3 ) (1.08) use PVIFA table or = $2,783.26 ) 08 . 1 ( 08 . 0 ) 08 . 1 ( 1 1 000 , 1 ) 1 ( ) 1 ( 1 1 3                               PV i i i PMT PV n
  • 52. Other Cash Flow Patterns 0 1 2 3
  • 53. Perpetuities  Suppose you will receive a fixed payment every period (month, year, etc.) forever. This is an example of a perpetuity.  You can think of a perpetuity as an annuity that goes on forever.
  • 54. Present Value of a Perpetuity  When we find the PV of an annuity, we think of the following relationship: PV = PMT (PVIFA i, n )
  • 55. Mathematically, (PVIFA i, n ) = We said that a perpetuity is an annuity where n = infinity. What happens to this formula when n gets very, very large? 1 - 1 (1 + i)n i
  • 56. When n gets very large, this becomes zero. So we’re left with PVIFA = 1 - 1 (1 + i)n i 1 i
  • 57. PMT i PV =  So, the PV of a perpetuity is very simple to find: Present Value of a Perpetuity
  • 58. What should you be willing to pay in order to receive $10,000 annually forever, if you require 8% per year on the investment? PMT $10,000 i .08 PV = = = $125,000
  • 59.  Is this an annuity?  How do we find the PV of a cash flow stream when all of the cash flows are different? (Use a 10% discount rate). Uneven Cash Flows 0 1 2 3 4 -10,000 2,000 4,000 6,000 7,000
  • 60.  Sorry! There’s no quickie for this one. We have to discount each cash flow back separately. 0 1 2 3 4 -10,000 2,000 4,000 6,000 7,000 Uneven Cash Flows
  • 61.  Sorry! There’s no quickie for this one. We have to discount each cash flow back separately. 0 1 2 3 4 -10,000 2,000 4,000 6,000 7,000 Uneven Cash Flows
  • 62.  Sorry! There’s no quickie for this one. We have to discount each cash flow back separately. 0 1 2 3 4 -10,000 2,000 4,000 6,000 7,000 Uneven Cash Flows
  • 63.  Sorry! There’s no quickie for this one. We have to discount each cash flow back separately. 0 1 2 3 4 -10,000 2,000 4,000 6,000 7,000 Uneven Cash Flows
  • 64.  Sorry! There’s no quickie for this one. We have to discount each cash flow back separately. 0 1 2 3 4 -10,000 2,000 4,000 6,000 7,000 Uneven Cash Flows
  • 65. period CF PV (CF) 0 -10,000 -10,000.00 1 2,000 1,818.18 2 4,000 3,305.79 3 6,000 4,507.89 4 7,000 4,781.09 PV of Cash Flow Stream: $ 4,412.95 0 1 2 3 4 -10,000 2,000 4,000 6,000 7,000
  • 66. Annual Percentage Yield (APY) Which is the better loan:  8% compounded annually, or  7.85% compounded quarterly?  We can’t compare these nominal (quoted) interest rates, because they don’t include the same number of compounding periods per year! We need to calculate the APY.
  • 68. Annual Percentage Yield (APY) APY = ( 1 + ) m - 1 quoted rate m
  • 69. Annual Percentage Yield (APY)  Find the APY for the quarterly loan: APY = ( 1 + ) m - 1 quoted rate m
  • 70. Annual Percentage Yield (APY)  Find the APY for the quarterly loan: APY = ( 1 + ) m - 1 quoted rate m APY = ( 1 + ) 4 - 1 .0785 4
  • 71. Annual Percentage Yield (APY)  Find the APY for the quarterly loan: APY = ( 1 + ) m - 1 quoted rate m APY = ( 1 + ) 4 - 1 APY = .0808, or 8.08% .0785 4
  • 72. Annual Percentage Yield (APY)  Find the APY for the quarterly loan:  The quarterly loan is more expensive than the 8% loan with annual compounding! APY = ( 1 + ) m - 1 quoted rate m APY = ( 1 + ) 4 - 1 APY = .0808, or 8.08% .0785 4
  • 73. Spreadsheet Example  Use the following formulas for TVM calculations  FV(rate,nper,pmt,pv)  PV(rate,nper,pmt,fv)  RATE(nper,pmt,pv,fv)  NPER(rate,pmt,pv,fv)  The formula icon is very useful when you can’t remember the exact formula  Click on the Excel icon to open a spreadsheet containing four different examples.
  • 74. Practice Problems  2002, Prentice Hall, Inc.
  • 75. Example  Cash flows from an investment are expected to be $40,000 per year at the end of years 4, 5, 6, 7, and 8. If you require a 20% rate of return, what is the PV of these cash flows?
  • 76. Example 0 1 2 3 4 5 6 7 8 $0 0 0 0 40 40 40 40 40  Cash flows from an investment are expected to be $40,000 per year at the end of years 4, 5, 6, 7, and 8. If you require a 20% rate of return, what is the PV of these cash flows?
  • 77. Retirement Example  If you invest $400 at the end of each month for the next 30 years at 12%, how much would you have at the end of year 30?
  • 78. House Payment Example If you borrow $100,000 at 7% fixed interest for 30 years in order to buy a house, what will be your monthly house payment?
  • 79. Example  Baim ingin menabung untuk biaya wisata keluar negeri. Bajuri ingin keluar negeri pada akhir tahun 2012. Ia ingin memulai tabungan ini pada awal tahun 2008. Biaya yang diperlukan untuk keluar negeri adalah Rp. 150.000.000,-.  Berapakah yang harus ditabung oleh Baim bila ia ingin menabung sekali saja pada awal tahun 2008 bila suku bunga adalah 15 % ?  Berapakah yang harus ditabung Baim bila ia ingin menabung setiap awal tahun mulai tahun 2008 hingga tahun 2012, asumsi suku bunga 15 % ?