1. © 2003 McGraw-Hill Ryerson Limited 9 9 Chapter The Time Value of Money-Part 2 McGraw-Hill Ryerson©2003 McGraw-Hill Ryerson Limited Based on: Terry Fegarty Seneca College Carol Edwards British Columbia Institute of Technology

2. © 2003 McGraw-Hill Ryerson Limited Chapter 9 - Outline  Time Value of Money  Future Value and Present Value  Compounding and Discounting  Compounding More Frequently Than Annually  Nominal and Effective Interest Rates  Multiple Cash Flows  Annuity and Annuity Due  Amortization Problems  Summary and Conclusions PPT 9-2

3. © 2003 McGraw-Hill Ryerson Limited Multiple Cash Flows  So far, we have looked at problems involving only a single cash flow.  This is unrealistic – most business investments will involve multiple cash flows over time.  We need a method for coping with such streams of cash flows!

4. © 2003 McGraw-Hill Ryerson Limited Multiple Cash Flows  Future Value Calculations  EXAMPLE  Assume interest rates are 8%.  You make 3 deposits to your bank account:  $1,200 today  $1,400 one year later.  $1,000 two years later.  How much money will you have in your account 3 years from now?

5. © 2003 McGraw-Hill Ryerson Limited Multiple Cash Flows  Doing Future Value Calculations  Calculate what each cash flow will be worth at the specified future date and add up these future values. 0123 $1,200$1,400$1,000 $1,080.00 = $1,000 x 1.08 $1,632.96 = $1,400 x (1.08) 2 $4,224.61 FV in Year 3: $1,511.65 = $1,200 x (1.08) 3

6. © 2003 McGraw-Hill Ryerson Limited Multiple Cash Flows  Present Value Calculations  Suppose we need to calculate the PV of a stream of future cash flows.  We use basically the same procedure as for working with the FV of multiple cash flows:  Calculate what each cash flow would be worth today, i.e. get its PV.  Add up these present values.

7. © 2003 McGraw-Hill Ryerson Limited Multiple Cash Flows  Present Value Calculations  EXAMPLE  Assume interest rates are 8%.  You wish to buy a car making three installments:  $8,000 today  $4,000 one year later.  $4,500 two years later.  How much money would you have to place in an account today to generate this stream of cash flows?

8. © 2003 McGraw-Hill Ryerson Limited Multiple Cash Flows  Present Value Calculations  You would need to place $15,561.32 in an account today to generate the desired cash flows: 012 -$8,000-$4,000-$4,500 $8,000.00 $4,000 / (1.08) = $3,703.30 $4,500 / (1.08) 2 = $3,858.02 $15,561.32 PV today:

9. © 2003 McGraw-Hill Ryerson Limited Multiple Cash Flows  Special Situations  In the previous examples, we worked with multiple cash flows of different sizes.  We will have also situations in which a series of equal cash flows is involved:  How much should you deposit now to be able to withdraw $1,000 per year over 10 years, if interest rates are 4%?  If you were to deposit $2,500 per year for 5 years at an interest rate of 7%, how much will your account balance be at the end of the 5 th year?

10. © 2003 McGraw-Hill Ryerson Limited Multiple Cash Flows  Special Situations Any sequence of equally spaced, level cash flows is called an  An Annuity occurs at the end of a period.  An Annuity Due (or Annuity in Advance) occurs at the beginning of the period. Annuity.

11. © 2003 McGraw-Hill Ryerson Limited PV of an Annuity  PV of an Annuity: the Long Method  So far, we have worked with multiple cash flows of different sizes.  Suppose we now need to calculate the PV of a stream of level future cash flows.  We could use the same procedure as before:  Calculate what each cash flow would be worth today, i.e. get its PV.  Add up these present values.

12. © 2003 McGraw-Hill Ryerson Limited PV of an Annuity  PV of an Annuity: the Long Method  EXAMPLE  Assume interest rates are 10%.  You wish to buy a car making three installments:  $4,000 a year from now.  $4,000 two years later.  $4,000 three years later.  How much money would you have to place in an account today to generate this stream of cash flows?

13. © 2003 McGraw-Hill Ryerson Limited PV of an Annuity  PV of an Annuity: the Long Method  You would need to place $9,947.41 in an account today to generate the desired cash flows: $4,000 / (1.10) = $3,636.36 $4,000 / (1.10) 2 = $3,305.79 $9,947.41 PV today: 0123 -$4,000 -4,000 $4,000 / (1.10) 3 = $3,005.26

14. © 2003 McGraw-Hill Ryerson Limited PV of an Annuity  PV of an Annuity: the Short Cut!  We have calculated that we need to put aside $9,947.41 to fund the following cash flows:  $4,000 a year from now.  $4,000 two years later.  $4,000 three years later.  However, is there an easier way to reach this answer? Yes! When you have level cash flows there is a short cut you can use …

15. © 2003 McGraw-Hill Ryerson Limited PV of an Annuity  PV of an Annuity: the Short Cut! = A x PVIFA i,n  Using the PV of an annuity calculation, we get the same answer as before: Put aside $9,947.41 to fund the cash flows. PV annuity = $4,000 x [1- (1/1.10) 3 /0.10] = $4,000 x 2.48685 = $9,947.41

16. © 2003 McGraw-Hill Ryerson Limited PV of an Annuity Due  PV of an Annuity Due: the Short Cut! = A x PVIFA i,n x (1+i) PV annuity = $4,000 x [1- (1/1.10) 3 /0.10] x (1.10) = $4,000 x 2.48685 x 1.10 = $4,000 x 2.7355 = $10,942.14

17. © 2003 McGraw-Hill Ryerson Limited PV of an Annuity  Our First Question …  You now have all the tools necessary to answer the very first question we asked!  Give it a try:  Assume interest rates are 4.3884%. You have just won a lottery and must choose between the following two options  Receive a cheque for $150,000 today.  Receive $10,000 a year for the next 25 years. Which option gives you the biggest “winnings”?

18. © 2003 McGraw-Hill Ryerson Limited PV of an Annuity  Our First Question …  Option 1 is worth $150,000.  To value Option 2, find the PV of $10,000 per year for 25 years at 4.3884%: Both options are worth $150,000! PV= $10,000 x [1/0.043884 – 1/0.043884 (1 +.043884) 25 ] = $10,000 x 15.000 = $150,000

19. © 2003 McGraw-Hill Ryerson Limited FV of an Annuity  Calculating the FV of an Annuity  Suppose interest rates are 10% and you decide to save $4,000 per year for 20 years. How much will you have saved for your retirement?  This is a FV problem.  We could use the same procedure as we used for multiple cash flows of different sizes:  Calculate what each cash flow would be worth in, 20 years, i.e. get its FV.  Add up these future values. Can you see the problem with using this method?

20. © 2003 McGraw-Hill Ryerson Limited FV of an Annuity  Calculating the FV of an Annuity  Calculating the FV this way would mean working out the FV for 20 separate cash flows... Yes! When you have level cash flows there is a short cut you can use …   Is there an easier way?

21. © 2003 McGraw-Hill Ryerson Limited FV of an Annuity  FV of an Annuity: the Short Cut! = A x FVIFA i,n  Using the FV of an annuity calculation, we see that you will have $229,100 in your account when you retire in 20 years. FV annuity = $4,000 x [ ((1 + 0.10) 20 – 1) / 0.10 ] = $4,000 x 57.27499949 = $229,100

22. © 2003 McGraw-Hill Ryerson Limited FV of an Annuity Due  FV of an Annuity Due: the Short Cut! = A x FVIF i,n x (1+i) FV annuity = $4,000 x [ ((1 + 0.10) 20 – 1) / 0.10 ] x (1.10) = $4,000 x 57.27499949 x (1.10) = $4,000 x 63.0025 = $252,009.9978

23. © 2003 McGraw-Hill Ryerson Limited Loan Amortization  Loan Amortization is the determination of the equal annual loan payments necessary to provide a lender with a specified interest return and to repay the loan principal over a specified period.  From the Formula = A x PVIFA i,n  We rearrange it as follows: A = PV A / PVIFA i,n

24. © 2003 McGraw-Hill Ryerson Limited Loan Amortization  For example, you borrow $6,000 at 10% and agree to pay equal annual end-of-year payments over the 4-year period.  Create an Amortization Schedule that shows the yearly payment, including the interest portion, principal and the loan balance.  Using Appendix D and the re-arranged formula A = PV A / PVIFA, we obtain PVIFA at i=10% and n=4 is 3.170. Since PV A = $6,000, we have A = PMT as A = 6,000 / 3.170 = $1,892.94

25. © 2003 McGraw-Hill Ryerson Limited Loan Amortization Loan Amortization Schedule End- of-year Loan Payment (1) Beginning- of-year- balance (2) Interest [0.10 x (2)] (3) Principal [(1)-(3)] (4) End-of-year balance [(2)-(4)] (5) 1$1,892.74$6,000.00$600.00$1,292.74 $4,707.26 2$1,892.74 4,707.26 470.73 1,422.01 3,285.25 3$1,892.74 3,285.25 328.53 1,564.21 1,721.04 4$1,892.74 1,721.04 172.10 1,720.64 -

26. © 2003 McGraw-Hill Ryerson Limited Effective Interest Rates Revisited  Effective Annual Rate (EAR) vs Stated Annual Rate (i)  So far, we have used annual interest rates applied to annual cash flows.  But interest can be applied daily, weekly, monthly, semi-annually – or for any other convenient time period.

27. © 2003 McGraw-Hill Ryerson Limited Effective Interest Rates Revisited  Stated Annual Rate (i)  is an interest rate that is annualized using simple interest.  For example: Your credit card charges 1.5% per month. What is the Stated Annual Rate? APR = Quoted rate x Number of Periods Per Year = 1.5% x 12 = 18%

28. © 2003 McGraw-Hill Ryerson Limited Effective Interest Rates Revisited  Effective Annual Interest Rate (EAR)  The effective annual interest rate (EAR) is an interest rate that is annualized using compound interest.  For example: Your credit card charges 1.5% per month. What is the Effective Annual Rate? EAR = (1 + Quoted rate) Number of Periods Per Year - 1 = (1 + 0.015) 12 - 1 = 19.5%

29. © 2003 McGraw-Hill Ryerson Limited Effective Interest Rates Revisited  Calculating the EAR  Convert the Stated Annual Rate (i) to a period rate (i/m) and then apply the equation: (1 + i/m) m - 1 (m = Number of periods per year) Stated Annual Rate (i) : 12% Compounding Period Periods per Year (m) Period Rate (= i/m) EAR = (1+ i/m) m - 1 1 year112.0000% Semiannually26.0000%12.3600% Quarterly43.0000%12.5509% Monthly121.0000%12.6825% Daily3650.0329%12.7475%

30. © 2003 McGraw-Hill Ryerson Limited 1....1.0101.0201.0301.0401.0601.0801.100 2....1.0201.0401.0611.0821.1241.1661.210 3....1.0301.0611.0931.1251.1911.2601.331 4....1.0411.0821.1261.1701.2621.3601.464 5....1.0511.1041.1591.2171.3381.4691.611 10....1.1051.2191.3441.4801.7912.1592.594 20....1.2201.4861.8062.1913.2074.6616.727 An expanded table is presented in Appendix A Future value of $1 (FV IF ) Periods1%2%3%4%6%8%10% PPT 9-6

31. © 2003 McGraw-Hill Ryerson Limited 1..... 0.9900.9800.9710.9620.9430.9260.909 2.....0.9800.9610.9430.9250.8900.8570.826 3.....0.9710.9420.9150.8890.8400.7940.751 4.....0.9610.9240.8880.8550.7920.7350.683 5.....0.9510.9060.8630.8220.7470.6810.621 10.....0.9050.8200.7440.6760.5580.4630.386 20.....0.8200.6730.5540.4560.3120.2150.149 Present value of $1 (PV IF ) Periods1%2%3%4%6%8%10% An expanded table is presented in Appendix B PPT 9-7

32. © 2003 McGraw-Hill Ryerson Limited Formula Appendix Future value—–single amount.. (9-1) A Present value—–single amount. (9-3) B Future value—–annuity....... (9-4a) C Future value—–annuity in advance................... (9-4b) – Present value—annuity....... (9-5a) D Determining the Yield on an Investment (a) PPT 9-15

33. © 2003 McGraw-Hill Ryerson Limited Formula Appendix Present value—annuity in advance................ (9-5b) – Annuity equalling a future value.................. (9-6a) C Annuity in advance equalling a future value............ (9-6b) – Annuity equalling a present value.................. (9-7a) D Annuity in advance equalling a present value........... (9-7b) – Determining the Yield on an Investment (b) PPT 9-16

34. © 2003 McGraw-Hill Ryerson Limited Summary and Conclusions  The financial manager uses the time value of money approach to value cash flows that occur at different points in time  A dollar invested today at compound interest will grow a larger value in future. That future value, discounted at compound interest, is equated to a present value today PPT 9-20