© 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

© 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

© 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!

© 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?

© 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 $1,200$1,400$1,000 $1, = $1,000 x 1.08 $1, = $1,400 x (1.08) 2 $4, FV in Year 3: $1, = $1,200 x (1.08) 3

© 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.

© 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?

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

© 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?

© 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.

© 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.

© 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?

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

© 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, 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 …

© 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, to fund the cash flows. PV annuity = $4,000 x [1- (1/1.10) 3 /0.10] = $4,000 x = $9,947.41

© 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 x 1.10 = $4,000 x = $10,942.14

© 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 %. 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”?

© 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 %: Both options are worth $150,000! PV= $10,000 x [1/ – 1/ ( ) 25 ] = $10,000 x = $150,000

© 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?

© 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?

© 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 [ (( ) 20 – 1) / 0.10 ] = $4,000 x = $229,100

© 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 [ (( ) 20 – 1) / 0.10 ] x (1.10) = $4,000 x x (1.10) = $4,000 x = $252,

© 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

© 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 Since PV A = $6,000, we have A = PMT as A = 6,000 / = $1,892.94

© 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, $4, $1, , , , $1, , , , $1, , ,

© 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.

© 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%

© 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 = ( ) = 19.5%

© 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 year % Semiannually % % Quarterly % % Monthly % % Daily % %

© 2003 McGraw-Hill Ryerson Limited An expanded table is presented in Appendix A Future value of $1 (FV IF ) Periods1%2%3%4%6%8%10% PPT 9-6

© 2003 McGraw-Hill Ryerson Limited Present value of $1 (PV IF ) Periods1%2%3%4%6%8%10% An expanded table is presented in Appendix B PPT 9-7

© 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

© 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

© 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