© 2003 McGraw-Hill Ryerson Limited 9 9 Chapter The Time Value of Money-Part 1 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 Time Value of Money  Money Problems …  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. KEY QUESTIONS FOR YOU: Which option gives you the bigger “winnings” How should you tackle this kind of problem?

© 2003 McGraw-Hill Ryerson Limited Time Value of Money  Money Problems …  As a financial manager you will often have to compare cash payments which occur at different dates:  Cash flows now, versus …  … cash flows later.  To make optimal decisions, you must understand the relationship between a dollar received (paid) today and a dollar received (paid) in the future.

© 2003 McGraw-Hill Ryerson Limited Time Value of Money  In short, we must incorporate the concept of the time value of money.  The basic idea behind the concept of time value of money is:  $1 received today is worth more than $1 in the future OR  $1 received in the future is worth less than $1 today Why?  because interest can be earned on the money  The connecting piece or link between present (today) and future is the interest or discount rate

© 2003 McGraw-Hill Ryerson Limited Time Value of Money  Money Problems …  As a financial manager you will face two basic types of cash flow problems:  Present Value (PV) problems.  Future Value (FV) problems.

© 2003 McGraw-Hill Ryerson Limited Future Value and Present Value  Future Value (FV) is what money today will be worth at some point in the future  Present Value (PV) is what money at some point in the future is worth today PPT 9-4

© 2003 McGraw-Hill Ryerson Limited Figure 9-1 Relationship of present value and future value $1,000 present value $1, future value Number of periods $ 10% interest PPT 9-5

© 2003 McGraw-Hill Ryerson Limited Present Value (PV)  Present Value (PV) Problems  PV problems involve calculating the value today of future cash flow(s).  For example:  Interest rates are 7%. If I need to have $100,000 saved in 10 years, how much money must I put aside today to create that cash flow?  Interest rates are 12%. If I need to create an income of $5,000 per year for 10 years, how much money must I put aside today to create that cash flow?

© 2003 McGraw-Hill Ryerson Limited Future Value (FV)  Future Value (FV) Problems  FV problems involve calculating the value an investment will grow to after earning interest.  For example:  Interest rates are 5%. If I invest $1,000 today, how much will it be worth in 8 years?  Interest rates are 10%. If I open an account and invest $2,500 per year, how much will it be worth in 12 years?

© 2003 McGraw-Hill Ryerson Limited Future Values  Compound Interest vs Simple Interest  Future value is the amount to which an investment will grow after earning interest.  There are two types of interest you may receive:  Compound interest.  Simple interest.

© 2003 McGraw-Hill Ryerson Limited Future Values  Simple Interest  Simple interest means that interest is earned only on your original investment:  No interest is earned on the interest.  Example:  Assume interest rates are 6%.  You invest $100 in an account paying simple interest.  How much will the account be worth in 5 years?

© 2003 McGraw-Hill Ryerson Limited Future Values  Simple Interest  You earn interest only on the amount invested.  Therefore you would earn:  $100 x 6% = $6.00 per year for 5 years.  Answer – you would have $130 after 5 years: Period (n) $100$106$112$118$124$130 Balance in your account:

© 2003 McGraw-Hill Ryerson Limited Future Values  Compound Interest  Most financial problems you will deal with will involve compound interest.  Compound interest means that interest is earned on interest.  The result: the income you earn would be higher than it would be with simple interest.  Can you see why?

© 2003 McGraw-Hill Ryerson Limited Future Values  Compound Interest  Your income would be higher than it would be with simple interest because you earn interest on both the original investment and the interest earned in previous years.  Try the example again using compound interest:  Interest rates are 6%. You invest $100 in an account paying compound interest. How much will the account be worth in 5 years?

© 2003 McGraw-Hill Ryerson Limited Future Values  Compound Interest  You earn interest on your interest:  $100 x 6% = $6.00 the first year.  $106 x 6% = $6.36 the second year.  $ x 6% = $6.74 the third year … etc.  After 5 years you would have $ : Period (n) $100$106$112.36$119.10$126.25$ Balance in your account:

© 2003 McGraw-Hill Ryerson Limited Future Values  Formula for Calculating FV  FV n = PV  (1+i) n = PV  FVIF i,n  Try the example again using the formula above:  Interest rates are 6%. You invest $100 in an account paying compound interest. How much will the account be worth in 5 years? FV= $100 x ( ) 5 = $100 x = $133.82

© 2003 McGraw-Hill Ryerson Limited Present Value  More Money Problems …  Assume interest rates are 10%.  You have just won a lottery and must choose between the following two options:  Receive $1,000,000 today.  Receive $1,000,000 five years from now. KEY QUESTIONS FOR YOU: Which option gives you the bigger “winnings” How should you tackle this kind of problem?

© 2003 McGraw-Hill Ryerson Limited Present Value  More Money Problems …  This is an example of a present value problem.  You shouldn’t even have to do a calculation to get the correct answer.  Obviously the first option is the better choice!  You would want to take the money today so that you could immediately start earning interest on your winnings.

© 2003 McGraw-Hill Ryerson Limited Present Value  More Money Problems …  This example demonstrates a basic financial principle:  A dollar received today is worth more than a dollar received tomorrow.  The key question is:  How much less valuable is a dollar received tomorrow as versus a dollar received today?  That question is answered by using the interest rate (also known as the discount rate) to calculate the PV of the second option.

© 2003 McGraw-Hill Ryerson Limited Present Value  Formula for Calculating PV  PV = FV x 1/(1 + i) n = FV  PVIF i,n  You have been offered $1 million five years from now. Interest rates are 10%.  What is that worth to you in today’s dollars? PV= $1.0 million x 1/ ( ) 5 = $1.0 million x = $620,921

© 2003 McGraw-Hill Ryerson Limited Present Value  More Money Problems …  Thus, you could have $1 million today.  Or you could have the second option, which equates to $620,921 in today’s dollars. $1 million now The equivalent of $620,921 now vs  You knew before that the first option was better, but now you can calculate exactly how much better off you are: $379,079 better off!

© 2003 McGraw-Hill Ryerson Limited Present Value vs Future Value  PV and FV are related!  Have you noticed that $620,921 becomes $1 million (and that $1 million requires $620,921) if you have a time period of 5 years and a discount rate of 10%? $620,921 $1,000,000 FV at 10% PV at 10%

© 2003 McGraw-Hill Ryerson Limited Present Value vs Future Value  PV and FV are related!  $620,921 invested for 5 years at 10% grows to $1 million.  Or, working it in reverse, if rates are 10%, and you need $1 million in 5 years, you must put aside $620,921 right now. FV = PV x (1 + i) n = $620,921 x ( ) 5 = $620,921 x = $1 million PV = FV x 1/(1 + i) n = $1 million x 1/ ( ) 5 = $1 million x = $620,921

© 2003 McGraw-Hill Ryerson Limited Present Value vs Future Value  PV and FV are related!  To calculate the FV of money which is available now (PV) to be invested for n years at an interest rate i, multiply the PV by (1+i) n.  To calculate the PV of a future payment, run the process in reverse and divide the FV by (1+i) n.

© 2003 McGraw-Hill Ryerson Limited Present Value vs Future Value  PV and FV are related!  Note that:  (1+i) n is called the future value factor.  1/(1+i) n is called the present value factor.  i is called the discount rate.  n is the number of periods.  Finding the FV is called compounding.  Finding the PV is called discounting.

© 2003 McGraw-Hill Ryerson Limited Present Value vs Future Value  Finding the Unknown … FV = PV x (1 + i) n PV = FV x 1/(1 + i) n The FV and PV formulas have many applications. Note that the variables used in these two equations are: FV PV i n Given any three variables in the equation, you can always solve for the remaining variable!

© 2003 McGraw-Hill Ryerson Limited Compounding More Frequently Than Annually  Interest is often compounded quarterly, monthly, daily or semiannually in the real world  To be able to use the time value of money tables correctly, an adjustment must be made:  the number of years n is multiplied by the number of compounding periods m  the annual interest rate i is divided by the number of compounding periods m FVn = PV  (1+i/m) m*n  Semiannual Compounding involves two compounding periods within the year.  Quarterly Compounding involves four compounding periods within the year.

© 2003 McGraw-Hill Ryerson Limited Nominal and Effective Annual Rates of Interest  The Nominal, or Stated Annual Rate is that charged by a lender or promised by a borrower.  The Effective Annual Rate (EAR) is the interest actually paid or earned due to compounding. EAR = (1+i/m) m - 1 where m = number of compounding periods per year i = stated annual rate

© 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