© 2003 The McGraw-Hill Companies, Inc. All rights reserved. Introduction to Valuation: The Time Value of Money Chapter Five

5.1 Time Value of Money  Time value of money: $1 received today is not the same as $1 received in the future.  How do we equate cash flows received or paid at different points in time?  Time value of money uses compounding of interest (i.e., interest is reinvested and receives interest).

5.2 Basic Definitions  Present Value – earlier money on a time line  Future Value – later money on a time line  Interest rate – “exchange rate” between money received today and money to be received in the future  Discount rate  Cost of capital  Opportunity cost of capital  Required return

5.3 Future Value – Example 1 – 5.1  Suppose you invest $1000 for one year at 5% per year. What is the future value in one year? The future value is determined by two components:  Interest =  Principal = $1,000  Future Value (in one year) = principal + interest  Future Value (FV) =  Suppose you leave the money in for another year. How much will you have two years from now?  FV =

5.4 Future Value: General Formula  FV = future value  PV = present value  r = period interest rate, expressed as a decimal  T = number of periods  Future value interest factor = FVIF= (1 + r) t

5.5 Effects of Compounding  Simple interest – earn interest on principal only  Compound interest – earn interest on principal and reinvested interest  Consider the previous example  FV with simple interest =  FV with compound interest =  The extra $2.50 comes from the interest of 5% earned on the $50 of interest paid in year one.

5.6 Calculator Keys  Financial calculator function keys  FV = future value (amount at which a cash flow or series of cash flows will grow over a given period of time when compounded at a given interest rate).  PV = present value (value today of a cash flow or series of cash flows)  PMT = the periodic payment for an annuity or perpetuity  I/Y = period interest rate  Interest is entered as a percent, not a decimal  N = number of interest periods in investment horizon, number of times interest is paid  Most calculators are similar in format

5.7 Calculator Approach: PV PMT N I/Y FV Future Value – Example 2  Suppose you invest $1000 for 5 years at 5% interest. How much will you have at the end of the five years?  Formula Approach:

5.8 Future Value – Example 2 continued  The effect of compounding is small for a small number of periods, but increases as the number of periods increases.  For example, if were to invest $1,000 for five years at 5% simple interest, the future value would be $1,250 versus $1, when the interest is compounded (see calculation on the previous page).  The compounding effect makes a difference of $26.28.

5.9 Future Value  Note 1: The longer the investment horizon, the greater the FV of a present amount.  The reason for that is there is more interest and interest on interest (compounded interest).  FV is PV discounted at an appropriate rate

5.10 Calculator Approach PV PMT N I/Y FV Future Value – Example 3  Compounding over long periods of time makes a huge difference  Suppose you had a relative who deposited $10 at 5.5% interest 200 years ago.  How much would the investment be worth today?  Formula Approach

5.11 Future Value – Example 3 continued  What is the effect of compounding?  Using simple interest  Using compound interest

5.12 Future Value as a General Growth Formula  Suppose your company expects to increase unit sales of widgets by 15% per year for the next 5 years. If you currently sell 3 million widgets in one year, how many widgets do you expect to sell in 5 years?  Formula Approach Calculator Approach PV PMT N I/Y FV units

5.13 Present Values – 5.2  How much do I have to invest today to have some specified amount in the future? Start with the formula for FV and rearrange  Rearrange to solve for PV:  When we talk about discounting, we mean finding the present value of some future amount.  When we talk about the “value” of something, we are talking about the present value unless we specifically indicate that we want the future value.

5.14 Present Value – One Period Example  Suppose you need $10,000 in one year for the down payment on a new car. If you can earn 7% annually, how much do you need to invest today?  Formula Approach Calculator Approach FV PMT N I/Y PV

5.15 Present Value – Example 2  You want to begin saving for you daughter’s college education and you estimate that she will need $150,000 in 17 years. If you feel confident that you can earn 8% per year, how much do you need to invest today?  Formula Approach Calculator Approach FV PMT N I PV

5.16 Present Value – Example 3  Your parents set up a trust fund for you 10 years ago that is now worth $19, If the fund earned 7% per year, how much did your parents invest?  Formula Approach Calculator Approach FV PMT N I/Y PV

5.17 Present Value – Important Relationships  For a given interest rate:  The longer the time period, the lower the present value  For a given time period  The higher the interest rate, the smaller the present value  What is the present value of $500  To be received in 5 years? Discount rate 10% and 15% FV=500, N=5, I/Y=10 » PV= FV=500, N=5, I/Y=15 » PV=  To be received in 10 years? Discount rate 10% and 15% FV=500, N=10, I/Y=10 » PV= FV=500, N=10, I/Y=15 » PV=

5.18 The Basic PV Equation - Refresher  There are four parts to this equation  PV, FV, r and t  If we know any three, we can solve for the fourth  If you are using a financial calculator, the calculator views cash inflows as positive numbers and cash outflows as negative numbers.  When solving for time (N) or rate (I/Y) using a financial calculator, you must have both an inflow (positive number) and an outflow (negative number), or you will receive an error message

5.19 Discount Rate – 5.3  We often want to know the implied interest rate for an investment  Rearrange the basic PV equation and solve for r

5.20 Discount Rate – Example 1  You are considering at an investment that will pay $1200 in 5 years if you invest $1000 today. What is the implied rate of interest?  Formula Approach Calculator Approach PV FV PMT N I/Y %

5.21 Discount Rate – Example 2  Suppose you are offered an investment that will allow you to double your money in 6 years. You have $10,000 to invest. What is the implied rate of interest?  Formula Approach Calculator Approach FV PV PMT N I/Y %

5.22 Discount Rate – Example 3  Suppose you have a 1-year old son and you want to provide $75,000 in 17 years towards his college education. You currently have $5,000 to invest. What interest rate must you earn to have the $75,000 when you need it?  Formula Approach Calculator Approach FV PV PMT N I/Y %

5.23 Finding the Number of Periods  Start with basic equation and solve for t

5.24 Number of Periods – Example 1  You want to purchase a new car and you are willing to pay $20,000. If you can invest at 10% per year and you currently have $15,000, how long will it be before you have enough money to pay cash for the car?  Formula Approach Calculator Approach FV PV PMT I/Y N years

5.25 Number of Periods – Example 2  Suppose you want to buy a new house. You currently have $15,000 and you figure you need to have a 10% down payment. If the type of house you want costs about $20,000 and you can earn 7.5% per year, how long will it be before you have enough money for the down payment?  Formula Approach Calculator Approach FV PV PMT I/Y N years

5.26 Table 5.4