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## Introduction to Valuation: The Time Value of Money

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**Chapter**Five Introduction to Valuation: The Time Value of Money**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).**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**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 = $1,000(.05) = $50 • Principal = $1,000 • Future Value (in one year) = principal + interest = $1,000 + $50 = $1,050 • Future Value (FV) = $1,000(1 + .05) = $1,050 • Suppose you leave the money in for another year. How much will you have two years from now? • FV = $1,000(1.05)(1.05) = $1,000(1.05)2 = $1,102.50**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**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 = $1,000 + $50 + $50 = $1,100 • FV with compound interest = $1,000(1.05)(1.05) = $1,000(1.05)2 = $1,102.50 • The extra $2.50 comes from the interest of 5% earned on the $50 of interest paid in year one.**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 (simple interest paid each period) • 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**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: • Calculator Approach: • 1,000 PV • 0 PMT • 5 N • 5 I/Y • FV -1,276.28**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 $1,276.28 when the interest is compounded (see calculation on the previous page). • The compounding effect makes a difference of $26.28.**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**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 • Calculator Approach • 10 PV • 0 PMT • 200 N • 5.5 I/Y • FV -447,189.84**Future Value – Example 3 continued**• What is the effect of compounding? • Using simple interest • Using compound interest**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 • 3,000,000 PV • 0 PMT • 5 N • 15 I/Y • FV -6,034,072 units**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.**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 • 10,000 FV • 0 PMT • 1 N • 7 I/Y • PV -9,345.79**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 • 150,000 FV • 0 PMT • 17 N • 8 I • PV -$40,540.34**Present Value – Example 3**• Your parents set up a trust fund for you 10 years ago that is now worth $19,671.51. If the fund earned 7% per year, how much did your parents invest? • Formula Approach • Calculator Approach • 19,671.51 FV • 0 PMT • 10 N • 7 I/Y • PV -$10,000**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=-310.46 FV=500, N=5, I/Y=15 » PV=-248.59 • To be received in 10 years? Discount rate 10% and 15% FV=500, N=10, I/Y=10 » PV=-192.77 FV=500, N=10, I/Y=15 » PV=-123.59**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**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**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 • -1000 PV • 1200 FV • 0 PMT • 5 N • I/Y 3.714%**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 • 20,000 FV • -10,000 PV • 0 PMT • 6 N • I/Y 12.25%**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 • 75,000 FV • -5,000 PV • 0 PMT • 17 N • I/Y 17.27%**Finding the Number of Periods**• Start with basic equation and solve for t**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 • 20,000 FV • -10,000 PV • 0 PMT • 10 I/Y • N 3.02 years**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 $200,000 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 • 20,000 FV • - 15,000 PV • 0 PMT • 7.5 I/Y • N 3.98 years