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Introduction to valuation: The time value of money Chapter 4 4 -1

Key concepts and skills Be able to compute the following: • The future value of an investment made today • The present value of cash to be received at some future date • The return on an investment Be able to predict how long it will take for an investment to reach a desired value Be able to solve time value of money problems using: • formulas • a financial calculator • a spreadsheet Copyright 2011 Mc. Graw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2 e by Ross et al. Slides prepared by David E. Allen and Abhay K. Singh 4 -2

Chapter outline • Future value and compounding • Present value and discounting • More on present and future values Copyright © 2011 Mc. Graw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2 e by Ross et al. Slides prepared by David E. Allen and Abhay K. Singh 4 -3

Basic definitions • Present value (PV) – The current value of future cash flows discounted at the appropriate discount rate. – Value at t=0 on a time line • Future value (FV) – The amount an investment is worth after one or more periods. – ‘Later’ money on a time line Copyright 2011 Mc. Graw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2 e by Ross et al. Slides prepared by David E. Allen and Abhay K. Singh 4 -4

Basic definitions (cont. ) • Interest rate (r) – Discount rate – Cost of capital – Opportunity cost of capital – Required return – Terminology depends on usage Copyright 2011 Mc. Graw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2 e by Ross et al. Slides prepared by David E. Allen and Abhay K. Singh 4 -5

Future values: General formula • FV = PV(1 + r)t – FV = Future value – PV = Present value – r = Period interest rate, expressed as a decimal – T = Number of periods • FVIF(r, t)=Future value interest factor for $1 invested at r% for t periods • Future value interest factor = (1 + r)t • Calculator note: yx key Copyright 2011 Mc. Graw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2 e by Ross et al. Slides prepared by David E. Allen and Abhay K. Singh 4 -6

Time line of cash flows • Tick marks at ends of periods • Time 0 is today • Time 1 is the end of Period 1 0 1 2 3 r% CF 0 CF 1 CF 2 +CF = Cash INFLOW -CF = Cash OUTFLOW PMT = Constant CF Copyright 2011 Mc. Graw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2 e by Ross et al. Slides prepared by David E. Allen and Abhay K. Singh 4 -7

Time line for a $100 lump sum due at the end of year 2 0 r% 1 2 Year 100 Copyright 2011 Mc. Graw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2 e by Ross et al. Slides prepared by David E. Allen and Abhay K. Singh 4 -8

Future values: Example 1 • Investing for a single period – Suppose you invest $100 for one year at 10% (r) per year. What is the future value in one year? • Interest = 100(. 1) = $10 • Value in one year = principal + interest = 100 + 10 = $110 • Future value (FV) = 100(1 +. 1) = $110 • Investing for more than one period – Suppose you leave the money in for another year. How much will you have two years from now? • FV = 100(1. 1) = 1000(1. 1)2 = $121 Copyright 2011 Mc. Graw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2 e by Ross et al. Slides prepared by David E. Allen and Abhay K. Singh 4 -9

Future values: Effects of compounding • Simple interest – Interest earned only on the original principal • Compound interest – Interest earned on principal and on interest received – ‘Interest on interest’—interest earned on reinvestment of previous interest payments Copyright 2011 Mc. Graw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2 e by Ross et al. Slides prepared by David E. Allen and Abhay K. Singh 4 -10

Future values: Effects of compounding • Consider the previous example: – FV w/simple interest = 100 + 10 = 120 – FV w/compound interest =100(1. 10)2 = 121. 00 – The extra 1. 00 comes from the interest of. 10(10) = 1. 00 earned on the first interest payment. Copyright 2011 Mc. Graw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2 e by Ross et al. Slides prepared by David E. Allen and Abhay K. Singh 4 -11

Future values of $100 at 10% Copyright 2011 Mc. Graw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2 e by Ross et al. Slides prepared by David E. Allen and Abhay K. Singh 4 -12

Future values: Simple interest and compound Interest Copyright 2011 Mc. Graw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2 e by Ross et al. Slides prepared by David E. Allen and Abhay K. Singh 4 -13

Future value of $1 for different interests and rates Copyright 2011 Mc. Graw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2 e by Ross et al. Slides prepared by David E. Allen and Abhay K. Singh 4 -14

Calculating future value • Using a calculator – Calculator key: yx • Using the Future Value Factor Table – Table 4. 2 Copyright 2011 Mc. Graw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2 e by Ross et al. Slides prepared by David E. Allen and Abhay K. Singh 4 -15

Future value: Calculator hints • Texas Instruments BA-II Plus – FV = Future value – PV = Present value – I/Y = Period interest rate • PV must equal 1 for the I/Y to be the period rate • Interest is entered as a percentage, not a decimal – N = Number of periods – Remember to clear the registers (CLR TVM) after each problem. – Other calculators are similar in format. Copyright 2011 Mc. Graw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2 e by Ross et al. Slides prepared by David E. Allen and Abhay K. Singh 4 -16

Future value: Example 2 • Suppose you invest the $100 from the previous example for 5 years. How much would you have? – Formula solution: • FV=PV(1+r)t=100(1. 10)5=161. 05 • The effect of compounding is small for a small number of periods, but increases as the number of periods increases. – Key strokes : • • • N=5, I/Y=10 PV=-100 CPT-FV FV=161. 05 Copyright 2011 Mc. Graw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2 e by Ross et al. Slides prepared by David E. Allen and Abhay K. Singh 4 -17

Future value: Example 3 • Suppose you had a relative deposit $5 at 6% interest 200 years ago. How much would the investment be worth today? – FV = 5(1. 06)200 = $575 629. 50 • What is the effect of compounding? – Simple interest = 5 + 200(5)(. 06) = 5+60=65 – Compounding added $575564. 5 to the value of the investment. Copyright 2011 Mc. Graw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2 e by Ross et al. Slides prepared by David E. Allen and Abhay K. Singh 4 -18

Quick quiz: Part 1 • What is the difference between simple interest and compound interest? • Suppose you have $500 to invest and you believe that you can earn 8% per year over the next 15 years. – How much would you have at the end of 15 years using compound interest? – How much would you have using simple interest? Copyright 2011 Mc. Graw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2 e by Ross et al. Slides prepared by David E. Allen and Abhay K. Singh 4 -19

Present value and discounting • The current value of future cash flows discounted at the appropriate discount rate • Value at t=0 on a time line • Answers the questions: – How much do I have to invest today to have a particular amount in the future? – What is the current value of a particular amount to be received in the future? Copyright 2011 Mc. Graw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2 e by Ross et al. Slides prepared by David E. Allen and Abhay K. Singh 4 -20

Present value • Present value = the current value of an amount to be received in the future. • Why is it worth less than face value? – Opportunity cost – Risk and uncertainty – Discount rate = ƒ (time, risk) Copyright 2011 Mc. Graw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2 e by Ross et al. Slides prepared by David E. Allen and Abhay K. Singh 4 -21

Present value (cont. ) FV = PV(1 + r)t • Rearrange to solve for basic present value equation PV = FV / (1+r)t PV = FV(1+r)-t • ‘Discounting’ = finding the present value of one or more future amounts. • (1+r)-t is the present value factor. Copyright 2011 Mc. Graw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2 e by Ross et al. Slides prepared by David E. Allen and Abhay K. Singh 4 -22

Present value Example 1—Single period • Suppose you need $400 to buy textbooks next year. You can earn 7% on your money. How much do you have to put up today? – Formula solution: • PV=FV(1+r)-t=400/(1+0. 07) • PV= 373. 83 – Calculator keys: • • 1 N 7 I/Y 400 FV CPT PV = -373. 83 Copyright 2011 Mc. Graw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2 e by Ross et al. Slides prepared by David E. Allen and Abhay K. Singh 4 -23

Present value Example 2—Multiperiod • You would like to buy a new car. You have $50 000, but the car costs $68 500. If you can earn 9%, how much do you have to invest today to buy the car in two years? Do you have enough? Assume the price will stay the same. – PV=FV/(1+r) t – =68500/(1. 09)2 – 57655. 08, so still short of the required amount by $7655 • Calculator keys: • • 2 N 9 I/Y 68500 FV CPT PV = -57655. 08 Copyright 2011 Mc. Graw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2 e by Ross et al. Slides prepared by David E. Allen and Abhay K. Singh 4 -24

Present value Example 3—Multiperiod • 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? – PV = 19 671. 51 / (1. 07)10 = $10 000 • Calculator keys: • • 10 N 7 I/Y 19671. 51 FV CPT PV = -10000 Copyright 2011 Mc. Graw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2 e by Ross et al. Slides prepared by David E. Allen and Abhay K. Singh 4 -25

Present value: Important relationship I For a given interest rate: – The longer the time period, the lower the present value. For a given r, as t increases, PV decreases. Copyright 2011 Mc. Graw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2 e by Ross et al. Slides prepared by David E. Allen and Abhay K. Singh 4 -26

Present value: Important relationship I (cont. ) • What is the present value of $500 to be received in 5 years? 10 years? The discount rate is 10%. – 5 years: PV = 500 / (1. 1)5 = $310. 46 – 10 years: PV = 500 / (1. 1)10 = $192. 77 • Calculator: – 5 years: N = 5; I/Y = 10; FV = 500; CPT PV = 310. 46 – 10 years: N = 10; I/Y = 10; FV = 500; CPT PV = 192. 77 Copyright 2011 Mc. Graw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2 e by Ross et al. Slides prepared by David E. Allen and Abhay K. Singh 4 -27

Present value: Important relationship II For a given time period: – The higher the interest rate, the smaller the present value. For a given t, as r increases, PV decreases. Copyright 2011 Mc. Graw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2 e by Ross et al. Slides prepared by David E. Allen and Abhay K. Singh 4 -28

Present value: Important relationship II (cont. ) • What is the present value of $500 received in 5 years if the interest rate is 10%? 15%? – Rate = 10%: PV = 500 / (1. 1)5 = $310. 46 – Rate = 15%: PV = 500 / (1. 15)5 = $248. 58 • Calculator: – 10%: N = 5; I/Y = 10; FV = 500; CPT PV = 310. 46 – 15%: N = 5; I/Y = 15; FV = 500; CPT PV = 248. 58 Copyright 2011 Mc. Graw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2 e by Ross et al. Slides prepared by David E. Allen and Abhay K. Singh 4 -29

Present value of $1 for different periods and rates Copyright 2011 Mc. Graw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2 e by Ross et al. Slides prepared by David E. Allen and Abhay K. Singh 4 -30

Quick quiz: Part 2 • What is the relationship between present value and future value? • Suppose you need $15 000 in 3 years. If you can earn 6% annually, how much do you need to invest today? • If you could invest the money at 8%, would you have to invest more or less than at 6%? How much more or less? Copyright 2011 Mc. Graw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2 e by Ross et al. Slides prepared by David E. Allen and Abhay K. Singh 4 -31

The basic PV equation: Refresher PV = FV / (1 + r)t There are 4 parts to this equation: – PV, FV, r and t – Know any 3, solve for the 4 th • Be sure to remember the sign convention +CF = Cash INFLOW -CF = Cash OUTFLOW Copyright 2011 Mc. Graw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2 e by Ross et al. Slides prepared by David E. Allen and Abhay K. Singh 4 -32

Present vs future value— Evaluating investments • Your company proposes to buy an asset for $335. This investment is very safe. You will sell the asset in 3 years for $400. You know you could invest the $335 elsewhere at 10% with very little risk. What do you think of the proposed investment? – Not a good investment. – FV of 335= 335(1. 1)3 = 445. 89 – Future value of $335 is more than the value of asset in three years. Copyright 2011 Mc. Graw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2 e by Ross et al. Slides prepared by David E. Allen and Abhay K. Singh 4 -33

Discount rate • To find the implied interest rate, rearrange the basic PV equation and solve for r: FV = PV(1 + r)t r = (FV / PV)1/t – 1 • If using formulas with a calculator, make use of both the yx and the 1/x keys. Copyright 2011 Mc. Graw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2 e by Ross et al. Slides prepared by David E. Allen and Abhay K. Singh 4 -34

Discount rate: Example 1 • You are looking at an investment that will pay $1200 in 5 years if you invest $1000 today. What is the implied rate of interest? – r = (1200 / 1000)1/5 – 1 =. 03714 = 3. 714% – Calculator – the sign convention matters! • • N=5 PV = -1000 (you pay $1000 today) FV = 1200 (you receive $1200 in 5 years) CPT I/Y = 3. 714% Copyright 2011 Mc. Graw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2 e by Ross et al. Slides prepared by David E. Allen and Abhay K. Singh 4 -35

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: – r = (20 000 / 10 000)1/6 – 1 =. 122462 = 12. 25% • Calculator: – N=6 – FV = 20 000 – PV = 10 000 – CPT I/Y = 12. 25% Copyright 2011 Mc. Graw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2 e by Ross et al. Slides prepared by David E. Allen and Abhay K. Singh 4 -36

Discount rate: Example 3 • Suppose you have a 1 -year-old son and you want to provide $75 000 in 17 years towards his university education. You currently have $5000 to invest. What interest rate must you earn to have the $75 000 when you need it? – Formula: • r = (75 000 / 5 000)1/17 – 1 =. 172688 = 17. 27% – – – Calculator: N = 17 FV = 75 000 PV = 5 000 CPT I/Y = 17. 27% Copyright 2011 Mc. Graw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2 e by Ross et al. Slides prepared by David E. Allen and Abhay K. Singh 4 -37

Quick quiz: Part 3 • What are some situations in which you might want to compute the implied interest rate? • Suppose you are offered the following investment choices: – You can invest $500 today and receive $600 in 5 years. The investment is considered low risk. – You can invest the $500 in a bank account paying 4%. • What is the implied interest rate for the first choice and which investment should you choose? Copyright 2011 Mc. Graw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2 e by Ross et al. Slides prepared by David E. Allen and Abhay K. Singh 4 -38

Finding the number of periods • Start with basic equation and solve for t: FV = PV(1 + r)t Calculator: CPT N Copyright 2011 Mc. Graw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2 e by Ross et al. Slides prepared by David E. Allen and Abhay K. Singh 4 -39

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: • t = ln(20 000 / 15 000)/ln(1. 1) = 3. 02 years – Calculator: • • I/Y = 10 FV = 20 000 PV = 15 000 CPT N = 3. 02 years Copyright 2011 Mc. Graw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2 e by Ross et al. Slides prepared by David E. Allen and Abhay K. Singh 4 -40

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% deposit plus an additional 5% in legal fees. If the type of house you want costs about $150 000 and you can earn 7. 5% per year, how long will it be before you have enough money for the deposit and legal fees? Copyright 2011 Mc. Graw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2 e by Ross et al. Slides prepared by David E. Allen and Abhay K. Singh 4 -41

Number of periods: Example 2 (cont. ) • How much do you need to have in the future? – Deposit =. 1(150 000) = $15 000 – Legal fees =. 05(150 000 – 15 000) = $6 750 – Total needed = 15 000 + 6 750 = $21 750 • Compute the number of periods: – PV = -15 000 – FV = 21 750 – I/Y = 7. 5 – CPT N = 5. 14 years • Using the formula: – t = ln(21 750 / 15 000) / ln(1. 075) = 5. 14 years Copyright 2011 Mc. Graw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2 e by Ross et al. Slides prepared by David E. Allen and Abhay K. Singh 4 -42

Example: Spreadsheet strategies • The formula icon is very useful when you can’t remember the exact formula. • Double-click on the Excel icon to open a spreadsheet containing four different examples. Copyright 2011 Mc. Graw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2 e by Ross et al. Slides prepared by David E. Allen and Abhay K. Singh 4 -43

Example: Work the Web • Many financial calculators are available online. • Click on the web surfer icon to go to the present value portion of the moneychimp website and work the following example: – You need $40 000 in 15 years. If you can earn 9. 8% interest, how much do you need to invest today? – You should get $9 841. Copyright 2011 Mc. Graw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2 e by Ross et al. Slides prepared by David E. Allen and Abhay K. Singh 4 -44

Summary of TVM calculations Copyright 2011 Mc. Graw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2 e by Ross et al. Slides prepared by David E. Allen and Abhay K. Singh 4 -45

Quick quiz: Part 4 • When might you want to compute the number of periods? • Suppose you want to buy some new furniture for your family room. You currently have $500 and the furniture you want costs $1600. If you can earn 6%, how long will you have to wait if you don’t add any additional money? Copyright 2011 Mc. Graw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2 e by Ross et al. Slides prepared by David E. Allen and Abhay K. Singh 4 -46

Chapter 4 END 4 -47