After completing this unit, you should be able to compute the future value of an investment made today, be able to compute the present value of cash to be received at some future date, be able to compute the return on an investment.
Introduction to valuation: The time value of money Chapter 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 2011 McGraw-Hill Australia Pty Ltd • aCopyright spreadsheet PPTs t/a Essentials of Corporate Finance 2e 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 McGraw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2e 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 McGraw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2e 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 McGraw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2e 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 x • Calculator note: y key Copyright 2011 McGraw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2e 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 is today • Time is the end of Period 1 r% CF0 CF1 CF2 +CF = Cash INFLOW -CF = Cash OUTFLOW Copyright 2011 McGraw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2e by Ross et al Slides prepared by David E Allen and Abhay K Singh PMT = Constant CF 4-7 Time line for a $100 lump sum due at the end of year r% Year 100 Copyright 2011 McGraw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2e by Ross et al Slides prepared by David E Allen and Abhay K Singh 4-8 Future values: Example • 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)(1.1) = 1000(1.1)2 = $121 Copyright 2011 McGraw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2e 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 McGraw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2e by Ross et al Slides prepared by David E Allen and Abhay K Singh 4-10 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 years for $400 You know you could invest the $335 elsewhere at 10% with very little risk What 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 McGraw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2e 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 – • If using formulas with a calculator, make use of both the yx and the 1/x keys Copyright 2011 McGraw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2e by Ross et al Slides prepared by David E Allen and Abhay K Singh 4-34 Discount rate: Example • You are looking at an investment that will pay $1200 in years if you invest $1000 today What is the implied rate of interest? – r = (1200 / 1000)1/5 – = 03714 = 3.714% – Calculator – the sign convention matters! • • • • N=5 PV = -1000 (you pay $1000 today) FV = 1200 (you receive $1200 in years) CPT I/Y = 3.714% Copyright 2011 McGraw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2e by Ross et al Slides prepared by David E Allen and Abhay K Singh 4-35 Discount rate: Example • Suppose you are offered an investment that will allow you to double your money in years You have $10 000 to invest What is the implied rate of interest? • Formula: – r = (20 000 / 10 000)1/6 – = 122462 = 12.25% • Calculator: –N=6 – FV = 20 000 – PV = 10 000 – CPT I/Y = 12.25% Copyright 2011 McGraw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2e by Ross et al Slides prepared by David E Allen and Abhay K Singh 4-36 Discount rate: Example • 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 / 000)1/17 – = 172688 = 17.27% – – – – – Calculator: N = 17 FV = 75 000 PV = 000 CPT I/Y = 17.27% Copyright 2011 McGraw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2e by Ross et al Slides prepared by David E Allen and Abhay K Singh 4-37 Quick quiz: Part • 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 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 Copyright 2011 McGraw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2e 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 = FV PV(1 + r)t ln t PV ln(1 r ) Calculator: CPT N Copyright 2011 McGraw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2e by Ross et al Slides prepared by David E Allen and Abhay K Singh 4-39 • Number of periods: Example 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 McGraw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2e by Ross et al Slides prepared by David E Allen and Abhay K Singh 4-40 Number of periods: Example • 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 McGraw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2e by Ross et al Slides prepared by David E Allen and Abhay K Singh 4-41 Number of periods: Example (cont.) • How much 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 + 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 McGraw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2e 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 McGraw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2e 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 you need to invest today? – You should get $9 841 Copyright 2011 McGraw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2e by Ross et al Slides prepared by David E Allen and Abhay K Singh 4-44 Summary of TVM calculations Copyright 2011 McGraw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2e by Ross et al Slides prepared by David E Allen and Abhay K Singh 4-45 Quick quiz: Part • 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 McGraw-Hill Australia Pty Ltd PPTs t/a Essentials of Corporate Finance 2e by Ross et al Slides prepared by David E Allen and Abhay K Singh 4-46 Chapter END 4-47 ... rate • Value at t=0 on a time line • Answers the questions: – How much I have to invest today to have a particular amount in the future? – What is the current value of a particular amount to be... 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... 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