McGraw-Hill/Irwin ©2001 The McGraw-Hill Companies All Rights Reserved 4.0 Chapter 4 Introduction to Valuation: “The Time Value of Money”

McGraw-Hill/Irwin ©2001 The McGraw-Hill Companies All Rights Reserved 4.1 Introduction to Valuation: “The Time Value of Money” Almost all business activities, whether they originate in marketing, management, operations, or strategy, involve comparing outlays made today to benefits projected for the future. “Time Value of Money” refers to the fact that a dollar today is worth more than a dollar at some time in the future. Because: you can earn interest between today and the future A dollar today could grow to more than a dollar at some time in the future

McGraw-Hill/Irwin ©2001 The McGraw-Hill Companies All Rights Reserved 4.2 Essential Points to Learn from this Chapter: How to determine the future value of an investment made today How to determine the present value of cash to be received at a future date How to determine the return on investment How to determine the number of periods

McGraw-Hill/Irwin ©2001 The McGraw-Hill Companies All Rights Reserved Future Value and Compounding Investing for a Single Period Investing for More Than One Period

McGraw-Hill/Irwin ©2001 The McGraw-Hill Companies All Rights Reserved 4.4 Future Value Future Value: The amount an investment is worth after one or more periods of time.

McGraw-Hill/Irwin ©2001 The McGraw-Hill Companies All Rights Reserved 4.5 Investing for a “Single” Period Suppose you invest $100 for one year at 10% per year. What is the Future Value (FV) in one year? Original principal = 100 Interest = 100(.10) = 10 FV in one year = 100 (original principle) + 10 (interest earned) = 110 FV = 100(1 +.10) = 110 Therefore: 110 is the future value of 100 invested for 1 year at 10% 100 today is worth 110 in one year, given a 10% interest rate

McGraw-Hill/Irwin ©2001 The McGraw-Hill Companies All Rights Reserved 4.6 Investing for More Than One Period Regarding the previous example: suppose you leave the money in for another year. How much will you have in two years? Begin year one with (1.10) = 110 at the end of year one Begin year two with (1.10) = 121 at the end of year two Therefore: 121 is the future value of 100 invested for 2 years at 10% 100 today is worth 121 in two years, given a 10% interest rate

McGraw-Hill/Irwin ©2001 The McGraw-Hill Companies All Rights Reserved 4.7 Investing for More Than One Period Compounding: The process of “accumulating interest” in an investment over time to earn more interest. Interest on Interest: Interest earned on the reinvestment of previous interest payments Compound Interest: Interest earned on both the initial principal and the interest reinvested from prior periods Simple Interest: Interest earned only on the original principal amount invested.

McGraw-Hill/Irwin ©2001 The McGraw-Hill Companies All Rights Reserved 4.8 Investing for More Than One Period Compounding: The process of leaving the original principal and any accumulated interest in the investment for more than one period Reinvesting any interest earned along with the original principal Earning interest on interest The effect of compounding is not great over short time periods, but starts to add up as the time horizon grows

McGraw-Hill/Irwin ©2001 The McGraw-Hill Companies All Rights Reserved 4.9 Investing for More Than One Period Table 4.1 Page 88: Using Compound Interest, total interest earned = $61.05 Simple interest of $10 would accumulate to only $50.00 Note: with “simple interest”, the interest is not reinvested, so interest is earned each period only on the original principal. The increase of $11.05 is from reinvesting or compounding the interest.

McGraw-Hill/Irwin ©2001 The McGraw-Hill Companies All Rights Reserved 4.10 Future Value Formula FV = 100(1.10)(1.10) = 121 FV = 100(1.10) 2 = 121 FV = PV(1 + r) t [Future Value Formula 4.1- pg 87] Where: FV = future value PV = present value r = period interest rate, expressed as a decimal t = number of periods Note: Disregard future value interest factor info on pgs 87 & 88 for this course Real World  Financial Calculators

McGraw-Hill/Irwin ©2001 The McGraw-Hill Companies All Rights Reserved 4.11 Using a Financial Calculator FV = PV(1 + r) t Basic Financial Calculator Keys of Interest Regarding: FV = PV(1 + r) t N= # of periods ( t = number of periods ) I = interest rate ( r = period interest rate ) PV= Present Value FV= Future Value

McGraw-Hill/Irwin ©2001 The McGraw-Hill Companies All Rights Reserved 4.12 Using a Financial Calculator Future Value Example: What is the Future Value of $100 at 10 percent for five years? We’re solving for the FV of our PV amt of $100 invested for 5 years at a 10% interest rate n= # of periods ( t )= 5 i = interest rate (r)= 10 PV= Present Value = (Pg 92) FV= Future Value= Note: Enter the PV as a negative amount – it is a cash “out flow” – the amount you are giving up (investing) today in exchange for cash inflows in the future. Read “How to Get the Wrong Answer Using a Financial Calculator” pg 92

McGraw-Hill/Irwin ©2001 The McGraw-Hill Companies All Rights Reserved 4.13 Using a Financial Calculator Future Value Example Chapter Review & Self-Test: 4.1, Page 104 Assume you deposit $1,000 today in an account that pays 8 percent interest. How much will you have in four years? n= # of periods ( t )= 4 i = interest rate (r)= 8 PV= Present Value = - 1,000 FV= Future Value= 1, Note: Enter the PV as a negative amount – it is a cash “out flow” – the amount you are giving up (investing) today in exchange for cash inflows in the future. Read “How to Get the Wrong Answers Using a Financial Calculator” pg 92 Always clear your financial calculator between problems!

McGraw-Hill/Irwin ©2001 The McGraw-Hill Companies All Rights Reserved 4.14 Using a Financial Calculator Future Value Example Questions And Problems: #2., Page 105 Interest PVYearsRateFV PVNIFV (Keys) $ 2,250418%= 4, ,3109 6= 15, , = 417, , = 925,198.50

McGraw-Hill/Irwin ©2001 The McGraw-Hill Companies All Rights Reserved Present Value and Discounting The Single-Period Case Present Values for Multiple Periods

McGraw-Hill/Irwin ©2001 The McGraw-Hill Companies All Rights Reserved 4.16 Present Value and Discounting Present Value: The “current value” of “future cash flows” discounted at the appropriate discount rate. Discount Rate – The rate used to calculate the Present Value of Future cash flows (i.e. interest rate)

McGraw-Hill/Irwin ©2001 The McGraw-Hill Companies All Rights Reserved 4.17 Present Value and Discounting Present Value is the reverse of Future Value Instead of compounding money forward into the future, we “discount” it back to the present To Discount - Calculate the “Present Value” of some “Future Amount” (i.e. reduce or discount the Future Amount back to the Present) Calculating the Present Value of a future cash flow is commonly called “Discounted Cash Flow (DCF) Valuation”

McGraw-Hill/Irwin ©2001 The McGraw-Hill Companies All Rights Reserved 4.18 Present Value and Discounting We know that: FV = PV (1 + r) t Therefore: PV = FV/(1 + r) t Solve for PV [Present Value Formula 4.2- pg 94] Where: FV = future value PV = present value r = period interest rate, expressed as a decimal t = number of periods

McGraw-Hill/Irwin ©2001 The McGraw-Hill Companies All Rights Reserved 4.19 The Single Period Case How much do we have to invest “today” (the Present Value) at 10% to get $1 in one year? PV = FV/(1 + r) t PV = 1 / (1 +.10) 1 PV = 1 / 1.10 PV =.91 Proof:.91 x 1.10 = 1

McGraw-Hill/Irwin ©2001 The McGraw-Hill Companies All Rights Reserved 4.20 Present Values for Multiple Periods What is the Present Value of $1,000 in two years at 7 percent? The amount invested must grow to $1,000 over two years PV = FV/(1 + r) t PV = 1000 / (1 +.07) 2 PV = 1000 / (1.07) 2 PV = 1000 / 1.14 PV =

McGraw-Hill/Irwin ©2001 The McGraw-Hill Companies All Rights Reserved 4.21 Present Values for Multiple Periods If you want $1,000 in three years and can earn 15% on your money, how much do you have to invest today? PV = FV / (1 + r) t Present Value Formula PV = $1,000 / (1 +.15) 3 PV = $1,000 / (1.15) 3 PV = $1,000 / PV = $ Therefore: $ is the present value of $1,000 to be received in three years at 15%

McGraw-Hill/Irwin ©2001 The McGraw-Hill Companies All Rights Reserved 4.22 Using the Financial Calculator to Calculate Present Values Real World  Financial Calculators Remember the Basic keys of interest N= # of periods ( t ) I = interest rate (r) PV= Present Value FV= Future Value

McGraw-Hill/Irwin ©2001 The McGraw-Hill Companies All Rights Reserved 4.23 Using a Financial Calculator Present Value Example (Pg 95) If you want $1,000 in three years and can earn 15% on your money, how much do you have to invest today? n= # of periods ( t )= 3 i= interest rate (r)=15 PV= Present Value = FV= Future Value=1,000 The answer has a negative sign; as we discussed previously, since this present value represents an outflow of cash today in exchange for the $1,000 inflow later. Note: A minor rounding difference in the answer

McGraw-Hill/Irwin ©2001 The McGraw-Hill Companies All Rights Reserved 4.24 Using a Financial Calculator Present Value Example Chapter Review & Self-Test: 4.2, Page 104 Suppose you have just celebrated your 19 th birthday. A rich uncle set up a trust fund for you that will pay you $100,000 when you turn 25. If the relevant discount rate is 11 percent, how much is this fund worth today? n=6 (25 – 19) i=11 PV=-$53,464 (Rounding diff from book using calc.) FV=$100,000

McGraw-Hill/Irwin ©2001 The McGraw-Hill Companies All Rights Reserved 4.25 Using a Financial Calculator Present Value Example Questions and Problems: #11., Page 106 You have just received notification that you have won the $2 million first prize in the Millennium Lottery. However, the prize will be awarded on your 100 th birthday (assuming you’re around to collect), 80 years from now. What is the present value of your windfall if the appropriate discount rate is 14 percent? n=80 i=14 PV=-$56.06 FV=$2,000,000

McGraw-Hill/Irwin ©2001 The McGraw-Hill Companies All Rights Reserved 4.26 Present Value Info: As the length of time until payment grows, present values decline. Present Values become smaller as the time horizon grows. For a given length of time, the higher the discount rate, the lower the Present Value Present Values and Discount Rates are inversely related Increasing the discount rate decreases the PV and vice-versa

McGraw-Hill/Irwin ©2001 The McGraw-Hill Companies All Rights Reserved More On Present and Future Values Determining the Discount Rate Finding the Number of Periods

McGraw-Hill/Irwin ©2001 The McGraw-Hill Companies All Rights Reserved 4.28 Determining the Discount Rate The Discount Rate may also be referred to as: Interest rate Cost of capital Opportunity cost of capital Required return

McGraw-Hill/Irwin ©2001 The McGraw-Hill Companies All Rights Reserved 4.29 Determining the Discount Rate Using the Financial Calculator Example: Pg 100 & 101 You will need $80,000 to send your child to college in eight years. You have $35,000 now. What interest rate will it take to reach your goal? n = 8 i = 10.89% PV = -35,000 FV = 80,000

McGraw-Hill/Irwin ©2001 The McGraw-Hill Companies All Rights Reserved 4.30 Determining the Discount Rate Using the Financial Calculator Example: 4.11, Pg 101 You would like to retire in 50 years as a millionaire. If you have $10,000 today, what rate of return do you need to earn to achieve your goal? n = 50 i = 9.65% PV = -10,000 FV = 1,000,000

McGraw-Hill/Irwin ©2001 The McGraw-Hill Companies All Rights Reserved 4.31 Determining the Number of Periods Using the Financial Calculator Example: Bottom of Pg 101 Suppose we were interested in purchasing an asset that costs $50,000. We currently have $25,000. If we can earn 12 percent on this $25,000, how long until we have the $50,000? n = 7 i =12% PV = -25,000 FV = 50,000

McGraw-Hill/Irwin ©2001 The McGraw-Hill Companies All Rights Reserved 4.32 Determining the Number of Periods Using the Financial Calculator Example: Example 4.12, Pg 102 You’ve been saving up to buy the Godot Company. The total cost will be $10 million. You currently have about 2.3 million. If you can earn 16 percent on your money, how long will you have to wait? n = 10 i =16% PV = -2,300,000 FV = 10,000,000

McGraw-Hill/Irwin ©2001 The McGraw-Hill Companies All Rights Reserved 4.33 Formulas Table 4.4: Summary of time value of money calculations You should know/understand these formulas However, in general, in this class we will use the financial calculator to solve time value of money problems. Use your “Solutions Manual” to check your answers. We will “not” use “present value/future value interest factors to solve time value of money problems.

McGraw-Hill/Irwin ©2001 The McGraw-Hill Companies All Rights Reserved 4.34 Chapter 4: Suggested Homework Know chapter theories, concepts, and definitions Re-read the chapter and review the Power Point Slides Suggested Homework: The Chapter Review and Self-Test Problems: Page , 4.2, 4.3, and 4.4 Answers are provided in the book just below the problems Critical Thinking and Concepts Review: Page 105 1, 2, 3, 4, and 6 Answers are provided in your Solutions Manual Questions and Problems: Page through 15 Answers are provided in your Solutions Manual Note: Use your financial calculator to work the problems. Do not use interest factors – they will “not” be provided on a test! Refer to the Solutions Manual to confirm your answers.