Chapter 5: The Time Value Of Money PDF

Chapter 5: The Time Value of Money Copyright© 2015 John Wiley & Sons, Inc. Learning Objectives 1. Explain time value of money and its importance to the field of finance. 2. Explain the concept of future value, including the meaning of the terms principal, simple interest and compound inte...

Chapter 5: The Time Value of Money Copyright© 2015 John Wiley & Sons, Inc. Learning Objectives 1. Explain time value of money and its importance to the field of finance. 2. Explain the concept of future value, including the meaning of the terms principal, simple interest and compound interest, and use the future value formula to make business decisions. 3. Explain the concept of present value and how it relates to future value, and use the present value formula to make business decisions. 4. Discuss why the concept of compounding is not restricted to money, and use the future value formula to calculate growth rates. Copyright© 2015 John Wiley & Sons, Inc. What is the Time Value of Money? The choice between spending today and spending tomorrow A dollar can be spent now, or invested to earn interest A dollar invested and earning interest increases wealth and the ability to consumer later The rate of interest determines the trade-off between consumption today and investing for the future The time value of money (TVM) is the difference between a dollar in hand today and a dollar promised in the future A dollar today is worth more than a dollar in the future because we can earn interest on today’s dollar Copyright© 2015 John Wiley & Sons, Inc. Future Value vs. Present Value Future Value (FV) measures the value of an investment after it earns interest for one or more periods Present Value (PV) measures the current value of future cash flows discounted at the appropriate interest rate Compounding is the process of increasing cash flows to a future value Discounting is the process of reducing future cash flows to a present value Copyright© 2015 John Wiley & Sons, Inc. Timelines Aid Problem Solving Timelines are an effective way to visualize cash flows Cash outflows are negative values, cash inflows are positive Copyright© 2015 John Wiley & Sons, Inc. Single-Period Investment We can determine the balance in an account at the end of a period if we know the interest rate earned on the principal If principal of $P0 is loaned for one period at the interest rate i, the account balance will increase to $P0(1 + i)1 The term (1+ i)n is the future value interest factor, or future value factor 𝐹𝑉1 = 𝑃0 ×1 + 𝑖 = $100 ×1 + 0.10 Copyright© 2015 John Wiley & Sons, Inc.  = $100 × 1.10 = $110 Two-Period Investment A two-period loan is two consecutive single-period loans Interest earned is added to the account at the end of the first period and the new account balance is the amount that earns the interest rate i during the second period The account balance is $ P0(1 + i)1 at the end of the first period and $ P0(1 + i)2 at the end of the second period. 𝐹𝑉2= 𝑃0 × 1 + 𝑖 2 Copyright© 2015 John Wiley & Sons, Inc.  = $100 × 1 + 0.10 2 = $100 × 1.21 = $121 Copyright© 2015 John Wiley & Sons, Inc. How Compound Interest Grows on $100 at 10 Percent Future Value and Compounding Future Value Equation Copyright© 2015 John Wiley & Sons, Inc.  The general equation to find a future value Equation 5.1 𝐹𝑉𝑛 = 𝑃𝑉 × 1 + 𝑖 𝑛 where: FVn = future value of investment at end of period n PV = original principle (P0), or present value i = the rate of interest per period n = the number of periods; for example a year, month, or day (1+i)n = the future value factor Copyright© 2015 John Wiley & Sons, Inc. Future Value and Compounding Future Value example You deposit $100 in a savings account earning 10% compounded annually for five years. How much is in the account at the end of that time? 𝐹𝑉5 = $100 × 1 + 0.10 5 = $100 × 1.10 5 = $100 × 1.6105 = $161.05 Copyright© 2015 John Wiley & Sons, Inc. Future Value of $1 for Different Periods and Interest Rates The Power of Compounding Compounding more than once a year Copyright© 2015 John Wiley & Sons, Inc.  The more frequently interest is compounded, the larger the future value of $1 at the end of a given time period If compounding occurs m times within a period, the future value equation becomes Equation 5.2  i mn FVn  PV1   m Compounding Semi-Annually You deposit $100 in an account that pays 5% annually with semi-annual compounding for two years. What is the ending account balance? Copyright© 2015 John Wiley & Sons, Inc.  2×2 0.05 𝐹𝑉2 = $1001 + 2 = $100 1 + 1.025 4 = $100 × 1.1038 = $110.38 Continuous Compounding When compounding occurs on a continuous basis, the future value equation becomes Equation 5.3 𝐹𝑉∞ = 𝑃𝑉 × 𝑒𝑖×𝑛, Copyright© 2015 John Wiley & Sons, Inc.  where 𝑒 ≈ 2.71828 Investments which compound continuously exhibit exponential growth Continuous Compounding Example Your grandmother wants to put $10,000 in a savings account. How much money will she have at the end of five years if the bank pays 5% interest compounded continuously? 𝐹𝑉∞ = $10,000 × 𝑒0.05×5 Copyright© 2015 John Wiley & Sons, Inc.  = $10,000 × 1.284025 = $12,840.25 Financial Calculators Future value calculations can be done easily on a financial calculator Given any four of the following five inputs, the financial calculator will solve for the fifth: Enter Nper i PV PMT FV Answer Copyright© 2015 John Wiley & Sons, Inc. Where N is the number of periods, i is the interest rate per period, PV is the present value, PMT is the amount of any recurring payments, and FV is the future value Calculator Example – Future Value Suppose we lend $5,000 at 15% for 10 years. How much money will we have at the end of that time? 10 15 -5,000 0 Enter Nper i PV PMT FV Copyright© 2015 John Wiley & Sons, Inc.  20,227.79 Answer Using Excel for TVM Calculations Suppose we lend $5,000 at 15% for 10 years. How much money will we have at the end of that time? Copyright© 2015 John Wiley & Sons, Inc. Copyright© 2015 John Wiley & Sons, Inc. Present Value and Discounting A present value calculation takes end of period cash flows and reverses the effect of compounding to determine the equivalent beginning of period cash flows This is discounting and the interest rate i is called the discount rate Present value (PV) is often referred to as the discounted value of future cash flows Time and the discount rate affect present value The greater the amount of time before a cash flow is to occur, the smaller the present value of the cash flow The higher the discount rate, the smaller the present value of a future cash flow Copyright© 2015 John Wiley & Sons, Inc. Present Value and Discounting Present Value Equation Equation 5.4 𝐹𝑉𝑛 𝑃𝑉 = 1+𝑖 𝑛 This equation has the same elements as Equation 5.1, the future value equation. They differ only in the arrangement of the elements. Here, 1 + 𝑖 𝑛 is used for division and is called the present value factor or discount factor. Copyright© 2015 John Wiley & Sons, Inc. Comparing Future Value & Present Value Calculations The future value and present value formulas are one and the same; the present value factor, 1Τ 1 + 𝑖 𝑛 , is just the reciprocal of the future value factor, 1 + 𝑖 𝑛 Copyright© 2015 John Wiley & Sons, Inc. Present Value Calculation Example You intend to buy a new BMW a year from now and estimate the car will cost $40,000. If your local bank pays 5% interest on savings, how much money will you need to save in order to buy the car as planned? 𝐹𝑉1 𝑃𝑉 = 1+𝑖 1 $40,000 = 1+0.05 = $38,095.24 Copyright© 2015 John Wiley & Sons, Inc. Present Value of $1 for Different Periods and Discount Rates Copyright© 2015 John Wiley & Sons, Inc.  Futur e value How much to invest today? Copyright© 2015 John Wiley & Sons, Inc. Future Value and Present Value Compared Copyright© 2015 John Wiley & Sons, Inc. Copyright© 2015 John Wiley & Sons, Inc. Calculator Example – Present Value What is the present value of $1,000 to be received 10 years from now if the discount rate is 9%? 10 9 0 1,000 Enter Nper i PV PMT FV -422.41 Answer Copyright© 2015 John Wiley & Sons, Inc. Finding the Interest Rate Many situations require using a time value of money calculation to determine a rate of change or growth rate An investor or analyst may want The growth rate in sales The rate of return on an investment The effective interest rate on a loan Calculator Example – Compound Growth Rate A firm’s sales increased from $20 million to $35 million in three years. What was the average annual growth rate in sales? Copyright© 2015 John Wiley & Sons, Inc.  3 -20 0 35 Enter N i PV PMT FV 20.51 Answer Calculator Example – Compound Growth Rate The house at 1245 Maple St. was appraised at $247,000 in 2006 and at $173,000 in 2011. What is the average annual change in its value? Copyright© 2015 John Wiley & Sons, Inc.  5 -247000 0 173000 Enter N i PV PMT FV -6.874 Answer Calculator Example – Number of Periods You would like to purchase a new motorcycle. The dealer requires a down payment of $1,175 but you only have $1,000. If you can earn 5% by investing your money, how long will it take for your $1,000 to grow to $1,175? Copyright© 2015 John Wiley & Sons, Inc.  5 -1000 0 1175 Enter N i PV PMT FV 3.31 Answer The Rule of 72 The Rule of 72 is a shortcut to estimate the number of periods it takes for an amount to double This shortcut is fairly accurate for interest rates between 5% and 20% The rule of 72 is a linear approximation of a non-linear functions The time to double your money (TDM) where i is the rate of return is: Copyright© 2015 John Wiley & Sons, Inc. Equation 5.5 72 𝑇𝐷𝑀 = 𝑖 Example – Rule of 72 If you can earn 8% compounded annually, how long will it take for your money to double? 72 72 𝑇𝐷𝑀 = = = 9 𝑦𝑒𝑎𝑟𝑠 𝑖 8 Using a financial calculator, Copyright© 2015 John Wiley & Sons, Inc.  8 -1 0 2 Enter N i PV PMT FV 9.006 Answer Copyright© 2015 John Wiley & Sons, Inc.