Chapter 5 Calculators Introduction to Valuation: The Time Value of Money McGraw-Hill/Irwin Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved. Real Life Example: Car Financing Key Concepts and Skills • 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 • Be able to compute the number of periods that equates a present value and a future value given an interest rate • Be able to use a financial calculator and a spreadsheet to solve time value of money problems Time Value of Money You are given 2 options: 1.Option A: You receive $10,000 now. 2.Option B: You will receive $10,000 in three years. Which option will you choose? Why? Why is there a time value of money? 5C-2 Time Value of Money Time Value Purposes It can be used to determine: a) Damages in Court Cases b) Contract Values c) Investment Values In this chapter, our primary focus is to identify the value of an investment, either now or in the future. Basic Definitions • Present Value – earlier money on a time line • Future Value – later money on a time line • Interest rate – “exchange rate” between earlier money and later money – – – – Discount rate Cost of capital Opportunity cost of capital Required return Basic Definitions (cont.) Present Value T = 0 (Y2015) r = 10 % Future Value T = 1 (Y2016) t Ex: What is the future value of your investment in a year if you put $ 100 in a saving account that earns r = 10% interest annually? FV = PV + r*PV = PV (1+ r) = 100(1.1) = $ 110 Basic Definitions (cont.) Types of interest rate: 1. Simple Interest Rate: Interest earned only on the original principal amount invested 2. Compound Interest Rate: – Compound means you reinvest your interest earned over time. – In this case, you earn interest on both the principal amount and the reinvested interest amount accumulated from the previous periods. Basic Definitions (cont.) Ex: What is the future value of your investment in 2 years if you put $ 100 in a saving account that earns r = 10% interest annually and r is a simple interest rate? FV2 = Principal + Interest earned in Y1 + Interest earned in Y2 FV2 = 100 + 10%*100 + 10%*100 = $120 Or, FV2 = 100 + 2*10%*100 = $120 Simple interest formula FV = PV(1 + r*t) In which, FV is the future value of the investment PV is the present value of the investment r is the simple interest rate t is the number of periods Basic Definitions (cont.) Ex: What is the future value of your investment in 2 years if you put $ 100 in a saving account that earns r = 10% interest annually and r is a compound interest rate? Y1: you earn 100 + 10%*100 = $ 110 A compound interest rate means that in Y2 you earn 10% interest on both the $100 principal and the $10 interest from Y1. Y2: you get 110 + 10%*(100+10) = $ 121 Compound interest formula FV = PV(1 + t r) In which, FV is the future value of the investment PV is the present value of the investment r is the compound interest rate t is the number of periods Previous Example Ex: What is the future value of your investment in 2 years if you put $ 100 in a saving account that earns r = 10% interest annually and r is a compound interest rate? Answer: FV2 = PV(1+r)2 = 100(1+0.1)2 = $ 121 Calculator Keys Texas Instruments BA-II Plus (APPENDIX D) § FV = future value; PV = present value (+/-) § I/Y = period interest rate • P/Y must equal 1 for the I/Y to be the period rate – [2nd] [I/Y] 1 [ENTER] • Interest is entered as a percent, not a decimal (Ex: 15%, how would you enter the number? § N = number of periods § Clear the registers (CLR TVM) after each problem § Should show 9 decimal places on your calculator when you are performing calculations. • [2nd] [Format] 9 [ENTER] Periods vs. Years If you earn interest annually for the next 3 years. How many times will you get the interest out of your investment? Now, if you earn interest every six months for the next 3 years. How many times will you get the interest out of your investment? FUTURE VALUE Simple vs. Compound Suppose you had a relative deposit $10 at 5.5% interest 200 years ago. How much would the investment be worth today? § 200 N; 5.5 I/Y; -10 PV § CPT FV = 447,189.84 What is the effect of compounding? § Simple interest = 10 + 200(10)(.055) = 120.00 § Compounding added $447,069.84 to the value of the investment Simple vs. Compound Interests on Interests $447,069.84 $110.00 $10.00 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 sell 3 million widgets in the current year, how many widgets do you expect to sell in the fifth year? • 5 N;15 I/Y; 3,000,000 PV • CPT FV = -6,034,072 units (remember the sign convention) APPENDIX D Example: Compute the future value of $ 2,250 at a 17% annual rate for 30 years. Answer: $ 249,895.46 Example: Compute the future value of $ 2,250 at a 17% semi-annual rate for 30 years. Answer: $ 300,584.66 PRESENT VALUE The one and only formula If the Future value: FV = PV(1 + r)t How do you compute the PV? PV = t FV/[(1+r) ] Present Value • 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. APPENDIX D (Cont.) Example: What is the present value of your investment now if you expect to have $ 75,000 in 18 years and the annual interest rate is 14.08%? Answer: $ 7,003.10 Present Value – Important Relationship I • For a given interest rate – the longer the time period, the lower the present value § What is the present value of $500 to be received in 5 years? 10 years? The discount rate is 10% § 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 Present Value – Important Relationship II • For a given time period – the higher the interest rate, the smaller the present value § What is the present value of $500 received in 5 years if the interest rate is 10%? 15%? • Rate = 10%: N = 5; I/Y = 10; FV = 500 CPT PV = -310.46 • Rate = 15%; N = 5; I/Y = 15; FV = 500 CPT PV = -248.59 DISCOUNT RATE OR INTEREST RATE The one and only formula If the Future value: FV = PV(1 + r)t How do you compute the interest rate r? r = (FV/PV)1/t - 1 Example 1 You are looking at an investment that will pay $1,200 in 5 years if you invest $1,000 today. What is the implied rate of interest? • r = (1,200 / 1,000)1/5 – 1 = .03714 = 3.714% • Calculator – the sign convention matters!!! § § § § N=5 PV = -1,000 (you pay 1,000 today) FV = 1,200 (you receive 1,200 in 5 years) CPT I/Y = 3.714% APPENDIX D (Cont.) Example: Assume that the total cost of a college education will be $ 75,000 when your child enters college in 18 years. Now you have $ 7,000 to invest. What rate of interest you must earn on your investment to cover the cost of your child’s college education in 18 years? Answer: 14.08% NUMBER OF PERIODS The one and only formula If the Future value: FV = PV(1 + r)t How do you compute the # of periods? t = ln(FV / PV) / ln(1 + r) 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? § I/Y = 10; PV = -15,000; FV = 20,000 § CPT N = 3.02 years APPENDIX D (Cont.) Example: How many year does it take for you to have $ 250,000 out of a $ 5,000 investment that earns 10% interest per year? Answer: 41.05 years Spreadsheet Example • Use the following formulas for TVM calculations – – – – FV(rate,nper,pmt,pv) PV(rate,nper,pmt,fv) RATE(nper,pmt,pv,fv) NPER(rate,pmt,pv,fv) • The formula icon is very useful when you can’t remember the exact formula • Click on the Excel icon to open a spreadsheet containing four different examples. Table 5.4 5C-36 Comprehensive Problem • You have $10,000 to invest for five years. • How much additional interest will you earn if the investment provides a 5% annual return, when compared to a 4.5% annual return? • How long will it take your $10,000 to double in value if it earns 5% annually? • What annual rate has been earned if $1,000 grows into $4,000 in 20 years? Comprehensive Problem N=5 PV = -10,000 At I/Y = 5, the FV = 12,762.82 At I/Y = 4.5, the FV = 12,461.82 The difference is attributable to interest. That difference is 12,762.82 – 12,461.82 = 301 To double the 10,000: I/Y = 5 PV = -10,000 FV = 20,000 CPT N = 14.2 years Note, the rule of 72 indicates 72/5 = 14 years, approximately. N = 20 PV = -1,000 FV = 4,000 CPT I/Y = 7.18% End of Chapter