Lecture 5: Time Value of Money Money has a time value. It can be expressed in multiple ways: A dollar today held in savings will grow. A dollar received in a year is not worth as much as a dollar received today. Chapter 5 Outline Interest and Future Value Future Value Interest: Simple vs. Compound Present Value Discount Rates and Present Values Multiple Cash Flows Level Cash Flows: Perpetuities and Annuities Annuities Due Effective Annual Interest Rates Inflation and the Time Value of Money Basic idea of this chapter: Money has a time value and it can be expressed by interest rates. 2

Future Values Future Value: Amount to which an investment will grow after earning interest. Let r = annual interest rate Let t = # of years Simple Interest - Interest earned only on the original investment. Compound Interest - Interest earned on interest. Simple Interest Compound Interest Future Value - Amount to which an investment will grow after earning interest. Compound Interest - Interest earned on interest. Simple Interest - Interest earned only on the original investment. 3

Simple Interest: Example Interest earned at a rate of 7% for five years on a principal balance of $100. Example - Simple Interest Today Future Years 1 2 3 4 5 Interest Earned Value 100 Value at the end of Year 5: $135 Future Value - Amount to which an investment will grow after earning interest. Simple Interest - Interest earned only on the original investment. 7 107 7 114 7 121 7 128 7 135 5

Compound Interest: Example Interest earned at a rate of 7% for five years on the previous year’s balance. Example - Compound Interest Today Future Years 1 2 3 4 5 Interest Earned Value 100 Future Value - Amount to which an investment will grow after earning interest. Compound Interest - Interest earned on interest. 7 107 7.49 114.49 8.01 122.50 8.58 131.08 9.18 140.26 Value at the end of Year 5 = $140.26 12

The Power of Compounding Interest earned at a rate of 7% for the first forty years on the $100 invested using simple and compound interest. 24

Present Value Present Value – Value today of a future cash flow What is it? Present Value – Value today of a future cash flow Why is it useful? Present Value – Value today of a future cash flow 57

Present Value Discount Rate: Interest rate used to compute present values of future cash flows Discount Factor: Present value of a $1 future payment Discount Rate – Interest rate used to compute present values of future cash flows Discount Factor – Present value of a $1 future payment Present Value – Value today of a future cash flow Present Value: Value today of a future cash flow Recall: t = number of years 31

Present Value: Example Always ahead of the game, Tommy, at 8 years old, believes he will need $100,000 to pay for college. If he can invest at a rate of 7% per year, how much money should he ask his rich Uncle GQ to give him? Note: Ignore inflation/taxes 34

Time Value of Money (applications) The PV formula has many applications. Given any variables in the equation, you can solve for the remaining variable. 38

Present Values: Changing Discount Rates The present value of $100 to be received in 1 to 20 years at varying discount rates: Discount Rates 24

PV of Multiple Cash Flows The present value of multiple cash flows can be calculated: Note: The present value of an entire investment can be computed by summing the present values of each individual cash flow. Recall: r = the discount rate 43

Multiple Cash Flows: Example Your auto dealer gives you the choice to pay $15,500 cash now or make three payments: $8,000 now and $4,000 at the end of the following two years. If your cost of money (discount rate) is 8%, which do you prefer? The total present value of the financed transaction is $15,133.06. Therefore you would save money should you choose to make payments instead of paying it all up front in cash. * The initial payment occurs immediately and therefore would not be discounted. 42

Perpetuities Let C = Yearly Cash Payment PV of Perpetuity: What are they? Perpetuity – A stream of level cash payments that never ends. Let C = Yearly Cash Payment PV of Perpetuity: Perpetuity – A stream of level cash payments that never ends. Recall: r = the discount rate 44

Perpetuities: Example In order to create an endowment, which pays $185,000 per year forever, how much money must be set aside today if the rate of interest is 8%? What if the first payment won’t be received until 3 years from today? 47

Annuities What are they? Annuities are equally-spaced, level streams of cash flows lasting for a limited period of time. Why are they useful? Annuity – Equally spaced level stream of cash flows lasting for a limited period of time 57

Present Value of an Annuity Let: C = yearly cash payment r = interest rate t = number of years cash payment is received Annuity – Equally spaced level stream of cash flows lasting for a limited period of time Annuity Factor - The present value of $1 paid every year for each of t years. The terms within the brackets are collectively called the “annuity factor.” 50

Annuities: Example Example: You are purchasing a home and are scheduled to make 30 annual installments of $10,000 per year. Given an interest rate of 5%, what is the price you are paying for the house (i.e. what is the present value)? 53

Future Value of Annuities Example - Future Value of annual payments You plan to save $4,000 every year for 20 years and then retire. Given a 10% rate of interest, how much will you have saved by the time you retire? 56

Annuity Due What is it? How does it differ from an ordinary annuity? Annuity Due: Level stream of cash flows starting immediately. How does it differ from an ordinary annuity? Annuity Due: Level stream of cash flows starting immediately. How does the future value differ from an ordinary annuity? Recall: r = the discount rate 44

Annuities Due: Example Example: Suppose you invest $429.59 annually at the beginning of each year at 10% interest. After 50 years, how much would your investment be worth? Annuity Due: Level stream of cash flows starting immediately. 54

Interest Rates: EAR & APR What is EAR? Effective annual interest rate: - Interest rate that is annualized using compound interest What is APR? Annual percentage rate: Interest rate that is annualized using simple interest. How do they differ? Effective annual interest rate: - Interest rate that is annualized using compound interest. Annual percentage rate: Interest rate that is annualized using simple interest. 26

EAR & APR Calculations Effective Annual Interest Rate (EAR): Annual Percentage Rate (APR): Effective annual interest rate: - Interest rate that is annualized using compound interest. Annual percentage rate: Interest rate that is annualized using simple interest. *where MR = monthly interest rate 26

EAR and APR: Example Example: Given a monthly rate of 1%, what is the Effective Annual Rate(EAR)? What is the Annual Percentage Rate (APR)? 27

Inflation What is it? What determines inflation rates? Inflation: The rate at which prices as a whole are increasing. What is deflation? 57

Inflation and Real Interest Exact calculation: Inflation: The rate at which prices as a whole are increasing. Nominal interest rate: The rate at which money invested grows. Real interest rate: The rate at which the purchasing power of an investment increases. Approximation: 57

Inflation: Example Example If the nominal interest rate on your interest-bearing savings account is 2.0% and the inflation rate is 3.0%, what is the real interest rate? 62

Appendix A: Inflation Annual U.S. Inflation Rates from 1900 - 2010