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- McGraw-Hill/Irwin Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.
- 5- 2 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.
- FV = Initial investment (1 ) Compound ´ + r Simple ´ + r ´t 5- 3 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 Compound Interest FV = Initial investment (1 )t
- 5- 4 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 7 107 7 114 7 121 Value at the end of Year 5: $135 7 128 7 135
- 5- 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 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
- 5- 6 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.
- 5- 7 Present Value What is it? Why is it useful?
- 5- 8 Present Value Discount Rate: Discount Factor: Present Value: 1 (1 r )t DF + = 1 (1 r )t PV FV + = ´ Recall: t = number of years r
- 5- 9 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? FV =$100,000 t =10 yrs r =7% PV = FV ´ 1 = $100,000 ´ 1 » $50,835 (1 + r )t (1.07) 10 Note: Ignore inflation/taxes
- 5- 10 Time Value of Money (applications) The PV formula has many applications. Given any variables in the equation, you can solve for the remaining variable. 1 (1 r )t PV FV + = ´
- Present Values: Changing Discount Rates 5- 11 The present value of $100 to be received in 1 to 20 years at varying discount rates: 120 100 80 60 40 20 0 Discount Rates 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Number of Years PV of $100 0% 5% 10% 15%
- Denote C C C : The cash flow in year 1 The cash flow in year 2 The cash flow in year t (with any number of cash flows in between) t 5- 12 PV of Multiple Cash Flows The present value of multiple cash flows can be calculated: 1 2 .... t (1 )1 (1 )2 (1 ) t C C C r r r PV + + + = + + + 1 2 = = = Recall: r = the discount rate
- 5- 13 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? Initial Payment* 8,000.00 PV of C PV of C = = = = 1 (1 .08) 1 4,000 2 + 4,000 2 (1 .08) 3,703.70 3,429.36 + = Total PV $15,133.06 * The initial payment occurs immediately and therefore would not be discounted.
- 5- 14 Perpetuities Let C = Yearly Cash Payment PV of Perpetuity: Cr PV = What are they? Recall: r = the discount rate
- 5- 15 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%? PV = 185,000 = $2,312,500 .08 What if the first payment won’t be received until 3 years from today? PV = 2,312,500 = $1,982,596 (1 + .08) 2
- 5- 16 Annuities What are they? Annuities are equally-spaced, level streams of cash flows lasting for a limited period of time. Why are they useful?
- 5- 17 Present Value of an Annuity Let: C = yearly cash payment r = interest rate t = number of years cash payment is received = ´ é - ù ë û 1 1 r r (1 r)t PV C ´ + The terms within the brackets are collectively called the “annuity factor.”
- 5- 18 Annuities: 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)? = é ù ë - 30 û = 1 1 .05 .05(1 .05) $10,000 $153,724.51 PV PV +
- 5- 19 Future Value of Annuities: Example 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? = é - ù´ + ë û = $4,000 1 1 (1 .10) 20 .10 .10(1 .10) 20 $229,100 FV FV +
- 5- 20 Annuity Due What is it? How does it differ from an ordinary annuity? (1 ) AnnuityDue Annuity PV = PV ´ + r How does the future value differ from an ordinary annuity? (1 ) AnnuityDue FV = FVAnnuity ´ + r Recall: r = the discount rate
- 5- 21 Annuities Due: Example FV FV (1 r) AD Annuity = ´ + 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? FV = FV ´ + r AD Annuity = ´ ($500,000) (1.10) $550,000 (1 ) = AD AD FV FV
- 5- 22 Interest Rates: EAR & APR What is EAR? What is APR? How do they differ?
- 5- EAR & APR Calculations Effective Annual Interest Rate (EAR): EAR = (1+MR)12 -1 Annual Percentage Rate (APR): APR = MR´12 MR = monthly interest rate 23
- 5- 24 EAR and APR: Example Given a monthly rate of 1%, what is the Effective Annual Rate(EAR)? What is the Annual Percentage Rate (APR)? (1.01)12 1 12.68% = ´ = = - = (0.01) (12) 12.00% EAR APR
- 5- 25 Inflation What is it? What determines inflation rates? What is deflation?
- 5- 26 Inflation and Real Interest 1+nominal interest rate Exact calculation: 1 + real interest rate= 1+inflation rate Approximation: Real interest rate » nominal interest rate - inflation rate
- 5- 27 Inflation: 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? 1+.02 + + 1 real interest rate= 1+.03 1 real interest rate= 0.9903 real interest rate = -.0097 or -.97% Approximation = .02-.03 = - .01 = - 1%
- 5- 28 Appendix A: Inflation Annual U.S. Inflation Rates from 1900 - 2010

- Chapter 5 Learning Objectives 1. Calculate the future value to which money invested at a given interest rate will grow. 2. Calculate the present value of a future payment. 3. Calculate present and future values of a series of cash payments. 4. Find the interest rate implied by present and future values. 5. Compare interest rates quoted over different time intervals—for example, monthly versus annual rates. 6. Understand the difference between real and nominal cash flows and between real and nominal interest rates.
- 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.
- 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.
- Future Value - Amount to which an investment will grow after earning interest. Simple Interest - Interest earned only on the original investment.
- Future Value - Amount to which an investment will grow after earning interest. Compound Interest - Interest earned on interest.
- Present Value – Value today of a future cash flow
- 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
- Note: The present value of an entire investment can be computed by summing the present values of each individual cash flow.
- 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.
- Perpetuity – A stream of level cash payments that never ends.
- Annuity – Equally spaced level stream of cash flows lasting for a limited period of time
- 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.
- Annuity Due: Level stream of cash flows starting immediately.
- Annuity Due: Level stream of cash flows starting immediately.
- Effective annual interest rate: - Interest rate that is annualized using compound interest. Annual percentage rate: Interest rate that is annualized using simple interest.
- Effective annual interest rate: - Interest rate that is annualized using compound interest. Annual percentage rate: Interest rate that is annualized using simple interest.
- Inflation: The rate at which prices as a whole are increasing.
- 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.

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