3- 1 Fundamentals of Corporate Finance Third Edition Chapter 3 The Time Value of Money Brealey Myers Marcus slides by Matthew Will Irwin/McGraw-Hill ©The McGraw-Hill Companies, Inc.,2001 3- 2 Topics Covered Future Values Present Values Multiple Cash Flows Perpetuities and Annuities Inflation & Time Value Effective Annual Interest Rate Irwin/McGraw-Hill ©The McGraw-Hill Companies, Inc.,2001 3- 3 Future Values 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. Irwin/McGraw-Hill ©The McGraw-Hill Companies, Inc.,2001 3- 4 Future Values Example - Simple Interest Interest earned at a rate of 6% for five years on a principal balance of $100. Irwin/McGraw-Hill ©The McGraw-Hill Companies, Inc.,2001 3- 5 Future Values Example - Simple Interest Interest earned at a rate of 6% for five years on a principal balance of $100. Interest Earned Per Year = 100 x .06 = $ 6 Irwin/McGraw-Hill ©The McGraw-Hill Companies, Inc.,2001 3- 6 Future Values Example - Simple Interest Interest earned at a rate of 6% for five years on a principal balance of $100. Irwin/McGraw-Hill ©The McGraw-Hill Companies, Inc.,2001 3- 7 Future Values Example - Simple Interest Interest earned at a rate of 6% for five years on a principal balance of $100. Today 1 Future Years 2 3 4 5 Interest Earned Value 100 Irwin/McGraw-Hill ©The McGraw-Hill Companies, Inc.,2001 3- 8 Future Values Example - Simple Interest Interest earned at a rate of 6% for five years on a principal balance of $100. Today Interest Earned Value 100 Irwin/McGraw-Hill 1 6 106 Future Years 2 3 4 5 ©The McGraw-Hill Companies, Inc.,2001 3- 9 Future Values Example - Simple Interest Interest earned at a rate of 6% for five years on a principal balance of $100. Today Interest Earned Value 100 Irwin/McGraw-Hill 1 6 106 Future Years 2 3 6 112 4 5 ©The McGraw-Hill Companies, Inc.,2001 3- 10 Future Values Example - Simple Interest Interest earned at a rate of 6% for five years on a principal balance of $100. Today Interest Earned Value 100 Irwin/McGraw-Hill 1 6 106 Future Years 2 3 6 6 112 118 4 5 ©The McGraw-Hill Companies, Inc.,2001 3- 11 Future Values Example - Simple Interest Interest earned at a rate of 6% for five years on a principal balance of $100. Today Interest Earned Value 100 Irwin/McGraw-Hill 1 6 106 Future Years 2 3 4 6 6 6 112 118 124 5 ©The McGraw-Hill Companies, Inc.,2001 3- 12 Future Values Example - Simple Interest Interest earned at a rate of 6% for five years on a principal balance of $100. Today Interest Earned Value 100 1 6 106 Future Years 2 3 4 5 6 6 6 6 112 118 124 130 Value at the end of Year 5 = $130 Irwin/McGraw-Hill ©The McGraw-Hill Companies, Inc.,2001 3- 13 Future Values Example - Compound Interest Interest earned at a rate of 6% for five years on the previous year’s balance. Irwin/McGraw-Hill ©The McGraw-Hill Companies, Inc.,2001 3- 14 Future Values Example - Compound Interest Interest earned at a rate of 6% for five years on the previous year’s balance. Interest Earned Per Year =Prior Year Balance x .06 Irwin/McGraw-Hill ©The McGraw-Hill Companies, Inc.,2001 3- 15 Future Values Example - Compound Interest Interest earned at a rate of 6% for five years on the previous year’s balance. Today 1 Future Years 2 3 4 5 Interest Earned Value 100 Irwin/McGraw-Hill ©The McGraw-Hill Companies, Inc.,2001 3- 16 Future Values Example - Compound Interest Interest earned at a rate of 6% for five years on the previous year’s balance. Today Interest Earned Value 100 Irwin/McGraw-Hill 1 6.00 106.00 Future Years 2 3 4 5 ©The McGraw-Hill Companies, Inc.,2001 3- 17 Future Values Example - Compound Interest Interest earned at a rate of 6% for five years on the previous year’s balance. Today Interest Earned Value 100 Irwin/McGraw-Hill Future Years 1 2 3 6.00 6.36 106.00 112.36 4 5 ©The McGraw-Hill Companies, Inc.,2001 3- 18 Future Values Example - Compound Interest Interest earned at a rate of 6% for five years on the previous year’s balance. Today Interest Earned Value 100 Irwin/McGraw-Hill Future Years 1 2 3 6.00 6.36 6.74 106.00 112.36 119.10 4 5 ©The McGraw-Hill Companies, Inc.,2001 3- 19 Future Values Example - Compound Interest Interest earned at a rate of 6% for five years on the previous year’s balance. Today Interest Earned Value 100 Irwin/McGraw-Hill Future Years 1 2 3 4 6.00 6.36 6.74 7.15 106.00 112.36 119.10 126.25 5 ©The McGraw-Hill Companies, Inc.,2001 3- 20 Future Values Example - Compound Interest Interest earned at a rate of 6% for five years on the previous year’s balance. Today Interest Earned Value 100 Future Years 1 2 3 4 5 6.00 6.36 6.74 7.15 7.57 106.00 112.36 119.10 126.25 133.82 Value at the end of Year 5 = $133.82 Irwin/McGraw-Hill ©The McGraw-Hill Companies, Inc.,2001 3- 21 Future Values Future Value of $100 = FV F V $ 1 0 0 (1 r ) Irwin/McGraw-Hill t ©The McGraw-Hill Companies, Inc.,2001 3- 22 Future Values F V $ 1 0 0 (1 r ) t Example - FV What is the future value of $100 if interest is compounded annually at a rate of 6% for five years? Irwin/McGraw-Hill ©The McGraw-Hill Companies, Inc.,2001 3- 23 Future Values F V $ 1 0 0 (1 r ) t Example - FV What is the future value of $100 if interest is compounded annually at a rate of 6% for five years? FV $ 100 (1 . 06 ) $ 133 . 82 5 Irwin/McGraw-Hill ©The McGraw-Hill Companies, Inc.,2001 3- 24 Future Values with Compounding Interest Rates 70 0% 60 5% FV of $1 50 10% 15% 40 30 20 10 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 0 0 Number of Years Irwin/McGraw-Hill ©The McGraw-Hill Companies, Inc.,2001 3- 25 Manhattan Island Sale Peter Minuit bought Manhattan Island for $24 in 1626. Was this a good deal? To answer, determine $24 is worth in the year 2000, compounded at 8%. FV $ 24 (1 . 08 ) 374 $ 75 . 979 trillion FYI - The value of Manhattan Island land is well below this figure. Irwin/McGraw-Hill ©The McGraw-Hill Companies, Inc.,2001 3- 26 Present Values Present Value Value today of a future cash flow. Irwin/McGraw-Hill ©The McGraw-Hill Companies, Inc.,2001 3- 27 Present Values Present Value Discount Factor Value today of a future cash flow. Present value of a $1 future payment. Irwin/McGraw-Hill ©The McGraw-Hill Companies, Inc.,2001 3- 28 Present Values Present Value Discount Factor Value today of a future cash flow. Present value of a $1 future payment. Discount Rate Interest rate used to compute present values of future cash flows. Irwin/McGraw-Hill ©The McGraw-Hill Companies, Inc.,2001 3- 29 Present Values P re se n t V a lu e = P V PV = Irwin/McGraw-Hill F u tu re V a lu e a fte r t p e rio d s (1 + r) t ©The McGraw-Hill Companies, Inc.,2001 3- 30 Present Values Example You just bought a new computer for $3,000. The payment terms are 2 years same as cash. If you can earn 8% on your money, how much money should you set aside today in order to make the payment when due in two years? PV Irwin/McGraw-Hill 3000 ( 1 . 08 ) 2 $ 2 ,572 ©The McGraw-Hill Companies, Inc.,2001 3- 31 Present Values Discount Factor = DF = PV of $1 DF 1 (1 r ) t Discount Factors can be used to compute the present value of any cash flow. Irwin/McGraw-Hill ©The McGraw-Hill Companies, Inc.,2001 Time Value of Money 3- 32 (applications) The PV formula has many applications. Given any variables in the equation, you can solve for the remaining variable. PV FV Irwin/McGraw-Hill 1 (1 r ) t ©The McGraw-Hill Companies, Inc.,2001 Time Value of Money 3- 33 (applications) Value of Free Credit Implied Interest Rates Internal Rate of Return Time necessary to accumulate funds Irwin/McGraw-Hill ©The McGraw-Hill Companies, Inc.,2001 3- 34 PV of 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 is 8%, which do you prefer? Immediate payment PV 1 PV 2 Total PV Irwin/McGraw-Hill 1 3, 703 . 70 2 3, 429 . 36 4 , 000 ( 1 . 08 ) 4 , 000 ( 1 . 08 ) 8,000.00 $15,133.06 ©The McGraw-Hill Companies, Inc.,2001 3- 35 PV of Multiple Cash Flows PVs can be added together to evaluate multiple cash flows. PV Irwin/McGraw-Hill C1 (1 r ) 1 C2 (1 r ) 2 .... ©The McGraw-Hill Companies, Inc.,2001 3- 36 Perpetuities & Annuities Perpetuity A stream of level cash payments that never ends. Annuity Equally spaced level stream of cash flows for a limited period of time. Irwin/McGraw-Hill ©The McGraw-Hill Companies, Inc.,2001 3- 37 Perpetuities & Annuities PV of Perpetuity Formula PV C r C = cash payment r = interest rate Irwin/McGraw-Hill ©The McGraw-Hill Companies, Inc.,2001 3- 38 Perpetuities & Annuities Example - Perpetuity In order to create an endowment, which pays $100,000 per year, forever, how much money must be set aside today in the rate of interest is 10%? Irwin/McGraw-Hill ©The McGraw-Hill Companies, Inc.,2001 3- 39 Perpetuities & Annuities Example - Perpetuity In order to create an endowment, which pays $100,000 per year, forever, how much money must be set aside today in the rate of interest is 10%? PV Irwin/McGraw-Hill 100 ,000 .1 0 $ 1, 0 0 0 , 0 0 0 ©The McGraw-Hill Companies, Inc.,2001 3- 40 Perpetuities & Annuities Example - continued If the first perpetuity payment will not be received until three years from today, how much money needs to be set aside today? Irwin/McGraw-Hill ©The McGraw-Hill Companies, Inc.,2001 3- 41 Perpetuities & Annuities Example - continued If the first perpetuity payment will not be received until three years from today, how much money needs to be set aside today? PV Irwin/McGraw-Hill 1,0 0 0 ,0 0 0 ( 1 .1 0 ) 3 $ 7 5 1, 3 1 5 ©The McGraw-Hill Companies, Inc.,2001 3- 42 Perpetuities & Annuities PV of Annuity Formula PV C 1 r 1 r (1 r ) t C = cash payment r = interest rate t = Number of years cash payment is received Irwin/McGraw-Hill ©The McGraw-Hill Companies, Inc.,2001 3- 43 Perpetuities & Annuities PV Annuity Factor (PVAF) - The present value of $1 a year for each of t years. PVAF Irwin/McGraw-Hill 1 r 1 r (1 r ) t ©The McGraw-Hill Companies, Inc.,2001 3- 44 Perpetuities & Annuities Example - Annuity You are purchasing a car. You are scheduled to make 3 annual installments of $4,000 per year. Given a rate of interest of 10%, what is the price you are paying for the car (i.e. what is the PV)? P V 4 ,0 0 0 1 .1 0 1 .1 0 ( 1 .1 0 ) 3 P V $ 9 , 9 4 7 .4 1 Irwin/McGraw-Hill ©The McGraw-Hill Companies, Inc.,2001 3- 45 Perpetuities & Annuities Applications Value of payments Implied interest rate for an annuity Calculation of periodic payments Mortgage payment Annual income from an investment payout Future Value of annual payments F V C P V A F (1 r ) Irwin/McGraw-Hill t ©The McGraw-Hill Companies, Inc.,2001 3- 46 Perpetuities & 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, what will be the FV of your retirement account? F V 4 ,0 0 0 1 .1 0 1 .1 0 ( 1 .1 0 ) 20 (1 .1 0 ) 20 F V $ 2 2 9 ,1 0 0 Irwin/McGraw-Hill ©The McGraw-Hill Companies, Inc.,2001 3- 47 Inflation Inflation - Rate at which prices as a whole are increasing. Nominal Interest Rate - Rate at which money invested grows. Real Interest Rate - Rate at which the purchasing power of an investment increases. Irwin/McGraw-Hill ©The McGraw-Hill Companies, Inc.,2001 3- 48 Inflation 1 + n o m in a l in te re s t ra te 1 re a l in te re st ra te = 1 + in fla tio n ra te approximation formula R e a l in t. ra te n o m in a l in t. ra te - in fla tio n ra te Irwin/McGraw-Hill ©The McGraw-Hill Companies, Inc.,2001 3- 49 Inflation Example If the interest rate on one year govt. bonds is 5.0% and the inflation rate is 2.2%, what is the real interest rate? 1 + real interest rate = 1 + .050 1 + .022 Savings Bond 1 + real interest rate = 1 .0 27 R eal interest rate = .0 27 or 2 .7 % A p p rox im atio n = .05 0 - .022 o r .028 o r 2 .8 % Irwin/McGraw-Hill ©The McGraw-Hill Companies, Inc.,2001 3- 50 Effective Interest Rates Effective Annual Interest Rate - Interest rate that is annualized using compound interest. Annual Percentage Rate - Interest rate that is annualized using simple interest. Irwin/McGraw-Hill ©The McGraw-Hill Companies, Inc.,2001 3- 51 Effective Interest Rates example Given a monthly rate of 1%, what is the Effective Annual Rate(EAR)? What is the Annual Percentage Rate (APR)? Irwin/McGraw-Hill ©The McGraw-Hill Companies, Inc.,2001 3- 52 Effective Interest Rates example Given a monthly rate of 1%, what is the Effective Annual Rate(EAR)? What is the Annual Percentage Rate (APR)? E A R = (1 + .0 1 ) 12 - 1 = r E A R = (1 + .0 1 ) 12 - 1 = .1 2 6 8 o r 1 2 .6 8 % A P R = .0 1 x 1 2 = .1 2 o r 1 2 .0 0 % Irwin/McGraw-Hill ©The McGraw-Hill Companies, Inc.,2001 3- 53 Web Resources invest-faq.com/articles/analy-fut-prs-val.html www.bankrate.com/brm/default.asp www.financenter.com www.financialplayerscenter.com/Overview.html Click to access web sites Internet connection required Irwin/McGraw-Hill ©The McGraw-Hill Companies, Inc.,2001

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