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Chapter 4 – Time Value of Money Topics Covered Future Values Present Values Multiple Cash Flows Perpetuities and Annuities Inflation & Time Value Effective Annual Interest Rate 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. Future Values Example - Compound Interest Interest earned at a rate of 6% for five years on the previous year’s balance. Today Future Years 1 2 3 4 5 Interest Earned 6.00 6.36 6.74 7.15 7.57 Value 100 106.00 112.36 119.10 126.25 133.82 Value at the end of Year 5 = $133.82 Future Value Future Value = Present Value of the investment times (1 plus the interest rate) raised to the number of periods FV = PV(1+r)t Future Value with Compounding 7000 Interest Rates 6000 0% 5% 5000 10% FV of $100 4000 15% 3000 2000 1000 0 20 22 24 26 28 30 10 12 14 16 18 0 2 4 6 8 Number of Years Future Value Example - FV What is the future value of $100 if interest is compounded annually at a rate of 6% for five years? Future Value Example - FV What is the future value of $100 if interest is compounded annually at a rate of 6% for five years? FV = PV(1+r) t = $100 (1.06) 5 = $100 x 1.3382 = $133.82 Future Values – Using Excel Example - FV What is the future value of $100 if interest is compounded annually at a rate of 6% for five years? =FV(rate, nper, pmt, type) =FV(6%, 5, 0, -100) = $133.82 Present Value Present Value (PV) = Future Value/(1+r)t Present Value 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? Present Value Present Value (PV) = Future Value/(1+r)t =$3,000/(1.08)2 =$3,000/1.1664=$2,572.02 Present Value or by Excel=PV(rate, nper, pmt, FV, type) =PV(8%,2,0,3000) PV = -$2,572.02 (negative because you give it up) How to be a Millionaire How much does a 21 year old have to save each year and invest at 11% (historical return on stocks) to have $1 million at age 40? How to be a Millionaire How much does a 21 year old have to save each year and invest at 11% (historical return on stocks) to have $1 million at age 40? =PMT(rate, nper, pv, fv, type) =PMT (11%,19,0,$1,000,000) -$17,562.50 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. Inflation Example If the (nominal) 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+ nominal rate)/(1 + inflation) Savings Bond 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+ nominal rate)/(1 + inflation) 1 real interestrate = 1+.022 1+.050 Savings Bond 1 real interestrate = 1.027 real interestrate = .027 or 2.7% 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. Effective Interest Rates example Given a monthly rate of 1%, what is the Effective Annual Rate(EAR)? What is the Annual Percentage Rate (APR)? Effective Interest Rates Example Given a monthly rate of 1%, what is the Effective Annual Rate(EAR)? What is the Annual Percentage Rate (APR)? EAR = (1 + .01)12 - 1 = r EAR = (1 + .01)12 - 1 = .1268 or 12.68% APR = .01 x 12 = .12 or 12.00% A zero-coupon bond that will pay $1,000 in 10 years is selling for $422.41 today. What interest rate does the bond offer? =RATE(nper, pmt, pv, fv) = RATE(10,0, -422.41, 1,000) 9% Present Value of Future Payments You have won the New York State Lottery (can’t win if you don’t play). You get $1 million per year (at the end of each year) for 10 years. If the cost of money is 9%, what is the present value of your prize? Present Value of Future Payments You have won the New York State Lottery. You get $1 million per year (at the end of each year) for 10 years. If the cost of money is 9%, what is the present value of your prize? =PV(rate, nper,pmt,FV) =PV(9%,10,1000000,0) =$6,417,657.70 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. (the lottery example is an annuity) PV of a Perpetuity = Payment/r A share of preferred stock pays $4 per year forever. If the cost of funds is 8.5%, what is a share worth? PV of a Perpetuity A share of preferred stock pays $4 per year forever. If the cost of funds is 8.5%, what is a share worth? PV = 4/.085 = $47.06 Future Value of Annual Payments You plan to save $4,000 every year for 40 years and then retire. Given a 10% rate of interest, what will be the FV of your retirement account? Future Value of Annual Payments You plan to save $4,000 every year for 40 years and then retire. Given a 10% rate of interest, what will be the FV of your retirement account? =FV(rate,nper,pmt) =FV(10%,40,-4000) $1,770,370.22 PV of a Perpetuity A share of preferred stock pays $4 per year forever. If the cost of funds is 8.5%, what is a share worth? PV = 4/.085 = $47.06 PROBLEMS Chapter 4: Question 4 You deposit $1,000 in your bank account. If the bank pays 4 percent simple interest, how much will you accumulate in your account after 10 years? What if the bank pays compound interest? How much of your earnings will be interest on interest? You deposit $1,000 in your bank account. If the bank pays 4 percent simple interest, how much will you accumulate in your account after 10 years? What if the bank pays compound interest? How much of your earnings will be interest on interest? With simple interest, you earn 4% of $1,000 $1,000 X .04 = $40 each year. There is no interest on interest (only with compounding) After 10 years, you earn total interest of $400, and your account accumulates to $1,400. You deposit $1,000 in your bank account. If the bank pays 4 percent simple interest, how much will you accumulate in your account after 10 years? What if the bank pays compound interest? How much of your earnings will be interest on interest? With compound interest =FV(rate, nper,pmt,pv)) =FV(4%,10,0,-1000) Over 10 years your account grows to: $148.24 x 10 = $1480.24 Interest on interest = $80.24 Chapter 4: Question 5 You will require $700 in 5 years. If you earn 5 percent interest on your funds, how much will you need to invest today in order to reach your savings goal? You will require $700 in 5 years. If you earn 5 percent interest on your funds, how much will you need to invest today in order to reach your savings goal? =PV(rate, nper,pmt,FV) =PV(5%,5,0,700) = -$548.47 Chapter 4: Question 10 How long will it take for $400 to grow to $1000 at the interest rate specified? a. 4 percent b. 8 percent c. 16 percent How long will it take for $400 to grow to $1000 at the interest rate specified? a. 4 percent b. 8 percent c. 16 percent In these problems, you can either solve the equation provided directly, or you can use Excel. =NPER(rate, pmt,pv,fv) =NPER(rate,0,-400,1000) =23.36 @4% =11.91@8% =6.17 @16% How long will it take for $400 to grow to $1000 at the interest rate specified? b. 8 percent PV = ()400 FV = 1000 PMT = 0 i as specified by the problem. Then compute n on the calculator. $400 (1.08)t = $1,000 t = 11.91 periods How long will it take for $400 to grow to $1000 at the interest rate specified? c. 16 percent In these problems, you can either solve the equation provided directly, or you can use your financial calculator. Setting: PV = ()400 FV = 1000 PMT = 0 i as specified by the problem. Then compute n on the calculator. $400 (1.16)t = $1,000 t = 6.17 periods Chapter 4: Question 19 A zero-coupon bond that will pay $1,000 in 10 years is selling for $422.41 today. What interest rate does the bond offer? A zero-coupon bond that will pay $1,000 in 10 years is selling for $422.41 today. What interest rate does the bond offer? =RATE(nper,pmt,pv , fv) =RATE(10,0,- 421.41,1000) = 9% Chapter 4: Question 22 If you take out an $8,000 car loan that calls for 48 monthly payments at an APR of 10 percent, what is your monthly payment? What is the effective annual interest rate on the loan? If you take out an $8,000 car loan that calls for 48 monthly payments at an APR of 10 percent, what is your monthly payment? What is the effective annual interest rate on the loan? =PMT(rate,nper,pv,fv) =PMT(10%/12,48,8000) =-$202.90 Effective annual interest = (1+.00833)12 – 1 = .1047 = 10.47% Chapter 4: Question 24 Professor’s Annuity Corp. offers a lifetime annuity to retiring professors. For a payments of $80,000 at age 65, the firm will pay the retiring professor $600 per month until death. a.If the professor’s remaining life expectancy is 20 years, what is the monthly rate on this annuity? b.If the monthly interest rate is .5 percent, what monthly annuity payment can the firm offer to the retiring professor? Professor’s Annuity Corp. offers a lifetime annuity to retiring professors. For a payment of $80,000 at age 65, the firm will pay the retiring professor $600/ month until death. a. If the professor’s remaining life expectancy is 20 years, what is the monthly rate on this annuity? =RATE(nper,pmt,pv,fv) =RATE(20*12,600,-80000) = 0.548% Professor’s Annuity Corp. offers a lifetime annuity to retiring professors. For a payment of $80,000 at age 65, the firm will pay the retiring professor $600/ month until death. b. If the monthly interest rate is .5 percent, what monthly annuity payment can the firm offer to the retiring professor? =PMT(rate,nper,pv,fv) =PMT(.5%,20*12,- 80000) CPT PMT = $573.14 Chapter 4: Question 25 You want to buy a new car, but you can make an initial payment of only $2,000 and can afford monthly payments of at most $400. A. If the APR on auto loans is 12 percent and you finance the purchase over 48 months, what is the maximum price you can pay for the car? B. How much can you afford if you finance the purchase over 60 months? You want to buy a new car, but you can make an initial payment of only $2,000 and can afford monthly payments of at most $400. If the APR on auto loans is 12 percent and you finance the purchase over 48 months, what is the maximum price you can pay for the car? =PV(rate,nper,pmt,pv,fv) =PV(12%/12,48,-400) =$15,189.58 Your monthly payments of $400 can support a loan of $15,189.58 With a down payment of $2,000, you can pay at most $17,189.58 for the car. You want to buy a new car, but you can make an initial payment of only $2,000 and can afford monthly payments of at most $400. How much can you afford if you finance the purchase over 60 months? =PV(rate,nper,pmt,pv,fv) =PV(12%/12,60,-400) CPT PV = $17,982.02 Your monthly payments of $400 can support a loan of $17,982.02 With a down payment of $2,000, you can pay at most $19,982.02 for the car. Chapter 4: Question 32 A store offers two payment plans. Under the installment method, you pay 25 percent down and 25 percent of the purchase price in each of the next three years. If you pay the entire bill immediately, you can take a 10 percent discount from the purchase price. Which is a better deal if you can borrow or lend funds at a 5 percent interest rate? A store offers two payment plans. Under the installation method, you pay 25 percent down and 25 percent of the purchase price in each of the next three years. If you pay the entire bill immediately, you can take a 10 percent discount from the purchase price. Which is a better deal if you can borrow or lend funds at a 5 percent interest rate? Compare the present value of the payments. Assume the product sells for $100. Installment plan: Pay in full: •=PV(rate,nper,pmt,pv,fv) Payment net of discount = $90 •=PV(5%,3,-25) •= $68.08 •+ $25 downpayment = $93.08 Choose the second plan for lower present value of payments. Chapter 4: Question 39 You’ve borrowed $4,248.68 and agreed to pay back the loan with monthly payment of $200. If the interest rate is 12 percent stated as an APR, how long will it take you to pay back the loan? What is the effective annual rate on the loan? You’ve borrowed $4,248.68 and agreed to pay back the loan with monthly payment of $200. If the interest rate is 12 percent stated as an APR, how long will it take you to pay back the loan? What’s the effective annual rate on the loan? The loan repayment is an annuity with PV equal to $4,248.68. Payments are made monthly and the monthly interest rate is 1%. We need to solve for the number of months, NPER =NPER(1%,-200,4248.68) = 24. Therefore, the solution is n = 24 months, or 2 years. You’ve borrowed $4,248.68 and agreed to pay back the loan with monthly payment of $200. If the interest rate is 12 percent stated as an APR, how long will it take you to pay back the loan? What’s the effective annual rate on the loan? The effective annual rate on the loan is: (1.01)12 1 = 0.1268 = EAR = 12.68% Chapter 4: Question 43 You’ve borrowed $100,000 to buy a condo. You will repay the loan in equal monthly payments of $804.62 over the next 30 years. What monthly interest rate are you paying on the loan? What is the effective annual interest rate on the loan? What rate is the lender more likely to quote on the loan? You’ve borrowed $100,000 to buy a condo. You will repay the loan in equal monthly payments of $804.62 over the next 30 years. What monthly interest rate are you paying on the loan? =RATE(12*30,-804.62,100000) CPT %i = .75% You’ve borrowed $100,000 to buy a condo. You will repay the loan in equal monthly payments of $804.62 over the next 30 years. What monthly interest rate are you paying on the loan? What is the effective annual interest rate on the loan? The effective annual rate is: (1.00750)12 1 = 0.0938 = 9.38% You’ve borrowed $100,000 to buy a condo. You will repay the loan in equal monthly payments of $804.62 over the next 30 years. What monthly interest rate are you paying on the loan? What is the effective annual interest rate on the loan? What rate is the lender more likely to quote on the loan? The effective annual rate is: (1.00750)12 1 = 0.0938 = 9.38% The lender is more likely to quote the APR 0.750% 12 = 9% which is lower than the effective annual rate of 9.38% (and is required by the Truth-in-Lending Law Chapter 4: Question 49 A local bank will pay you $100 a year for your lifetime if you deposit $2,500 in a bank today. If you plan to live forever, what interest rate is the bank paying? A local bank will pay you $100 a year for your lifetime if you deposit $2,500 in a bank today. If you plan to live forever, what interest rate is the bank paying? If you live forever, you will receive a $100 perpetuity that has present value equal to: $100/r Therefore: $100/r = $2,500 r = 4 percent Chapter 4: Question 60 You believe you will need to have saved $500,000 by the time you retire in 40 years in order to live comfortably. If the interest rate is 6 percent per year, how much must you save each year to meet your retirement goals? You believe you will need to have saved $500,000 by the time you retire in 40 years in order to live comfortably. If the interest rate is 6 percent per year, how much must you save each year to meet your retirement goals? =PMT(6%,40,0,500000) =$3,230.77 Chapter 4: Question 65 An engineer in 1950 was earning $6,000 per year. Today, she earns $60,000 per year. However, on average, goods today cost 6 times what they did in 1950. What is her real income today in terms of constant 1950 dollars? An engineer in 1950 was earning $6,000 per year. Today, she earns $60,000 per year. However, on average, goods today cost 6 times what they did in 1950. What is her real income today in terms of constant 1950 dollars? $60,000/6 = $10,000. Her real income increased from $6,000 to $10,000. Chapter 4 – Time Value of Money