380 CHAPTER Mathematics of Finance | EXERCISES | 6.2 COMPOUND INTEREST For each investment situation in Problems 5–8, identify (a) the annual interest rate, (b) the length of the investment in years, (c) the periodic interest rate, and (d) the number of periods of the investment 8% compounded quarterly for years 12% compounded monthly for years 9% compounded monthly for years 10% compounded semiannually for years Find the future value if $8000 is invested for 10 years at 12% compounded annually 10 What is the future value if $8600 is invested for years at 10% compounded semiannually? 11 What are the future value and the interest earned if $3200 is invested for years at 8% compounded quarterly? 12 What interest will be earned if $6300 is invested for 3 years at 12% compounded monthly? 13 What lump sum parents need to deposit in an account earning 10%, compounded monthly, so that it will grow to $80,000 for their son’s college fund in 18 years? 14 What lump sum should be deposited in an account that will earn 9%, compounded quarterly, to grow to $1.2 million for retirement in 25 years? 15 What present value amounts to $10,000 if it is invested for 10 years at 6% compounded annually? 16 What present value amounts to $300,000 if it is invested at 7%, compounded semiannually, for 15 years? 17 Find the future value if $5100 is invested for years at 9% compounded continuously 18 Find the interest that will result if $8000 is invested at 7%, compounded continuously, for years 19 What is the compound interest if $410 is invested for 10 years at 8% compounded continuously? 20 If $8000 is invested at 8.5% compounded continuously, find the future value after 421 years 21 Grandparents want to make a gift of $100,000 for their grandchild’s 20th birthday How much would have to be invested on the day of their grandchild’s birth if their investment could earn (a) 10.5% compounded continuously (b) 11% compounded continuously? (c) Describe the effect that this slight change in the interest rate makes over the 20 years of this investment 22 Suppose an individual wants to have $200,000 available for her child’s education Find the amount that would 23 24 25 26 In Problems 27 and 28, rank each interest rate and compounding scheme in order from highest yield to lowest yield 27 8% compounded quarterly, 8% compounded monthly, 8% compounded annually 28 6% compounded continuously, 6% compounded semiannually, 6% compounded monthly 29 Two different investment companies offer college savings plans, one at 8.2% compounded continuously and the other at 8.4% compounded quarterly Which is the better investment? 30 For life insurance policies, some of the premium pays for the cost of the insurance, and the remainder goes toward the cash value of the policy and earns interest like a savings account Suppose that, on the cash value of their policies, one insurance company pays 4.8% compounded monthly and another pays 4.82% compounded semiannually Which company offers a higher yield? 31 The figure shows a graph of the future value of $100 at 8% compounded annually, along with the graph of $100 at 8% compounded continuously Which is which? Explain $ Dollars In Problems 1–4, find the requested value and identify each of the other values as the periodic rate, the number of periods, the principal, or the future value Find S: S ϭ 2000(1 ϩ 0.02)24 Find S: S ϭ 15,000(1 ϩ 0.005)360 Find P: 25,000 ϭ P(1 ϩ 0.03)48 Find P: 100,000 ϭ P(1 ϩ 0.06)25 have to be invested at 12%, compounded continuously, if the number of years until college is (a) years (b) 14 years (c) Does leaving the money invested twice as long mean that only half as much is needed initially? Explain why or why not Which investment will earn more money, a $1000 investment for years at 8% compounded annually or a $1000 investment for years compounded continuously at 7%? How much more interest will be earned if $5000 is invested for years at 7% compounded continuously, instead of at 7% compounded quarterly? Find the annual percentage yield for an investment at (a) 7.3% compounded monthly (b) 6% compounded continuously What is the annual percentage yield (or effective annual rate) for a nominal rate of (a) 8.4% compounded quarterly and (b) 10% compounded continuously? time Years Copyright 2016 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it SECTION 6.2 32 The figure shows a graph of the future value of $350 at 6% compounded monthly, along with the graph of $350 at 6% compounded annually Which is which? Explain Dollars $ time Years 33 Microsoft’s stock price peaked at 6118% of its I.P.O price more than 13 years after the I.P.O (Source: Bloomberg Financial Markets) Suppose that $10,000 invested in Microsoft at its I.P.O price had been worth $600,000 (6000% of the I.P.O price) after exactly 13 years What interest rate, compounded annually, does this represent? 34 If $10,000 had been invested in the Sagamore Capital Opportunity Fund on September 30, 2004, it would have been worth $46,649.55 on September 30, 2014 What interest rate, compounded annually, did this investment earn? 35 How long (in years) would $700 have to be invested at 11.9%, compounded continuously, to earn $300 interest? 36 How long (in years) would $600 have to be invested at 8%, compounded continuously, to amount to $970? 37 At what nominal rate, compounded quarterly, would $20,000 have to be invested to amount to $26,425.82 in years? 38 At what nominal rate, compounded annually, would $10,000 have to be invested to amount to $14,071 in 7 years? 39 For her 1st birthday, Ruth’s grandparents invested $1000 in an 18-year certificate for her that pays 8% compounded annually How much will the certificate be worth on Ruth’s 19th birthday? 40 To help their son buy a car on his 16th birthday, a boy’s parents invest $7500 on his 10th birthday If the investment pays 9% compounded continuously, how much is available on his 16th birthday? 41 (a) A 40-year-old man has $432,860 in an IRA account He decides to make no additional contributions to the account but expects it to grow at 7.5% compounded annually How much does he expect to have in the account when he retires at age 62? (b) How much more money would the man have if his investments earned 8.5% compounded annually? Compound Interest; Geometric Sequences 381 42 (a) The purchase of Alaska cost the United States $7 million in 1869 If this money had been placed in a savings account paying 6% compounded annually, how much money would be available from this investment in 2015? (b) If the $7 million earned 7% compounded annually since 1869, how much would be available in 2015? (c) Do you think either amount would purchase Alaska in 2015? Explain in light of the value of Alaska’s resources or perhaps the price per acre of land 43 A couple needs $45,000 as a down payment for a home If they invest the $30,000 they have at 8% compounded quarterly, how long will it take for the money to grow into $45,000? 44 How long does it take for an account containing $8000 to be worth $15,000 if the money is invested at 9% compounded monthly? 45 Mary Stahley invested $2500 in a 36-month certificate of deposit (CD) that earned 8.5% annual simple interest When the CD matured, she invested the full amount in a mutual fund that had an annual growth equivalent to 18% compounded annually How much was the mutual fund worth after years? 46 Suppose Patrick Goldsmith deposited $1000 in an account that earned simple interest at an annual rate of 7% and left it there for years At the end of the 4 years, Patrick deposited the entire amount from that account into a new account that earned 7% compounded quarterly He left the money in this account for years How much did he have after the 10 years? In Problems 47 and 48, use a spreadsheet or financial program on a calculator or computer 47 Track the future values of two investments of $5000, one at 6.3% compounded quarterly and another at 6.3% compounded monthly for each interest payment period for 10 years (a) How long does it take each investment to be worth more than $7500? (b) What are the values of each investment after 3 years, years, and 10 years? 48 Track the future values of two investments of $1000, one at 6.0% compounded semiannually and one at 6.6% compounded semiannually for each interest payment period for 25 years (a) How long before the difference between these investments is $50? (b) How much sooner does the 6.6% investment reach $1500? GEOMETRIC SEQUENCES For each geometric sequence given in Problems 49 and 50, write the next three terms 49 (a) 3, 6, 12, (b) 81, 54, 36, Copyright 2016 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 382 CHAPTER Mathematics of Finance 50 (a) 4, 12, 36, (b) 32, 40, 50, 68 In Problems 51–60, write an expression that gives the requested term or sum 51 The 13th term of the geometric sequence with first term 10 and common ratio 52 The 11th term of the geometric sequence with first term and common ratio 53 The 16th term of the geometric sequence with first term and common ratio 23 54 The 20th term of the geometric sequence with first term and common ratio Ϫ2 55 The sum of the first 17 terms of the geometric sequence with first term and common ratio 56 The sum of the first 14 terms of the geometric sequence with first term and common ratio 57 The sum of the first 35 terms of the geometric sequence 1, 3, 9, 58 The sum of the first 14 terms of the geometric sequence 16, 64, 256, 59 The sum of the first 18 terms of the geometric sequence 6, 4, 83 , 60 The sum of the first 31 terms of the geometric sequence 9, Ϫ6, 4, 69 70 71 72 73 74 APPLICATIONS 61 Inflation A house that 20 years ago was worth $160,000 has increased in value by 4% each year because of inflation What is its worth today? 62 Inflation If inflation causes the cost of automobiles to increase by 3% each year, what should a car cost today if it cost $25,000 years ago? 63 Population growth Suppose a country has a population of 20 million and projects a growth rate of 2% per year for the next 20 years What will the population of this country be in 10 years? 64 Spread of AIDS Suppose a country is so devastated by the AIDS epidemic that its population decreases by 0.5% each year for a 4-year period If the population was originally 10 million, what is the population at the end of the 4-year period? 65 Population growth If the rate of growth of a population continues at 2%, in how many years will the population double? 66 Population If a population of million begins to increase at a rate of 0.1% each month, in how many months will it be 10 million? 67 Ball rebounding A ball is dropped from a height of 128 feet If it rebounds 34 of the height from which it 75 76 77 falls every time it hits the ground, how high will it bounce after it strikes the ground for the fourth time? Water pumping A pump removes 13 of the water in a container with every stroke What amount of water is still in a container after strokes if it originally contained 81 cm3? Depreciation A machine is valued at $10,000 If the depreciation at the end of each year is 20% of its value at the beginning of the year, find its value at the end of years Profit Suppose a new business makes a $1000 profit in its first month and has its profit increase by 10% each month for the next years How much profit will the business earn in its 12th month? Bacterial growth The size of a certain bacteria culture doubles each hour If the number of bacteria present initially is 5000, how many would be present at the end of hours? Bacterial growth If a bacteria culture increases by 20% every hour and 2000 are present initially, how many will be present at the end of 10 hours? Profit If changing market conditions cause a company earning $8,000,000 in 2015 to project a loss of 2% of its profit in each of the next years, what profit does it project in 2020? Profit Suppose a new business makes a $1000 profit in its first month and has its profit increase by 10% each month for the next years How much profit will it earn in its first year? Chain letters Suppose you receive a chain letter with six names on it, and to keep the chain unbroken, you are to mail a dime to the person whose name is at the top, cross out the top name, add your name to the bottom, and mail it to five friends If your friends mail out five letters each, and no one breaks the chain, you will eventually receive dimes How many sets of mailings before your name is at the top of the list to receive dimes? How many dimes would you receive? (This is a geometric sequence with first term 5.) Chain letters Mailing chain letters that involve sending money has been declared illegal because most people would receive nothing while a comparative few would profit Suppose the chain letter in Problem 75 were to go through 12 unbroken progressions (a) How many people would receive money? (b) How much money would these people receive as a group? Chain letters How many letters would be mailed if the chain letter in Problem 75 went through 12 unbroken progressions? Copyright 2016 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it ... investment pays 9% compounded continuously, how much is available on his 16th birthday? 41 (a) A 40-year-old man has $432,860 in an IRA account He decides to make no additional contributions to the... population decreases by 0.5% each year for a 4-year period If the population was originally 10 million, what is the population at the end of the 4-year period? 65 Population growth If the rate... $15,000 if the money is invested at 9% compounded monthly? 45 Mary Stahley invested $2500 in a 36-month certificate of deposit (CD) that earned 8.5% annual simple interest When the CD matured,