Homework 4 Solutions

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1 Homework 4 Solutions Chapter 4B Does it make sense? Decide whether each of the following statements makes sense or is clearly true) or does not make sense or is clearly false). Explain your reasoning. 11. The bank that pays the highest annual percentage rate APR) is always the best deal. No, it does not make sense. The frequency of compounding can affect the rate of accumulation e.g. a rate of 5.9% compounded daily will accumulate interest faster than than a rate of 6% compounded annually the APY of the first is greater than 6%). Review of Powers. Use the skills covered in the Brief Review on p. 210 to evaluate or simplify the following expressions = 1 / /2 = 4 Algebra Review. Use the skills covered in the Brief Review on p. 211 to solve the following equations a + 4 = 6 + 4a 3a + 4 = 6 + 4a 3a a = 6 + 4a 4 4a 3a 4a = 6 4 a = 2 a = 2 Simple Interest. Calculate the amount of money you will have in the following accounts after five years, assuming that you earn simple interest. 45. You deposit $3200 in an account with an annual interest rate of 2.5%. Amount after five years = $ $ ) = $3600. Compound Interest. Use the compound interest formula to compute the balance in the following accounts after the stated period of time, assuming interest is compounded annually. 49. $10,000 is invested at an APR of 4% for 10 years. $10, ) 10 = $14,

2 Compounding More Than Once a Year. Use the appropriate compound interest formula to compute the balance in the following accounts after the stated period of time. 58. $10,000 is invested for 5 years with an APR of 2.75% and monthly compounding. $10, ) 5 = $11, $2000 is invested for 15 years with an APR of 5% and monthly compounding. $ ) 15 = $ Annual Percentage Yield APY). Find the annual percentage yield to the nearest 0.01%) in the following situations. 64. A bank offers an APR of 4.5% compounded monthly. APY = ) 1 = or 4.59% Continuous Compounding. Use the formula for continuous compounding to compute the balance in the following accounts after 1, 5, and 20 years. Also, find the APY for each account. 68. A $2000 deposit in an account with an APR of 3.1% after 1 year: $2000 e ) = $ after 5 years: $2000 e ) = $ after 20 years: $2000 e ) = $ APY: e ) 1 = or 3.15% Planning Ahead. How much must you deposit today into the following accounts in order to have $25,000 in 8 years for a down payment on a house? Assume no additional deposits are made 75. An account with monthly compounding and an APR of 6% P = $25, 000 ) = $15, Comparing Annual Yields. Consider an account with an APR of 6.6%. Find the APY with quarterly compounding, monthly compounding, and daily compounding. Comment on how changing the compounding period affects the annual yield. quarterly APY: 6 / 4 ) 4 1 = or 6.77% monthly APY: 6 / ) 1 = or 6.80% daily APY: 6 / 360 ) = or 6.82% APY increases as the compounding period increases.

3 84. Comparing Annual Yields. Consider an account with an APR of 5%. Find the APY with quarterly compounding, monthly compounding, and daily compounding. Comment on how changing the compounding period affects the annual yield. quarterly APY: 0.05 / 4 ) 4 1 = or 5.09% monthly APY: 0.05 / ) 1 = 0.05 or 5.% daily APY: 0.05 / 360 ) = or 5.13% APY increases as the compounding period increases. 87. Comparing Investment Plans. Rosa invests $3000 in an account with an APR of 4% and annual compounding. Julian invests $2500 in an account with an APR of 5% and annual compounding. a. Compute the balance in each account after 5 and 20 years. Rosa: $ ) 5 = $ $ ) 20 = $ Julian: $ ) 5 = $ $ ) 20 = $ b. Determine, for each account and for 5 and 20 years, the percentage of the balance that is interest. Rosa: Julian: $ $3000 $ $ $2500 $ % 21.6% $ $3000 $ $ $2500 $ c. Comment on the effect of interest rates and patience. The longer the investment time, the higher the overall return. 54.4% 62.3% 88. Comparing Investment Plans. Paula invests $4000 in an account with an APR of 4.8% and continuous compounding. Petra invests $3600 in an account with an APR of 5.6% and continuous compounding. a. Compute the balance in each account after 5 and 20 years. Paula: $4000 e ) = $ $4000 e ) = $10, Petra: $3600 e ) = $ $3600 e ) = $11, b. Determine, for each account and for 5 and 20 years, the percentage of the balance that is interest. Paula: Petra: $ $4000 $ $ $3600 $ % 24.4% $10, $4000 $10, $11, $3600 $11, c. Comment on the effect of interest rates and patience. The longer the investment time, the higher the overall return. 61.7% 67.4%

4 Chapter 4C Solving with Powers and Roots. Solve the following equations for the unknown quantity. 17. x 4) 2 = v = 68 x 4) 2 = 36 x 4) 2 ] 1 /2 = 36 1 /2 x = x = 10 v = 68 v = 68 4 v 3 ) 1 /3 = 64 1 /3 v = 4 Savings Plan Formula. Assume monthly deposits and monthly compounding in the following savings plans. 24. Find the savings plan balance after 5 years with an APR of 2.5% and monthly payments of $ )] 5) 1 A = $100 ) = $ Find the savings plan balance after 3 years with an APR of 4% and monthly payments of $ )] 3) 1 A = $400 ) = $15, Investment Plans. Use the savings plan formula to answer the following questions. 27. At age 25, you set up an IRA individual retirement account) with an APR of 5%. At the end of each month, you deposit $75 in the account. How much will the IRA contain when you retire at age 65? Compare that amount to the total deposits made over the time period. You save for 40 years, so the value of the IRA is 0.05 )] 40) 1 A = $75 ) = $114, Since you deposit $75 each month for the 40 years, your will have deposited a total of $75 40 = $36, 000, or just under a third of the final value of the IRA.

5 Planning for the Future. Use the savings plan formula to answer the following questions. 32. At age 35 you start saving for retirement. If your investment plan pays an APR of 6% and you want to to have $2 million when you retire in 30 years, how much should you deposit monthly? ) P MT = $2, 000, 000 )] 30) 1 = $ Comfortable Retirement. Suppose you are 30 years old and would like to retire at age 60. Furthermore, you would like to have a retirement fund from which you can draw an income of $100,000 per year forever! How can you do it? Assume a constant APR of 6%. In order to be able to draw $100,000 in interest each year your retirement account balance must be $100, 000/0.06 = $1, 666, Hence, your monthly deposit should be ) P MT = $1, 666, Alternate better) answer: )] 30) 1 = $ In order to be able to draw $100,000 in interest each year your retirement account balance must be $100, 000/0.06 = $1, 666, Hence, your annual deposit should be ) P MT = $1, 666, )] 1 30) 1 = $21, Who Comes Out Ahead? Consider the following pairs of savings plans. Compare the balances in each plan after 10 years. In each case, which person deposited more money in the plan? Which of the two investments strategies do you believe was better? Assume that the compounding and payment periods are the same. 65. Yolanda deposits $200 per month in an account with and APR of 5%, while Zach deposits $2400 at the end of each year in an account with an APR of 5%. After ten years, the balance in Yolanda s account is 0.05 )] 10) 1 A = $200 ) = $31, She deposits $200 each month, so her total deposits are $ = $24, 000. Zach s account is worth 0.05 )] 1 10) 1 1 A = $2400 ) = $30, He deposits $2400 each year, so his total deposits are $ = $24, 000. Yolanda s strategy is better because, since her account compounds monthly, she enjoys a higher APY.

6 76. Cigarettes to Dollars. The website MSN Money claims, A 40-year-old who quits smoking puts the savings into a 401k) earning 9% a year would have nearly $250,000 by age 70. Check the accuracy of this statement for a pack-a-day smoker who spends $4.50 per pack. Assume monthly deposits and monthly compounding. The monthly deposit would be $ = $135, so the final value of the savings plan would be: ) A = $135 ) = $247, 150, 0.09 which means the article was correct. 77. Get Started Early! Mitch and Bill are the same age. When Mitch is 25 years old, he begins depositing $1000 per year into a savings account. He makes deposits for 10 years, at which point he is forced to stop making deposits. However, he leaves his money in the account for the next 40 years where it continues to earn interest). Bill doesn t start saving until he is 35 years old, but for the next 40 years he makes annual deposits of $1000. Assume that both accounts earn interest at an annual rate of 7% and interest in both accounts is compounded once a year. a. How much money does Mitch have in his account at age 75? At age 35, the balance in Mitch s account is A = $ ) ) = $13, For the next 40 years he is not making any more deposits so we should use the compound interest formula using this balance as the principle. A = $13, ) 40 = $206, b. How much money does Bill have in his account at age 75? At age 75, the balance in Bill s account is A = $ ) ) = $199, c. Compare the amounts of money that Mitch and Bill deposit into their accounts. Mitch deposited $1000 per year for ten years, or $10,000. Bill deposited $1000 per year for forty years, which is $40,000, or four times as much as Mitch.