SECTION 5.7 Financial Models 353 54. Analyzing Interest Rates on a Mortgage Demetrius and Aniyah have just purchased a house for $650,000, with the seller holding a second mortgage of $100,000. They promise to pay the seller $100,000 plus all accrued interest 5 years from now. The seller offers them three interest options on the second mortgage: (a) Simple interest at 6% per annum (b) 5.5% interest compounded monthly (c) 5.25% interest compounded continuously Which option is best? That is, which results in paying the least interest on the loan? 55. Comparing Bank Accounts Two bank accounts are opened at the same time.The first has a principal of $1000 in an account earning 5% compounded monthly.The second has a principal of $2000 in an account earning 4% interest compounded annually. Determine the number of years, to the nearest tenth, at which the account balances will be equal. 56. Per Capita Federal Debt In 2024, the federal debt was about $34.8 trillion. In 2024, the U.S. population was about 327 million. Assuming that the federal debt is increasing about 5.5% per year and the U.S. population is increasing about 0.7% per year, determine the per capita debt (total debt divided by population) in 2040. 51. Comparing IRA Investments Will invests $2000 of the money in his IRA in a bond trust that pays 9% interest compounded semiannually. His friend Henry invests $2000 in his IRA in a certificate of deposit that pays 8 1 2 % compounded continuously. Who has more money after 20 years,Will or Henry? 52. Comparing Two Alternatives Suppose that April has access to an investment that will pay 10% interest compounded continuously. Which is better: to be given $1000 now so that she can take advantage of this investment opportunity or to be given $1325 after 3 years? 53. College Costs The average annual cost of attending a 4-year private college was $41,540 in the 2023–2024 academic year. This was a 4% increase from the previous year. (a) If the cost of college increases by 4% each year, what will be the average cost of attending a 4-year private college for the 2043–2044 academic year? (b) College savings plans, such as a 529 plan, allow individuals to put money aside now to help pay for college later. If one such plan offers a rate of 6% compounded continuously, how much should be put in a college savings plan in 2025 to pay for 1 year of the cost of college at a 4-year private college for an incoming freshman in 2043? Source: The College Board Problems 57–62 require the following discussion. Inflation is a term used to describe the erosion of the purchasing power of money. In other words, if r is the inflation rate, expressed as a percent, then an item that costs $P0 now will cost ( ) = + P 1 r P 1 0 after one year at that inflation rate. For example, if the annual inflation rate is 100%, then an item that costs $500 today will cost ( ) + = 1 1 0 500 $1000 . in one year.That is, at 100% inflation, the item will cost twice as much. Inflation is sometimes described using the term purchasing power. We can ask the question: What will $500 I have today be worth in one year with 100% interest? The answer is ( ) = + = A 500 1 1.0 500 2. That is, at an inflation rate of 100%, $500 will be worth half that amount in one year. In general, if the rate of inflation averages r% per annum over n years, the amount A that $P will purchase after n years is ( ) = + A P r 1 n 57. Inflation If the inflation rate averages 3%, what will be the purchasing power of $1000 in 2 years? 58. Inflation If the inflation rate averages 8%, what will be the purchasing power of $1000 in 3 years? 59. Inflation If the purchasing power of $1000 is only $950 after 2 years, what was the average inflation rate? 60. Inflation If the purchasing power of $1000 is only $930 after 2 years, what was the average inflation rate? 61. Inflation If the average inflation rate is 2%, how long is it until the purchasing power is cut in half? 62. Inflation If the average inflation rate is 4%, how long is it until the purchasing power is cut in half? Problems 63–66 involve zero-coupon bonds. A zero-coupon bond is a bond that is sold now at a discount and will pay its face value at the time when it matures; no interest payments are made. 63. Zero-Coupon Bonds A zero-coupon bond can be redeemed in 20 years for $10,000. How much should you be willing to pay for it now if you want a return of: (a) 5% compounded monthly? (b) 5% compounded continuously? 64. Zero-Coupon Bonds A child’s grandparents are considering buying an $80,000 face-value, zero-coupon bond at her birth so that she will have enough money for her college education 17 years later. If they want a rate of return of 6% compounded annually, what should they pay for the bond? 65. Zero-Coupon Bonds How much should a $10,000 facevalue, zero-coupon bond, maturing in 10 years, be sold for now if its rate of return is to be 4.5% compounded annually? 66. Zero-Coupon Bonds If Pat pays $15,334.65 for a $25,000 face-value, zero-coupon bond that matures in 8 years, what is his annual rate of return? 67. Time to Double or Triple an Investment The formula ( ) = + t m n r n ln ln 1 can be used to find the number of years t required to multiply an investment m times when r is the per annum interest rate compounded n times a year. (a) How many years will it take to double the value of an IRA that compounds annually at the rate of 6%? (b) How many years will it take to triple the value of a savings account that compounds quarterly at an annual rate of 5%? (c) Give a derivation of this formula.
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