Chapter 5 The 'I'Irne Value of Money 157 41. 42. 43. 45. 46. 47. 48. Loan Year-End Interest on. on rent YIItvEl'ld Amot'tluol'l of Loan Bel-nee l$l Payment [5} l0." l3) lietlrement Savings. A couple will retire in 50 years; they Plan 10 SW9d "'0'" "um I 3"!" (tn current dollars} in retirement. which should last about 2.5 years. 1116:! Mic": "'5' "'3' can earn a real interest rate of 8% on retirement savings. (L053) :1. if they nuke annuai payments into a savings plan. how much will they "05d to 53"" "Ch ital"? Assume the rst payment comes in 1 year. I:- How would the answer to pan (a) change it 01.: couple also realize that in 20 years they will need '0 Spend $60,000 in current dollars on their child's college education? iletirement Savings. You believe you will need to have saved 5500.000 by the time 3'0" mire In 40 years to order to live comfortably. If the interest rate is 6% per year. 110'" much mm you save each year to meet your retirement goal? (L054) Batirement Savings. You believe you will need to have saved $500,000 by the Lime you retire in 40 years in order to live comfortably. You also believe that you will inherit 5100.000 in '0 years. If the interest rate is 6% per year, how much must you save each year to meet your retirement goal? (L05-3) Retirement Savings. You believe you will spend 540.000 a year for 20 years one: you retire in 40 years. If the interest rate is 6% per year. how much must you save each year until retirement to meet your retirement goal? (L053) Retirement Savings. A couple thinking about retirement decide to put aside $3.000 each year in a savings plan that earns 8% interest. in 5 years they will receive a gift of 510.000 that 5130 can be invested. (MS-3) a. How much money will they have accumulated 30 years from now? b. If their goal is to retire with $300.000 of savings. how much extra do they need to save every year? Perpenrllius and Effective Interest Rate. What is the value of a perpetuity that pays $100 every 3 months forever? The interest rate quoted on an APR basis is 6%. (L053) Amortizing Loans and Ination. Suppose you take out a 3100.000. 20-year mortgage loan to buy a condo. The interest rate on the loan is 6%. To keep things simple. we will assume you make payments on the loan annually at the end of each year. (005-3) a. What is your annual payment on the loan? it. Construct a mortgage amortization table in Excel similar to Table 5.5 in which you compute the interest payment each year. the amortization of the loan. and the loan balance each year. (Allow the interest rate to he an input that the user of the Spreadsheet can enter and change.) What fraction of your initial loan payment is interest? What fraction of your initial loan payment is amortization? e. What fraction of the loan has been paid off after 10 years (halfWay through the life of the loan)? Why is the answer not 50%? f. If the inflation rate is 2%. what is the real value of the rst (year~end) payment? If the ination rate is 2%. what is the real value of the last (yeanend) payment? Now assume the inflation rate is 8% and the real interest rate on the loan is unchanged. What must be the new nominal interest rate? i. Recompute the amortization table. What is the real value of the rst {year-end) payment in this high-ination scenario? j. What is the real value of the last payment in this high-ination scenario? so we Mortgage with Points. Home loans often involve "points." which are fees charged by the lender. Each. point charged means that the borrower must pay 1% of the loan amount as :1 ice. For example. if the loan is for $100,000 and 2 points are charged. the loan repayment schedule is calculated on a $100,000 loan but the net amount the borrower receives is only $93,000 Assume the interest rate is 1% per month. What is the effective annual interest rate charged on such a loan. assuming loan repayment occurs over 360 months? (L054) Part Two Value 49. Effective Interest Rate. A store will give you a 3% discount on the cost of your purchase if you pay cash today. Otherwise, you will be billed the full price with payment due in 1 month. Why month? (LO5-4) is the implicit borrowing rate being paid by customers who choose to defer payment for the 50. Effective Interest Rate. You've borrowed $4,248.68 and agreed to pay back the loan with monthly payments of $200. If the interest rate is 12% stated as an APR, how long will it take you to pay back the loan? What is the effective annual rate on the loan? (LO5-4) 51. Effective Interest Rate. You invest $1,000 at a 6% annual interest rate, stated as an APR. Inter. est is compounded monthly. How much will you have in 1 year? In 1.5 years? (105-4) 52. Effective Interest Rate. If a bank pays 6% interest with continuous compounding, what is the effective annual rate? (LOS-4) 53. Effective Interest Rate. In a discount interest loan, you pay the interest payment up front. For example, if a 1-year loan is stated as $10,000 and the interest rate is 10%%, the borrower "pays" .10 x $10,000 = $1,000 immediately, thereby receiving net funds of $9.000 and repaying $10,000 in a year. (LO5-4) a. What is the effective interest rate on this loan? b. What is the effective annual rate on a 1-year loan with an interest rate quoted on a discount basis of 20%? 54. Effective Interest Rate. Banks sometimes quote interest rates in the form of "add-on inter- est." In this case, if a 1-year loan is quoted with a 20% interest rate and you borrow $1,000, then you pay back $1,200. But you make these payments in monthly installments of $100 each. (LO5-4) a. What is the true APR on this loan? b. What is the effective annual rate on the loan? 55. Effective Interest Rate. You borrow $1,000 from the bank and agree to repay the loan over the next year in 12 equal monthly payments of $90. However, the bank also charges you a loan initiation fee of $20, which is taken out of the initial proceeds of the loan. What is the effective annual interest rate on the loan, taking account of the impact of the initia- tion fee? (LO5-4) 56. Effective Interest Rate. First National Bank pays 6.2% interest compounded semiannually. Second National Bank pays 6% interest compounded monthly. Which bank offers the higher effective annual interest rate? (LOS-4) 57. Loan Payments. You take out an $8,000 car loan that calls for 48 monthly payments starting after 1 month at an APR of 10%. (LO5-4) a. What is your monthly payment? b. What is the effective annual interest rate on the loan? c. Now assume the payments are made in four annual year-end installments. What annual pay- ment would have the same present value as the monthly payment you calculated? 58. Continuous Compounding. How much will $100 grow to if invested at a continuously com- pounded interest rate of 10% for 8 years? What if it is invested for 10 years at 8%? (LOS-4) 59. Effective Interest Rate. Find the effective annual interest rate for each case. (LO5-4) APR Compounding Period 12% 1 month 8 10 a W 60. Effective Interest Rate. Find the APR (the stated interest rate) for each case. (LO5-4) Effective Annual Compounding Interest Rate Period 10.00% 1 month 6.09 8.24 W O