• Document: Chapter 2 The Time Value of Money
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Chapter 2 The Time Value of Money 2-1 The effective interest rate is 19.56%. If there are 12 compounding periods per year, what is the nominal interest rate? Solution ieff = (1 + (r/m))m − 1 ⇒ r/m = (1 + ieff)1/m − 1 = (1.1956)1/12 − 1 = 1.5% r = 12 × 1.5 = 18% 2-2 What is the effective interest rate on a continuously compounded loan that has a nominal interest rate of 25%? a. e1.25 b. e0.25 c. e1.25 - 1 d. e.25 - 1 Solution ieff = er - 1 ieff = e.25 - 1 The answer is d. 2-3 Given a situation where the annual interest rate is 5%, when continuous compounding is used rather than monthly compounding, the nominal interest rate a. increases. b. remains the same. c. decreases. Solution 43 44 Chapter 2 Time Value of Money The answer is b: Remains the same 2-4 A local bank is advertising they pay savers 6% compounded monthly, yielding an effective annual rate of 6.168%. If \$2,000 is placed in savings now and no withdrawals are made, how much interest (to the penny) will be earned in one year? Solution Interest = Effective annual rate × principal = 0.06168 × 2,000 = \$123.36 Monthly compounding is irrelevant when the effective rate is known. 2-5 A man decides to put \$100 per month beginning one month from today into an account paying 12% compounded monthly. Determine how much (to the nearest penny) will be in the account immediately after the fourth deposit using only basic concepts. Solution Beginning Ending Month Balance Interest @ 1% Deposit Balance 1 \$ 0.00 0.00 \$100 \$100.00 2 100.00 1.00 100 201.00 3 201.00 2.01 100 303.01 4 303.01 3.03 100 406.04 ←Answer 2-6 A small company borrowed \$10,000 to expand the business. The entire principal of \$10,000 will be repaid in 2 years but quarterly interest of \$330 must be paid every three months. What nominal annual interest rate is the company paying? Solution The \$330 is interest for one period, therefore i = 330/10,000 = 3.3% per quarter r = 3.3 × 4 = 13.2% nominal annual 2-7 A Cole’s Home Solutions policy is to charge 1¼% interest each month on the unpaid balance. What is the nominal interest the Cole’s is charging? What is the effective interest? Solution (a) r = mi = 12(1.25) = 15% (b) ieff = (1 + i)n - 1 = (1.0125)12 - 1 = 16.075% Chapter 2 Time Value of Money 45 2-8 E. Z. Marc received a loan of \$50 from the S.H. Ark Loan Company that he had to repay one month later with a single payment of \$60. What was the nominal annual interest rate for this loan? Solution Interest = \$10 for one month i = 10/50 = 20% i = im = 20 × 12 = 240% 2-9 A local college parking enforcement bureau issues parking tickets which must be paid within one week. The person receiving the ticket may pay either \$5 immediately or \$7 if payment is deferred one week. What nominal interest rate is implied in the arrangement? Solution i = (7 - 5)/5 = 40% per week r = im = 52(40) = 2,080% 2-10 A deposit of \$300 was made one year ago into an account paying monthly interest. If the account now has \$320.52, what was the effective annual interest rate? Give answer to 1/100 of a percent. Solution ieff = 20.52/300 = 6.84% 2-11 Which is the better investment, a fund that pays 15% compounded annually, or one that pays 14% compounded continuously? Solution ieff = 15% compounded annually = [(1 + .15)1 - 1] = 15% ieff = 14% compounded continuously = [e.14 - 1] = 15.03% Therefore 14% compounded continuously is slightly better. 2-12 Ten years ago, Jenna C. deposited \$2,000 into an account that paid 5% simple interest for the first four years and then it paid 6% compounded monthly for the remaining six years. The amount in the account at the end of the ten-year period is closest to a. \$3,420 b. \$3,440 c. \$3,460 d. \$3,480 46 Chapter 2 Time Value of Money Solution Simple interest earned at end of four years = 2,000 × .05 × 4 = \$400 i = 6/12 = ½% n = (12)(6) = 72 F = 2,400(1.005)72 = \$3,436.90 The answer is b. 2-13 You borrowed \$25 from a friend and after five months repaid \$27. What nominal interest rate did you pay? What was the effective interest rate? Solution F = P(1 + i)n 27 = 25(1 + i)5 (27/25)1/5 = 1 + i = 1.0155 i = 1.55% per month or 18.6% per year ieff = (1 + .0155)12 - 1 = 20.27% 2-14 A local Qwik Kash will loan a person \$2,000 with a payment of \$2,200 due in four weeks.

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