Chapter 4: Homework1. A $50,000 loan is to be amortized over 7 years, with annual end-of-year payments. Which ofthese statements is CORRECT?a. The annual payments would be larger if the interest rate were lower.b. If the loan were amortized over 10 years rather than 7 years, and if the interest rate were thesame in either case, the first payment would include more dollars of interest under the 7-yearamortization plan.c. The proportion of each payment that represents interest as opposed to repayment ofprincipal would be lower if the interest rate were lower.d. The last payment would have a higher proportion of interest than the first payment.e. The proportion of interest versus principal repayment would be the same for each of the 7payments.Answer:C is correctA is incorrect. The annual payments would be smaller if the interest rate were lower.For example, PV = -50,000, N =7, I = 10%, then PMT = $10,270If I = 8%, then PMT = $9,603 onlyB is incorrect.If the loan were amortized over 10 years rather than 7 years, and if the interest ratewere the same in either case, the first payment would be the same dollars of interest under the 7-year amortization plan.D is incorrect. The last payment would have a lower proportion of interest than the first payment.E is incorrect. The proportion of interest vs. principal repayment would not be the same for eachof the 7 payments.
2. Which of the following statements is CORRECT?a. If you have a series of cash flows, each of which is positive, you can solve for I, where thesolution value of I causes the PV of the cash flows to equal the cash flow at Time 0.b. If you have a series of cash flows, and CF0 is negative but each of the following CFs ispositive, you can solve for I, but only if the sum of the undiscounted cash flows exceeds the cost.c. To solve for I, one must identify the value of I that causes the PV of the positive CFs toequal the absolute value of the PV of the negative CFs. This is, essentially, a trial-and-errorprocedure that is easy with a computer or financial calculator but quite difficult otherwise.d. If you solve for I and get a negative number, then you must have made a mistake.e. If CF0 is positive and all the other CFs are negative, then you cannot solve for I.Answer: C is correct.3. Riverside Bank offers to lend you $50,000 at a nominal rate of 6.5%, compounded monthly.The loan (principal plus interest) must be repaid at the end of the year. Midwest Bank also offersto lend you the $50,000, but it will charge an annual rate of 7.0%, with no interest due until theend of the year. How much higher or lower is the effective annual rate charged by Midwestversus the rate charged by Riverside?a. 0.52%b. 0.44%c. 0.36%d. 0.30%e. 0.24%Answer: D is correctEFF% of Riverside Bank = [1 + INOM/M] M – 1.0 = [1 + 0.065/12] 12 – 1.0 = [1 + 0.00542] 12 – 1.0 = 1.06697 – 1.0 = 0.06697 = 6.697%EFF% of Midwest Bank = 7%, the difference = 7% - 6.697% = 0.30%So, EFF% charged by Midwest is 0.30% higher
4. Steve and Ed are cousins who were both born on the same day, and both turned 25 today.Their grandfather began putting $2,500 per year into a trust fund for Steve on his 20th birthday,and he just made a 6th payment into the fund. The grandfather (or his estate’s trustee) will make40 more $2,500 payments until a 46th and final payment is made on Steves 65th birthday. Thegrandfather set things up this way because he wants Steve to work, not be a "trust fund baby,"but he also wants to ensure that Steve is provided for in his old age.Until now, the grandfather has been disappointed with Ed, hence has not given him anything.However, they recently reconciled, and the grandfather decided to make an equivalent provisionfor Ed. He will make the first payment to a trust for Ed today, and he has instructed his trustee tomake 40 additional equal annual payments until Ed turns 65, when the 41st and final paymentwill be made. If both trusts earn an annual return of 8%, how much must the grandfather put intoEds trust today and each subsequent year to enable him to have the same retirement nest egg asSteve after the last payment is made on their 65th birthday?a. $3,726b. $3,912c. $4,107d. $4,313e. $4,528Answer: A is correct.
SteveInterest 8% $ 2,500.00 per year 40 paymentsAge 20 46 65 1 2 3 4 5 6 26 46 | | | | | | | | | | | | | | | | | | | | | | | | | |Beginning $ 2,500.00 $ 2,500.00 Paid by Steve Make payment Final paymentN 46PV $ -PMT $ 2,500.00FV $1,129,750.38EdInterest 8%Age 25 41 65 1 5 10 15 16 20 30 41 | | | | | | | | | | | | | | | | First payment Make payment Final paymentN 41PV 0PMT ($3,725.55)FV $ 1,129,750.385. John and Daphne are saving for their daughter Ellens college education. Ellen just turned 10at (t = 0), and she will be entering college 8 years from now (at t = 8). College tuition andexpenses at State U. are currently $14,500 a year, but they are expected to increase at a rate of3.5% a year. Ellen should graduate in 4 years--if she takes longer or wants to go to graduateschool, she will be on her own. Tuition and other costs will be due at the beginning of eachschool year (at t = 8, 9, 10, and 11).So far, John and Daphne have accumulated $15,000 in their college savings account (at t = 0).Their long-run financial plan is to add an additional $5,000 in each of the next 4 years (at t = 1,2, 3, and 4). Then they plan to make 3 equal annual contributions in each of the following years,t = 5, 6, and 7. They expect their investment account to earn 9%. How large must the annualpayments at t = 5, 6, and 7 be to cover Ellens anticipated college costs?a. $1,965.21b. $2,068.64c. $2,177.51
d. $2,292.12e. $2,412.76Answer: E is correct.Interest rate 9%Periods: Ending 0 1 2 3 4 5 6 7 | | | | | | | | | | | | | | | | $ 15,000.00 $ 5,000.00 $ 5,000.00 $ 5,000.00 $ 5,000.00 X X X FV @ T = 7 $ 27,420.59 $ 8,385.50 $ 7,693.12 $7,057.91 $6,475.15 $ 57,032.26College tuition currently $ 14,500.00 per year 3.50% increase each yearInterest 9%Periods: 7 8 9 10 11 | | | | | | | | | | $ 19,093.73 $ 19,762.01 $ 20,453.68 $ 21,169.56 $ 17,517.18 $ 16,633.29 $ 15,794.00 X(1.09)^2 + X(1.09) + X = $ 7,906.26 $ 14,997.05 1.1881X + 1.09X + X = $ 7,906.26NPV @ T = 7 $64,941.52 3.2781X = $ 7,906.26 $ 57,032.26 X = 7906.26/3.2781Additional payment $7,909.26 $2,412.76