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1209

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Apr 3, 2024

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Serena Le Professor Provost FINA1209 1/23/23 Assignment 1 – Time Value of Money Problems Prepare solutions to the following problems and submit them as instructed by your section instructor. Show all calculations, or identify the sequence of steps and buttons to your financial calculator, to support the solutions you submit. i 1. John is purchasing a house for $500,000. He plans to make a down payment of $100,000 and take out a 30-year mortgage for $400,000. A. If the interest rate on the house is 5.25 per cent per year, how much will his monthly payment be for principal and interest. His monthly payment would be $2,208.81. Steps and calculations: 1) Identify number of periods (N) => 30 years x 12 months/year = 360 months 2) Identify interest per period (I) => 5.25% per year / 12 months/year = 0.4375% per month 3) Identify present value (PV) => $400,000 4) Future Value (FV) => $0 5) CPT => PMT = $2,208.81 B. If the interest rate is the same, how much would his monthly payment for principal and interest be if he took out a 15-year mortgage? Note: Normally, a 15-year mortgage would carry a lower interest rate. The interest rate here is held constant to simplify the problem. His monthly payment would be $3,215.51 Steps and calculations: 1) Adjust number of periods (N) => 15 years x 12 months/year = 180 months 2) All other variables remain the same. 3) CPT => PMT = $3,215.51 C. How much interest will John save in total with the 15-year mortgage? He would be saving $216,379.80. Steps and calculations: 1) Identify total amount paid in 30-year mortgage => payment per period x number of periods => $2,208.81 per month x 360 months = $795,171.60 2) Identify total amount paid in 15-year mortgage => payment per period x number of periods => $3,215.51 per month x 180 months = $578,791.80 3) Subtract 15-year amount from 30-year amount => $795,171.60 - $578,791.80 = $216,379.80
2. In a recent large PowerBall lottery, the prize was reported to be worth $590 million, which could be taken in 25 equal annual installments of $23.6 million beginning today or as a single payment today of $334 million. A. What interest rate is the lottery commission using? A. 4.0% B. 4.8% C. 5.5% D. 6.1% C) 5.5%; The lottery commission is using an interest rate of 5.5%. Steps and calculations: Part 1: Identify Knowns 1) Identify number of periods (N) => 25 years 2) Identify payments per period (PMT) => $23,600,000 3) Identify present value => $334,000,000 Part 2: Adjust and Calculate 1) N = 25, PMT = $23,600,000, and PV = -$334,000,000 because it is not yes “disbursed” and “owed,” and FV = 0 when it is completely disbursed. 2) CPT => (I/Y) ≈ 5.5%. B. How would you decide whether to take the 25 year annuity or the lump sum? If you can take the lump sum and invest it into something with an annual return greater than 5.5%, you should take the lump sum. If not, you should take the annuity. 3. Twins, Megan and Melissa, graduated from Northeastern and began working at age 23. Megan decided to put $2,000 per year into a retirement account for 10 years, planning to stop at that time to raise a family. She made no additional contributions until she retired at the age of 67. Melissa did not begin making contributions to her retirement account until she was 33, ten years later, and then continued to make contributions of $2,000 each year until she reached her normal full retirement age of 67. In both cases all contributions were to be made at the end of the year, and both earned 10% on their money. A. How much did Megan contribute in total and how much will she have when she retires at 67? Megan contributed $20,000 and will have $814,328.14 by retirement. Steps and calculations: Part 1: Identify how much she will have by 33 1) Identify periods (N) => 10 years 2) Identify interest/year (I/Y) => 10% 3) Identify payment per period (PMT) => $2,000 per year 4) Identify present value (PV) => $0 5) CPT => FV = $31,874 (at 33 years old) Part 2: Identify how much interest she will accumulate from 33 to 67 (retirement) 1) Identify periods (N) => 67 years - 33 years = 34 years 2) Identify interest/year (I/Y) => 10% 3) Identify payment per period (PMT) => $0 per year 4) Identify present value (PV) => $31,874.85
5) CPT => FV = $814,328.14 B. How much did Melissa contribute in total and how much will she have when she retires at 67? Melissa contributed $68,000 and Part 1: Identify how much she contributed 1) Identify the number of periods (N) => 67 years – 33 years = 34 years 2) Identify the payment per period (PMT) => $2,000 per year 3) Contribution = N x PMT = 34 years x $2,000 per year = $68,000 Part 2: Identify FV of annuity 1) Identify number of periods (N) => 67 years – 33 years = 34 years 2) Identify the payment per period (PMT) => $2,000 per year 3) Identify interest rate (I/Y) => 10% 4) Identify present value (PV) => $0 5) CPT => FV = $490,953.40 C. How do you explain the difference? Megan contributed early and was able to accumulate a good amount in her retirement account so that by the time she reached 33 and stopped contributing, she could take advantage of compound interest. The amount she earns on interest is bigger on a larger sum than it is on a smaller, so she builds up a considerable sum of $31,874.85 in 10 years and allowed that amount to compound to over $800,000 in the 34 years she didn’t contribute. On the other hand, Melissa started contributing later and contributed more over a longer period. However, by contributing up until retirement, she didn’t have time to let the interest just sit and compound, leaving her with less in her account by retirement. She ended up having a much worse return on investment and ended up contributing $48,000 more than her sister and having a little over half the amount in her account by retirement than Megan.
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i All calculations made using Pearson MyLab Financial Calculator.