Week 7 homework 2024_chaitanya

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School

Webster University *

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Course

5200

Subject

Finance

Date

Apr 3, 2024

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docx

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Uploaded by LieutenantRockLemur10

Week 7 Homework 2024 1. Thomas recently graduated with his Master’s degree, and landed a great job. He decided that he wants to begin investing and he has asked you for advice. He wants to know that if he invests 4,000 today and leaves it invested for 10 years at a rate of 8% compounded annually, plus he invests 200 per month at the end of each month with the same rate of return and compounding frequency, how much will his investment be worth at the end of 10 years? To calculate the future value of Thomas's investment, we can use the formula for compound interest: A=P× ( 1 + r n ) nt Where:  A =future value of the investment.  P = initial investment.  r =interest rate (as a decimal).  n is the number of times the interest is compounded per year.  t is the time the money is invested for in years. Given: P = $4000(initial investment)

r = 0.08(8% annual interest rate) n = 12(compounded monthly) t = 10 years First, let’s calculate the future value of the initial investment: A initial = 4000 × ( 1 + 0.08 12 ) 12 × 10 = 4000 × ( 1.00666 ) 120 = 4000 × 2.2178 Now, let’s calculate the future value of the monthly investments. A annuity = PMT × ( 1 + r n ) nt − 1 r n

Where: PMT = $200 r = 0.08 n = 12 t = 10 years A annuity = 200 × ( 1 + 0.08 12 ) 12 × 10 − 1 0.08 12 = 200 × 2.217 − 1 0.00666 = 200 × 184.39 = 36,878.78 Now, Summing both: A total = A initial + A annuity = 45,749.9 So, Thomas’s investment will be worth $45,749.9 at the end of 10 years. 2. Given the same information in question #1, how much will Thomas earn in interest over the 10 year period? To find out how much interest Thomas will earn over the 10-year period, we can subtract the total amount he invested from final value of his investment.  Initial investment: $4000  Monthly investment: $200 * 12 months * 10 years = $24,000 Total amount invested = $4000 + $24,000 = 28,000 Total future value of his investment after 10 years is $45,749.9 Interest earned = Total future value – Total amount invested ⸫

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= 45,749.9 – 28,000 = 17,749.9 3. Payday loans are very short-term loans that charge very high interest rates. You can borrow $800 today and repay $850 in two weeks. What is the compound annual rate implied by this percent rate charged for only two weeks? To calculate: 𝐴 = P ( 1 + r n ) nt Where, P = $800, A = $850, n=52 (assuming interest is compounded weekly, as there are 52 weeks in a year), t=2/52years (since it's a two-week period) so, r= n ( ( A P ) 1 / nt − 1 ) ❑ 𝑟 = 52 ( ( 850 800 ) 1 / 52 ∗ 0.038 − 1 ) ❑ 𝑟 =52(0.03115) r = 1.628 or 162% The compound annual rate implied by this percent rate charged for only two weeks is 162% 4. Payday loans are very short-term loans that charge very high interest rates. You can borrow $650 today and repay $700 in 10 days. What is the compound annual rate implied by this percent rate charged for only 10 days? Interest charged for 10 days is $700 – 650 = $50 $50 / 650 = 1 13 = 0.0769 364 days / 10 days = 36.4 Total interest earned = 0.07 × 36.4 = 2.548* 100 = 254.8%

5. Tammy wants to have $1,000,000 when she retires. If she deposits $50,000 in the bank today and earns 12% compounded annually, how many years will it take her to reach her investment goals? Answer: To calculate this problem, we can use the formula of t:

t = ⸫ log ( FV PV ) log ( 1 + r ) Given: PV = $50,000(present value) FV = $1,000,000(future value) r = 12% or 0.12(annual interest rate) t =? (number of years) t = log ( 1,000,000 50,000 ) log ( 1 + 0.12 ) = log ( 20 ) log ( 1.12 ) = 1.3016 0.0492 t = 26.4 Therefore, it will take Tammy 27 years to reach her goal. 6. Given the same information in #5, what was the total amount of interest Tammy earned on her investment. we’ve already calculated that it will take approximately 26.4 years for Tammy to reach her investment goal. Now, we can calculate the total interest earned by subtracting the principal amount from the future value. Total Interest = Final Amount – Initial principal = $1,000,000 - $50,000 = $950,000 So, Tammy earned a total interest of $950,000 on her investment.

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7. Mr. Jones is considering purchasing a used car for $25,000. Mr. Jones has an excellent credit rating, so the dealer offers to finance the car at 5% interest over a 4 year period. What is the monthly payment amount that Mr. Jones would be expected to pay? To calculate the monthly payment for a loan, formula for a fixed-rate loan payment: M= P ∗ r ( 1 + r ) n ( 1 + r ) n − 1 Given, P = $25,000, r = 5%/12= 0.0041, n = 4×12=48. M= 25000 ∗ 0.0041 ( 1 + 0.0041 ) 48 ( 1 + 0.0041 ) 48 − 1 M= 25000 ∗ 0.0041 ( 1 + 0.0041 ) 48 ( 1 + 0.0041 ) 48 − 1 M = 25000 ∗ 0.0041 ∗ 1.217 1.217 − 1 M = 25000 ∗ 0.0049 0.217 M= 574.850 The monthly payment amount that Mr. Jones would pay is approximately $575 8. Given the same information in #7, Mr Jones decides that the terms result in a payment that he cannot afford. What will be the payment amount if he finances over a 5 year period instead of 4 years, using the same interest rate and purchase price of the car? M= P ∗ r ( 1 + r ) n ( 1 + r ) n − 1 Given, P = $25,000, r = 5%/12= 0.0041, n = 5×12=60. M= 25000 ∗ 0.0041 ( 1 + 0.0041 ) 60 ( 1 + 0.0041 ) 60 − 1 M= 25000 ∗ 0.0041 ( 1 + 0.0041 ) 60 ( 1 + 0.0041 ) 60 − 1

M = 25000 ∗ 0.0041 ∗ 1.278 1.278 − 1 M = 25000 ∗ 0.00523 0.278 M= 470.323 9. Given the same information in #7 and #8, how much more in interest would Mr. Jones pay if he chooses the 5 year option over the 4 year option? the total interest paid for each option can be calculated as: lets say monthly payment amount for 4 year =M4 , monthly payment amount for 5 year =M5 Then the total interest paid for each option can be calculated as: Total interest for 4 year I4= M4*48-25000 Total interest for 5 year I5= M5*60-25000 I4=574.850*48-25000 I4=27592.81-25000 I4=2592.8 I5=470.32*60-25000 I5=28219.2-25000 I5=3219.2 Difference in Interest=I5-I4 = 3219.2- 2592.8 =626.4 So, Mr. Jones would pay approximately $626.4 more in interest if he chooses the 5-year option over the 4-year option.

10. Mr. Webster is planning to retire in 30 years, and his goal is to have $1,500,000 in his retirement fund when he retires. He will start by depositing $35,000 into his fund today and $1,000 at the end of each month for the 30 year period. What is the annual rate of return that Mr. Webster will need to accomplish his goal? Future Value (FV) = 1,500,000 Present Value (PV) = 35,000 PMT = 1,000 Number of periods (N) = 30yrs = 360months FV = PV × ( 1 + r ) n r + PMT × ( 1 + r ) n − 1 r 1,500,000 = 35,000 * (1+r) ^360 + 1000((1+r) ^ 360 – 1)/r) = 7.34 Therefore, the annual rate of interest Mr. Webster gets around 7.34%. 11. Given the same information in #10, what is the total amount of interest that Mr. Webster will earn on his investment assuming he meets his retirement goals/ Total amount of his goal = $1,500,000 Initial principal = $35,000 Monthly payments = $1,000 * 30 * 12 = $360,000 Total invested = $35,000 + $360,000 = $395,000 Total Interest = $1,500,000 - $395,000 Total interest = $1,105,000 Therefore, Mr. Webster earned $1,105,000 in total for his investment goals. 12. A car loan is offered with monthly payments and 10 percent APR. What is the loan's effective annual rate (EAR)? EAR = [1+r/n]^n – 1 EAR = [1+0.1/12]^12 – 1

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EAR = [1.0083]^12 – 1 EAR = 0.1047 = 10.47% So, the effective annual rate (EAR) of the loan is approximately 10.47%. 13. Mr. Roberts wants to buy a car for $35,000. The bank will provide terms of 5 years at 6%. Mr. Roberts can afford a payment of up to $625 per month. Based on these terms, can Mr. Roberts afford to make the payments? Explain your answer. M= P ∗ r ( 1 + r ) n ( 1 + r ) n − 1 P= 35000, r= 0.06/12= 0.005, n=60 M= 35000 ∗ 0.005 ( 1.005 ) 60 ( 1.005 ) 60 − 1 M= 35000 ∗ 0.005 ∗ 1.348 0.348 M= 35000 ∗ 0.00674 0.348 M= 677.87 As Mr. Roberts can afford an amount of $625 but the calculated EMI with given details is $677.87, he will not be able to afford the EMI for car. 14. Given the same information in #13, it’s now 1 year later. For purposes of question #14 only, assume that Mr. Roberts was able to purchase the car. He is now considering selling his new vehicle after making the payments for 1 year (12 months). What is the selling price Mr. Roberts will need to get for his vehicle in order to pay off the principle? Payments made period till date (t) = 12 B = P * [(1+r) ^ n – (1+r) ^ t] / [(1+r) ^ n -1] Where: B is the remaining principal balance P is the initial amount = $30,000

r is the monthly interest rate divided by 12 n is the total number of payments made-12 payments B = 35,000 * [(1 + 0.005) ^ 60 – (1 + 0.005) ^ 12] / [(1 + 0.005) ^ 60 – 1] B = 35,000 * [(1.005) ^ 60 – (1.005) ^ 12] / [(1.005) ^ 60 -1] B = 35,000 * [1.3489 – 1.0617]/ [1.3489 – 1] B = 35,000 * (0.2872/0.3489) B = 35,000 * 0.823 B = 28, 810 Hence, Mr. Roberts should sell the car for a minimum of $28,810 to recover the cost of car. 15. Susan realizes that she has charged too much on her credit card and has racked up $22,000 in debt. If Susan can pay $500 each month and the card charges 23 percent APR (compounded monthly), how long will it take her to pay off the debt? A = $22,000 P = $500 Annual Interest Rate = 23% Monthly Interest Rate (r) = 23%/12 = 0.23/12 = 0.01967 n = -log (1 – (r *A)/(P)) / log (1 + r) n = -log (1 – (0. 01967*22000)/500) /log (1+0. 01967) n = - log (1 – 0.843) / log (1.01967) n = 0.804/0.00824 n = 97.57 Therefore, it takes around 97.57 months for Susan to pay off the debt. 16. Given the same information in #15 and that Susan makes the $500 payment each month until the debt is paid off. How much interest will Susan pay? Approximate months taken = 97.57 Amount paid per month = $500 Total amount to be paid = 500 * 97.57 = $48,785 Actual amount (A) = $22,000 Total Interest = Total paid – Actual amount Total Interest = $48,785 - $22,000

Total Interest = $26,785 Therefore, Susan will pay $26,785 as interest on the credit card bill.

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