Question

Having been working for two years, Sarah has decided to purchase a car for daily commute and leisure. After hearing the advices and suggestions from friends and family, she has visited several auto dealerships, and chosen the new car she would like to purchase. She now wants to research her financing options to choose the best way to pay for the car. Sarah knows that with taxes, licence, delivery, and dealer preparation fees, the car will cost $27,650. She has $7500 from deposit account and $5000 from parents toward the purchase price but must borrow the rest. She has narrowed her financing choices to three options: dealer financing, credit union financing, and bank financing. (i) The car dealer has offered 48-month financing at 8.5% compounded monthly. (ii) The credit union has offered 36-month financing at 9% compounded quarterly. It has also offered 48-month financing at 9.3% compounded quarterly. (iii) The bank has offered 36-month financing at 8.8% compounded semi-annually. It has also offered 48-month financing at 9.1% compounded semi-annually. BUSI1003 Math for Business_Project 2 Sarah desires the financing option that offers the best interest rate. However, she also wants to explore the financing options that allow her to pay off her car loan more quickly. QUESTIONS 1. Sarah wants to compare the 48-month car loan options offered by the car dealer, the credit union, and the bank. (a) What is the effective annual rate of interest for each 48-month option? – 6 marks (b) How much interest will Sarah save by choosing the best option compared to the worst option? – 6 marks 2. Suppose Sarah wants to pay off her car loan within three years. (a) What is the effective annual rate of interest for both of the 36-month options? – 6 marks (b) How much interest will Sarah save by choosing the better option?

Answer 1

To Solve the given question you just need to apply the formula for effective annual interest rate i.e. (EAR)

Which is;

EAR (effective annual rate)

=

(1 + (nominal interest rate / number of compounding periods)) ^ (number of compounding periods) – 1

Check the below mentioned data for detailed solution.

Thanks
1. (a) Calculation of effective annual interest rate for each 48-month option :- EAR (effective annual rate) (1 + (nominal interest rate / number of compounding periods)) ^ (number of compounding periods) - 1 Car dealer (48 month) 8.50% Nominal interest rate Number of compunding periods 12 EAR (1 + (0.085 / 12)) ^ (12) - 1 EAR 8.84% Credit union (48 month) 9.30% Nominal interest rate Number of compunding periods 4 EAR (1 + (0.093/4))^(4) - 1 EAR 9.63% Bank (48 month) 9.10% Nominal interest rate Number of compunding periods 2 EAR (1 + (0.091 / 2)) ^ (2) - 1 EAR 9.31%

(b) Calculation of savings made by choosing best option over the worst option :- Since the EAR of car dealer is least it is the best option for 48 month period & similarly Credit union has highest EAR it is worst Loan amount = Cost of car-Deposit account-borrowing from parents 27650-7500-5000 15140 =(loan amount)* (EAR)*(years) Total interest to be paid under credit union option for 48 month financing (A) =(15140)*(0.0963)*(4) 5835.78 =(loan amount)* (EAR)*(years) Total interest to be paid under car dealer option for 48 month financing (B) =(15140)*(0.0884)*(4) 5357.04 Saving from choosing best option over worst (A-B) 478.74

2. (a) Calculation of effective annual interest rate for each 36-month option :- EAR (effective annual rate) (1 + (nominal interest rate / number of compounding periods)) ^ (number of compounding periods) - 1 Credit union (36 month) 9.00% Nominal interest rate Number of compunding periods 4 EAR (1 + (0.090 / 4))^(4) - 1 EAR 9.31% Bank (48 month) 8.80% Nominal interest rate Number of compunding periods 2 EAR (1 + (0.091 / 2)) ^ (2) - 1 EAR 8.99%

(b) Calculation of savings made by choosing better option :- Since the EAR of Credit union is less than the Bank hence, Credit union is better option for 36 month period Loan amount = Cost of car-Deposit account-borrowing from parents 27650-7500-5000 15140 I =(loan amount)*(EAR)*(years) Total interest to be paid under Credit union option for 36 month financing (C) =(15140)*(0.0931)*(3) II 4231.4 = =(loan amount)*(EAR)*(years) Total interest to be paid under Bank option for 48 month financing (D) 11 =(15140)*(0.0899)*(3) = 4085.96 Saving from choosing better option (C-D) II 145.44