Business Maths Question - help appreciated.
James and Jamie have just finished repaying a 5 year, $8000 reducing balance loan.
During the first year interest was debited monthly at 8.5% p.a.; during the next 2 years the rate fell to 8% p.a. but rose to 8.3% p.a. for the remaining period of the loan.
Before signing their contract James and Jamie were also given the option of fixing the interest rate at 8.25% p.a. (debited monthly) for the term of the loan.
By considering the total amount of interest that they paid, determine whether James and Jamie made the right decision in choosing the variable interest rate for the loan rather than the fixed rate.
Answers
Fixed: Interest = $1790.20
Variable: Interest = $1784.51 so correct decision is the variable, by $5.69
First of all, this is a ridiculously long question. It's highly unlikely you'll get a question this difficult in the exam unless I've taken a really long approach (haven't done business maths in ages....) The finance solver entries work off each other, so take your time looking through what I've changed in each calculation.
Let's first calculate the amount of interest using the fixed rate of interest, then calculate the interest using the variable interest rate.
Fixed Interest Rate (using Finance Solver for Ti-nspire): We need to find the monthly payment, so input all the given information.
n=60
(5 years multiplied by 12 months per year)I=8.25
Pv=8000
Fv=0
Ppy=12
Pmt=-163.17To find out how much interest we are paying, multiply the payment by the amount of payments and subtract the value of the loan:
Interest (fixed rate)= 163.17x60-8000=1790.20
So using the fixed interest rate, we pay $1790.20 interest.
Variable Interest Rate: We again need to find the monthly payment, but we need to split this loan up into three different sections. 1: Interest of 8.5% for 1 year. 2: Interest of 8% for 2 years. 3: Interest of 8.3% for 2 years.
1a: first find the payment if the interest stayed at 8.5% for five years.
n=60
I=8.5
Pv=8000
Fv=0
Ppy=12
Pmt=-164.13231b: find the future value if that payment lasted only one year.
n=12
I=8.5
Pv=8000
Ppy=12
Pmt=-164.1323
Fv=6658.96752a: find the payment for the next four years with interest rate 8% if the payments went for all the years.
n=48
I=8
Pv=6658.9675
Fv=0
Ppy=12
Pmt=-162.56492b: find the future value if the $162.56 payment only lasted 2 years.
n=24
I=8.5
Pv=6658.9675
Ppy=12
Pmt=-162.56
Fv=3594.39733: find the final payment to complete the loan (2 years) at 8.3%.
n=24
I=8.3
Pv=3594.3973
Fv=0
Ppy=12
Pmt=-163.0571So how much interest does he pay for the 5 year loan with the variable interest rate?
164.1323 x 12 + 162.5649 x 24 + 163.0571 x 24 - 8000 =
1784.52Fixed Interest: 1790.20
Variable Interest: 1784.52
So they save $5.68 by choosing the Variable Interest loan and hence it was the right decision.Note: I got a slightly different answer (1 cent) because of the large amount of calculations, so I think they rounded in between steps.
Hope that helps!