serena wants to borrow $15000 and pay it back in 10 years. the bank gives her 2 options
1: borrow the money at 10% compounded quarterly for the full term
option 2: borrow the money at 12%compounded quarterly for 5 years and after 5 years that interest rate will be 6% compounded quarterly how much will serena save by choosing the second option?
To find out how much Serena will save by choosing the second option, we need to calculate the total amount she would have to pay back under each option and then compare the results.
Option 1: Borrowing at 10% compounded quarterly for the full 10-year term.
To calculate the total amount Serena would have to pay back under this option, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = Total amount to be paid back
P = Principal (initial loan amount)
r = Annual interest rate (in decimal form, so 10% becomes 0.10)
n = Number of times interest is compounded per year (quarterly means 4 times per year)
t = Number of years
In this case, P = $15,000, r = 0.10, n = 4, and t = 10.
A = $15000(1 + 0.10/4)^(4*10)
A = $15000(1 + 0.025)^40
A = $15000(1.025)^40
A ≈ $15000(1.200673)
A ≈ $18010.10
Therefore, Serena would have to pay back approximately $18,010.10 under option 1.
Option 2: Borrowing at 12% compounded quarterly for 5 years, and then at 6% compounded quarterly for the remaining 5 years.
Under this option, we need to calculate two separate amounts:
1. The total amount Serena would have to pay back in the first 5 years at 12% compounded quarterly.
2. The total amount Serena would have to pay back in the next 5 years at 6% compounded quarterly.
For the first 5 years:
P = $15,000, r = 0.12, n = 4, and t = 5.
A1 = $15000(1 + 0.12/4)^(4*5)
A1 = $15000(1 + 0.03)^20
A1 = $15000(1.03)^20
A1 ≈ $15000(1.806112)
A1 ≈ $27,091.68
For the remaining 5 years:
P = The remaining principal from the first 5 years = $27,091.68.
r = 0.06, n = 4, and t = 5.
A2 = $27091.68(1 + 0.06/4)^(4*5)
A2 = $27091.68(1 + 0.015)^20
A2 = $27091.68(1.015)^20
A2 ≈ $27091.68(1.345724)
A2 ≈ $36,428.92
Therefore, Serena would have to pay back approximately $27,091.68+$36,428.92 = $63,520.60 under option 2.
To find the savings, we subtract the total payment under option 2 from option 1:
Savings = Total payment under option 1 - Total payment under option 2
Savings = $18,010.10 - $63,520.60
Savings ≈ -$45,510.50
Therefore, Serena would save approximately $45,510.50 by choosing the second option.
15000*(1+.10/4)^(4*10)-(15000*(1+.12/4)^(4*5))*(1+.06/4)^(4*5)
= $3787.41