Serena wanst to borrow $15 000 an pay it back in 10 years. Interest rates are so high, so the bank makes her 2 offers
option 1 - borrow the money at 12%/a compounded quarterly for the full term
option 2 -Borrow the money at 12%/a compunded quarterly for 5 years and then renegotiate the loan based on the mew balance for he last 5 years
if in 5 years the interest rate wil be 6%/a compunded quarterly , how much will Serena save by choosing the second option
option 1
amount after 10 years = 15000(1.03)^40 = 48930.57
option 2
amount after 5 years at first rate = 15000(1.03)^20 = 27091.67
amount of that 5 years later at new rate
= 27091.67(1.015)^20 = 36488.55
take the difference
To determine how much Serena will save by choosing the second option, we need to calculate the total amount she would pay under each option and compare them. Let's break down the calculations step by step:
Option 1: Borrowing the money at 12% per annum compounded quarterly for the full term of 10 years.
First, we need to calculate the quarterly interest rate for this option:
Quarterly interest rate = Annual interest rate / Number of compounding periods per year
Quarterly interest rate = 12% / 4 = 0.03 or 3%
Next, we calculate the total compounding periods for 10 years:
Total compounding periods = Number of years * Number of compounding periods per year
Total compounding periods = 10 * 4 = 40
Now, we can calculate the future value (amount to be repaid) of the loan using the formula for compound interest:
Future value = Principal * (1 + interest rate)^n
Future value = $15,000 * (1 + 0.03)^40
Option 2: Borrowing the money at 12% per annum compounded quarterly for the first 5 years, then renegotiating at a new interest rate of 6% per annum compounded quarterly for the remaining 5 years.
For the first 5 years, we follow the same steps as in Option 1 to calculate the future value. However, the interest rate in the remaining 5 years changes:
First, we calculate the new quarterly interest rate for the remaining 5 years (6% per annum):
New quarterly interest rate = 6% / 4 = 0.015 or 1.5%
Now, we calculate the future value for the first 5 years using the original interest rate of 12% (same as in Option 1):
Future value for the first 5 years = Principal * (1 + interest rate)^n
Future value for the first 5 years = $15,000 * (1 + 0.03)^20
Finally, we calculate the future value for the remaining 5 years using the new interest rate of 6%:
Future value for the remaining 5 years = (Future value for the first 5 years) * (1 + new interest rate)^n
Future value for the remaining 5 years = (Future value for the first 5 years) * (1 + 0.015)^20
To find the amount Serena would save by choosing the second option, we subtract the total amount payable under Option 2 from the total amount payable under Option 1:
Amount saved = (Future value under Option 1) - (Future value under Option 2)
By plugging in the values into these formulas and performing the calculations, you will arrive at the answer, which represents the amount that Serena would save by choosing the second option.