In: Finance
Four years ago, your friend took out a 10-year loan for $200,000 from BOA Bank at a stated interest rate of 10% p.a. with interest compounded quarterly. She has been making equal, quarterly payments on the loan during this time and now wishes to repay the loan in full. The amount that she needs to repay the bank today is closest to:
a) $72,524.
b) $104,012.
c) $142,494.
d) $187,678.
Step-1. Calculation of the Quarterly Loan Payment
Loan Amount (P) = $200,000
Interest Rate (n) = 2.50% per Quarter [10% / 4 Quarters]
Number of years (n) = 40 Years [10 Years x 4 Quarters]
Quarterly Loan Payment = [P x {r (1+r)n} ] / [( 1+r)n – 1]
= [$200,000 x {0.025 x (1 + 0.025)40}] / [(1 + 0.025)40 – 1]
= [$200,000 x {0.025 x 2.685063}] / [2.685063 – 1]
= [$200,000 x 0.067127] / 1.685063
= $7,967.25 per Quarter
Step-2, The Remaining balance of the loan after 4 years
Remaining Balance of a Loan is calculated by using the following formula
Remaining Balance = [Amount Borrowed x (1 + r) n] – [Monthly Payment x {{(1 + r)n -1}/ r]
Loan Amount (P) = $200,000
Quarterly Interest Rate (n) = 2.50% per Quarter
Number of Quarters (n) = 16 Quarters [4 Years x 4 Quarters]
Remaining Balance = [Amount Borrowed x (1 + r) n] – [Monthly Payment x {{(1 + r)n -1}/ r]
= [$20,000 x (1 + 0.025)16] – [$7,967.25 x {{(1 + 0.025)16 -1}/ 0.025]
= [$200,000 x 1.484506] – [$7,967.25 x {(1.484506 - 1) / 0.025]
= [$200,000 x 1.484506] – [$7,967.25 x (0.484506 / 0.025)]
= $296,901 - $154,407
= $142,494
“Therefore, she needs to repay $142,494 to the bank to repay the loan in full”