A $8700 personal loan at 5.5% compounded monthly is to be repaid over a 4 year term by equal monthly payments.
a) calculate the interest and principle component of the 18th payment
b) how much interest will be paid in the third year of the loan?
To calculate the interest and principal component of the 18th payment, we first need to determine the monthly payment.
To calculate the monthly payment, we can use the formula for the fixed monthly payment on a loan:
\[ PMT = \dfrac{P \cdot r \cdot (1 + r)^n}{(1+r)^n - 1} \]
Where:
P = Principal amount (loan amount) = $8700
r = Monthly interest rate = Annual interest rate / 12 / 100 = 5.5% / 12 / 100 = 0.0045833 (approx.)
n = Total number of months = 4 years * 12 months/year = 48
Substituting the values into the formula, we get:
\[ PMT = \dfrac{8700 \cdot 0.0045833 \cdot (1 + 0.0045833)^{48}}{(1 + 0.0045833)^{48} - 1} \]
Using a calculator, we find that PMT ≈ $201.67 (rounded to the nearest cent).
a) Now, to calculate the interest and principal components of the 18th payment, we need to determine the interest and principal parts of the payment schedule.
The interest part of the payment can be calculated using the formula:
\[ Interest = Principal Balance \cdot Monthly interest rate \]
The principal part of the payment can be calculated using the formula:
\[ Principal = Monthly payment - Interest \]
For the 18th payment, we first need to find the Principal Balance.
Principal Balance after (n) months can be calculated using the formula:
\[ Principal Balance = P \cdot (1 + r)^n - \left[ \dfrac{(1 + r)^n - 1}{r} \cdot PMT \right] \]
Substituting the values into the formula, we get:
\[ Principal Balance = 8700 \cdot (1 + 0.0045833)^{18} - \left[ \dfrac{(1 + 0.0045833)^{18} - 1}{0.0045833} \cdot 201.67 \right] \]
Using a calculator, we find that Principal Balance ≈ $5018.14 (rounded to the nearest cent).
Now, we can calculate the interest and principal components of the 18th payment:
\[ Interest = Principal Balance \cdot Monthly interest rate \]
\[ Interest = 5018.14 \cdot 0.0045833 \]
\[ Interest ≈ $22.98 (rounded to the nearest cent) \]
\[ Principal = Monthly payment - Interest \]
\[ Principal = 201.67 - 22.98 \]
\[ Principal ≈ $178.69 (rounded to the nearest cent) \]
b) To calculate the interest paid in the third year of the loan, we need to find the interest portion of each monthly payment for the third year and sum them up.
To find the interest portion of each monthly payment for the third year, we can use the formula:
\[ Interest = Principal Balance \cdot Monthly interest rate \]
First, let's find the Principal Balance at the beginning of the third year. Since the loan has a 4-year term, the Principal Balance at the beginning of the third year is the Principal Balance after 24 months.
\[ Principal Balance = 8700 \cdot (1 + 0.0045833)^{24} - \left[ \dfrac{(1 + 0.0045833)^{24} - 1}{0.0045833} \cdot 201.67 \right] \]
Using a calculator, we find that Principal Balance ≈ $3520.80 (rounded to the nearest cent).
Now, let's calculate the interest portion of each monthly payment for the third year.
For the third year, there are 12 monthly payments. We find the interest portion for each month and sum them up.
\[ Total Interest = \sum_{i=1}^{12} (Principal Balance_i \cdot Monthly interest rate) \]
Where Principal Balance_i is the Principal Balance at the beginning of the ith month.
Using a calculator, we find that the Total Interest ≈ $256.32 (rounded to the nearest cent).
Therefore, the amount of interest paid in the third year of the loan is approximately $256.32.