Final answer:
The monthly payment for the loan is approximately $668.33. The amount of interest paid for the 4th month is approximately $421.11. The APY for this loan is approximately 13.80%.
Step-by-step explanation:
To calculate the monthly payment for a loan, we can use the formula:
Monthly Payment = P*(r*(1+r)^n)/((1+r)^n-1)
Where P is the principal amount, r is the monthly interest rate, and n is the number of months.
In this case, we have P = $40,000, r = (13%/12) = 0.01083, and n = 8*12 = 96.
Plugging in these values into the formula, we get:
Monthly Payment = $40,000*(0.01083*(1+0.01083)^96)/((1+0.01083)^96-1)
Calculating this expression gives us a monthly payment of approximately $668.33
b. To find the amount of interest paid for the 4th month, we can calculate the interest for each month using the formula:
Interest = P*r
Where P is the remaining principle after each payment, and r is the monthly interest rate.
In this case, the interest for the 4th month would be:
Interest = (Remaining Principal after 3rd payment)*(Monthly Interest Rate)
Since we already know the monthly payment, we can calculate the remaining principal after 3 payments using the formula:
Remaining Principal after n payments = P*(1+r)^n - ((1+r)^n-1)*monthly payment / r
Substituting the values, we find:
Remaining Principal after 3 payments = $38,933.02
So the interest for the 4th month would be:
Interest = $38,933.02*(0.01083) = $421.11
c. To calculate the APY (Annual Percentage Yield) for this loan, we can use the formula:
APY = (1+r)^n - 1
Where r is the annual interest rate and n is the number of compounding periods in a year.
In this case, the annual interest rate is 13% and there are 12 compounding periods in a year.
Substituting the values, we find:
APY = (1+(0.13/12))^12 - 1
Calculating this expression gives us an APY of approximately 13.80%