Answer:
Monthly Payment: $1,879
Annual Payment: $13,975
Explanation:
To find the answer, we will use the present value of an annuity formula:
The formula is:
PV = A (1 - (1 + i)^-n) / i
Where:
- PV = Present value of the investment (in this case, of the loan)
- A = Value of the annuity (will be our incognita)
- i = interest rate
- n = number of compounding periods
The reason why we use this formula is because both the annual payments, and the monthly payments are annuities: payments that have regular time intervals, and have the same interest rate, which means that the value of each payment is the same.
To find the monthly payment, we first convert the annual interest rate of 6.2% to a monthly rate. The result is a 0.5% monthly rate.
Next, the number of compounding periods changes, because the monthly rate compounds each month, not once every year. For these reason, we use the number of months that there are in 15 years, which is 180 months (15 x 12 = 180).
Third, we divide the interest rate by 100 to obtain the decimal value: 0.5 / 100 = 0.005
Finally, we plug the correct amounts into the formula:
225,000 = X (1 - (1 + 0.005)^-180) / 0.005
225,000 = X (118.5)
225,000 / 118.5 = X
1,899 = X
Now, for the annual payment, we simply use the annual rate of 6.2% (divided by 100) instead of the monthly rate, and the compounding periods are now 15 years, instead of 180 months:
225,000 = X (1 - (1 + 0.062)^-15 / 0.062
225,000 = X (16.1)
225,000 / 16.1 = X
13,975 = X
