Please provide detailed explanation

Suppose you obtain a $250,00 conventional mortgage for 30 years at an annual interest rate of 3%. What is your monthly payment What would your monthly payment be for a 15 year mortgage at the same interest rate? (Normally, you would receive a better interest rate for a 15 year mortgage than a 30 year mortgage.) Background: Suppose we put $100 dollars in the bank, and the effective interest rate is I per period. After 1period, we would have 100(1 + i) dollars n our account. After one more we would have 100(1 + i)(1 + i) = 100(1 + i)^2 dollars. After n we would have Alternatively, suppose we wanted to know how much money we need to put in the bank right it now to have $500 after 3 periods. Let b denote the unknown amount, then we need b(1 + i)^3 = 500. We could rewrite this equation as b = 500d^3 where d = I/(1 _ i). once we know the interest rate of we can determine the "discount rate" d, and compute b One way to describe this is that the amount b is the "present value of $500 received 3 periods from now, To solve the first mortgage question, we need a stream of 360 monthly payments (30 years) to have a present value of $250,000. Thus, we need to find the monthly payment amount a so that the present value of those 360 monthly payments is $250,000. That is, 250.000 = sigma_k = 1^360 sd^k The tricky part of dealing with real estate loans is that you're supposed to know that an annual interest of 3% per year does not mean that the effective interest per year is 3%. It means that the effective interest rate is 3/12%per month. That is, the interest rate is i = 03/12 per month. (This would that the effective interest rate for one year is 03/12^12) know the effective interest rate for can determine the discount rate d per month and use that to solve for the monthly payment. If you or someone you know has a mortgage, you might check to see that the monthly payment amount correct; mistakes have been made. If the amount looks incorrect, make sure that you're looking at the monthly payment for principal and interest. Sometimes, the payment includes additional money that is escrowed for homeowners insurance and/or taxes