9 Suppose you obtain a 20000 conventional mortgage for 30 ye

9. Suppose you obtain a $200,00 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 1 period, we would have 100 (1+i) dollars in our account. After one more period, we would have 100(1 + i) (1 + i) = 100(1 + i)2 dollars. After n periods, we would have Alternatively, suppose we wanted to know how much money we need to put in the bank right 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 = 500d3 where d = 1/(1 + i). Once we know the interest rate of i per period, 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

Solution

To find out the monthly payment we would use the following formula:

Present value of annuity(PVA) = EMI X (1-(1+r)^-n)/r

Here given PVA = $200,000, r = 3%/12 = 0.25%, n = 30 x 12 = 360

200000 = EMI x (1-(1+.0025)^-360)/.0025

200000 = EMI x 237.1894

EMI = 200000/237.1894 => $843.21

Same way for 15 years

Here given PVA = $200,000, r = 3%/12 = 0.25%, n = 15 x 12 = 180

200000 = EMI x (1-(1+.0025)^-180)/.0025

200000 = EMI x 144.8055

EMI = 200000/144.8055 => $1,381.16

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