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S Firefox File Edit View History Bookmarks Tools Window Help Mail Wethington, Francis 2027 hmw 3.pdf Search A https//files t-square. gatech.edl nt/attachment/gtc-b068-5cf9-5606-bfc5-e6581ff2959t/ anon beccb869-ba2 t Page: 2 of 3 130% is simply the number of heads among 100 tosses of a coin, either real or falsified 7. Suppose you obtain a S250,00 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+ij dollars in our account. After one more period, we would have 100(1 +i)(1 +i) 100 (1 +i) 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 S500 after 3 periods. Let b denote the unknown amount, then we need b(1 500. We could rewrite this equation as b 500d where d 1/(1 i). O we know the interest 3 nce 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. 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 S250,000. That is, 360 250,000 Tue 2:40 PM

Explanation / Answer

Solution

Monthly payment for the conventional mortgage of $250000 for 30 years at 3% interest per annum.

Let the monthly payment be b. Given the annual interest rate of 3%, interest rate per period (month) is r =0.25% or 0,0025.

The discounting factor, d = 1/(1 + r) = 1/1.0025.

The present value of b’s over 360 months(30 years),

P = (bd) + (bd2) + (bd3) + …… + (bd360).

This is a GP of 360 terms with first term, a = b/d and common ratio, R = d.

Hence, this sum, P = (bd){(d360 – 1}/(d - 1)}

                               = 582.7368b [substituting the values and simplifying].

Now, we should have 582.7368b = 250000 or b = 42.9010 = 42.90 (rounded)

Answer;

Monthly payment for the conventional mortgage of $250000 for 30 years at 3% interest per annum. = $42.90

[NOTE: In the question given, at the beginning the mortgage amount is given as $25000, but in illustration later it has been taken as $250000, which is what we have used here.]

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