The Monthly Payment. ( ) ( ) n. P r M = r 12. k r. 12C, which must be rounded up to the next integer.

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1 MATH 116 Amortization One of the most useful arithmetic formulas in mathematics is the monthly payment for an amortized loan. Here are some standard questions that apply whenever you borrow money to buy a car or a house: 1. If you borrow $ P (principal) at a fixed interest rate r to be paid back in n monthly payments, then what is the required monthly payment M? 2. After k monthly payments of $ M, what is your balance due? Of what you have paid, how much of the loan have you paid off and how much have you paid in interest? 3. How do you create an amortization chart that shows how much goes to interest, how much goes to principal, and the remaining balance after each payment. 4. If you pay $ C each month, then how long will it take to pay off the balance? The Monthly Payment For a principal balance of P with fixed interest rate r (in decimal), the amount due each month to pay off the loan in n monthly payments is given by n. P r M = r 12 The balance due after k payments is B k = P 1 r k 12 12M 1 r k r 12 1 Let B 0 be the initial balance and let B i the balance after the i th payment. The interest on i th payment is given by I i = B i 1 r / 12. The principal on the i th payment is given by P i = M I i. The balance after the i th payment is B i = B i 1 P i. Using a payment of $C on a balance of $B, the number if required payments will be log 1 Br m = 12C, which must be rounded up to the next integer. log(1 r /12) (The final payment will actually be less than $C.)

2 Example 1. Suppose you are buying a house. After your downpayment, closing costs, etc., you are to finance $125,000 at 5.25% to be paid back over 30 years. (a) Find the required monthly payment (which excludes any amount paid extra for taxes and insurance). (b) How much will you pay over the course of 30 years? How much goes to interest? (c) On the first few payments, show how much of your payment goes to interest, how much goes to principle, and the balance due. (d) After just 10 years of payments: (i) How much have you paid in? (ii) What is the balance due? (iii) How much of the loan have you paid off? (iv) How much of your payments have gone to interest? (e) If you paid $1000 a month, then how many payments would be needed? Solution. (a) For 30 years, there are 360 payments. The monthly payment is 360 = M = 125, = $ (b) After 30 years, or 360 payments, you will pay $ = $248,490. The amount paid to interest is $248,490 $125,000 = $123,490. (c) The interest on the first payment is I 1 = B 0 r / 12 = 125, /12 = $ So the amount to principal is $ $ = $143.37, which gives a balance of $124, Payment To Interest To Principal Balance $125, $ $ $ $124, $ $ $ $124, $ $ $ $124, (d) (i) After 10 years, or 120 payments, you have paid in $ = $82,830.

3 (ii) But the balance due after 120 payments is B 120 = 125, = 125, = $102, ( 1 ( / 12) 120 ) ( ) (iii) So you have paid off only $125,000 $102, = $22, (iv) The amount paid for interest is then $82,830 $22, = $60, (e) For monthly payments of $1000, the loan will be paid off in m = log log( / 12) payments. Thus, there will be 181 payments of $1000 and a final payment of a little over 0.33 $1000 = $330. By paying $1000 a month rather than just $690.25, you end up paying less than $181,400 total over the course of 182/12 = 15 years, 2 months, rather than paying $248,490 over the course of 30 years.

4 Exercise You are buying a house and you finance $150,000 at 5.4% over a 25 year period. (a) What is the required monthly payment? (b) How much will you pay over the course of 25 years? How much goes to interest? (c) For the first three payments, make a table that shows how much of your payment goes to interest, how much goes to principal, and the balance due. (d) After 5 years of payments: (i) How much have you paid in? (ii) What is the balance due? (iii) How much of the loan have you paid off? (iv) How much of your payments have gone to interest? Consider the balance after 5 years of payments. At this point, you start paying $1500 a month. (e) How many more payments will be required? (f) Considering the total amount paid in over the first 5 years, and the amount paid in using $1500 payments, how many total years of payments are made and about how much total is paid in?

5 Solutions 1. (a) There are 60 monthly payments over 5 years. The payment is / = = $413.15; (b) Total paid is = $24,789, so $2789 will go to interest. (c) Payment 1 To interest: / 12 = $88; To principle: = $ New Balance after 1st payment = 22, = $21, Payment 2 To interest: 21, / 12 = $86.70; To principle: = $ Balance after 2nd payment = 21, = $21, (d) (i) After 24 payments, you have paid in = $ (ii) The balance due is B 24 = 22, = 22, = $13, (iii) You have paid off 22,000 13, = $ ( 1 ( / 12) 24 ) (iv) So = $ has gone to interest after 24 payments. (e) If you pay $ per month (an extra $150), then after 24 payments the balance due is B 24 = 22, = 22, = $10, ( 1 ( 1.048/ 12) 24 ) 2. (a) There are 360 monthly payments over 30 years. The payment is / = = $ (b) Total paid is = $303,228;, so $153,228 will go to interest. (c) Payment 1 To interest: / 12= $675; To principle: = $ New Balance after 1st payment = 150, = $149,

6 Payment 2 To interest: 149, / 12 = $674.25; To principle: = $ Balance after 2nd payment = 149, = $149, (d) (i) You have paid in = $101,076 after 10 years. (ii) After 120 payments (10 years), the balance due is B 120 = 150, = 150, = $123, (iii) You have paid off 150, , = $26, ( 1 ( / 12) 120 ) (iv) So 101,076 26, = $74, has gone to interest after 10 years of payments. (e) If you pay $ per month (an extra $200), then after 120 payments the balance due is B 120 = 150, = 150, = $91, ( 1 ( /12) 120 )

7 Monthly Payments For a principal balance of P with fixed interest rate r (in decimal), the amount due each month to pay off the loan in n monthly payments is given by n. P r M = r 12 The balance due after k payments is: B k = P 1 r k 12 12M r 1 r k Suppose you finance $150,000 at 6.6% over a 30 year period. (a) What is the monthly payment and the total amount paid over the 30 years? (b) What is the balance due after 5 years of payments? (c) How much have you paid in over the first 5 years? (d) How much have you paid off over the first 5 years? (e) How much have you paid in interest over the first 5 years?