Lecture 31 Amortization of Debts Ana Nora Evans 403 Kerchof Math 1140 Financial Mathematics

Math Financial Mathematics The exam 2 was A)Easy B)Just right C)Hard D)Too hard 2

Math Financial Mathematics Terms The cash price of an object is the price actually paid for that object. Sometimes, the lender requires the borrower to pay some part of the object and only borrow the rest of the money. This payment is called down payment. The borrower borrows only the difference between the cash price and the down payment. 3

Math Financial Mathematics To amortize a debt means to pay a sequence of equal-size payments at equal time intervals. What is this sequence of payments? An ordinary annuity whose price is the amount borrowed. 4

Math Financial Mathematics Suppose that you take out a mortgage to buy a house. You borrow $210,000, and agree to repay the loan with monthly payments for 30 years, the first coming a month from now. If the interest rate is 6.25% convertible monthly, what is the monthly payment? P = $210,000 n = 30 × 12 = 360 i = /12 5

Math Financial Mathematics Each payment has two components 1. the interest due on the balance right after the previous payment 2. part of the principal 6

Math Financial Mathematics How much is the interest part of the payment? The interest part of the payment is the interest on the previous month balance for one month. More precisely, the interest part of the payment is the previous balance times the interest rate per month. The interest part of the payment is iB, where B is the previous month balance and i is the interest rate per month. 7

Math Financial Mathematics How much of the payment reduces the principal? The rest of the payment. Monthly payment minus the interest paid on the balance. The part of the payment that goes to the payment is R-iB. The new balance is B(1+i)-R. 8

Math Financial Mathematics Balance before the first payment: $210,000. The interest part of the first payment is: $210,000 × /12 = $1, The principal reduction is $1, $1, = $ The balance after the first payment is $210,000 - $ = $209, $210,000( /12) - $1, = $209,

Math Financial Mathematics Balance right after the first payment: $209, The interest part of the second payment is: $209, × /12 = $1, The principal reduction is $1, $1, = $ The balance after the second payment is $209, $ = $209,

Math Financial Mathematics Amortization Schedule 11

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Math Financial Mathematics Recall s(n,i) = (1+i) n-1 +(1+i) n-2 +…+(1+i) 2 +(1+i)+1 s(1,i)=1 s(n+1,i) = (1+i) n +(1+i) n-1 +…+(1+i) 2 +(1+i)+1 = [(1+i) n-1 +(1+i) n-2 +…+(1+i) 2 +(1+i)+1](1+i)+1 = s(n,i)(1+i)+1 13

Math Financial Mathematics s(n,i) = (1+i) n-1 +(1+i) n-2 +…+(1+i) 2 +(1+i)+1 s(12,i) = A)(1+i) 12 +(1+i) 11 +…+(1+i) 2 +(1+i)+1 B)(1+i) 11 +(1+i) 10 +…+(1+i) 2 +(1+i)+1 C)(1+i) 13 +(1+i) 12 +…+(1+i) 2 +(1+i)+1 D)(1+i) 10 +(1+i) 9 +…+(1+i) 2 +(1+i)+1 14

Math Financial Mathematics Goal: Calculate the balance after k th payment. Balance after first payment: P(1+i)-R= P(1+i)-Rs(1,i). Balance after second payment: [P(1+i)-Rs(1,i)](1+i)-R = P(1+i) 2 - R[s(1,i)(1+i)+1] = P(1+i) 2 - R s(2,i) 15

Math Financial Mathematics Balance after third payment: (P(1+i) 2 - R s(2,i))(1+i)-R = P(1+i) 3 - R [s(2,i)(1+i)+1] = P(1+i) 3 - R s(3,i) Balance after fourth payment: (P(1+i) 3 - R s(3,i))(1+i)-R = P(1+i) 4 - R [s(3,i)(1+i)+1] = P(1+i) 4 - R s(4,i) 16

Math Financial Mathematics Balance after k th payment: P(1+i) k - R s(k,i) How much of the (k+1) th payment goes to the interest? [P(1+i) k - R s(k,i)](1+i) How much of the (k+1) th payment goes to the principal? R - [P(1+i) k - R s(k,i)](1+i) 17

Math Financial Mathematics Suppose that you take out a mortgage to buy a house. You borrow $210,000, and agree to repay the loan with monthly payments for 30 years, the first coming a month from now. If the interest rate is 6.25% convertible monthly, what is the 12 th line of the amortization table? P=$210,000 R =$1, i = /12 18

Math Financial Mathematics The part of the (k+1) th payment goes to the interest is [P(1+i) k - R s(k,i)](1+i). The part of the 12 th payment that goes to the interest is: A)[P(1+i) 10 - R s(10,i)](1+i) B)[P(1+i) 11 - R s(11,i)](1+i) C)[P(1+i) 12 - R s(12,i)](1+i) D)[P(1+i) 11 - R s(12,i)](1+i) 19

Math Financial Mathematics How much of the (k+1) th payment goes to the principal? R - [P(1+i) k - R s(k,i)](1+i) The part of the 12 th payment that goes to the principal is A)R - [P(1+i) 10 - R s(10,i)](1+i) B)R - [P(1+i) 11 - R s(11,i)](1+i) C)R - [P(1+i) 12 - R s(12,i)](1+i) D)R - [P(1+i) 11 - R s(12,i)](1+i) 20

Math Financial Mathematics Balance after k th payment: P(1+i) k - R s(k,i) Balance after 12 th payment: A)P(1+i) 10 - R s(10,i) B) P(1+i) 11 - R s(11,i) C) P(1+i) 12 - R s(12,i) D) P(1+i) 11 - R s(12,i) 21

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Math Financial Mathematics Discount Points A point is 1% of the amount of money being lent. Discount points is an up-front charge on the amount being lent. The discount points are similar to discount interest. Compound interest is also charged as usual. The borrower receives the amount being borrowed less the discount points. The borrower pays back the amount borrowed, plus the interest. 23

Math Financial Mathematics Example Alice takes out a 15 years, $100,000 loan from a bank charging 2 points and 7.25%(12) interest. How much money Alice gets on the day of the loan? $100, × $100,000 = $98,000 How much does Alice has to pay back? $100,000 and the compound interest on $100,

Math Financial Mathematics Alice takes out a 15 years, $100,000 loan from a bank charging 2 points and 7.25%(12) interest. If Alice has to pay equal monthly payments, how much is the payment? i = /12 n = 15 × 12 P = $100,000 R = $

Math Financial Mathematics Alice takes out a 15 years, $100,000 loan from a bank charging 2 points and 7.25%(12) interest. What is the actual interest paid by Alice? Alice receives $98,000 from the bank. Alice pays this loan by making monthly payments of $ Use Wolfram Alpha to determine the interest rate % per month or %(12). 26

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Math Financial Mathematics What if Alice does not have the money for the discount points? Alice needs P dollars to buy the item she wants. The bank charges p discount points. How much does Alice need to borrow to pay both the item and the discount points charge? 28

Math Financial Mathematics The discount charge is pP. Alice needs to borrow P+pP. Let’s check it! 29

Math Financial Mathematics Alice borrows P+pP. The discount points charge is p(P+pP). The amount Alice gets from the bank is P+pP - p(P+pP) = P – p 2 P 30

Math Financial Mathematics What if Alice does not have the money for the discount points? Alice needs P dollars to buy the item she wants. The bank charges p discount points. How much does Alice need to borrow to pay both the item and the discount points charge? We denote the amount borrowed by X. The discount charge is pX. The amount Alice gets is X – pX. This is the amount she uses to pay for the item. P = X – pX = X(1-p) 31

Math Financial Mathematics Alice takes out a 15 years, $100,000 loan from a bank charging 2 points and 7.25%(12) interest. The points are added to the loan. Alice has to pay equal monthly payments. What is the actual interest Alice is being charged? Step 1: Calculate the amount borrowed X = $100,000 / (1 – 0.02) X = $102,

Math Financial Mathematics Alice takes out a 15 years, $100,000 loan from a bank charging 2 points and 7.25%(12) interest. The points are added to the loan. Alice has to pay equal monthly payments. What is the actual interest Alice is being charged? Step 2: Calculate the monthly payment P = $102, i = /12 n = 15 × 12 R = $

Math Financial Mathematics Alice takes out a 15 years, $100,000 loan from a bank charging 2 points and 7.25%(12) interest. The points are added to the loan. Alice has to pay equal monthly payments. What is the actual interest Alice is being charged? Step 3: Calculate the monthly interest rate P = $100,000 i = /12 n = 15 × 12 R = $ Monthly interest rate is % The nominal interest rate is %. 34

Math Financial Mathematics Monday Read sections 6.1 and 6.2 Nov 11 (next Friday) Project Progress Report Dec 8 (a little over a month) FINAL!!!! Charge 35