1. Lecture 31 Amortization of Debts Ana Nora Evans 403 Kerchof AnaNEvans@virginia.edu http://people.virginia.edu/~ans5k Math 1140 Financial Mathematics

2. Math 1140 - Financial Mathematics The exam 2 was A)Easy B)Just right C)Hard D)Too hard 2

3. Math 1140 - 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

4. Math 1140 - 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

5. Math 1140 - 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 = 0.0625/12 5

6. Math 1140 - 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

7. Math 1140 - 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

8. Math 1140 - 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

9. Math 1140 - Financial Mathematics Balance before the first payment: $210,000. The interest part of the first payment is: $210,000 × 0.0625/12 = $1,093.75 The principal reduction is $1,293.01 - $1,093.75 = $199.26 The balance after the first payment is $210,000 - $199.26 = $209,800.74 $210,000(1+ 0.0625/12) - $1,293.01 = $209,800.74 9

10. Math 1140 - Financial Mathematics Balance right after the first payment: $209,800.74. The interest part of the second payment is: $209,800.74 × 0.0625/12 = $1,092.71 The principal reduction is $1,293.01 - $1,092.71 = $200.03 The balance after the second payment is $209,800.74 - $200.03 = $209,600.44 10

11. Math 1140 - Financial Mathematics Amortization Schedule 11

12. Math 1140 - Financial Mathematics 12

13. Math 1140 - 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

14. Math 1140 - 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

15. Math 1140 - 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

16. Math 1140 - 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

17. Math 1140 - 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

18. Math 1140 - 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 =$1,293.01 i = 0.0625/12 18

19. Math 1140 - 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

20. Math 1140 - 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

21. Math 1140 - 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

22. Math 1140 - Financial Mathematics 22

23. Math 1140 - 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

24. Math 1140 - 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? $100,000 - 0.02 × $100,000 = $98,000 How much does Alice has to pay back? $100,000 and the compound interest on $100,000. 24

25. Math 1140 - 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 = 0.0725/12 n = 15 × 12 P = $100,000 R = $912.86 25

26. Math 1140 - 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 $912.86. Use Wolfram Alpha to determine the interest rate 0.6316% per month or 7.5792%(12). 26

27. Math 1140 - Financial Mathematics 27

28. Math 1140 - 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

29. Math 1140 - Financial Mathematics The discount charge is pP. Alice needs to borrow P+pP. Let’s check it! 29

30. Math 1140 - 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

31. Math 1140 - 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

32. Math 1140 - 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 = $102,040.82 32

33. Math 1140 - 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 = $102,040.82 i = 0.0725/12 n = 15 × 12 R = $931.49 33

34. Math 1140 - 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 = 0.0725/12 n = 15 × 12 R = $931.49 Monthly interest rate is 0.6316% The nominal interest rate is 7.5792%. 34

35. Math 1140 - 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