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Lecture 7 Partial Payments Discount Interest Ana Nora Evans 403 Kerchof AnaNEvans@virginia.edu http://people.virginia.edu/~ans5k/ Math 1140 Financial Mathematics

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Math 1140 - Financial Mathematics Covers everything we do in class, including Friday. You must know: definitions, formulas, applications. Check page 39 from the textbook. Exam practice. Credit card interest will not be on the test (lecture 1 and part of lecture 2). Quiz on Monday 2

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Math 1140 - Financial Mathematics Survey Results The pace is fast. No one feels overworked for now. I should speak a little louder for the people in the back and not so loudly for the people in the front. 3

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Math 1140 - Financial Mathematics I will add more details to the examples we solve in class. I will give you more time to take notes. I changed the slides style so the question is up while we work on it. But … You must come prepared to class! Go through examples by yourself and come to office hours. Start homework early! Examples 4

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Math 1140 - Financial Mathematics Someone hates that I “don’t do enough practice problems like the ones in the homework”. The homework should help you understand the material, not test your ability to recognize patterns and use a calculator. Try the problems yourself and come to office hours! 5

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Math 1140 - Financial Mathematics “Some things are not explained well. I feel like the majority of the class doesn’t understand some of the stuff you try to explain.” I will try to add more checkpoints to make sure we are all on the same page. Do something about it: Come prepared for class. Ask questions in class. Come to office hours. 6

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Math 1140 - Financial Mathematics How to prepare for class Read the previous class. Remember the definitions and the formulas. Solve the examples from class by yourself. 7

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Math 1140 - Financial Mathematics Questions? 8

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Math 1140 - Financial Mathematics How much money will I have in 6 months if I invest $100 at 5% simple interest? S = P + I = P + Pit = P(1+it) Future value at simple interest 9

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Math 1140 - Financial Mathematics I have $100 today. The value of my $100 three days ago at 5% simple interest was A)$100 B)Less than $100 C)More than $100 10

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Math 1140 - Financial Mathematics How much money do I have to invest now at 5% simple interest to have $100 in 6 months? How much money do I need to invest today using simple interest to have a given amount at a future date? Present value at simple interest 11

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Math 1140 - Financial Mathematics For a simple interest loan, the borrower is not required to make any payments until the due date. The borrower and the lender may agree how any partial payments will reduce the interest charges. We are given: the principal the term the partial payments the dates the partial payments are made We calculate: the balance on the due date Partial Payments 12

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Math 1140 - Financial Mathematics The balance of a simple interest loan with interest rate i is the sum of the payments, Q 1, …, Q n moved to the due date using simple interest i, minus of the principal P moved to the due date using simple interest I. Merchant’s Rule 13 Given loan and due date principal P the payments Q 1, …, Q n the dates of the payments Want to calculate balance on due date, which is how much the borrower has left to pay at the due date.

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Math 1140 - Financial Mathematics Merchant’s Rule 14 Alice borrows $10 from Bob for three years at an annual interest rate 10%. After one year Alice makes a partial payment of $5 dollars. After two year Alice makes a partial payment of $2 dollars. How much does Alice have to pay at the due date? Use the US rule.

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Math 1140 - Financial Mathematics United States Rule The balance on the loan date is -P. The new balance(at the payment date) is the sum of the payment and the previous balance moved to the payment date. The balance on due date is the value of the balance moved to the due date. 15

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Math 1140 - Financial Mathematics United States Rule 16 Alice borrows $10 from Bob for three years at an annual interest rate 10%. After one year Alice makes a partial payment of $5 dollars. After two year Alice makes a partial payment of $2 dollars. How much does Alice have to pay at the due date? Use the US rule.

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Math 1140 - Financial Mathematics Using Merchant’s Rule Alice has to pay $4.80 at the due date. Using US Rule is Alice has to pay $5.06 at the due date. For a loan with interest rate i, principal P, partial payments Q 1, …, Q n, and term t : A)The borrower always pays more with the US Rule. B)The borrower sometimes pays more with the Merchant’s Rule. C)The borrower always pays more with the Merchant’s Rule. D)The borrower always pays more with the Mob’s Rule. 17

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Math 1140 - Financial Mathematics The correct answer is A. The borrower always pays more with the US Rule. When using the US Rule, at the first payment date we add the interest to the principal. That interest will accrue interest. When using the Merchant’s Rule, the interest is calculated at the end and no interest on interest is paid. 18

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Math 1140 - Financial Mathematics A)Crystal clear B)I’m fine C)Not so clear D)I have no idea what that is Merchant’s Rule 19 If you answered C, you should read section 1.7, Equations of Value, and read the posted solutions for exercise 14 at page 26 from homework 3.

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Math 1140 - Financial Mathematics A)Crystal clear B)I’m fine C)Not so clear D)I have no idea what that is US Rule 20

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Math 1140 - Financial Mathematics Discount Interest The interest is charged upfront. Alice borrows today $500 from Shady Bank with a 10% discount rate. Alice receives $450 from the back today. One year from today Alice pays the back $500. Why $450? $500 - 10% x$500 x1 = $450 21

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Math 1140 - Financial Mathematics Amount: S = $500 Discount: D = $50 Proceeds: P = $450 Discount rate: d =10% per year Term: t = 1 year Discount Interest 22 Amount = money borrowed Discount = charge for the use of money Proceeds = money received from the loan Discount rate = percent of the amount used to calculate the discount Term = length of the loan in units of time

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Math 1140 - Financial Mathematics The Basic Discount Interest Formula D = Sdt Where D is the discount S is the amount (future value) d is the discount rate t is the term This is the formula for calculating the discount! 23

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Math 1140 - Financial Mathematics P = S - D Where P is the proceeds D is the discount S is the amount (future value) P = S - D = S - Sdt = S(1-dt) Discount Proceeds Formula 24

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Math 1140 - Financial Mathematics Shady Bank charges 10% for short-term discount loans. What are the proceeds for a six months loan for $5,000? Example 25

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Math 1140 - Financial Mathematics A)Crystal clear B)I’m fine C)Not so clear D)I have no idea what that is Discount Interest 26

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Math 1140 - Financial Mathematics Next class More on discount interest Quiz Monday Everything we covered up then except credit card interest. Next Wednesday Homework 4 is due. First Exam (max 15 points): 26 September 2011 at 7pm Location to be announced Charge 27

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