3.1 Mathematic of Finance: Simple Interest

Transcription

1 3.1 Mathematic of Finance: Simple Interest Introduction Part I This chapter deals with Simple Interest, and teaches students how to calculate simple interest on investments and loans. The Simple Interest Formula I Prt I B interest P B principal amount r B annual simple interest rate (as a decimal*) t B time in years * Note: interest rates, for the purpose of this chapter, must be converted to decimal form before they are used in the calculation. Definitions Principal The Principal is a sum of money that you borrow from a lending institution, or deposit in a savings account. Interest The Interest is the fee charged for borrowing money. Think of this fee as rent charged for using someone elses money. Interest rates are a computed percentage of the principal over a certain amount of time. Interest rates are usually annual.

2 Example The interest on a loan of $100 at 12% for 9 months is calculated as follows: I Prt (100) (0.12*) (0.75*) $9 * Note : 12% is converted to a decimal (0.12) 9 months is converted to years (9/ ) After 9 months the borrower has to repay the lending institution both the principal and the interest. $100 (principal) + $9 (interest) $109 (total to be repaid) Page -2-

3 Introduction Part II Theorem 1: We can expand on the simple interest formula we just covered. If we wanted to figure out how much we would owe someone who lent us money at the end of a time period, including the principal amount and the interest, we can use the formula: AP + Prt then factor out a P to get: A P(1+rt) This is called the Amount: Simple Interest Formula. The variables P, r and t are the same as they were in the simple interest formula, where: P the principal or present value (the amount borrowed) r the annual simple interest rate (in decimal form) t the time (in years) A stands for the future value, or the amount a borrower would owe a lender after t years. If you are given any three of the four variables A, P, r, or t, you can solve for the fourth variable using this formula. Throughout the rest of the lecture, we will be showing you examples of several situations where you would be able to use this formula. Page -3-

4 Examples Example #1 Total Amount Due on a Loan Review Formula 2 A P + Prt A P(1 + rt) Work Through: You receive a loan of $1200 at 7% simple interest to be paid in 6 months. What is the total amount due? A? (future value, or total amount due at the end of the time period) P 1200 (principle, or present value) r 0.07 (remember, 7% is to be written as a decimal) t 6/12 (simple interest is calculated annually, so the number of months in which the loan is due should be written as a fraction) t 2 (the fraction is simplified) Solution: Therefore, at the end of 6 months the total amount of your loan will be: Matched Problem Since your student loan barely pays tuition and books, you decided to negotiate a loan with a family member. You borrow $5000 at the Page -4-

5 interest rate of 3%, to be repaid when you can afford to. How much would you owe if you repaid the loan in 2 years? In 5 years? In 10 years? Example #2 Present Value of an Investment If you want to earn an annual rate of 10% on your investments, how much (to the nearest cent) should you pay for a note that will be worth $5,000 in 9 months? Use the amount simple interest formula (A P + Prt) A amount or future value P (1 + rt) P principle or present value r annual simple interest rate *(always written as a decimal) t time in years Wanting to solve for the principle P (present value) Given, A $5,000 r 0.1 t years 12 A P(1 + rt) Page -5-

6 Matched Problem With a period of 6 months? Example #3 Interest Rate Earned On A Note T - bills (Treasury Bills) are one of the instruments the US Treasury Department uses to finance the public debt. If you buy a day T - bill with a maturity value of $10,000 for $9,893.78, what annual simple interest rate will you earn? (Express answer as a percentage, correct to three decimal places.) Solution We use the formula A P(1 + rt ) Where A amount, or future value, P principle or present value, r annual simple interest rate (written as a decimal ) and t time in years, and we are interested in finding the value of r, since we are given P $9, , A $10,000 and t 0.5 years (180 days x 1 year/360 days) A P(1+rt) Page -6-

7 Example #4 Interest Rate Earned on an Investment Example: Suppose that after buying a new car you decide to sell your old car to a friend. You accept a 270-day note for $3, 500 at 10% simple interest as payment. (Both principal and interest will be paid at the end of 270 days). Sixty days later you find that you need the money and sell the note to a third party for $3, 550. What annual interest rate will the third party receive for the investment? (Express the answer as a percentage, correct to three decimal places). First we will need to find the amount that your friend will owe the holder of the note after his/her 270 days are up. A P Page -7-

8 r t AP(1+rt) Your friend will owe the holder of the note: the end of 270 days. at Next we will need to find the annual rate of interest (r) needed, for the third party, to make $3,550 to grow to $3, in 210 days (270-60). A P r t AP+Prt r A-P/Pt r The annual rate of interest needed to make $3, 550 grow to $3, is: Example #5 Interest on an Investment As an investor you buy 50 shares in a company at $47.52 a share. After 200 days you sell the shares for $52.19 a share. What is the annual rate of interest earned by this investment? (write your answer as a percentage correct to three decimal places) Page -8-

9 Use the following commission schedule to calculate your answer: Transaction size Commission $0 - $2,499 $ % of principal $2,500 - $9,999 $ % of principal $10,000 + $ % of principal Solution Then use the 2 nd amount simple interest formula to perform the final calculation: A P (1 + rt) Matched Problem Repeat Example 5 if you bought 250 shares for $28.60 each and sold them 320 days later for $36.40 each. Page -9-

10 Solution Use the 2 nd amount simple interest formula to complete the calculation: A P (1 + rt) Assignment for Section 3-1: Pages ; Questions - #1-16 ALL, #19-35 odd, & 38 Page -10-