Simple Interest (for One Year)

Transcription

1 Simple Interest (for One Year) Suppose you invest $ at 3.22% interest per year. How much will you have at the end of one year? Solution: 3.22% interest means that over the course of one year, one dollar will grow to = $ Thus, $1500 will grow to = $ The amount earned (i.e., the interest earned) is = $48.30 Paul Koester () MA 111, Simple Interest February / 12

2 Simple Interest (Multiple Years) Suppose you invest $ at 3.22% interest per year. How much will you have at the end of two years? Solution: It depends... We will see several different notions of interest. The simplest is called simple interest. To determine how much we have at the end of two years using simple interest, we can argue as follows: Over the course of the first year you earned $48.30 in interest. Thus, for two years you will earn = $96.60 Thus, the amount you will have after two years is = $ Paul Koester () MA 111, Simple Interest February / 12

3 Simple Interest (Multiple Years) Suppose you invest $ at 3.22% interest per year. How much will you have at the end of 4 months? In a full year you will earn $48.30 in interest. 4 months corresponds to 4/12 = 1/3 of a year. Therefore, you will only earn 1/3 of the full year s interest: = $16.10 Thus, the amount you will have after 4 months is = $ Paul Koester () MA 111, Simple Interest February / 12

4 Simple Interest With simple interest, the amount of interest earned is directly proportional to the time it is invested. Suppose a savings bond earns $400 in interest over the course of 8 years. How much did it earn over the first three years? (Assuming simple interest) Since the investment earned $400 over the course of 8 years, it earned 400/8 = $50 per year. Therefore, the investment earned 3 50 = $150 in three years. Paul Koester () MA 111, Simple Interest February / 12

5 Simple Interest One method for determining the total value of the investment using simple interest: (1) Determine amount of interest earned in ONE year. (2) Multiply that amount by the number of years invested. (3) Add to the initial amount invested. An alternative method: (1) Multiply the number of years invested by the interest rate (expressed as a decimal). (2) Add that amount to 1 to get a growth factor. (3) Multiply that growth factor with the initial amount invested. Paul Koester () MA 111, Simple Interest February / 12

6 Present Value Problem You would like to spend a semester in a study abroad program a year and a half from now. You would like to have $ to help pay for the trip. You can invest money at a simple interest rate of 4%. How much must you invest today to have $ for your trip a year and a half from now? This is a Present Value problem: We wish to find how much we need now in order to obtain the Future Value of $ Compare to the earlier examples in which the present value was known and the future value was to be found. Paul Koester () MA 111, Simple Interest February / 12

7 Present Value Problem You would like to spend a semester in a study abroad program a year and a half from now. You would like to have $ to help pay for the trip. You can invest money at a simple interest rate of 4%. How much must you invest today to have $ for your trip a year and a half from now? Solution: Suppose you invest an amount P today. The future value is the present value plus what ever is earned in interest. In one year, one dollar will earn $0.04 in interest. Therefore, in a year and a half, one dollar will earn = $0.06 Therefore, in a year and a half, $P will earn $P (0.06) Thus, the future value of $P in a year and a half will be P + P 0.06 = P (1.06) Paul Koester () MA 111, Simple Interest February / 12

8 Present Value Problem You would like to spend a semester in a study abroad program a year and a half from now. You would like to have $ to help pay for the trip. You can invest money at a simple interest rate of 4%. How much must you invest today to have $ for your trip a year and a half from now? Now, the future value is supposed to be $ so 3000 = P 1.06 so P = = $ Paul Koester () MA 111, Simple Interest February / 12

9 Simple Interest Formula In the solution of the previous problem, we derived a general formula for simple interest: If the present value is P, the future value is F, the annual percentage rate (or APR) is R% and the time of the investment is t, then, (for simple interest) F = P (1 + r t) where r = R 100 This formula has FOUR variables. In a given application problem, any THREE of the variables could be given, and we could be asked to solve for the FOURTH. The first example asked us to solve for F, the most recent problem asked us to solve for P. The APR is sometimes referred to as the yield, or APY, although this term is more correctly associated with a different notion of interest which we will discuss in future lessons. The length of time of an investment is often called the duration. Paul Koester () MA 111, Simple Interest February / 12

10 An unknown APR Suppose $ is invested for two and a half years. At the end of the two and a half years, the investment is worth $ What is the APR? Ad Hoc Solution: The interest earned over two and a half years is Thus, the interest earned over one year is = = $320 Now, 320 is = 8% of $ Thus, the APR is R = 8% Paul Koester () MA 111, Simple Interest February / 12

11 An unknown APR Suppose $ is invested for two and a half years. At the end of the two and a half years, the investment is worth $ What is the APR? Algebraic Solution: P = , F = , and t = 2.5 so so so 4800 = 4000 ( r) r = = r = = 0.2 so r = = 0.08 But r = R/100 so R = 100 r = = 8. Paul Koester () MA 111, Simple Interest February / 12

12 An unknown duration You would like to buy a car which sells for $10, 000. You have $ which you can invest at 3.5% APR. How long must your $ be invested for in order to grow to $10, 000? Algebraic Solution: Using the simple interest formula, we have F = 10, 0000, P = 6000 and r = 3.5/100 = 0.035, so 10, 000 = 6000 ( t) We want to solve to t. Divide both sides by 6000: so Subtract 1 from both sides: 10, = t = t = 0.035t Divide both sides by 0.035: = t Therefore, t = It will take over 19 years; You need to find a better investment! Paul Koester () MA 111, Simple Interest February / 12