4.7 Compound Interest

Transcription

1 4.7 Compound Interest 4.7 Compound Interest Objective: Determine the future value of a lump sum of money. 1

2 Simple Interest Formula: InterestI = Prt Principal interest rate time in years 2

3 A credit union pays interest of 8% per annum compounded quarterly on a savings account. If $1,000 is deposited and the interest is left to accumulate, how much is in the account after 1 year? I = Prt 1 I = ($1000)(0.08)( ) 4 I = $20 3

4 At the end of one quarter, we have earned $20 in interest. My principal is now $1020. So at the end of the second quarter, how much do I earn? 1 I = ($1020)(0.08)( ) 4 I = $20.40 Adding that interest to the money in my account, I now have $ months into the year. 4

5 At the end of the third quarter, how much do I have? 1 I = ($ )(0.08)( ) 4 I = $20.81 This brings my principal to $1,

6 At the end of the fourth quarter, or at the end of the year what do I have? 1 I = ($ )(0.08)( ) 4 I = $21.22 This brings the total in my bank account to $1, just by collecting interest. 6

7 This process can be very lengthy especially if the interest were being compounded daily. This leads to a more general formula for calculating interest that is compounded. Compounded interest formula: r A = P(1 + ) nt n n = the number of compounds per year 7

8 Compounded interest is interest that is paid on principal and previously earned interest. Common forms of compounding are: annually: once per year semiannually: twice per year quarterly: four times per year monthly: 12 times per year daily: 365 times per year 8

9 Lets compare investments using different compounding periods. r A = P(1 + ) nt n Find the resulting amount of an investment of $3000 when it is compounded annually, semiannually, quarterly, monthly and daily at a rate of 4% for a period of 3 years A = (3000)(1 + ) A = (3000)( ) 3 A = (3000)(1.04) 3 A = $3, n = the number of compounds per year (1 3) 9

10 Lets compare investments using different compounding periods. r A = P(1 + ) nt n Find the resulting amount of an investment of $3000 when it is compounded annually, semiannually, quarterly, monthly and daily at a rate of 4% for a period of 3 years A = (3000)(1 + ) A = (3000)( ) 6 A = (3000)(1.02) 6 A = $3, n = the number of compounds per year (2 3) 10

11 Lets compare investments using different compounding periods. r A = P(1 + ) nt n Find the resulting amount of an investment of $3000 when it is compounded annually, semiannually, quarterly, monthly and daily at a rate of 4% for a period of 3 years A = (3000)(1 + ) (4 3) A = (3000)( ) 12 A = (3000)(1.01) 12 A = $3, n = the number of compounds per year 11

12 Lets compare investments using different compounding periods. r A = P(1 + ) nt n Find the resulting amount of an investment of $3000 when it is compounded annually, semiannually, quarterly, monthly and daily at a rate of 4% for a period of 3 years A = (3000)(1 + ) (12 3) A = (3000)( ) A = (3000)( ) A = $3, n = the number of compounds per year

13 Lets compare investments using different compounding periods. r A = P(1 + ) nt n Find the resulting amount of an investment of $3000 when it is compounded annually, semiannually, quarterly, monthly and daily at a rate of 4% for a period of 3 years A = (3000)(1 + ) (365 3) A = (3000)( ) 1095 A = (3000)( ) A = $3, n = the number of compounds per year

14 What if the investment is compounded more often than daily? Compounding Interest Continuously: A = Pe rt Find the resulting amount of an investment of $3000 when it is compounded continuously at a rate of 4% for a period of 3 years. A = Pe rt A = (3000)e A = (3000)e 0.12 A = $3, (0.04 3) 14

15 How much principle is needed now to have $4000 after 2 years at 6% compounded quarterly? r A = P(1 + ) nt n n = the number of compounds per year = P(1 + ) (4 2) 4000 = P( ) = P(1.015) = P( ) P = $3,

16 How long will it take for an investment $250 to reach $675 in value if it earns 9% compounded monthly? A = P(1 + r n ) nt n = the number of compounds per year = (250)(1 + ) (12 t) 675 = (250)( ) 675 = (250)(1.0075) 2.7 = (1.0075) 12t log = 12t 12t 12t log 2.7 log = 12t = 12t t years 16

17 How long will it take for an investment to double in value if it earns 5% compounded continuously? A = Pe rt 2P = Pe 0.05t 2 = e 0.05t ln 2 = ln e 0.05t ln 2 = 0.05t ln e ln 2 = 0.05t t = years 17

18 Homework page 322 (4, 5, 8, 9, 12, 14, 16, 18, 19, 28, 32, 36, 37, 48) 18