example 4 Continuous Versus Annual Compounding of Interest

Chapter 5.5

a. For each of 9 years, compare the future value of an investment of $1000 at 8% compounded annually and of $1000 at 8% compounded continuously.

b. Graph the functions for annual compounding and for continuous compounding on the same axes.

c. What conclusion can be made regarding compounding annually and compounding continuously?

2009 PBLPathways

2009 PBLPathways

a. For each of 9 years, compare the future value of an investment of $1000 at 8% compounded annually and of $1000 at 8% compounded continuously.

b. Graph the functions for annual compounding and for continuous compounding on the same axes.

c. What conclusion can be made regarding compounding annually and compounding continuously?

2009 PBLPathways

a. For each of 9 years, compare the future value of an investment of $1000 at 8% compounded annually and of $1000 at 8% compounded continuously.

Future Value of an Investment with Periodic Compounding

If $P is invested for t years at the annual interest rate r, where the interest is compounded k times per year, then the interest rate per period is , the number of compounding periods is kt, and the future value that results is given by

1 dollarsk t

rS P

k

rk

10.08

1000 11

1000 1.08

t

t

S

0.081000 tS e

Compounded annually Compounded continuously

2009 PBLPathways

a. For each of 9 years, compare the future value of an investment of $1000 at 8% compounded annually and of $1000 at 8% compounded continuously.

Future Value of an Investment with Periodic Compounding

If $P is invested for t years at the annual interest rate r, where the interest is compounded k times per year, then the interest rate per period is , the number of compounding periods is kt, and the future value that results is given by

1 dollarsk t

rS P

k

rk

10.08

1000 11

1000 1.08

t

t

S

0.081000 tS e

Compounded annually Compounded continuously

2009 PBLPathways

a. For each of 9 years, compare the future value of an investment of $1000 at 8% compounded annually and of $1000 at 8% compounded continuously.

Future Value of an Investment with Periodic Compounding

If $P is invested for t years at the annual interest rate r, where the interest is compounded k times per year, then the interest rate per period is , the number of compounding periods is kt, and the future value that results is given by

1 dollarsk t

rS P

k

rk

10.08

1000 11

1000 1.08

t

t

S

0.081000 tS e

Compounded annually Compounded continuously

2009 PBLPathways

a. For each of 9 years, compare the future value of an investment of $1000 at 8% compounded annually and of $1000 at 8% compounded continuously.

Future Value of an Investment with Periodic Compounding

If $P is invested for t years at the annual interest rate r, where the interest is compounded k times per year, then the interest rate per period is , the number of compounding periods is kt, and the future value that results is given by

1 dollarsk t

rS P

k

rk

10.08

1000 11

1000 1.08

t

t

S

0.081000 tS e

Compounded annually Compounded continuously

2009 PBLPathways

a. For each of 9 years, compare the future value of an investment of $1000 at 8% compounded annually and of $1000 at 8% compounded continuously.

Future Value of an Investment with Continuous Compounding

If $P is invested for t years at the annual interest rate r compounded continuously, then the future value S is given by

dollarsr tS Pe

10.08

1000 11

1000 1.08

t

t

S

0.081000 tS e

Compounded annually Compounded continuously

2009 PBLPathways

a. For each of 9 years, compare the future value of an investment of $1000 at 8% compounded annually and of $1000 at 8% compounded continuously.

Future Value of an Investment with Continuous Compounding

If $P is invested for t years at the annual interest rate r compounded continuously, then the future value S is given by

dollarsr tS Pe

0.081000 tS e

Compounded annually Compounded continuously

10.08

1000 11

1000 1.08

t

t

S

2009 PBLPathways

a. For each of 9 years, compare the future value of an investment of $1000 at 8% compounded annually and of $1000 at 8% compounded continuously.

t

0 1000.00 1000.00

1 1080.00 1082.29

2 1166.40 1173.51

3 1259.71 1271.25

4 1360.49 1377.13

5 1469.33 1491.82

6 1586.87 1616.07

7 1713.82 1750.67

8 1850.93 1896.48

9 1999.00 2054.43

1000 1.08t 0.081000 te

$2054.43 $1999.00 $55.43

2009 PBLPathways

a. For each of 9 years, compare the future value of an investment of $1000 at 8% compounded annually and of $1000 at 8% compounded continuously.

t

0 1000.00 1000.00

1 1080.00 1083.29

2 1166.40 1173.51

3 1259.71 1271.25

4 1360.49 1377.13

5 1469.33 1491.82

6 1586.87 1616.07

7 1713.82 1750.67

8 1850.93 1896.48

9 1999.00 2054.43

1000 1.08t 0.081000 te

$2054.43 $1999.00 $55.43

2009 PBLPathways

a. For each of 9 years, compare the future value of an investment of $1000 at 8% compounded annually and of $1000 at 8% compounded continuously.

t

0 1000.00 1000.00

1 1080.00 1083.29

2 1166.40 1173.51

3 1259.71 1271.25

4 1360.49 1377.13

5 1469.33 1491.82

6 1586.87 1616.07

7 1713.82 1750.67

8 1850.93 1896.48

9 1999.00 2054.43

1000 1.08t 0.081000 te

$2054.43 $1999.00 $55.43

2009 PBLPathways

a. For each of 9 years, compare the future value of an investment of $1000 at 8% compounded annually and of $1000 at 8% compounded continuously.

t

0 1000.00 1000.00

1 1080.00 1083.29

2 1166.40 1173.51

3 1259.71 1271.25

4 1360.49 1377.13

5 1469.33 1491.82

6 1586.87 1616.07

7 1713.82 1750.67

8 1850.93 1896.48

9 1999.00 2054.43

1000 1.08t 0.081000 te

$2054.43 $1999.00 $55.43

2009 PBLPathways

a. For each of 9 years, compare the future value of an investment of $1000 at 8% compounded annually and of $1000 at 8% compounded continuously.

t

0 1000.00 1000.00

1 1080.00 1083.29

2 1166.40 1173.51

3 1259.71 1271.25

4 1360.49 1377.13

5 1469.33 1491.82

6 1586.87 1616.07

7 1713.82 1750.67

8 1850.93 1896.48

9 1999.00 2054.43

1000 1.08t 0.081000 te

$2054.43 $1999.00 $55.43

2009 PBLPathways

b. Graph the functions for annual compounding and for continuous compounding on the same axes.

2009 PBLPathways

b. Graph the functions for annual compounding and for continuous compounding on the same axes.

1000 1.08 t

0.081000 te

t

S

2009 PBLPathways

c. What conclusion can be made regarding compounding annually and compounding continuously?

1000 1.08 t

0.081000 te

t

S