example 2 Daily Versus Annual Compounding of Interest

Chapter 5.5

a. Write the equation that gives the future value of $1000 invested for t years at 8% compounded annually.

b. Write the equation that gives the future value of $1000 invested for t years at 8% compounded daily.

c. Graph the equations from parts (a) and (b) on the same axes with t between 0 and 30.

d. What is the additional amount of interest earned in 30 years from compounding daily rather than annually?

2009 PBLPathways

2009 PBLPathways

a. Write the equation that gives the future value of $1000 invested for t years at 8% compounded annually.

b. Write the equation that gives the future value of $1000 invested for t years at 8% compounded daily.

c. Graph the equations from parts (a) and (b) on the same axes with t between 0 and 30.

d. What is the additional amount of interest earned in 30 years from compounding daily rather than annually?

2009 PBLPathways

a. Write the equation that gives the future value of $1000 invested for t years at 8% compounded annually.

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

$1000

0.08

1

P

r

k

1000 1 0.08

1000 1.08

t

t

S

2009 PBLPathways

a. Write the equation that gives the future value of $1000 invested for t years at 8% compounded annually.

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

$1000

0.08

1

P

r

k

1000 1 0.08

1000 1.08

t

t

S

2009 PBLPathways

a. Write the equation that gives the future value of $1000 invested for t years at 8% compounded annually.

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

$1000

0.08

1

P

r

k

1000 1 0.08

1000 1.08

t

t

S

2009 PBLPathways

a. Write the equation that gives the future value of $1000 invested for t years at 8% compounded annually.

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

$1000

0.08

1

P

r

k

1000 1 0.08

1000 1.08

t

t

S

2009 PBLPathways

a. Write the equation that gives the future value of $1000 invested for t years at 8% compounded annually.

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

$1000

0.08

1

P

r

k

1000 1 0.08

1000 1.08

t

t

S

2009 PBLPathways

a. Write the equation that gives the future value of $1000 invested for t years at 8% compounded annually.

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

$1000

0.08

1

P

r

k

1000 1 0.08

1000 1.08

t

t

S

2009 PBLPathways

b. Write the equation that gives the future value of $1000 invested for t years at 8% compounded daily.

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

$1000

0.08

365

P

r

k

3650.08

1000 1365

t

S

2009 PBLPathways

b. Write the equation that gives the future value of $1000 invested for t years at 8% compounded daily.

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

$1000

0.08

365

P

r

k

3650.08

1000 1365

t

S

2009 PBLPathways

b. Write the equation that gives the future value of $1000 invested for t years at 8% compounded daily.

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

$1000

0.08

365

P

r

k

3650.08

1000 1365

t

S

2009 PBLPathways

b. Write the equation that gives the future value of $1000 invested for t years at 8% compounded daily.

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

$1000

0.08

365

P

r

k

3650.08

1000 1365

t

S

2009 PBLPathways

c. Graph the equations from parts (a) and (b) on the same axes with t between 0 and 30.

30

1000 1.08

1000 1.08

10,062.66

tS

S

365

365 30

0.081000 1

365

0.081000 1

365

11,020.28

t

S

S

Compounded annually

Compounded daily

2009 PBLPathways

c. Graph the equations from parts (a) and (b) on the same axes with t between 0 and 30.

30

1000 1.08

1000 1.08

10,062.66

tS

S

365

365 30

0.081000 1

365

0.081000 1

365

11,020.28

t

S

S

Compounded annually

Compounded daily

t

S

2009 PBLPathways

30

1000 1.08

1000 1.08

10,062.66

tS

S

365

365 30

0.081000 1

365

0.081000 1

365

11,020.28

t

S

S

Compounded annually

Compounded daily

d. What is the additional amount of interest earned in 30 years from compounding daily rather than annually?

t

S

2009 PBLPathways

30

1000 1.08

1000 1.08

10,062.66

tS

S

365

365 30

0.081000 1

365

0.081000 1

365

11,020.28

t

S

S

Compounded annually

Compounded daily

d. What is the additional amount of interest earned in 30 years from compounding daily rather than annually?

t

S

2009 PBLPathways

30

1000 1.08

1000 1.08

10,062.66

tS

S

365

365 30

0.081000 1

365

0.081000 1

365

11,020.28

t

S

S

Compounded annually

Compounded daily

d. What is the additional amount of interest earned in 30 years from compounding daily rather than annually?

t

S

2009 PBLPathways

30

1000 1.08

1000 1.08

10,062.66

tS

S

365

365 30

0.081000 1

365

0.081000 1

365

11,020.28

t

S

S

Compounded annually

Compounded daily

d. What is the additional amount of interest earned in 30 years from compounding daily rather than annually?

t

S

2009 PBLPathways

30

1000 1.08

1000 1.08

10,062.66

tS

S

365

365 30

0.081000 1

365

0.081000 1

365

11,020.28

t

S

S

Compounded annually

Compounded daily

d. What is the additional amount of interest earned in 30 years from compounding daily rather than annually?

t

S

2009 PBLPathways

30

1000 1.08

1000 1.08

10,062.66

tS

S

365

365 30

0.081000 1

365

0.081000 1

365

11,020.28

t

S

S

Compounded annually

Compounded daily

d. What is the additional amount of interest earned in 30 years from compounding daily rather than annually?

t

S

$11,020.28 $10,062.66 $957.62