11.1: The Constant e and Continuous Compound Interest

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# 11.1: The Constant e and Continuous Compound Interest - PowerPoint PPT Presentation

11.1: The Constant e and Continuous Compound Interest. Review (Mat 115). Just like π , e is an irrational number which can not be represented exactly by any finite decimal fraction. However, it can be approximated by for a sufficiently large x. e.

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### 11.1: The Constant e and Continuous Compound Interest

Review (Mat 115)
• Just like π, e is an irrational number which can not be represented exactly by any finite decimal fraction.
• However, it can be approximated by

for a sufficiently large x

e

e

e

The Constant e

Reminder:

Use your calculator, e = 2.718 281 828 459 …

DEFINITION OF THE NUMBER e

Review
• Simple interest: A = P + Prt = P(1 + rt)
• Compound Interest: A = P(1 + r)t

or

with n = 1 (interest is compounded annually

– once per year)

• Other compounding periods:

semiannually(2), quarterly(4), monthly(12),

weekly(52), daily(365), hourly(8760)…

• Continuous Compounding:(see page 589 for the proof)

A = Pert

A: future value

P: principal

r: interest rate

t: number of years

Example 1: Generous Grandma

Your Grandma puts \$1,000 in a bank for you, at 5% interest. Calculate the amount after 20 years.

Simple interest:

A = 1000 (1 + 0.0520) = \$2,000.00

Compounded annually:

A = 1000 (1 + .05)20 =\$2,653.30

Compounded daily:

Compounded continuously:

A = 1000 e(.05)(20) = \$2,718.28

Example 2: IRA
• After graduating from Barnett College, Sam Spartan landed a great job with Springettsbury Manufacturing, Inc. His first year he bought a \$3,000 Roth IRA and invested it in a stock sensitive mutual fund that grows at 12% a year, compounded continuously. He plans to retire in 35 years.
• What will be its value at the end of the time period?
• A = Pert = 3000 e(.12)(35) =\$200,058.99
• The second year he repeated the purchase of an identical Roth IRA. What will be its value in 34 years?
• A = Pert = 3000 e(.12)(34) =\$177,436.41
Example 3

What amount (to the nearest cent) will an account have

after 5 years if \$100 is invested at an annual nominal rate

of 8% compounded annually? Semiannually? continuously?

• Compounded annually
• Compounded semiannually
• Compounded continuously

A = Pert = 100e(.08*5)

= 149.18

If \$5000 is invested in a Union Savings Bank 4-year CD that earns 5.61% compounded continuously, graph the amount in the account relative to time for a period of 4 years.Example 4
• Press y=
• Type in 5000e^(x*0.0561)
• Press ZOOM, scroll down, then press ZoomFit
• You will see the graph
• To find out the amount after 4 years
• Press 2ND, TRACE, 1:VALUE
• Then type in 4, ENTER
How long will it take an investment of \$10000

to grow to \$15000 if it is invested at 9% compounded

continuously?

Formula: A =P ert

15000 = 10000 e .09t

1.5 = e .09t

Ln (1.5) = ln (e .09t)

Ln (1.5) = .09 t

So t = ln(1.5) / .09

t = 4.51

It will take about 4.51 years

Example 5
How long will it take money to triple if it is

invested at 5.5% compounded continuously?

Formula: A =P ert

3P = P e .055t

3 = e .055t

Ln 3 = ln (e .055t)

Ln 3 = .055t

So t = ln3 / .055

t = 19.97

It will take about 19.97 years

Example 6