11 1 the constant e and continuous compound interest
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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.

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Review mat 115
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
The Constant e

Reminder:

Use your calculator, e = 2.718 281 828 459 …

DEFINITION OF THE NUMBER e



Review
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
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
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
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


Example 4

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

  • Use your graphing calculator:

  • 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


Example 5

How long will it take an investment of $10000 that earns 5.61% compounded continuously, graph the amount in the account relative to time for a period of 4 years.

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


Example 6

How long will it take money to triple if it is that earns 5.61% compounded continuously, graph the amount in the account relative to time for a period of 4 years.

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



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