5.4

5.4 – The Number e and the

Function ex

Objectives: You should be able to… 1. Use compound interest formulas to solve real-life problems.

Compound Interest

To examine compound interest at different interest periods we use the formula

where i is the rate (expressed as a decimal), n is the period, and t is years it will grow to an amount A of the principal P.

1nt

iA Pn

For interest compounded continuously, we can use…

P(t) = P0ert where P0 is the initial amount, r is the annual rate, and t is the time in years.

**NOTE** this formula can be used for any quantity compounding continuously.

Evaluate:a) e– 0.06

b) 6

5e

Example:

Suppose you invest $1.00 at 12% annual interest. Calculate the amount that you would have after one year if the interest is compounded:

a) quarterly

b) monthly

c) continuously

Example:

Example: With which plan would an investor earn more, Plan A, B, or C?

Plan A: A 7.5% annual rate compounded monthly over a 4-year period.

Plan B: A 7.2% annual rate compounded daily over a 4-year period.

Plan C: A 7% annual rate compounded continuously over a 4-year period.

Effective Annual Yield – If semiannual is 4%, you cannot multiply by 2 for EAY because you wouldn’t be considering the interest earned in the first period.

initial

initialendEAY

100 deposited in a bank account that compounds interest quarterly yields $107.50 over 1 year. Find the effective annual yield.

Example:

Look at graph of y = ex.a. Graph y = ex + 1.

b. Graph y = ex + 1.

Example: