Compound Interest

Discussion

Example 1:

We have $1000 invested for 5 years at 7% interest.

Using simple interest
    i = prt
    i = 1000×0.07×5 = 350
    So my total is $1,350

What if I withdrew the money each year and redeposited it?
    Year 1 
       i = 1000×0.07×1 = $70
       A = $1,070
    Year 2
       i = 1070×0.07×1 = 74.90 
       (Where did the extra 4.90 come from?)
       A = $1,144.90
    Year 3
       i = 1144.90 ×0.07×1 = 80.14
       A = $1,225.04
    Year 4 
       i = 1225.04×0.07×1 = 85.75
       A = $1,310.80
    Year 5
       i = 1310.80×0.07×1 = 91.76
       A = $1,402.55

This is called compound interest.
In this case, we earn interest on our interest :)
We are compounding once a year, or annually
We are compounding for a period of 5 years
A formula for this is
- Where A = amount (interest + principal)
- p = principal
- r = rate (per years)
- n = the number of compounding periods per year
- t = the number of years.

Example 1 (continued)


    p = $1000
    i = 7%
    n = 1
    t = 5

    A = 1000(1+0.07/1)^1×5
      = 1000(1.07)⁵
      = 1000×1.402551
      = 1,402.55

On my calculator, I have an y^x key
To use this
- Enter the base (1.07)
- press y^x
- Enter the exponent (5)
- press =
Experiment with something you know, like 2⁵ = 32
If you do it backwards you will get 5² = 25

We can compound interest

4 times a year or quarterly
12 times a year or monthly
360 times a year or daily
Or anything else we wish.
Even continuously, but we will not deal with that here.

What simple interest rate would we need in the above example to equal annual compounding?

Example 1, continued again

A = $1,402.55
p = $1,000
i = 1,402.44 - 1,000 = 402.55

i = prt
r = i/pt

r = 402.55/(1000*5)
r = .0805
r = 8.05%

This is called the effective annual yield or the annual percentage yield

We can solve the compound interest formula for p

All of the variables are the same, but this tells us how much you need to invest now, at a given interest rate, given compounding rate, given time , to achieve a desired amount.

Example 2

    I would like to have $2,000 to buy a computer in 4 years.  
    The local bank will pay 3.5% interest, compounded monthly
    on a cd purchased now.  How much should I deposit now to
    have $2,000 in 4 years?

    A = 2,000
    r = 3.5
    n = 12
    t = 4

    p = 2000/(1+0.035/12)^12*4
      =  2000/(1.0029)⁴⁸
      = 2000/1.15
      = $1,739.07

Homework

Please do 7-27 odd, plus 41 and 43