Compound Interest

Section 5.2

 

Bank and savings accounts pay compounded interest; interest that is periodically paid out on existing accounts that include the principal and the previous interest payments. (Interest is not taken out so that the balance increases)

Compounding period can be monthly, daily, quarterly, semi-annually, weekly; which refers to the frequency with which interest is computed and deposited. 

Since interest rates are given annually, we compute the periodic rate which refers to the compounding period.

      i = periodic rate = r/periods in a year

      n = (periods in a year)(# of years) = total number of periods

      FV = P(1 + i)n

 

Exercise1:  

Find the Future Value of an account with $2000 deposited, compounded quarterly @ 9.8% for 5 years.

        i= .098/4 = .0245       n = (4)(5) = 20

        FV = $2000(1 + .0245)20

        FV = $3245.41

The longer that money is invested, the larger the difference between simple and compounded interest.

Exercise2:  

How much must you invest in order to have $10,000 @ 10% compounded monthly for 6 years?  Find P

        FV = $10,000            i = .10/12 = .008333333           n = (12)(6) = 72

        $10,000 = P(1 +.008333333)72 = P(1.817594276)

        P =  $ 10,000       =  $5501.78
               1.817594276

How much interest would you have earned?         

        $10,000 - $5501.78 = $4498.22

Annual Yield: simple interest rate that has the same Future Value as a compounded rate.  Itís based on one year.

In other words: 

        FV(compounded interest) = FV(simple interest)

 

Exercise 3:

What is the annual yield of an account with $1,000  @ 8.9 % compounded quarterly?  Find r.

        P (1 + i)n           =       P   (1 + rt)

        $1000(1 + .089/4)4 = $1000(1 + r)

        (1 + .02225)4  =  ( 1 + r)

        1.0920146 = 1 + r

        0.09202146 = r = 9.2%

 

What is better? 5.8% compounded daily or 5.9% compounded monthly?

        r1 = .0597 = 5.97%

        r2 = .06062 = 6.06%

 


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