Activities

Compound interest - modelling on the TI-83

Supppose $500 is invested at 9% pa and that interest is calculated yearly and added to the amount invested.
 
time (years)
0
1
2
3
4
interest ($)
-
45
49.05
53.46
58.28
amount ($)
500
545
594.05
647.51
705.79

At the end of 4 years, the investment will be worth $705.79.

Each year the amount of interest increases as interest is being paid on the interest (compound interest).

In practice, the amount of the investment can be found by repeated multiplication by 1.09.

Enter 500 on the calculator. Multiply by 1.09 and then keep pressing ENTER.

Problems
If $500 is invested at compound interest payable yearly, how long would it take for the money to grow to $2000 if the rate of interest is:
  • 9% pa
  • 6% pa
  • 3% pa?
    		•
    		$500 is invested at 9% pa compound interest payable yearly. At the end of each year an additional $100 is also invested. How long would it take for the money to grow to $2000?
    		$100000 is borrowed at 7% pa compound interest payable yearly. At the end of each year, a repayment of $12000 is made. How long does it take to repay the loan?

    Compound interest formula

    The concept of repeated multiplication leads to the formula 

    This formula lends itself to the use of the exponent key on the calculator. Consider the earlier example of $500 invested at 9% pa compound interest payable yearly.
     
    time (years)
    0
    1
    2
    3
    4
    calculation
    500
    500(1.09)
    500(1.09)2
    500(1.09)3
    500(1.09)4
    amount ($)
    500
    545
    594.05
    647.51
    705.79

    There are four basic problems i.e. find A, find P, find i and find n. Can you rearrange the formula so that it can be used for the last three types? (details)

    Problems
    		$5000 is to be invested for 6 years.
    Plan A pays 7.1% pa compound interest payable yearly.
    Plan B pays 5.8% pa compound interest payable yearly.
    How much greater is the final payout under plan A?
    		How much should be invested at 6% pa compound interest payable yearly in order to produce a final amount of $10000 in 8 years time?

    Compound growth - an example of exponential functions

    Problem
    		$1000 can be invested at 8%, 6% or 4% pa compound interest payable yearly.
    Create an equation to model each option.
    Use a graphical method to find the time under each option for the investment to grow to $2500?

    Compound interest payable other than yearly

    The formula is still appropriate but care needs to be taken with the values for the rate of interest i and the number of interest periods n.

    For example, consider $700 invested for 5 years at 9% pa compound interest payable quarterly.

    Use the compound interest formula .

    The interest period is a quarter. Therefore n is the number of quarters (3 months) and i is the rate per quarter.

    final amount
    If $700 invested for 5 years at 9% pa compound interest payable quarterly, the final amount is $1092.36.

    Problem
    		$2000 is invested at 7% pa compound interest.
    Investigate the effect of the interest being payable yearly, quarterly, monthly and weekly.
    What is the difference in the final amounts after 5 years?

    Investigation

    Suppose an investment of $100 earns 8% pa compound interest for one year.

    Calculate the final amount if the interest is added yearly, monthly, weekly, daily, hourly, ...

    Surprised?

    Time/Value/Money Solver on the TI-83

    The TVM Solver works on the principle that money paid out is negative and money received is positive.

    The variables are as follows.
     

    N
    		 number of payments
  • not the same as n in the compound interest formula
  • equals the product of the number of years and the value of P/Y
  • same as number of years if P/Y is 1
    		•
    I%
    		 % nominal rate of interest pa
  • not the same as i in the compound interest formula
    		•
    PV
    		 present value
  • amount at the beginning
    		•
    PMT
    		 amount of payment
  • 0 if no payments
    		•
    FV
    		 future value
  • amount at the end
    		•
    P/Y
    		 number of payments per year
  • 1 if no payments
    		•
    C/Y
    		 number of times per year that interest is payable
    PMT:END BEGIN
  • END if payments are paid at the end of the payment period
  • BEGIN if payments are paid at the beginning of the payment period
    		•

    When there are no regular payments, remember:
     
    N
    		 number of years
  • not the same as n in the compound interest formula
    		•
    I%
    		 % nominal rate of interest pa
  • not the same as i in the compound interest formula
    		•
    PV
    		 present value
    PMT
    		 0
    FV
    		 future value
    P/Y
    		 1
    C/Y
    		 number of times per year that interest is payable

    Use the TVM Solver to answer the seven problems above.