Constant Percentage and Exponentials

Key Questions

  • Answer:

    Constant percentage may be translated to a growth factor.


    Say something grows with a constant percentage of 10% every period, we may say that the growth factor #g=1.10#
    This means that after #t# periods, the something has grown to #g^t=1.10^t# its original value (as in compounded interest).

    If something dimished with a a constant percentage of say 10% every period, this means that the growth factor is #0.90#, and after #t# periods only #0.90^t# of it is left.

    You can rework these results to total percentage by multiplying by #100%#

  • Appreciation is a rate of positive change. So constant appreciation means the rate of change is constant. This implies a linear function, however, the appreciation occurs at certain time intervals, so it is more like an arithmetic sequence.

    An example of constant appreciation is simple interest. This is where interest is typically paid on the principal annually. For instance, $1000 is invested in a term deposit at 2% simple interest for 5 years . What is the value of the term deposit at the end of 5 years? 2% of $1000 is $20, so $20 is the constant appreciation. At the end of 5 years, the term deposit would be worth $1100: #$1000+5xx$20#.

    Note that most banks pay out using compound interest rather than simple interest.