Most bank accounts and loans employ some form of compounding—producing compound interest Again, consider an account that pays interest at a rate of r per year If interest is compounded yearly, then after 1 year, the first year's interest is added to the original principal to define a larger principal base for the second year. Thus during the second year, the account earns interest on interest, This is the compounding effect, which is continued year after year
Under yearly compounding, money left in an account is multiplied by (1 4- / ) after 1 year. After the second year, it grows by another factor of (1 + /) to (1 -f ' )2, After n years, such an account will grow to (1 4- / )" times its original value, and this is the analytic express ton for the account growth under compound interest. This expression is said to exhibit geometric growth because of its /ith-power form
As n increases, the growth due to compounding can be substantial. For example, Figure 2,1 shows a graph of a $100 investment over time when it earns 10% interest under simple and compound interest rules. The figure shows the characteristic shapes of linear growth for simple interest and of accelerated upward growth for compound interest, Note that under compounding, the value doubles in about 7 years.
There is a cute Iitde rule that can be used to estimate the effect of interest compounding,
FIGURE 2..1 Simple and compound interest Simple interest leads to linear growth over time, whereas compound interest leads !o an accelerated increase defined by geometric growth The figure shows both cases for an interest rate of 10%
The seven-ten rule Money invested at 7% per year doubles in approximately JO years Also, money invested at J0% per year doubles in approximately 7 years
(More exactly, at 7% and 10 years, an account increases by a factor of 1.97, whereas at 10% and 7 years it increases by a factor of 1.95.) The rule can be generalized, and slightly improved, to state that, for interest rates less than about 20%, the doubling time is approximately 72//, where i is the interest rate expressed as a percentage (that is, 10% interest corresponds to i — 10). (See Exercise 2 )
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