VALUATION
PRINCIPLES
Compound Interest and Future Value
The concept of the future value of a current amount of money
lies in the rate of earnings applied to it and the frequency of the application of the rate. For example,
the future value of a current $100 at an annually applied interest of 5% will be $105. That same $100
at an annually applied interest for 2 years will be $105 at the end of the first year as shown, and an
additional 5% the second year or $105 multiplied by 1.05 for a total sum of $110.25. This is called compounding
interest because we are applying interest to the principal plus the interest in each period.
Notice that if we applied simple interest of 5% for each year the sum would have been $5 for year
one and an additional $5 for the second year for a total of $110 or $.25 less that the annually compounded
sum.
Another way of expressing compounded interest is by multiplying
the principal amount by what is called a future value interest factor. The future value interest factor
is computed by taking 1 plus the interest rate and raising it to the appropriate power. For example, in
the above calculation 1 plus .05 = 1.05. For two years, then, the equation is (1.05)^2=1.025 multiplied
by the $100. This equals $110.25, which is the future value we computed the long way in the preceding
paragraph.
It is easier to do compounding by factoring for all of the
periods we are compounding. For example, if we were compounding the original $100 for ten years, multiplying
the rate and adding for each year would be a cumbersome process. It is much simpler to do (1.05)^10=1.6288
multiplied by 100 =$162.88. Many sources have pre-computed factoring tables to facilitate these calculations.
In addition, many people use financial mode line calculators to determine compounded future values.
The advantage of these calculators is that not only can you solve for future value, but you can solve
for any unknown in the future value series. For example, with a such a calculator, if you assumed a present
and future value and a time frame, you could instantly solve for the compounded interest rate that would
make it happen. This is a challenging calculation using long hand methods.
One more important point about compounded interest must be
made. Financial institutions and markets quote interest on an annual basis and when interest is applied
it becomes part of the principal amount.
Thus, virtually all interest applications are compounded.
Mathematical rules concerning compounding are that the more frequently you compound the same interest
rate, the higher the sum will be. In the example above , if we compounded a 5% annual rate every six months
for two years, the formula would be (1.025)^4, multiplied by $100, equals $110.38. Notice we divided the
interest by 2 and multiplied the periods by 2 to accommodate the semi-annual compounding, which resulted
in an additional 13 cents in total sum at the end of 2 years. If we compounded quarterly, the formula
would be (1.0125)^8 or $110.45, an additional 7 cents in total value.
Compounding frequency can be daily, even continuously:
every mini- micro second of every day.
