In general, when money is invested, be it in a bank, a company, property, bonds or any other investment, the value of the money is expected to increase over time. This expectation of the increased value of the money over time is referred to as its future value. If a sum of money is invested in the bank at a rate of interest ‘i’ for a period of one year, at the end of that year the future value of the money is expected to be:
Future Value = Present Value * (1 + i)
After two years, if the interest rate remains the same, the future value is expected to be:
Future Value = Present Value * (1 + i) * (1 + i)
Investing money over a period of time like this is known as compounding, as the interest compounds onto the principle every year and hence the value grows at a compounded, or exponential, rate. In general, after ‘n’ years, the future value of money invested at a rate ‘i’ will be:
Future Value = Present Value * (1 + i) ^ t
This equation can be used to calculate the future value of any investment, as well as the interest rate needed to achieve a given future value over a certain time period, or the amount of time money must be invested for to reach a desired future value at a given interest rate. There are a number of ways this can be achieved.
The simplest method is iteration, which simply involves repeatedly calculating the future value for different interest rates or times, until the desired future value is found. However, this is time consuming and not always accurate. Alternatively, many calculators or computer programmes will have a built in function for rapidly iterating to the solution. Also, it is possible to use interest rate tables, which demonstrate how interest rates and time periods will affect the future value of a present investment. However, the most exact way to calculate it is to use algebra to manipulate the equation to provide an exact solution.
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The first step in any algebraic solution is to divide both sides by PV. This gives:
FV / PV = (1 + i) ^ t
To find the interest rate, each side should be raised by the power of 1 / t:
(FV / PV) ^ (1 / t) = 1 + i
i = (FV / PV) ^ (1 / t) - 1
To find the required time period, take the logarithm of each side:
log (FV / PV) = log (( 1 + i ) ^ t)
As the logarithm of anything raised to a power is equal to the logarithm of that value raised to the power:
log (FV / PV) = t * log (1 + i)
t = log ( FV / PV )
log ( 1 + i )
Whilst it is not essential to understand the maths behind this, it can make future value questions easier to solve.
Note that many schools and universities will provide interest factor tables, providing the value of (1 + i) ^ t for a variety of time periods and interest rates. This is useful for answering future and present value questions, but it is wise to check whether your school or university offers these tables in any examinations.
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