Calculating the Time-Period (or Number of Years)

The most difficult part in compound interest is calculating the number of years to come up with the expected amount. The computation involves the use of logarithms. Most of the time, calculations for years are applied when determining the time when the amount invested will double or triple itself.

Here are the steps in order to get the total number of periods:

  • 1) Future amount, principal, nominal rate of interest and number of periods per year should be given.
  • 2) Divide the future amount by the principal amount.
  • 3) Transform the equation into logarithmic form.

Continuing, from Equation (II) in the derivation of nominal rate of Interest.

`text(Future Amount)/text(Principal)= (1+text(Nominal Rate)/text(Periods per Year) )^text(Number of Periods) --- (II)`

Taking log on both sides,

log `(text(Future Amount)/text(Principal))=(text(Number of Periods)) ` x log `(1+text(Nominal Rate)/text(Periods per Year))`
`text(Number of Periods)= (log (text(Future Amount)/text(Principal Amount)))/(log (1+text(Nominal Rate)/text(Periods per Year)))`

Note: Total number of periods should be a whole number.

In order to get the number of years;

`text(Number of Years)= text(Number of Periods)/text(Periods per Year)`

Note: When approximating or rounding off the number of years or periods into a whole number, consider the nearest larger whole number.

Example 1:
 5 Years
Explanation:

The principal amount is $60,000.

The future amount is $69,697.01.

Nominal rate of interest is 3%.

Number of periods per year is 12 (since monthly).

Using the formula for solving the total number of periods;

`text(Number of Periods) = log (text(Future Amount)/text(Principal))/(log (1+text(Nominal Rate)/text(Periods per Year)))`

`text(Number of Periods) = log ⁡69697.01/60000 / log⁡(1+ 0.03/12) `

`text(Number of Periods) = (log⁡1.162)/(log⁡(1+0.0025))`

`text(Number of Periods) = (log⁡ 1.162)/(log⁡1.0025) =0.065/0.00108`

`text(Number of Periods) = 59.94 = 60`

Determining the number of years;

`text(Number of Years) = text(Number of Periods)/text(Periods per Year)`

`text(Number of Years) = 60/12`

`text(Number of Years) = 5`

Therefore, the company paid the loan for 5 years.


Example 2:
 4
Explanation:

The future amount is twice the principal amount and thus their ratio is 1:2.

Let, Principal = x, then

Future Amount = 2x

Nominal rate of interest is 6%.

Number of periods per year is 1.

Using the formula for solving the total number of periods;

`text(Number of Periods) = log (text(Future Amount)/text(Principal))/(log (1+text(Nominal Rate)/text(Periods per Year)))`

`text(Number of Periods) = (log⁡2)/(log⁡(1+ 0.06/1))`

`text(Number of Periods) = (log⁡2)/(log⁡(1+0.06))`

`text(Number of Periods) = (log⁡2)/(log⁡1.06)`

`text(Number of Periods) = 11.9 ≈ 12`

Determining the number of years;

`text(Number of Years) = text(Number of Periods)/text(Periods per Year)`

`text(Number of Years) = 12/1 = 12`

Therefore, the amount will double after 12 years.







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