Why n in the compounded half-yearly replaced by 2n and not by $\frac n{2}$?

Question

Compound interest

Compounded Annually:

Amount after one year = $P(1+\frac R{100})^n$

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Compounded half-yearly:

Amount after half year = $P(1+\frac {R/2}{100})^{2n}$

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Why n in the compounded half-yearly replaced by 2n and not by $\frac n{2}$?

Answer 1

The exponent $n$ in the compound interest formula represents periods. The rate of interest must synchronize with the period. Generally, interest rate is given per annum (eg: 7% p.a). However, compounding period may be annual, half-yearly, quarterly, monthly, bi-weekly, daily or theoretically anything we want.

So, if the compounding is done half-yearly and rate of interest is given per annum, we need to normalize the rate of interest and the period in the exponent.

We do that by halving the rate of annual interest. This still leaves the exponent as annual periods. If compounding is done half-yearly, the periods goes up by a factor of 2. That is why we multiply the exponent by 2.

Here's a detailed worked example of Compound Interest:

Answer 2

Because it's compounded twice each year, not half a time each year. Thus each year contributes $$ \left(1+\frac{R/2}{100}\right)\cdot\left(1+\frac{R/2}{100}\right)= \left(1+\frac{R/2}{100}\right)^2 $$ to the total interest. Taken over $n$ years, this gives a total exponent of $2n$.