What is the effective annual rate of interest corresponding to a rate of 10% per annum compounded half-yearly?

Here the given rate of interest is 10% per annum compounded half-yearly.

compounded half-yearly means 5% rate of interest in six months(half-year).

Let's assume the principal amount is 100y.

interest after 1 year = 100y of (100+5)% of (100+5)% - 100y

= 100y of 105% of 105% - 100y

= $$100y\times1.05\times1.05-100y$$

= $$100y\times1.1025-100y$$

= $$100y\left(1.1025-1\right)$$

= $$100y\times\ 0.1025$$

= 10.25y

effective annual rate of interest = $$\frac{10.25y}{100y}\times\ 100$$

= 10.25%

Shortcut :

effective annual rate of interest = $$5+5+\frac{5\times\ 5}{100}$$

= $$10+\frac{25}{100}$$

= 10+0.25

= 10.25%