⏱️ Half-Yearly, Quarterly & Effective Rate

The interest can only use the rate that belongs to its time-slice. Over 6 months you cannot charge a full year's rate — you charge half of it. So:

• Half-yearly: rate becomes $\dfrac{R}{2}$ and the number of periods becomes $2T$, giving $A=P\left(1+\dfrac{R}{2\times100}\right)^{2T}$.

• Quarterly: rate becomes $\dfrac{R}{4}$ and periods become$4T$, giving $A=P\left(1+\dfrac{R}{4\times100}\right)^{4T}$.

The deadly trap is doing only HALF the swap — doubling the periods but forgetting to halve the rate (or the other way round). Both moves happen together, every time.

The effective annual rate is the real percent your money grows in one whole year once this extra compounding is counted. Take $R=10\%$ compounded half-yearly: that is$5\%$ then $5\%$. On ₹100 the first 6 months add ₹5 (money becomes ₹105), the next 6 months add $5\%$ of ₹105 $=$ ₹5.25. Total growth$=10.25$, so the effective rate is $10.25\%$, not 10%. Fast way: fold the two equal periods with $x+y+\dfrac{xy}{100}$ → $5+5+\dfrac{5\times5}{100}=10.25\%$. The effective rate is ALWAYS a little above the nominal rate.

That little extra has a name: interest on interest. For a single year, the gap between half-yearly and annual interest equals exactly the half-yearly interest multiplied by the half-yearly rate — $(\text{half-yearly interest})\times\dfrac{R}{2}$. Nothing more is happening; the second half-year simply earns a tiny bit of interest on the first half-year's interest.