Q.What is the effective annual rate of interest compounding equivalent to a nominal rate of interest 5% per annum compounded quarterly?

The key idea is that effective annual rate (EAR) converts a nominal rate with intra-year compounding into the equivalent annual rate that would produce the same final amount.

Step 1: Nominal rate $r = 5\% = 0.05$, compounded quarterly means $n = 4$ periods per year.

Step 2: The effective annual rate formula is:

Step 3: Substitute the values:

Step 4: Compute $(1.0125)^4$:

So $\text{EAR} \approx 1.050945 - 1 = 0.050945$.

The core idea here is that when interest compounds more than once a year, you earn "interest on interest" within the year itself. A nominal rate of 5% per annum compounded quarterly doesn't mean you get 5% at the end of the year — it means you get 1.25% (that's 5% ÷ 4) every three months. Because each quarter's interest starts earning its own interest in the next quarter, the actual yearly growth is slightly higher than 5%.

The effective annual rate (EAR) is the single annual rate that, compounded once per year, gives the same final amount as the nominal rate compounded multiple times. This is the rate you actually feel in your pocket.

Let’s work through it.

1. Identify the given data.

   Nominal annual rate: $r = 5\% = 0.05$

   Compounding frequency: $n = 4$ (quarterly)

2. Find the periodic interest rate per quarter.

   Since the nominal rate is spread evenly over the year, each quarter gets:

3. Think about growth over one year.

   If you start with ₹1, after one quarter you have $1 \times (1 + 0.0125) = 1.0125$.

   After two quarters: $1.0125 \times (1 + 0.0125) = (1.0125)^2$

   After three quarters: $(1.0125)^3$

   After four quarters (one full year): $(1.0125)^4$

   So the total growth factor for the year is $(1.0125)^4$.

4. Relate this to the effective annual rate.

   If the effective annual rate is $R$, then ₹1 grows to $1 + R$ after one year.

   Setting this equal to the quarterly compounding growth:

5. Compute the value.

Let’s do it step by step:

• $1.0125 \times 1.0125 = 1.02515625$
• $1.02515625 \times 1.0125 = 1.037970703125$
• $1.037970703125 \times 1.0125 = 1.0509453369140625$

So $(1.0125)^4 \approx 1.050945$.

6. Solve for $R$.

As a percentage: $R \approx 5.0945\%$.