Q.Find the effective rate of interest equivalent to a nominal rate of 6% compounded

Concept: Effective Interest Rate — the actual annual rate that accounts for compounding within the year.

Step 1: Formula for discrete compounding

If nominal rate $r$ is compounded $n$ times per year, the effective rate $R$ is

Step 2: Apply for each case

• (i) Semi-annually: $n = 2$, $r = 0.06$

• (ii) Quarterly: $n = 4$ …

The Core Idea: Why Effective Rate Matters

A nominal interest rate of 6% per year sounds straightforward — but if interest is compounded more than once a year, you actually earn more than 6%. Why? Because each compounding period adds interest on top of interest already earned. The effective annual rate (EAR) tells you the true annual growth, stripping away the compounding frequency so you can compare different loans or investments fairly.

Think of it this way: if a bank says "6% compounded semi-annually," they mean 3% every six months. After one year, your money grows by a factor of $(1.03)^2 = 1.0609$, which is a 6.09% increase — not 6%. The effective rate captures that extra 0.09%.

The general formula for EAR when compounding $n$ times per year at a nominal rate $r$ is:

For continuous compounding, we take the limit as $n \to \infty$, which gives:

Now let's apply this to each case.

Step-by-Step Solutions

1. Semi-annual compounding ($n = 2$)

The nominal rate $r = 0.06$ (6%). Each half-year, you earn $r/n = 0.06/2 = 0.03$ (3%). Over two periods:

Compute: $1.03^2 = 1.0609$, so:

As a percentage: 6.09%.

2. Quarterly compounding ($n = 4$)

Each quarter, the rate is $0.06/4 = 0.015$ (1.5%). Over four quarters:

Compute step by step:

• $1.015^2 = 1.030225$
• $1.015^4 = (1.015^2)^2 = (1.030225)^2 = 1.061363550625$

So:

Rounded to three decimal places: 6.136%. …