Question

The effective rate that is equivalent to a nominal rate of 6% per annum compounded semi-annually is

• A. 6%
• B. 6.09%
• C. 0.069%
• D. 9.06%

Hint:

Effective rate \(=(1+\frac{0.06}{2})^2-1\).

Answer 1

The Correct Option is B

Step 1: Understand the terms:
The nominal rate is the yearly rate as quoted, here 6% per year. It is compounded semi-annually, so interest is added twice a year.
The effective rate is the single yearly rate which gives the same final amount as this compounding.

Step 2: Recall the formula:
If the nominal rate is \(r\) per year and interest is compounded \(m\) times a year, then \[ \text{Effective rate}=\left(1+\frac{r}{m}\right)^{m}-1 \]
Here \(r=0.06\) and \(m=2\).

Step 3: Calculate:
The rate for each half-year is \(\frac{0.06}{2}=0.03\).
\[ (1.03)^2-1=1.0609-1=0.0609 \]
As a percentage, this is 6.09%.

Step 4: Check the options:
Option 1 (6%) would be right only for yearly compounding. Option 3 (0.069%) has a wrong decimal shift and is far too small. Option 4 (9.06%) is too large, since compounding twice cannot lift 6% by 3 points. Only 6.09% fits.

Final Answer:
The effective rate is 6.09% per annum, which is option 2. \[ \boxed{6.09\%} \]