Questions: Which one of the following has the highest effective annual rate? 6 percent compounded quarterly 6 percent compounded annually 6 percent compounded daily 6 percent compounded semiannually

Solution Steps

To determine which option has the highest effective annual rate (EAR), we need to calculate the EAR for each compounding frequency using the formula:
\[ \text{EAR} = \left(1 + \frac{r}{n}\right)^n - 1 \]
where \( r \) is the nominal interest rate and \( n \) is the number of compounding periods per year. We will compare the EARs for quarterly, annually, daily, and semiannually compounded interest.

To find the effective annual rate (EAR) for each compounding frequency, we use the formula:
\[ \text{EAR} = \left(1 + \frac{r}{n}\right)^n - 1 \]
where \( r = 0.06 \) is the nominal interest rate and \( n \) is the number of compounding periods per year.

• Quarterly Compounding: \( n = 4 \)
  \[ \text{EAR}_{\text{quarterly}} = \left(1 + \frac{0.06}{4}\right)^4 - 1 = 0.06136 \]

• Annually Compounding: \( n = 1 \)
  \[ \text{EAR}_{\text{annually}} = \left(1 + \frac{0.06}{1}\right)^1 - 1 = 0.06 \]

• Daily Compounding: \( n = 365 \)
  \[ \text{EAR}_{\text{daily}} = \left(1 + \frac{0.06}{365}\right)^{365} - 1 = 0.06183 \]

• Semiannually Compounding: \( n = 2 \)
  \[ \text{EAR}_{\text{semiannually}} = \left(1 + \frac{0.06}{2}\right)^2 - 1 = 0.06090 \]

Now, we compare the calculated EARs to determine which compounding frequency results in the highest effective annual rate.

• \(\text{EAR}_{\text{quarterly}} = 0.06136\)
• \(\text{EAR}_{\text{annually}} = 0.06\)
• \(\text{EAR}_{\text{daily}} = 0.06183\)
• \(\text{EAR}_{\text{semiannually}} = 0.06090\)

The highest EAR is \( \text{EAR}_{\text{daily}} = 0.06183 \).

Final Answer