Questions: What are the nominal and effective annual rates for an account paying 4% annual interest, compounded (round the answers to three decimal places, where necessary) a) annually? nominal: % effective: % b) quarterly? nominal: % effective: % c) daily? nominal: % effective: %

Solution Steps

To solve this problem, we need to calculate both the nominal and effective annual interest rates for different compounding periods. The nominal rate is simply the stated annual interest rate, while the effective annual rate (EAR) takes into account the compounding periods within the year.

1. Annually:

   • Nominal rate: The stated annual interest rate.
   • Effective rate: Since it is compounded annually, the effective rate is the same as the nominal rate.

2. Quarterly:

   • Nominal rate: The stated annual interest rate.
   • Effective rate: Use the formula for effective annual rate with quarterly compounding.

3. Daily:

   • Nominal rate: The stated annual interest rate.
   • Effective rate: Use the formula for effective annual rate with daily compounding.

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

For annual compounding:

• Nominal rate: \( r = 0.04 \)
• Effective rate: Since it is compounded annually, the effective rate is the same as the nominal rate.

\[ \text{Effective Annual Rate} = \left(1 + \frac{0.04}{1}\right)^1 - 1 = 0.04 \]

For quarterly compounding:

• Nominal rate: \( r = 0.04 \)
• Number of compounding periods per year: \( n = 4 \)

\[ \text{Effective Quarterly Rate} = \left(1 + \frac{0.04}{4}\right)^4 - 1 = \left(1 + 0.01\right)^4 - 1 \approx 0.040604 \]

For daily compounding:

• Nominal rate: \( r = 0.04 \)
• Number of compounding periods per year: \( n = 365 \)

\[ \text{Effective Daily Rate} = \left(1 + \frac{0.04}{365}\right)^{365} - 1 \approx 0.040808 \]

Final Answer