Practice Question
RBX currently offers an investment account with an interest rate of 28% compounded monthly. The bank wants to create a new account that offers interest compounded quarterly. In order for the effective rates to be equal, what interest rate should they quote for the new account?
To solve the problem of converting the monthly compounded interest rate to a quarterly compounded interest rate, we need to follow these steps:
Step 1: Define Variables
- Monthly nominal interest rate: \( r_m = 28\% \)
- Monthly compounding frequency: \( m = 12 \)
- Quarterly compounding frequency: \( q = 4 \)
Step 2: Convert Monthly Nominal Rate to Effective Annual Rate
\[ EAR = \left(1 + \frac{r_m}{m}\right)^m - 1 \] \[ EAR = \left(1 + \frac{0.28}{12}\right)^{12} - 1 = (1.023333)^{12} - 1 \approx 0.317 \]
The effective annual rate is approximately 31.7%.
Step 3: Convert Effective Annual Rate to Quarterly Nominal Rate
We now reverse-engineer the quarterly nominal rate \( r_q \) from the EAR:
\[ EAR = \left(1 + \frac{r_q}{4}\right)^4 - 1 \] \[ 0.317 = \left(1 + \frac{r_q}{4}\right)^4 - 1 \Rightarrow 1.317 = \left(1 + \frac{r_q}{4}\right)^4 \] \[ \left(1.317\right)^{\frac{1}{4}} = 1 + \frac{r_q}{4} \Rightarrow 1.071 = 1 + \frac{r_q}{4} \Rightarrow \frac{r_q}{4} = 0.071 \Rightarrow r_q = 0.284 \]
Conclusion
To match the effective annual rate of 28% compounded monthly, the new account should quote an interest rate of approximately 28.66% compounded quarterly.