4. In the last 5 year, ALB Mutual Fund grew at the rate of 10.4% per year compounded quarterly. Over the same period, SCG Mutual Fund grew at the rate of 10.6% per year compounded semi - annually. Which mutual fund has a better rate of return?

To determine which mutual fund has a better rate of return, we need to calculate the effective annual rate (EAR) for both mutual funds. The EAR accounts for the compounding periods and provides a comparable rate of return on an annual basis.

### 1. Effective Annual Rate for ALB Mutual Fund

The ALB Mutual Fund grows at an annual nominal rate of 10.4%, compounded quarterly. To calculate the EAR for this fund:

\[
EAR_{ALB} = \left(1 + \frac{r}{n}\right)^{n} - 1
\]

where:
- \( r = 0.104 \) (the nominal annual rate),
- \( n = 4 \) (quarterly compounding, so 4 compounding periods per year).

Substitute the values into the formula:

\[
EAR_{ALB} = \left(1 + \frac{0.104}{4}\right)^{4} - 1
\]

\[
EAR_{ALB} = \left(1 + 0.026\right)^{4} - 1
\]

\[
EAR_{ALB} = (1.026)^{4} - 1
\]

Calculating this:

\[
EAR_{ALB} \approx 1.106186 - 1 = 0.106186 \text{ or } 10.6186\%
\]

### 2. Effective Annual Rate for SCG Mutual Fund

The SCG Mutual Fund grows at an annual nominal rate of 10.6%, compounded semi-annually. The EAR for this fund is calculated as:

\[
EAR_{SCG} = \left(1 + \frac{r}{n}\right)^{n} - 1
\]

where:
- \( r = 0.106 \) (the nominal annual rate),
- \( n = 2 \) (semi-annual compounding, so 2 compounding periods per year).

Substitute the values into the formula:

\[
EAR_{SCG} = \left(1 + \frac{0.106}{2}\right)^{2} - 1
\]

\[
EAR_{SCG} = \left(1 + 0.053\right)^{2} - 1
\]

\[
EAR_{SCG} = (1.053)^{2} - 1
\]

Calculating this:

\[
EAR_{SCG} \approx 1.108809 - 1 = 0.108809 \text{ or } 10.8809\%
\]

### Conclusion

The **SCG Mutual Fund** has a better rate of return with an EAR of **10.8809%** compared to the ALB Mutual Fund's EAR of **10.6186%**.

Would you like more details or have any questions?

Here are 5 related questions:
1. How does the frequency of compounding affect the effective annual rate?
2. What is the formula for converting nominal interest rates to effective annual rates for different compounding frequencies?
3. Can you explain how continuous compounding works and how it would compare to quarterly or semi-annual compounding?
4. How would the EAR change if the compounding frequency were increased to monthly or daily?
5. What would the nominal rate need to be for the ALB Mutual Fund to have a higher EAR than the SCG Mutual Fund?

**Tip:** To compare interest rates effectively, always convert them to the same compounding period, ideally the effective annual rate (EAR), to get a true comparison of the returns.

Compare the rate of return between ALB Mutual Fund and SCG Mutual Fund using effective annual rates (EAR). Understand how compounding periods affect investment returns with detailed calculations. ALB Mutual Fund has an EAR of 10.6186% and SCG Mutual Fund has an EAR of 10.8809%.

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