Consider a mutual fund with $400 million in assets at the start of the year and 10 million shares outstanding. The fund invests in a portfolio of stocks that provides dividend income at the end of the year of $1.8 million. The stocks included in the fund's portfolio increase in price by 8%, but no securities are sold and there are no capital gains distributions. The fund charges 12b-1 fees of 1.00%, which are deducted from portfolio assets at year-end.
 
a. What is the fund's net asset value at the start and end of the year? (Round your answers to 2 decimal places.)
 

 
b. What is the rate of return for an investor in the fund? (Round your intermediate calculations to 3 decimal places and final answer to 2 decimal places.)
 

Let's break down the problem into steps to solve each part.

### Given:
- **Assets at the start of the year**: \$400 million
- **Shares outstanding**: 10 million
- **Dividend income at the end of the year**: \$1.8 million
- **Price increase of stocks**: 8%
- **12b-1 fees**: 1.00%

### Part (a) Net Asset Value (NAV)
#### NAV at the start of the year:
\[ \text{NAV}_{\text{start}} = \frac{\text{Total Assets}}{\text{Shares Outstanding}} = \frac{\$400\ \text{million}}{10\ \text{million shares}} \]
\[ \text{NAV}_{\text{start}} = \$40.00 \]

#### NAV at the end of the year:
To calculate this, we need to take into account the increase in the value of the stocks, the dividend income, and the fees.

1. **Stock value increase**:
   - Initial stock value: \$400 million
   - Increase: 8%
   - New stock value: 
   \[
   \text{New stock value} = \$400\ \text{million} \times 1.08 = \$432\ \text{million}
   \]

2. **Add Dividend Income**:
   - Total assets before fees:
   \[
   \text{Total assets before fees} = \$432\ \text{million} + \$1.8\ \text{million} = \$433.8\ \text{million}
   \]

3. **Deduct 12b-1 Fees**:
   - 12b-1 fees: 
   \[
   \text{Fees} = 1\% \times \$433.8\ \text{million} = \$4.338\ \text{million}
   \]
   - Total assets after fees:
   \[
   \text{Total assets after fees} = \$433.8\ \text{million} - \$4.338\ \text{million} = \$429.462\ \text{million}
   \]

4. **Calculate NAV at the end of the year**:
   \[
   \text{NAV}_{\text{end}} = \frac{\$429.462\ \text{million}}{10\ \text{million shares}} = \$42.95
   \]

So, the NAV at the start of the year is **$40.00** and the NAV at the end of the year is **$42.95**.

### Part (b) Rate of Return
The rate of return for an investor in the fund is calculated as:
\[
\text{Rate of Return} = \frac{\text{NAV}_{\text{end}} - \text{NAV}_{\text{start}} + \text{Dividend per Share}}{\text{NAV}_{\text{start}}}
\]

Where:
- Dividend per share:
\[
\text{Dividend per share} = \frac{\$1.8\ \text{million}}{10\ \text{million shares}} = \$0.18
\]

Substituting the values:
\[
\text{Rate of Return} = \frac{\$42.95 - \$40.00 + \$0.18}{\$40.00} = \frac{\$3.13}{\$40.00} = 0.07825 = 7.83\%
\]

### Final Answers:
- **(a) NAV at the start of the year**: \$40.00
- **NAV at the end of the year**: \$42.95
- **(b) Rate of return for the investor**: 7.83%

Would you like more details, or do you have any other questions?

### Related Questions:
1. How would the NAV change if the dividend income was reinvested?
2. What impact would selling some securities during the year have on the NAV?
3. How would an increase in the 12b-1 fees affect the rate of return?
4. What would the rate of return be if the fund had a capital gains distribution?
5. How do changes in the number of shares outstanding affect NAV?

### Tip:
Understanding how fees impact investment returns is crucial. Even small fees can significantly affect the long-term growth of your investment portfolio.

Learn how to calculate the net asset value (NAV) and rate of return for a mutual fund with assets of $400 million, dividend income of $1.8 million, and 10 million shares outstanding. Understand the impact of stock price increases, dividend income, and fees on NAV calculations. Suitable for advanced learners interested in finance and investment calculations.

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