During the past 10 years, the percent returns on two mutual funds (aggressive and passive) expressed in percentages were as follows:
 
Year	Aggressive Fund	Passive Fund
-10	0%	4%
-9	7%	4%
-8	0%	4%
-7	1%	2%
-6	7%	2%
-5	2%	3%
-4	7%	2%
-3	2%	3%
-2	1%	3%
Last Year	6%	2%
 
Note that this is a sample of returns.
a) Compute the expected return for the two funds.  Round your answers to two decimal places.
Aggressive =   
Passive =   
 
b) Compute the variance and standard deviation of the returns of the two funds.  Round your answers to two decimal places.
 
Variance:
Aggressive =   
Passive =   
 
Standard Deviation:
Aggressive =   %
Passive =   %

To solve this problem, we will proceed step by step. The steps to compute the expected return, variance, and standard deviation are outlined below.

### a) **Compute the Expected Return for Both Funds**

The expected return (\(E[R]\)) of a mutual fund is the average of its yearly returns. This can be calculated using the formula:

\[
E[R] = \frac{1}{n} \sum_{i=1}^{n} R_i
\]

Where:
- \(R_i\) is the return in year \(i\),
- \(n\) is the number of years (in this case, 10).

Let's calculate the expected return for both funds:

#### Aggressive Fund:

\[
E[R_{Aggressive}] = \frac{1}{10} \times (0 + 7 + 0 + 1 + 7 + 2 + 7 + 2 + 1 + 6) = \frac{1}{10} \times 33 = 3.30\%
\]

#### Passive Fund:

\[
E[R_{Passive}] = \frac{1}{10} \times (4 + 4 + 4 + 2 + 2 + 3 + 2 + 3 + 3 + 2) = \frac{1}{10} \times 29 = 2.90\%
\]

So, the expected returns are:
- **Aggressive Fund:** 3.30%
- **Passive Fund:** 2.90%

### b) **Compute the Variance and Standard Deviation**

#### Variance Calculation:

The variance (\(Var[R]\)) of returns is calculated using the formula:

\[
Var[R] = \frac{1}{n} \sum_{i=1}^{n} (R_i - E[R])^2
\]

Let's calculate the variance for both funds.

#### Aggressive Fund Variance:
First, we compute the squared deviations from the expected return (\(E[R_{Aggressive}] = 3.30\%\)):

\[
\begin{aligned}
(0 - 3.30)^2 &= 10.89 \\
(7 - 3.30)^2 &= 13.69 \\
(0 - 3.30)^2 &= 10.89 \\
(1 - 3.30)^2 &= 5.29 \\
(7 - 3.30)^2 &= 13.69 \\
(2 - 3.30)^2 &= 1.69 \\
(7 - 3.30)^2 &= 13.69 \\
(2 - 3.30)^2 &= 1.69 \\
(1 - 3.30)^2 &= 5.29 \\
(6 - 3.30)^2 &= 7.29 \\
\end{aligned}
\]

Now, sum up these squared deviations:

\[
\text{Total} = 10.89 + 13.69 + 10.89 + 5.29 + 13.69 + 1.69 + 13.69 + 1.69 + 5.29 + 7.29 = 83.10
\]

Now, compute the variance:

\[
Var[R_{Aggressive}] = \frac{83.10}{10} = 8.31
\]

#### Passive Fund Variance:
Now, for the passive fund, we compute the squared deviations from \(E[R_{Passive}] = 2.90\%\):

\[
\begin{aligned}
(4 - 2.90)^2 &= 1.21 \\
(4 - 2.90)^2 &= 1.21 \\
(4 - 2.90)^2 &= 1.21 \\
(2 - 2.90)^2 &= 0.81 \\
(2 - 2.90)^2 &= 0.81 \\
(3 - 2.90)^2 &= 0.01 \\
(2 - 2.90)^2 &= 0.81 \\
(3 - 2.90)^2 &= 0.01 \\
(3 - 2.90)^2 &= 0.01 \\
(2 - 2.90)^2 &= 0.81 \\
\end{aligned}
\]

Now, sum up these squared deviations:

\[
\text{Total} = 1.21 + 1.21 + 1.21 + 0.81 + 0.81 + 0.01 + 0.81 + 0.01 + 0.01 + 0.81 = 7.90
\]

Now, compute the variance:

\[
Var[R_{Passive}] = \frac{7.90}{10} = 0.79
\]

#### Standard Deviation Calculation:

The standard deviation (\(\sigma\)) is simply the square root of the variance:

\[
\sigma = \sqrt{Var[R]}
\]

So, for both funds:

\[
\sigma_{Aggressive} = \sqrt{8.31} \approx 2.88\%
\]

\[
\sigma_{Passive} = \sqrt{0.79} \approx 0.89\%
\]

### Final Answers:

#### a) Expected Returns:
- Aggressive Fund = 3.30%
- Passive Fund = 2.90%

#### b) Variance:
- Aggressive Fund = 8.31
- Passive Fund = 0.79

#### Standard Deviation:
- Aggressive Fund = 2.88%
- Passive Fund = 0.89%

Would you like further details or clarifications? 

Here are 5 related questions to expand your understanding:
1. How does higher variance affect the risk level of a mutual fund?
2. What is the difference between expected return and actual return over a period?
3. Can standard deviation alone fully describe a fund’s risk?
4. How would a different sample size (more years) affect the expected return?
5. What is the significance of using past returns to predict future returns?

**Tip:** The standard deviation gives you an idea of the volatility or risk; the higher it is, the more the returns deviate from the average.

During the past 10 years, the percent returns on two mutual funds (aggressive and passive) expressed in percentages were as follows: Year: Aggressive Fund, Passive Fund -10: 0%, 4%, -9: 7%, 4%, -8: 0%, 4%, -7: 1%, 2%, -6: 7%, 2%, -5: 2%, 3%, -4: 7%, 2%, -3: 2%, 3%, -2: 1%, 3%, Last Year: 6%, 2%. Compute the expected return for the two funds. Round your answers to two decimal places. Aggressive = Passive =. Compute the variance and standard deviation of the returns of the two funds. Round your answers to two decimal places.

Learn how to calculate the expected return, variance, and standard deviation for two mutual funds (Aggressive and Passive) over a 10-year period. This problem uses key statistical concepts, including expected return, variance, and standard deviation, with detailed formulas and step-by-step solutions. Suitable for college-level finance and statistics students.

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