An investment has probabilities 0.1, 0.2, 0.35, 0.25, and 0.1 of giving returns
equal to 40%, 30%, 15%, −5%, and −15%. What is the expected return and the
standard deviation of returns? show solution

To solve for the expected return and standard deviation of the returns for the given investment, we'll use the following steps:

### Step 1: Calculate the Expected Return
The expected return (\(E(R)\)) is calculated by multiplying each possible return by its corresponding probability and then summing up all these products.

Given:
- Probabilities: \(P_1 = 0.1\), \(P_2 = 0.2\), \(P_3 = 0.35\), \(P_4 = 0.25\), \(P_5 = 0.1\)
- Returns: \(R_1 = 40\%\), \(R_2 = 30\%\), \(R_3 = 15\%\), \(R_4 = -5\%\), \(R_5 = -15\%\)

The formula for the expected return is:

\[
E(R) = \sum_{i=1}^{n} P_i \times R_i
\]

Substituting the values:

\[
E(R) = (0.1 \times 40) + (0.2 \times 30) + (0.35 \times 15) + (0.25 \times -5) + (0.1 \times -15)
\]

Let's calculate this:

\[
E(R) = 4 + 6 + 5.25 - 1.25 - 1.5 = 12.5\%
\]

### Step 2: Calculate the Variance and Standard Deviation
Next, we calculate the variance (\(\sigma^2\)) using the formula:

\[
\sigma^2 = \sum_{i=1}^{n} P_i \times (R_i - E(R))^2
\]

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

First, we calculate each squared deviation from the expected return, multiply by the corresponding probability, and then sum:

\[
\sigma^2 = 0.1 \times (40 - 12.5)^2 + 0.2 \times (30 - 12.5)^2 + 0.35 \times (15 - 12.5)^2 + 0.25 \times (-5 - 12.5)^2 + 0.1 \times (-15 - 12.5)^2
\]

Let's calculate each term:

\[
\begin{aligned}
&\text{Term 1: } & 0.1 \times (27.5)^2 &= 0.1 \times 756.25 = 75.625 \\
&\text{Term 2: } & 0.2 \times (17.5)^2 &= 0.2 \times 306.25 = 61.25 \\
&\text{Term 3: } & 0.35 \times (2.5)^2 &= 0.35 \times 6.25 = 2.1875 \\
&\text{Term 4: } & 0.25 \times (-17.5)^2 &= 0.25 \times 306.25 = 76.5625 \\
&\text{Term 5: } & 0.1 \times (-27.5)^2 &= 0.1 \times 756.25 = 75.625 \\
\end{aligned}
\]

Summing these:

\[
\sigma^2 = 75.625 + 61.25 + 2.1875 + 76.5625 + 75.625 = 291.25
\]

Finally, the standard deviation:

\[
\sigma = \sqrt{291.25} \approx 17.07\%
\]

### Final Results:
- **Expected Return**: 12.5%
- **Standard Deviation**: 17.07%

Would you like any further details on this solution? Here are some related questions:

1. How would changing the probabilities affect the expected return?
2. What does the standard deviation tell us about the risk of the investment?
3. How would an increase in the negative returns impact the overall standard deviation?
4. What is the significance of having a high expected return with a high standard deviation?
5. How would you calculate the expected return for a portfolio of multiple investments?

**Tip:** The standard deviation is a key measure of risk in investments; the higher it is, the more variability (and hence, risk) you have in your returns.

Learn how to calculate the expected return and standard deviation of investment returns with a detailed step-by-step solution. This problem involves calculating probabilities and corresponding returns to find the expected return, then using variance and standard deviation formulas to assess risk. Suitable for advanced high school students and beyond.

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