(Expected **rate of return and**​ risk) ​ Syntex, Inc. is considering an investment in one of two common stocks.  Given the information that​ follows, which investment is​ better, based on the risk​ (as measured by the standard​ deviation) and​ return?

**Common Stock A**             

**Common Stock B**             

 **Probability**

**Return**

**Probability**

**Return**

 0.350.35

1212​%

0.150.15

negative 4−4​%

 0.300.30

1616​%

0.350.35

88​%

 0.350.35

2121​%

0.350.35

1313​%

0.150.15

2121​%

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Part 1

**a.**  Given the information in the​ table, the expected rate of return for stock A is 

16.3516.35​%. 

 ​(Round to two decimal​ places.)

Part 2

The standard deviation of stock A is 

3.773.77​%. 

 ​(Round to two decimal​ places.)

Part 3

**b.**  The expected rate of return for stock B is 

enter your response here​%. 

 ​(Round to two decimal​ places.)

To calculate the **expected rate of return** and **standard deviation** for Common Stock B, we can use the following steps. These steps involve calculating the weighted average return (expected rate of return) and standard deviation based on the probabilities provided.

### 1. **Expected Rate of Return (Stock B):**

The expected return is calculated as the sum of the probability-weighted returns.

The formula for the expected return \( E(R) \) is:

\[
E(R) = \sum (P_i \times R_i)
\]

Where:
- \( P_i \) is the probability of each return
- \( R_i \) is the return

For **Common Stock B**, we have:
- \( P_1 = 0.15 \), \( R_1 = -4\% \)
- \( P_2 = 0.35 \), \( R_2 = 8\% \)
- \( P_3 = 0.35 \), \( R_3 = 13\% \)
- \( P_4 = 0.15 \), \( R_4 = 21\% \)

Let's plug in these values:

\[
E(R) = (0.15 \times (-4)) + (0.35 \times 8) + (0.35 \times 13) + (0.15 \times 21)
\]

Calculating each term:
\[
E(R) = (0.15 \times -4) + (0.35 \times 8) + (0.35 \times 13) + (0.15 \times 21)
\]
\[
E(R) = -0.60 + 2.8 + 4.55 + 3.15 = 9.9\%
\]

So, the **expected rate of return** for Stock B is **9.90%**.

### 2. **Standard Deviation (Stock B):**

To calculate the standard deviation, we need to compute the variance first. The formula for variance is:

\[
\sigma^2 = \sum P_i \times (R_i - E(R))^2
\]

Where:
- \( P_i \) is the probability of each return
- \( R_i \) is the return
- \( E(R) \) is the expected return we calculated above

Let's calculate each term \( (R_i - E(R))^2 \) and multiply by the respective probabilities:

For \( R_1 = -4\% \):
\[
P_1 \times (R_1 - 9.9)^2 = 0.15 \times (-4 - 9.9)^2 = 0.15 \times (-13.9)^2 = 0.15 \times 193.21 = 28.9815
\]

For \( R_2 = 8\% \):
\[
P_2 \times (R_2 - 9.9)^2 = 0.35 \times (8 - 9.9)^2 = 0.35 \times (-1.9)^2 = 0.35 \times 3.61 = 1.2635
\]

For \( R_3 = 13\% \):
\[
P_3 \times (R_3 - 9.9)^2 = 0.35 \times (13 - 9.9)^2 = 0.35 \times 3.1^2 = 0.35 \times 9.61 = 3.3635
\]

For \( R_4 = 21\% \):
\[
P_4 \times (R_4 - 9.9)^2 = 0.15 \times (21 - 9.9)^2 = 0.15 \times 11.1^2 = 0.15 \times 123.21 = 18.4815
\]

Now, sum up these values to get the variance:
\[
\sigma^2 = 28.9815 + 1.2635 + 3.3635 + 18.4815 = 52.09
\]

To find the standard deviation, take the square root of the variance:

\[
\sigma = \sqrt{52.09} \approx 7.22\%
\]

So, the **standard deviation** for Stock B is approximately **7.22%**.

### Summary:
- The expected rate of return for Stock B is **9.90%**.
- The standard deviation (risk) for Stock B is **7.22%**.

Let me know if you would like further details or have any questions!

Here are 5 related questions:
1. How is the risk-return tradeoff evaluated in stock investments?
2. What does the standard deviation of returns indicate about investment risk?
3. How does expected return influence investment decisions?
4. How do we interpret negative returns in the context of risk and reward?
5. What are other measures of risk besides standard deviation?

**Tip:** When choosing between investments, higher returns usually come with higher risk. Evaluate your risk tolerance when making decisions.

Syntex, Inc. is considering an investment in one of two common stocks. Given the information that follows, which investment is better, based on the risk (as measured by the standard deviation) and return?

Common Stock A

Common Stock B

Probability Return Probability Return
0.35 12% 0.15 -4%
0.30 16% 0.35 8%
0.35 21% 0.35 13%
0.15 21%

The expected rate of return for stock A is 16.35%.
The standard deviation of stock A is 3.77%.

What is the expected rate of return and standard deviation for Stock B?

This page provides a step-by-step solution for calculating the expected rate of return (9.90%) and standard deviation (7.22%) for Common Stock B using probability-weighted returns. This problem covers key finance concepts, such as expected return and risk evaluation, suitable for college-level or advanced high school students.

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