A​ company's stock price rose 

4.24.2​% 

in​ 2011, and in​ 2012, it increased 

74.374.3​%.

a. nbspa. 

Compute the geometric mean rate of return for the​ two-year period 

2011minus−2012. 

​(Hint: Denote an increase of 

74.374.3​% 

by 

0.7430.743​.)

b. nbspb. 

If someone purchased​ $1,000 of the​ company's stock at the start of​ 2011, what was its value at the end of​ 2012?

c. nbspc. 

Over the same​ period, another company had a geometric mean rate of return of 

9.79.7​%. 

If someone purchased​ $1,000 of the other​ company's stock, how would its value compare to the value found in part​ (b)?

Let's go through each part of the problem step by step.

### Part (a): Geometric Mean Rate of Return

The formula for the geometric mean rate of return $ R $ over $ n $ periods is:

\[
R = \left( \prod_{i=1}^{n} (1 + r_i) \right)^{\frac{1}{n}} - 1
\]

Here, $ r_1 = 0.042 $ and $ r_2 = 0.743 $.

\[
R = \left( (1 + 0.042) \times (1 + 0.743) \right)^{\frac{1}{2}} - 1
\]

Calculating:

1. $ 1 + 0.042 = 1.042 $
2. $ 1 + 0.743 = 1.743 $
3. $ 1.042 \times 1.743 = 1.816206 $
4. $ \sqrt{1.816206} \approx 1.3475 $
5. $ R \approx 1.3475 - 1 = 0.3475 $ or **34.75%**

So, the geometric mean rate of return over the two-year period is approximately **34.75%**.

### Part (b): Value of the Stock at the End of 2012

The value of the investment over two years is calculated as follows:

\[
\text{Final Value} = \text{Initial Value} \times (1 + r_1) \times (1 + r_2)
\]

Given $ \text{Initial Value} = 1000 $, $ r_1 = 0.042 $, and $ r_2 = 0.743 $:

\[
\text{Final Value} = 1000 \times 1.042 \times 1.743
\]

Calculating:

1. $ 1000 \times 1.042 = 1042 $
2. $ 1042 \times 1.743 \approx 1816.21 $

So, the value of the stock at the end of 2012 is approximately **$1,816.21**.

### Part (c): Comparison with Another Company's Investment

If the other company has a geometric mean rate of return of $ 9.7\% $ over the same period, we can calculate the final value as follows:

\[
\text{Final Value} = 1000 \times (1 + R)
\]

Where $ R = 0.097 $:

\[
\text{Final Value} = 1000 \times (1 + 0.097)^2
\]

Calculating:

1. $ 1 + 0.097 = 1.097 $
2. $ 1.097^2 \approx 1.2049 $
3. $ 1000 \times 1.2049 \approx 1204.90 $

So, the final value of the other company’s stock is approximately **$1,204.90**.

### Summary of Answers
- **(a)** The geometric mean rate of return over the two-year period is approximately **34.75%**.
- **(b)** The value of the stock at the end of 2012 is approximately **$1,816.21**.
- **(c)** The value of the other company’s stock would be **$1,204.90**, which is lower than the value found in part (b) by approximately **$611.31**.

Would you like more details on any of these parts, or have any further questions?

#### Related Questions:
1. How is the geometric mean rate of return different from the arithmetic mean rate of return?
2. Can the geometric mean rate of return be used to measure risk in investment returns?
3. What are the limitations of using the geometric mean for evaluating investment performance?
4. How would the results change if the returns were compounded quarterly?
5. What is the effect of negative returns on the geometric mean rate of return?

#### Tip:
The geometric mean is generally preferred for returns over multiple periods as it takes into account the compounding effect.

A​ company's stock price rose 4.2% in​ 2011, and in​ 2012, it increased 74.3%. a. Compute the geometric mean rate of return for the​ two-year period 2011−2012. (Hint: Denote an increase of 74.3​% by 0.743​.) b. If someone purchased​ $1,000 of the​ company's stock at the start of​ 2011, what was its value at the end of​ 2012? c. Over the same​ period, another company had a geometric mean rate of return of 9.7​%. If someone purchased​ $1,000 of the other​ company's stock, how would its value compare to the value found in part​ (b)?

Learn how to calculate the geometric mean rate of return over a two-year period for a company's stock that grew by 4.2% in 2011 and 74.3% in 2012. This detailed solution covers the formula for geometric mean, investment growth, and compares it with another company's stock return of 9.7%. Suitable for finance or business students.

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