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.