Solution

Given Data:

Monthly Returns:
$r_1 = 3\%$
$r_2 = -1\%$
$r_3 = 2\%$
$r_4 = 6\%$
$r_5 = -4\%$

These values will be expressed in decimal form where necessary, i.e., $r = \frac{\text{percentage}}{100}$.

---

Part (a): Arithmetic Average Monthly Return

The arithmetic average return is given by: $R_a = \frac{\sum r_i}{n}$

Substituting the values: $R_a = \frac{3 + (-1) + 2 + 6 + (-4)}{5}$ $R_a = \frac{6}{5} = 1.2\%$

---

Part (b): Geometric Average Monthly Return

The geometric average return is given by: $R_g = \left( \prod_{i=1}^n (1 + r_i) \right)^{\frac{1}{n}} - 1$

Convert percentages to decimals: $3\% = 0.03, -1\% = -0.01,$ etc.

$R_g = \left( (1 + 0.03)(1 - 0.01)(1 + 0.02)(1 + 0.06)(1 - 0.04) \right)^{\frac{1}{5}} - 1$

Simplify: $R_g = \left( 1.03 \cdot 0.99 \cdot 1.02 \cdot 1.06 \cdot 0.96 \right)^{\frac{1}{5}} - 1$

Calculate: $R_g = \left( 1.058726 \right)^{\frac{1}{5}} - 1$ $R_g \approx 1.01149 - 1 = 0.01149 \text{ or } 1.15\%$

---

Part (c): Monthly Variance

The variance is given by: $\text{Variance} = \frac{\sum (r_i - R_a)^2}{n}$

Substitute $R_a = 1.2\% = 0.012$: $\text{Variance} = \frac{(0.03 - 0.012)^2 + (-0.01 - 0.012)^2 + (0.02 - 0.012)^2 + (0.06 - 0.012)^2 + (-0.04 - 0.012)^2}{5}$

Simplify: $\text{Variance} = \frac{(0.018)^2 + (-0.022)^2 + (0.008)^2 + (0.048)^2 + (-0.052)^2}{5}$ $\text{Variance} = \frac{0.000324 + 0.000484 + 0.000064 + 0.002304 + 0.002704}{5}$ $\text{Variance} = \frac{0.00588}{5} = 0.001176$

---

Part (d): Monthly Standard Deviation

The standard deviation is the square root of the variance: $\text{Standard Deviation} = \sqrt{\text{Variance}} = \sqrt{0.001176}$ $\text{Standard Deviation} \approx 0.0343 \text{ or } 3.43\%$

---

Final Results:

(a) Arithmetic Average Monthly Return: $1.2\%$
(b) Geometric Average Monthly Return: $1.15\%$
(c) Monthly Variance: $0.001176$
(d) Monthly Standard Deviation: $3.43\%$

---

Would you like detailed calculations for any step?

Here are 5 related questions:

1. How does the arithmetic average differ from the geometric average for these returns?
2. What would the results look like if there was an additional return added?
3. How is variance affected by increasing the number of data points?
4. What are the practical implications of using standard deviation in finance?
5. Can negative returns affect the geometric average more significantly?

Tip: Always double-check the decimal conversion when working with percentages