To solve the problem, we need to calculate two things:

1. The arithmetic average return over the 4 years.
2. The unknown return for the third year ($X$) using the given compound annual growth rate (CAGR).

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Definitions:

• Arithmetic Average Return: This is the simple average of the annual returns: $\text{Arithmetic Average} = \frac{\text{Sum of annual returns}}{\text{Number of years}}$

• Compound Annual Growth Rate (CAGR): This is given by: $(1 + \text{CAGR})^4 = (1 + R_1)(1 + R_2)(1 + R_3)(1 + R_4)$ where $R_1, R_2, R_3, R_4$ are the annual returns as decimals (e.g., 17.15% = 0.1715).

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Step 1: Set up the equation for the CAGR

The compound return is given as $1.03\% = 0.0103$. Using the formula: $(1 + 0.0103)^4 = (1 + 0.1715)(1 + 0.2099)(1 + X)(1 - 0.2151)$

Simplify: $1.0103^4 = 1.1715 \times 1.2099 \times (1 + X) \times 0.7849$

Calculate the left-hand side ($1.0103^4$): $1.0103^4 \approx 1.0416$

Substitute into the equation: $1.0416 = 1.1715 \times 1.2099 \times 0.7849 \times (1 + X)$

Calculate the product of the first three terms on the right-hand side: $1.1715 \times 1.2099 \times 0.7849 \approx 1.1107$

So: $1.0416 = 1.1107 \times (1 + X)$

Solve for $1 + X$: $1 + X = \frac{1.0416}{1.1107} \approx 0.9378$

Subtract 1 to find $X$: X = 0.9378 - 1 = -0.0622 \text{ (or -6.22%)}

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Step 2: Calculate the arithmetic average

Now, calculate the arithmetic average of the returns over the 4 years: $\text{Arithmetic Average} = \frac{17.15\% + 20.99\% - 6.22\% - 21.51\%}{4}$

Convert percentages to decimals: $\text{Arithmetic Average} = \frac{0.1715 + 0.2099 - 0.0622 - 0.2151}{4}$

Simplify the numerator: $0.1715 + 0.2099 - 0.0622 - 0.2151 = 0.1041$

Divide by 4: \text{Arithmetic Average} = \frac{0.1041}{4} \approx 0.0260 \text{ (or 2.60%)}

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Final Answers:

1. The arithmetic average annual return is 2.60%.
2. The unknown return for the third year ($X$) is -6.22%.

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Do you have any questions, or would you like a step-by-step breakdown of the calculations?

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Follow-up Questions:

1. What is the difference between the compound annual growth rate and the arithmetic average return?
2. How does a negative return affect both the CAGR and arithmetic average?
3. Why does the compound return account for compounding effects while the arithmetic average does not?
4. Can the arithmetic average be higher than the CAGR? Why?
5. What is the impact of volatility on these calculations?

Tip:

Always double-check compounding effects when calculating long-term returns, as they often lead to significant differences from simple averages.