Step 1: Understand the problem

We need to calculate two things:

1. The missing return for 3 years ago ($X$).
2. The arithmetic average annual return over the past 4 years.

Step 2: Definitions

1. Compound Return Formula: $(1 + \text{CR}) = \sqrt[4]{(1 + R_1)(1 + R_2)(1 + R_3)(1 + R_4)}$ where:

   • $R_1, R_2, R_3, R_4$ are the annual returns as decimals.
   • CR is the compound return as a decimal.

2. Arithmetic Average Annual Return: $\text{Arithmetic Average} = \frac{R_1 + R_2 + R_3 + R_4}{4}$

Step 3: Known values

• $R_1 = 16.51\% = 0.1651$
• $R_2 = 20.35\% = 0.2035$
• $R_4 = -22.15\% = -0.2215$
• Compound return ($CR$) = $1.83\% = 0.0183$.

Step 4: Solve for $X$ (Return 3 years ago)

Rearrange the compound return formula: $(1 + 0.0183)^4 = (1 + 0.1651)(1 + 0.2035)(1 + X)(1 - 0.2215)$

Simplify: $1.0746 = 1.1651 \times 1.2035 \times (1 + X) \times 0.7785$

Calculate the product of the known terms: $1.1651 \times 1.2035 \times 0.7785 = 1.0908$

Divide both sides by 1.0908 to isolate $1 + X$: $\frac{1.0746}{1.0908} = 1 + X$

$1 + X = 0.9851$

Solve for $X$: $X = 0.9851 - 1 = -0.0149 = -1.49\%$

Step 5: Calculate the Arithmetic Average Annual Return

Using the formula: $\text{Arithmetic Average} = \frac{R_1 + R_2 + R_3 + R_4}{4}$

Substitute: $\text{Arithmetic Average} = \frac{0.1651 + 0.2035 - 0.0149 - 0.2215}{4}$

Simplify: $\text{Arithmetic Average} = \frac{0.1322}{4} = 0.03305 = 3.31\%$

Final Answer:

The arithmetic average annual return is 3.31.

---

Let me know if you'd like further clarification or details!
Here are some related questions to expand understanding:

1. How does compound return differ from arithmetic average return?
2. What scenarios favor using the compound return instead of the arithmetic return?
3. How do negative returns impact compound returns versus arithmetic averages?
4. What is the significance of the time horizon in evaluating stock returns?
5. How does volatility affect the difference between compound and arithmetic returns?

Tip: The arithmetic average is often higher than the compound return when returns are volatile due to the geometric mean's sensitivity to variability.