To calculate the compound annual return (CAR), we follow these steps:

Step 1: Arithmetic Average Annual Return Formula

The arithmetic average annual return is given as: $\text{Arithmetic Average Return} = \frac{\text{Sum of yearly returns}}{\text{Number of years}}$ Given this is $-0.62\%$, we use it to find the missing return, $X$.

Step 2: Solve for $X$

Let the yearly returns be $R_1 = 20.03\%$, $R_2 = 23.89\%$, $R_3 = X$, $R_4 = -18.63\%$. Then: $\frac{20.03 + 23.89 + X - 18.63}{4} = -0.62$ $20.03 + 23.89 + X - 18.63 = -2.48$ $X = -2.48 - (20.03 + 23.89 - 18.63)$ $X = -2.48 - 25.29 = -27.77\%$

Step 3: Compound Annual Return Formula

The compound annual return (CAR) is calculated using: $\text{CAR} = \left( (1 + R_1)(1 + R_2)(1 + R_3)(1 + R_4) \right)^{\frac{1}{4}} - 1$ Convert percentages to decimals: $R_1 = 0.2003, \quad R_2 = 0.2389, \quad R_3 = -0.2777, \quad R_4 = -0.1863$

Substitute these values into the formula: $\text{CAR} = \left( (1 + 0.2003)(1 + 0.2389)(1 - 0.2777)(1 - 0.1863) \right)^{\frac{1}{4}} - 1$ $\text{CAR} = \left( 1.2003 \times 1.2389 \times 0.7223 \times 0.8137 \right)^{\frac{1}{4}} - 1$

Step 4: Compute the Result

First, calculate the product: $1.2003 \times 1.2389 \times 0.7223 \times 0.8137 = 0.8784$ Now, take the fourth root: $0.8784^{\frac{1}{4}} = 0.9679$ Finally, subtract 1 to find CAR: $\text{CAR} = 0.9679 - 1 = -0.0321$ Convert to a percentage: $\text{CAR} = -3.21\%$

Final Answer:

$\boxed{-3.21}$

Do you have any questions or need further clarification?

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Related Questions:

1. How does compound annual return differ from arithmetic average return?
2. What are the limitations of using arithmetic average for returns?
3. Can compound annual return be negative while arithmetic average is positive?
4. What is the significance of using percentage change rather than raw values?
5. How can geometric mean help in financial calculations?

Tip: Compound annual return accounts for the compounding effect, making it more accurate for multi-year financial performance evaluation.