Math Problem Statement

A stock had returns of 17.98 percent, −5.22 percent, and 20.45 percent for the past three years. What is the variance of the returns?

Multiple Choice

.14162

.02005

.01604

.00850

.02674

Solution

To calculate the variance of the returns, we need to follow these steps:

  1. Calculate the Mean (Average) Return: Mean=17.98+(5.22)+20.453=33.213=11.07%\text{Mean} = \frac{17.98 + (-5.22) + 20.45}{3} = \frac{33.21}{3} = 11.07\%

  2. Calculate Each Return's Deviation from the Mean and Square it:

    • For 17.98%17.98\%: (17.9811.07)2=6.912=47.7681(17.98 - 11.07)^2 = 6.91^2 = 47.7681
    • For 5.22%-5.22\%: (5.2211.07)2=16.292=265.0641(-5.22 - 11.07)^2 = -16.29^2 = 265.0641
    • For 20.45%20.45\%: (20.4511.07)2=9.382=87.9844(20.45 - 11.07)^2 = 9.38^2 = 87.9844
  3. Calculate the Variance: Variance=47.7681+265.0641+87.98443=400.81663=133.6055\text{Variance} = \frac{47.7681 + 265.0641 + 87.9844}{3} = \frac{400.8166}{3} = 133.6055

  4. Convert Variance to Decimal Form (if the options require it): Since the original returns are given as percentages, the variance here represents 13.36055%13.36055\%. Converting it to decimal form by dividing by 100100: Variance in decimal=0.02005\text{Variance in decimal} = 0.02005

The correct answer is: 0.02005

Would you like a deeper breakdown of these calculations or further explanation on variance?

Here are five related questions for further exploration:

  1. How would the variance change if another year's return is included?
  2. What is the standard deviation of the returns for these three years?
  3. How is variance interpreted in the context of financial risk?
  4. How does variance differ when we use population versus sample variance?
  5. What is the relationship between variance and the coefficient of variation?

Tip: Variance measures the spread of data points. In finance, a higher variance indicates greater risk or volatility in returns.

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Math Problem Analysis

Mathematical Concepts

Statistics
Variance
Mean
Deviation
Square of Deviation

Formulas

Variance formula: \text{Variance} = \frac{1}{n} \sum_{i=1}^{n} (x_i - \text{Mean})^2

Theorems

Variance theorem
Mean and Deviation theorem

Suitable Grade Level

Grades 9-12