Let's proceed step by step as you requested: --- ### **Step 1: Introduction of the Concept** In finance, the **return** of an asset measures its gain or loss over a period. The **volatility** of returns, usually measured by the standard deviation, tells us how much the returns fluctuate. The **covariance** shows how two assets move together, and the **correlation coefficient** standardizes this to a value between -1 and 1, showing the strength and direction of their relationship. --- ### **Step 2: Calculate the Volatility (Standard Deviation) of IBM's Returns** #### **IBM Returns:** - 2019: 4.5% - 2020: -27.3% - 2021: 2.5% 1. **Mean of IBM Returns:** \[ \text{Mean} = \frac{4.5 + (-27.3) + 2.5}{3} = \frac{-20.3}{3} = -6.77\% \] 2. **Variance:** \[ \text{Variance} = \frac{(4.5 + 6.77)^2 + (-27.3 + 6.77)^2 + (2.5 + 6.77)^2}{3} \] \[ = \frac{(11.27)^2 + (-20.53)^2 + (9.27)^2}{3} = \frac{127.03 + 421.52 + 85.94}{3} = \frac{634.49}{3} = 211.5 \] (All numbers are in percent squared) 3. **Standard Deviation (Volatility):** \[ \text{Volatility} = \sqrt{211.5} = 14.54\% \] --- ### **Step 3: Calculate the Volatility of Home Depot's Returns** #### **Home Depot Returns:** - 2019: 17.5% - 2020: -5.5% - 2021: 31.5% 1. **Mean of Home Depot Returns:** \[ \text{Mean} = \frac{17.5 + (-5.5) + 31.5}{3} = \frac{43.5}{3} = 14.5\% \] 2. **Variance:** \[ \text{Variance} = \frac{(17.5 - 14.5)^2 + (-5.5 - 14.5)^2 + (31.5 - 14.5)^2}{3} \] \[ = \frac{(3)^2 + (-20)^2 + (17)^2}{3} = \frac{9 + 400 + 289}{3} = \frac{698}{3} = 232.67 \] 3. **Standard Deviation (Volatility):** \[ \text{Volatility} = \sqrt{232.67} = 15.25\% \] --- ### **Step 4: Calculate the Covariance between IBM and Home Depot Returns** #### **Deviations from Means:** | Year | IBM | IBM Dev. | HD | HD Dev. | Product | |------|-----|----------|----|---------|---------| | 2019 | 4.5 | 11.27 | 17.5 | 3 | 33.81 | | 2020 | -27.3| -20.53 | -5.5| -20 | 410.60 | | 2021 | 2.5 | 9.27 | 31.5 | 17 | 157.59 | Sum of products: 33.81 + 410.60 + 157.59 = 602 Covariance: \[ \text{Covariance} = \frac{602}{3} = 200.67 \] --- ### **Step 5: Calculate the Correlation Coefficient** \[ \text{Correlation} = \frac{\text{Covariance}}{\text{(Std. Dev. IBM)} \times \text{(Std. Dev. HD)}} \] \[ = \frac{200.67}{14.54 \times 15.25} = \frac{200.67}{221.035} = 0.91 \] --- ### **Step 6: Final Answer in Few Sentences** - **IBM's volatility:** 14.54% - **Home Depot's volatility:** 15.25% - **Covariance:** 200.67 - **Correlation coefficient:** 0.91 --- ### **Step 7: Summary of Steps** 1. Introduced concepts of return, volatility, covariance, and correlation. 2. Calculated IBM's volatility (standard deviation). 3. Calculated Home Depot's volatility. 4. Found the covariance between IBM and Home Depot returns. 5. Computed the correlation coefficient. 6. Summarized all final answers in a concise format. Let me know if you need any clarifications or further details!