"""Create an educational video to explain the CFA Level 1 knowledge: Portfolio return statistics (mean/variance) 🎓 Content Requirements: Start with a clear, beginner-friendly definition of the concept Explain the core components and logic step by step Include simple numerical examples or visual analogies Add a short summary or key takeaways at the end Ensure the structure follows a logical teaching flow from concept to application 🎨 Visual and Layout Requirements: Full-screen visuals with centered, readable content Use smooth animations to transition between steps or sections Highlight important terms, formulas, and keywords with bright accent colors (e.g., yellow, red, blue) Avoid text crowding or overlap; leave clear visual spacing Use animated icons, graphs, or diagrams where appropriate (e.g., timelines, flowcharts, charts) Minimize blank space; keep each screen visually rich and balanced 🗣️ Tone and Style: Friendly, clear, and professional Focus on making the topic accessible for first-time learners Avoid excessive jargon; use plain language wherever possible Maintain alignment with CFA curriculum terminology and scope"""

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Investing involves risk and return, but how do we measure them for a portfolio? When you invest in multiple assets, understanding portfolio return statistics becomes crucial. Today we'll explore portfolio return, measured by the mean, and portfolio risk, measured by variance. These are fundamental concepts in investment decision-making and essential knowledge for CFA Level 1 candidates. Portfolio return is the weighted average of the returns of individual assets within the portfolio. Think of it as a blend where each asset contributes to the overall return based on its proportion in the portfolio. The key components are weights, which represent the proportion of total portfolio value invested in each asset, and expected returns of individual assets. The formula shows that portfolio expected return equals the sum of each weight multiplied by its corresponding expected return. This weighted average approach ensures that larger positions have greater influence on overall portfolio performance. Portfolio risk, measured by variance, represents the dispersion or volatility of returns around the expected return. Variance serves as our proxy for investment risk. Here's the crucial insight: portfolio risk is NOT simply the weighted average of individual asset risks. This is fundamentally different from how we calculate portfolio return. The reason is diversification. When assets don't move perfectly together, combining them can actually reduce overall portfolio risk. The key factor that makes this possible is covariance, which measures how assets move in relation to each other. This relationship between assets is what creates the diversification benefit. Now let's examine the portfolio variance formula for a two-asset portfolio. The formula has three components: first, the squared weights multiplied by individual variances, and second, the crucial covariance term. This covariance term, multiplied by two times the weights, captures how the two assets move together. Covariance can be expressed as correlation multiplied by the standard deviations of both assets. When correlation is positive, assets move together, increasing portfolio risk. When correlation is zero, assets move independently. When correlation is negative, assets move in opposite directions, providing maximum diversification benefit and reducing portfolio risk. Let's summarize the key takeaways about portfolio return statistics. First, portfolio return is simply the weighted average of individual asset returns. Second, portfolio risk is NOT a weighted average due to diversification benefits. Third, covariance and correlation are crucial for calculating portfolio risk accurately. Fourth, these formulas are essential tools for investment decision-making and fundamental knowledge for CFA Level 1 candidates. Remember to practice with numerical examples to master these concepts. Understanding portfolio return statistics is the foundation for more advanced portfolio management topics. Thank you for learning with us today!