**Subjects**:Portfolio Management **Module**:Portfolio Risk and Return: Part I **Knowledge Points**:Minimum-variance frontier and efficient frontier of risky assets **Subjects**:Portfolio Management **Module**:Portfolio Theory **Knowledge Points**:Minimum-Variance Frontier and Efficient Frontier --- ### **Minimum-Variance Frontier and Efficient Frontier of Risky Assets** --- **1. Minimum-Variance Frontier** **Definition**: The minimum-variance frontier represents the set of portfolios that minimize risk (variance) for a given level of expected return. It is derived from the mean-variance optimization framework and shows the lowest possible risk (standard deviation) for each level of return achievable with a combination of risky assets. - **Characteristics**: - Includes all portfolios that provide the lowest risk for each level of expected return. - The curve typically bulges to the left, indicating that as we include more assets with varying correlations, portfolios can achieve reduced risk compared to a weighted average of the individual asset risks. - The global minimum-variance portfolio is the point on this frontier that has the lowest risk of all possible portfolios. **Visual Representation**: The minimum-variance frontier can be graphically represented in a risk-return space, with risk on the x-axis and expected return on the y-axis. --- **2. Efficient Frontier** **Definition**: The efficient frontier frontier is the curve that lies above and to the right of the global minimum-variance portfolio on the minimum-variance frontier. It represents the set of portfolios that offer the highest expected return for a given level of risk or the lowest risk for a given level of expected return. - **Characteristics**: - Only includes efficient portfolios that rational, risk-averse investors would choose, as they provide optimal risk-return combinations. - Points below the efficient frontier are considered inefficient, as there exist portfolios that offer higher returns for the same level of risk or lower risk for the same level of return. - The efficient frontier is typically upward sloping, reflecting the positive relationship between risk and return: as risk increases, expected return also increases. **Visual Representation**: The efficient frontier is also illustrated in the risk-return space, lying above the minimum-variance frontier, where each point signifies an optimal portfolio. --- ### **Practical Application**: Understanding these frontiers is crucial for portfolio construction. Investors seek to optimize their portfolios by choosing assets that lie on the efficient frontier, as these represent the best best risk-return trade-offs based on their risk tolerance. --- **Relevant CFA Subject and Exam Weight**: The concepts of the minimum-variance frontier and the efficient frontier appear in the "Portfolio Management" section of the CFA Level I curriculum, typically weighted about 5%-10% of the exam. --- Grasping the concepts of minimum-variance and efficient frontiers allows investors and portfolio managers to make informed decisions in constructing diversified portfolios that align with their investment goals and risk preferences.---Exhibit 19: Minimum-Variance Frontier **Chart Description:** * **Type:** A two-dimensional graph illustrating portfolio risk and return. * **Coordinate Axes:** * X-axis: Labeled "Portfolio Standard Deviation". Origin is 0. * Y-axis: Labeled "Portfolio Expected Return", denoted as $E(R_p)$. Origin is 0. * **Curves:** * Minimum-Variance Frontier: A curve opening to the right, representing portfolios with the lowest risk for a given expected return. * Efficient Frontier: The upper part of the Minimum-Variance Frontier, representing portfolios with the highest expected return for a given risk. It is highlighted and labeled "Efficient Frontier" with an arrow pointing to the curve segment above the Global Minimum-Variance Portfolio. * **Points:** Several points are plotted as black dots: * Point Z: Labeled "Global Minimum-Variance Portfolio (Z)". It is the leftmost point on the Minimum-Variance Frontier, marking the portfolio with the lowest overall standard deviation. An arrow points to it from the label. * Point C: A point on the lower part of the Minimum-Variance Frontier, below the horizontal dashed line. An arrow points to it from the label "C". * Point A: A point located to the left of the Minimum-Variance Frontier, above the horizontal dashed line. An arrow points to it from the label "A". * Point X: A point located to the left of point A, above the horizontal dashed line. An arrow points to it from the label "X". * Point B: A point located on the Efficient Frontier, above the horizontal dashed line. An arrow points to it from the label "B". * Point D: A point located on the Efficient Frontier, to the right of B, above the horizontal dashed line. An arrow points to it from the label "D". * There are two additional unlabeled black dots plotted: one on the Minimum-Variance Frontier between Z and C, and one to the left of the Efficient Frontier between points A and B. * **Lines:** * Horizontal dashed line: Passes through the Global Minimum-Variance Portfolio (Z). * **Labels and Annotations:** * "Exhibit 19: Minimum-Variance Frontier" (Title) * "$E(R_p)$" (Y-axis label) * "Portfolio Expected Return" (Y-axis label) * "0" (Origin) * "Portfolio Standard Deviation" (X-axis label) * "Efficient Frontier" (Label with arrow pointing to the upper curve segment) * "Minimum-Variance Frontier" (Label with arrow pointing to the entire curve) * "Global Minimum-Variance Portfolio (Z)" (Label with arrow pointing to point Z) * Labels "X", "A", "B", "C", "D" with arrows pointing to their respective plotted points. **Chart Description:** * **Type:** A two-dimensional plot showing a curve representing portfolio possibilities. * **Coordinate Axes:** * X-axis: Labeled 'σ', representing risk (standard deviation), increasing from left to right starting at 0. * Y-axis: Labeled 'E(r)', representing expected return, increasing from bottom to top starting at 0. * **Elements:** * A curve connecting two points labeled "资产1" (Asset 1) and "资产2" (Asset 2). * The point labeled "资产1" is located in the upper right portion of the chart. * The point labeled "资产2" is located in the lower right portion of the chart. * The curve passes through both "资产1" and "资产2". * A specific point on the curve is labeled "最小方差组合" (Minimum Variance Portfolio) with an arrow pointing to it. This point is the leftmost point on the curve. * The curve is concave to the left, forming a shape resembling a sideways parabola opening to the right. **Textual Information:** * E(r) (Label for the Y-axis, Expected Return) * σ (Label for the X-axis, Standard Deviation) * 0 (Label for the origin) * 资产1 (Label for a point on the curve, Asset 1) * 资产2 (Label for another point on the curve, Asset 2) * 最小方差组合 (Label for a specific point on the curve, Minimum Variance Portfolio) **Chart Description:** * **Type:** 2D scatter/line plot. * **Coordinate Axes:** * Vertical axis (Y-axis) is labeled "E(r)", representing expected return. The arrow indicates increasing values upwards. * Horizontal axis (X-axis) is labeled "σ_P", representing portfolio standard deviation (risk). The arrow indicates increasing values to the right. * **Lines/Curves:** * A blue curve is plotted, starting from a point (A), going upwards and rightwards, and also downwards and rightwards from A, forming a shape resembling a parabola opening to the right. This curve is labeled "最小方差前沿" (Minimum Variance Frontier). * A red dashed horizontal line passes through point A, extending leftwards from the Y-axis. * **Points:** * A black solid point labeled "A" is located on the blue curve, specifically at the leftmost point of the curve (minimum σ_P). * **Labels and Annotations:** * "最小方差前沿" (Minimum Variance Frontier) is labeled near the blue curve, likely referring to the entire curve. * "有效前沿" (Efficient Frontier) is labeled with a red arrow pointing to the upper part of the blue curve, above point A. * The point A is explicitly labeled. * A watermark "知乎 @同机而P" appears in the bottom right corner. * **Relative Position and Direction:** * Point A is the point with the minimum σ_P on the "最小方差前沿". * The "有效前沿" is the segment of the "最小方差前沿" at or above point A. * The red dashed line represents the expected return E(r) at point A. **Extracted Text:** * E(r) * 有效前沿 * 最小方差前沿 * A * σ_P * 知乎 @同机而P Chart/Diagram Description: * **Type:** Scatter plot with curves representing portfolio frontiers. * **Coordinate Axes:** * **X-axis:** Labeled "Portfolio Standard Deviation". Starts from 0. * **Y-axis:** Labeled "Portfolio Expected Return" and denoted as $E(R_p)$. Starts from 0. * **Curves:** * A curved line extends upwards and to the right, forming a shape resembling a parabola or hyperbola segment opening to the right. The left-most point is labeled "Global Minimum-Variance Portfolio (Z)". This curve is labeled "Minimum-Variance Frontier". * The upper part of the Minimum-Variance Frontier, starting from the Global Minimum-Variance Portfolio (Z) and extending upwards and to the right, is labeled "Efficient Frontier". * **Points:** * Point X: Located above and to the left of the Minimum-Variance Frontier. Labeled "X" with an arrow pointing to the point. * Point A: Located to the left of the Global Minimum-Variance Portfolio (Z) on the Minimum-Variance Frontier. Labeled "A" with an arrow pointing to the point. * Point B: Located to the right of the Global Minimum-Variance Portfolio (Z) on the Minimum-Variance Frontier, below the Efficient Frontier segment. Labeled "B" with an arrow pointing to the point. * Point C: Located below the Global Minimum-Variance Portfolio (Z) on the Minimum-Variance Frontier. Labeled "C" with an arrow pointing to the point. * Point D: Located on the Efficient Frontier, to the right of point B. Labeled "D" with an arrow pointing to the point. * Point labeled "Global Minimum-Variance Portfolio (Z)": This is the left-most point of the Minimum-Variance Frontier. An arrow points to this point. * **Lines:** * A dashed horizontal line extends from the Y-axis across the chart, passing through the Global Minimum-Variance Portfolio (Z). * **Labels and Annotations:** * $E(R_p)$: Denotes Portfolio Expected Return. * Portfolio Expected Return: Label for the Y-axis. * Portfolio Standard Deviation: Label for the X-axis. * Efficient Frontier: Label for the upper segment of the Minimum-Variance Frontier. An arrow points to this segment. * Minimum-Variance Frontier: Label for the entire curve. An arrow points to the lower segment of the curve. * X, A, B, C, D: Labels for specific points on or off the frontiers, with arrows pointing to the respective points. * Global Minimum-Variance Portfolio (Z): Label for the left-most point on the frontier, corresponding to the minimum standard deviation portfolio. An arrow points to this point.