### 1: Introduction Portfolio management often requires hedging strategies to protect against anticipated market downturns. Beta (\(\beta\)) is a measure of a portfolio's sensitivity to market movements, indicating how much the portfolio's value is expected to change for a given change in the market index (e.g., the S&P 500). When the market is expected to drop, portfolio managers may use futures contracts based on the index to hedge the portfolio. The value of the futures position will change in response to movements in the underlying index. #### **Explanation:** Beta reflects systematic risk, quantifying the relationship between the portfolio's returns and the market index. A beta of 0.60 means the portfolio is expected to move 60% as much as the market index. When anticipating a drop in the market, futures contracts can be used to offset potential losses. The change in value of a futures position is directly linked to the change in the index, adjusted by the contract size and the number of contracts held. Understanding these relationships is essential for effective risk management. --- ### 2: Identifying the Correct Option Among the Choices The question asks: **For a 200-point drop in the S&P 500, by how much does the value of the futures position change?** #### **Explanation:** Determining this requires knowing the contract specifications (typically, the notional value of one S&P 500 futures contract is the index level multiplied by a fixed dollar amount per point, e.g., $250 per point for the standard contract). Since the notional value or contract size per point is not specified, the answer will assume the standard contract. --- ### A Step-by-Step Calculation #### **Step 1: Calculate the Change in the S&P 500 Index** - The anticipated drop = 1400 (current) - 1200 (anticipated) = **200 points** **Explanation:** The question explicitly mentions a 200-point fall, which matches the difference between the given index values. --- #### **Step 2: Determine the Value Change per Futures Contract** - Standard S&P 500 futures contract size = **$250 per index point** - Value change per contract = Change in points × Dollar value per point = 200 points × $250/point = **$50,000** **Explanation:** Each contract increases or decreases in value by $250 for every point change in the S&P 500. Thus, a 200-point movement results in a $50,000 change per contract. --- #### **Step 3: Calculate the Hedge Ratio (if needed)** - Portfolio Value = $1,000,000 - Portfolio Beta = 0.60 - Current Index Level = 1400 - Futures contract size = $250 × 1400 = $350,000 (notional value per contract) Number of contracts required to hedge: \[ \text{Number of Contracts} = \frac{\text{Portfolio Value} \times \text{Portfolio Beta}}{\text{Futures Contract Value}} \] \[ = \frac{1,000,000 \times 0.60}{350,000} \approx 1.71 \] **Explanation:** This step determines how many contracts are needed to fully hedge the portfolio's market risk, based on its beta. --- #### **Step 4: Calculate the Total Change in Value of the Futures Position** - Total change = Value change per contract × Number of contracts = $50,000 × 1.71 = **$85,500** **Explanation:** This is the total change in value of the futures position that would offset the loss in the portfolio due to the market decline, adjusted for the portfolio's beta. --- ### 3: Analysis of Why Other Options Are Incorrect If other options suggested different amounts (for example, $50,000, $120,000, $200,000, etc.), here is why they would be incorrect: - **$50,000:** This represents the change for only one contract, not for the number required to hedge the portfolio. - **$120,000/$200,000:** These values do not correspond to the calculated number of contracts or the actual value change required to hedge a portfolio of $1 million with a beta of 0.60. - **Any value not matching $85,500:** Fails to account for either the correct number of contracts or the correct contract size or portfolio beta. **Explanation:** Other values ignore the relationship among portfolio beta, portfolio value, contract size, and the number of contracts needed for a proper hedge. --- ### Conclusion For a 200-point drop in the S&P 500, the value of the futures position required to hedge a $1 million portfolio with a beta of 0.60 changes by **$85,500**. This is calculated by determining the number of futures contracts needed to hedge the portfolio and multiplying by the value change per contract for a 200-point drop. This comprehensive approach ensures that the portfolio manager effectively neutralizes the systematic risk associated with the anticipated market decline.

### 1: Introduction Hedging strategies are fundamental in portfolio management to mitigate potential losses arising from adverse market movements. Futures contracts are common tools for hedging, allowing investors to offset potential declines in the value of their portfolios by taking positions in derivatives based on the same underlying assets. The sensitivity of a portfolio to market fluctuations is quantified by its beta (\(\beta\)), which indicates how much the portfolio's returns are expected to change relative to market movements. A beta of 0.60 suggests that the portfolio's value moves 60% in response to changes in the market index, such as the S&P 500. When a decline in the market is anticipated, futures contracts can be employed to counteract the expected loss. The change in the futures position's value depends on both the magnitude of the market movement and the contract specifications, including the contract size and the beta-adjusted exposure. Understanding how to translate a change in the market index into a corresponding change in the futures position is essential for effective hedging. **Explanation:** This introduction emphasizes the importance of beta as a measure of systematic risk and the role of futures contracts in hedging against market downturns. Recognizing the relationship between market movements and futures position changes enables portfolio managers to determine the appropriate size of futures positions needed to hedge against expected declines, thereby minimizing potential portfolio losses. --- ### 2: Presentation of Relevant Formulas Required To Solve The Question **Formula 1: Change in Futures Contract Value** \[ \Delta V_{futures} = \text{Number of Contracts} \times \text{Value Change per Contract} \] **Formula 2: Value Change per Contract** \[ \text{Value Change per Contract} = \text{Point Change} \times \text{Dollar Value per Point} \] **Formula 3: Number of Contracts for Hedging** \[ \text{Number of Contracts} = \frac{\text{Portfolio Value} \times \beta}{\text{Futures Contract Notional Value}} \] **Formula 4: Futures Contract Notional Value** \[ \text{Futures Contract Notional Value} = \text{Index Level} \times \text{Dollar Value per Point} \] **Explanation:** These formulas link the market movement to the change in the futures position's value, scaled by the number of contracts held. The key is to determine how many contracts are needed to hedge the portfolio's market risk, considering its beta and size, and then calculate the resulting change in the futures position's value based on the anticipated market movement. --- ### 3: A Detailed Step-by-Step Solution #### **Step 1: Identify the expected change in the S&P 500 index** - The anticipated market decline = current index level - future index level = 1400 - 1200 = **200 points** **Explanation:** This quantifies the expected decline in the market index, which directly influences the futures position change. --- #### **Step 2: Calculate the notional value of one futures contract** - Standard futures contract size per point = **$250** \[ \text{Futures Contract Notional Value} = 1400 \times 250 = \$350,000 \] **Explanation:** This represents the total dollar amount covered by one futures contract at the current index level, serving as the basis for determining the number of contracts needed. --- #### **Step 3: Determine the number of contracts required to hedge the portfolio** \[ \text{Number of Contracts} = \frac{\text{Portfolio Value} \times \beta}{\text{Futures Contract Notional Value}} \] \[ = \frac{1,000,000 \times 0.60}{350,000} \approx 1.71 \] **Explanation:** This calculation finds the number of futures contracts needed to hedge 60% of the portfolio's systematic risk, scaled by the contract size. --- #### **Step 4: Calculate the change in value per futures contract due to the market decline** \[ \text{Value Change per Contract} = 200 \times 250 = \$50,000 \] **Explanation:** This shows the dollar change in value of one futures contract for a 200-point drop in the index. --- #### **Step 5: Calculate the total change in the futures position's value** \[ \text{Total Change} = 1.71 \times 50,000 \approx \$85,500 \] **Explanation:** Multiplying the number of contracts by the change per contract yields the total dollar change in the futures position, which offsets the anticipated decline in the portfolio. --- ### **Final Answer:** The value of the futures position changes by approximately **\$85,500** for a 200-point drop in the S&P 500. --- ### 4: Analysis of Why Other Options Are Incorrect - **Options suggesting a change of \$50,000 or \$100,000:** These values only account for the change per single contract or an incorrect number of contracts. They do not incorporate the portfolio beta or the required hedging ratio, leading to underestimation or overestimation. - **Options with values like \$200,000 or \$250,000:** These are inconsistent with the calculated notional values based on the contract size and market movement, ignoring the hedging ratio and portfolio specifics. **Explanation:** Such options fail to consider the number of contracts necessary to hedge the specific portfolio risk or misapply the contract size, resulting in inaccurate estimates. --- ### **Conclusion** A 200-point decline in the S&P 500 results in an approximate \$85,500 decrease in the value of the futures position required to hedge a \$1 million portfolio with a beta of 0.60. This calculation incorporates the contract size, the portfolio's systematic risk, and the magnitude of market movement, providing an effective hedge.

### 1. Introduction Hedging strategies are vital tools in portfolio management to protect against potential adverse market movements. Futures contracts, in particular, are commonly used derivatives that enable investors to offset possible losses in their portfolios by taking positions aligned with their expectations and risk exposure. The sensitivity of a portfolio to fluctuations in the market is quantified by its **beta (\(\beta\))**, which measures the degree to which the portfolio's returns are expected to change relative to changes in the market index, such as the S&P 500. When an investor anticipates a market decline, they may employ futures contracts to hedge the portfolio's exposure. The change in the value of these futures positions in response to market movements directly affects the overall hedging effectiveness. In particular, understanding how a specific change in the index level (e.g., a 200-point drop in the S&P 500) influences the futures position's value is essential for precise risk management. **Explanation:** This introduction delineates the fundamental concepts of beta, futures hedging, and their interconnected roles in managing systematic risk within a portfolio. Recognizing how market movements translate into futures position changes forms the basis for calculating the necessary hedge size and assessing potential gains or losses, thereby aligning risk mitigation strategies with market expectations. --- ### 2. Presentation of Relevant Formulas Required To Solve The Question **Formula 1: Market Index Change in Points** \[ \Delta \text{Index Points} = \text{Anticipated Index Level} - \text{Current Index Level} \] - **Application:** Quantifies the expected change in the index level in points, which directly influences the futures contract value. **Formula 2: Change in Futures Contract Value** \[ \Delta V_{\text{futures per contract}} = \Delta \text{Index Points} \times \text{Dollar Value per Point} \] - **Application:** Calculates how much the value of a single futures contract will change in response to a specified market movement. **Formula 3: Number of Contracts Needed for Hedging** \[ N = \frac{\text{Portfolio Value} \times \beta}{\text{Futures Contract Notional Value}} \] - **Application:** Determines the quantity of futures contracts required to hedge the market risk of the portfolio, considering its beta. **Formula 4: Total Change in Futures Position’s Value** \[ \Delta V_{\text{total}} = N \times \Delta V_{\text{futures per contract}} \] - **Application:** Provides the overall dollar change in the futures position, which offsets the portfolio's expected loss. **Explanation:** These formulas serve as the mathematical foundation for translating expected market movements into quantifiable impacts on futures positions. They connect the change in index levels with the number of contracts needed and the resulting dollar change, enabling precise hedging calculations tailored to the portfolio's risk profile. --- ### 3. A Detailed Step-by-Step Solution **Step 1: Calculate the market index change in points** \[ \Delta \text{Index Points} = 1200 - 1400 = -200 \text{ points} \] *The index is expected to fall by 200 points, which is the basis for subsequent calculations.* **Explanation:** This absolute change in index points directly influences the futures contract value change and the hedge size. --- **Step 2: Determine the dollar value change per futures contract** \[ \Delta V_{\text{per contract}} = \Delta \text{Index Points} \times \text{Dollar Value per Point} \] \[ = (-200) \times \$250 = -\$50,000 \] *For each futures contract, a 200-point decline in the index results in a \$50,000 gain (since the index drops, futures gain value).* **Explanation:** The dollar value per point for a standard S&P 500 futures contract is typically \$250. Multiplying by the index change in points gives the dollar change in the futures contract's value for that market movement. --- **Step 3: Calculate the number of futures contracts needed for hedging** \[ N = \frac{\text{Portfolio Value} \times \beta}{\text{Futures Contract Notional Value}} \] \[ = \frac{\$1,000,000 \times 0.60}{\$350,000} \approx 1.71 \] *Approximately 1.71 futures contracts are needed to hedge the systematic risk.* **Explanation:** The notional value of one futures contract at the current index level is: \[ \text{Futures Contract Notional} = 1400 \times \$250 = \$350,000 \] The number of contracts reflects the size of the hedge, scaled for the portfolio's beta to account for its sensitivity to market movements. --- **Step 4: Calculate the total change in the futures position** \[ \Delta V_{\text{futures}} = N \times \Delta V_{\text{per contract}} \] \[ = 1.71 \times \$50,000 \approx \$85,500 \] *The futures position will gain approximately \$85,500 as the market drops, offsetting the portfolio's expected loss.* **Explanation:** This dollar change in the futures position quantifies the hedge's effectiveness. Since the futures are short (for hedging a long position), a market decline causes a gain in futures, counteracting the portfolio's loss. --- ### **Final Answer:** **The value of the futures position changes by approximately \$85,500** for a 200-point decline in the S&P 500. --- ### 4. **Analysis of Why Other Options Are Incorrect** - Any value significantly lower than \$85,500 suggests an under-hedge, neglecting the number of contracts based on the portfolio's beta. - Values higher than \$85,500 imply over-hedging, which does not align with the calculated hedge ratio. - Smaller amounts, e.g., \$50,000, only account for one contract's change, ignoring the necessary number of contracts to adequately hedge the portfolio. - Larger amounts, e.g., \$200,000 or \$250,000, overstate the change, possibly due to miscalculations of contract size or ignoring the beta-adjusted hedge ratio. **Explanation:** Inaccurate options stem from neglecting the precise calculation of the hedge ratio, the number of contracts, or the contract's dollar value per point. Proper application of formulas ensures the correct hedge magnitude. --- ### **Summary:** For a 200-point decline in the S&P 500, the change in the futures position value necessary to hedge a \$1 million portfolio with a beta of 0.60 is approximately **\$85,500**. This calculation accounts for the market movement, the contract specifications, and the portfolio's sensitivity, providing a precise measure for effective risk management.