Question:

### 1. **Introduction:** In financial portfolio management, **systematic risk**, also referred to as **market risk**, is often measured using a metric called **beta**. Beta quantifies a portfolio's sensitivity to movements in the market index, such as the **S\&P 500**. A beta of 1 indicates that the portfolio's value moves in line with the market, whereas a beta less than 1 suggests reduced volatility relative to the market. When a market downturn is anticipated, investors often use **futures contracts** to hedge against potential losses in their portfolios. The value of these futures positions adjusts inversely to the changes in the underlying market index, providing an effective means of risk mitigation. In the context of a predicted **200-point drop** in the S\&P 500, it becomes critical to evaluate how much the **value of a hedging futures position** would change in response to such a market movement. This allows portfolio managers to quantify the hedge effectiveness and the extent of financial protection provided against anticipated losses. #### **Explanation:** Understanding beta and its application in market risk estimation is essential in portfolio risk management. Futures contracts serve as financial instruments that can be utilized to hedge against adverse market movements. When the market is expected to decline, futures are typically shorted, and the gain from the drop in index value offsets the portfolio losses. This introduction builds the foundation for determining how the change in the market translates into changes in portfolio value and how much compensation a futures position must provide to hedge that loss accurately. These foundational concepts ensure clarity in the subsequent steps of computation. --- ### 2. **Presentation of Relevant Formulas Required To Solve The Question:** To compute the change in the value of the futures position, the following formulas are required: #### **Formula 1: Percentage Change in the Index** $$ \text{Percentage Change in Index} = \frac{\text{Anticipated Index Value} - \text{Current Index Value}}{\text{Current Index Value}} \times 100 $$ #### **Formula 2: Portfolio Value Change Due to Beta Exposure** $$ \text{Change in Portfolio Value} = \text{Portfolio Beta} \times \text{Percentage Change in Index} \times \text{Portfolio Value} $$ --- #### **Explanation:** * **Formula 1** calculates how much the market is expected to decline in percentage terms, which serves as the basis for evaluating the portfolio's response. * **Formula 2** incorporates the portfolio's beta to estimate how sensitive the portfolio is to market fluctuations. Multiplying this adjusted percentage change by the total portfolio value quantifies the expected financial impact of the market decline. * These formulas are essential because they establish a direct link between market movements and portfolio performance. By applying these relationships, the value change in a hedging futures position needed to counteract portfolio losses can be determined accurately. --- ### 3. **A Detailed Step-by-Step Solution:** **Step 1: Compute the Percentage Change in the S\&P 500 Index** $$ \text{Percentage Change} = \frac{1200 - 1400}{1400} \times 100 = \frac{-200}{1400} \times 100 = -14.2857\% $$ #### **Explanation:** The index is expected to fall from 1400 to 1200, resulting in a 200-point drop. The formula translates this absolute change into a percentage to standardize it for beta-based application across different portfolio sizes and market levels. --- **Step 2: Compute the Expected Change in Portfolio Value** $$ \text{Change in Portfolio Value} = 0.60 \times (-14.2857\%) \times \$1{,}000{,}000 $$ $$ = -0.085714 \times \$1{,}000{,}000 = -\$85{,}714.29 $$ #### **Explanation:** Applying the portfolio's beta adjusts the raw market change to reflect the portfolio’s specific sensitivity. The result quantifies the expected decline in the portfolio's market value due to the anticipated market downturn. --- **Step 3: Interpret the Change in Futures Position Value** Since the question asks: **"For a 200-point drop in the S\&P 500, by how much does the value of the futures position change?"** Assuming the futures are used to hedge against the loss, the change in the value of the futures position will mirror the portfolio loss in magnitude but with an opposite sign: $$ \text{Change in Futures Value} = +\$85{,}714.29 $$ #### **Explanation:** The futures position gains value as the market drops, offsetting the portfolio loss. This step completes the hedge analysis by equating the loss from the equity portfolio to the gain from the futures position. It reflects proper risk management application and confirms that the hedging strategy is appropriately calibrated. --- ### **Conclusion:** The anticipated **200-point decline** in the S\&P 500 index results in a **14.29%** drop in the index. Given the portfolio's **beta of 0.60** and value of **\$1,000,000**, the portfolio is expected to lose **\$85,714.29** in value. Consequently, the value of the futures position changes by **+\$85,714.29**, thereby offsetting the loss and serving as an effective hedge.