To understand the equation and the hedging process, let's go through each line step-by-step. The goal of hedging with index futures is to reduce the systematic risk (market risk) of an existing equity portfolio. The systematic risk is often measured using the beta of the portfolio.
1. **Portfolio Beta before Hedging (Beta_P):**
- This is denoted as ẞ, and it represents the beta of the existing equity portfolio before implementing the hedging strategy.
- Beta is a measure of the portfolio's sensitivity to market movements. A beta of 1 indicates the portfolio moves in line with the market, while a beta greater than 1 suggests the portfolio is more volatile than the market, and a beta less than 1 indicates the portfolio is less volatile than the market.
2. **Target Beta after Hedging (Target Beta_*):**
- This is denoted as *, and it represents the desired beta after implementing the hedging strategy with index futures.
- The investor decides on the level of systematic risk they want to maintain in the portfolio. This could be lower or higher than the current beta depending on the investment objective.
3. **Portfolio Value (P):**
- This represents the total value of the existing equity portfolio.
4. **Value of the Underlying Asset (A):**
- This refers to the value of the stock index futures contract that will be used for hedging.
- Each stock index futures contract typically represents a specific dollar value of the underlying index.
The equation for computing the appropriate number of futures to achieve the target beta is as follows:
**Target Beta (* = Beta_Target) = (Portfolio Beta before Hedging (Beta_P) * Portfolio Value (P)) / Value of the Underlying Asset (A)**
Now, let's illustrate this with a numerical example:
Suppose you have an existing equity portfolio valued at $500,000 (P = $500,000), and its current beta is 1.2 (Beta_P = 1.2). You want to reduce the systematic risk by targeting a beta of 0.8 (Target Beta_* = 0.8). The stock index futures contract you are using for hedging (e.g., S&P 500 futures) has an underlying asset value of $100,000 (A = $100,000).
**Step 1:** Plug the values into the equation.
**Target Beta (* = Beta_Target) = (1.2 * $500,000) / $100,000**
**Step 2:** Calculate the target beta.
**Target Beta (* = Beta_Target) = $600,000 / $100,000**
**Step 3:** Determine the appropriate number of futures contracts.
**Target Beta (* = Beta_Target) = 6**
In this example, you would need to use 6 stock index futures contracts to achieve the desired beta of 0.8 in your portfolio after implementing the hedging strategy.
By holding these futures contracts, you are essentially adjusting the exposure of your portfolio to market movements, bringing it in line with your desired systematic risk level, and thus reducing the portfolio's overall beta to the target level. This is how hedging with index futures can be used to manage systematic risk in an equity portfolio.
Sure! Here are some multiple-choice questions related to hedging an existing equity portfolio with index futures:
**Question 1:**
Hedging an equity portfolio with index futures is primarily aimed at:
A) Increasing the portfolio's systematic risk.
B) Reducing the portfolio's systematic risk.
C) Eliminating all types of risks in the portfolio.
D) Speculating on the future direction of individual stocks.
**Answer:** B) Reducing the portfolio's systematic risk.
**Question 2:**
What does the beta of a portfolio measure in the context of the Capital Asset Pricing Model (CAPM)?
A) The total risk of the portfolio.
B) The unsystematic risk of the portfolio.
C) The sensitivity of the portfolio to market movements.
D) The average return of the portfolio.
**Answer:** C) The sensitivity of the portfolio to market movements.
**Question 3:**
The formula to compute the appropriate number of index futures contracts for hedging is:
A) * = (Beta_P * P) / A
B) * = (P * A) / Beta_P
C) * = (A * Beta_P) / P
D) * = (P + A) * Beta_P
**Answer:** A) * = (Beta_P * P) / A
**Question 4:**
If the target beta after implementing the strategy with index futures is 0.5, and the portfolio beta before hedging is 1.2, what does this indicate?
A) The portfolio will have more systematic risk after hedging.
B) The portfolio will have less systematic risk after hedging.
C) The portfolio beta remains the same after hedging.
D) The portfolio beta is not related to the use of index futures.
**Answer:** B) The portfolio will have less systematic risk after hedging.
**Question 5:**
Assume a portfolio value of $800,000, a portfolio beta of 1.5, and an index futures contract with an underlying asset value of $200,000. What is the target beta after hedging if the investor wants to reduce the portfolio's beta to 0.8?
A) 0.5
B) 0.8
C) 1.0
D) 1.2
**Answer:** B) 0.8
**Question 6:**
Which of the following statements is true regarding hedging with index futures?
A) Hedging increases the total risk of the portfolio.
B) Hedging involves buying/selling individual stocks to reduce systematic risk.
C) The appropriate number of futures contracts is calculated using * = (Beta_P * P) / A.
D) Hedging eliminates all types of risk in the portfolio.
**Answer:** C) The appropriate number of futures contracts is calculated using * = (Beta_P * P) / A.
**Question 7:**
If an equity portfolio has a beta of 0.9 and the market experiences a significant downturn, how is the portfolio likely to perform in comparison to the market?
A) The portfolio will perform better than the market.
B) The portfolio will perform worse than the market.
C) The portfolio's performance will not be affected by market movements.
D) The portfolio's performance will be the same as the market.
**Answer:** A) The portfolio will perform better than the market.
Note: The answers provided are based on the information given in the initial explanation. Make sure to verify the answers against the specific context and information provided in your learning materials or the scope of the course.
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