# Avery Fund Hedging Analysis ## Given Data | Variable | Value | |-------------------------------|-----------------| | Portfolio Value ($S$) | $14,000,000 | | Portfolio Alpha | $3.58\%$ | | Portfolio Beta | $1$ | | Expected Idiosyncratic Risk | $.00\%$ | | S&P 500 Index Level | $4,900$ | | S&P 500 Contract Multiplier | $250$ | | Expected Market Return | $10.49\%$ | | Risk-Free Rate | $2.88\%$ | --- ## 1. **Hedge Ratio to Fully Hedge Market Risk** To fully hedge market (systematic) risk, set the **portfolio beta to zero** using S&P 500 futures. **Hedge Ratio Formula:** \[ \text{Hedge Ratio} = -\frac{\text{Portfolio Beta} \times \text{Portfolio Value}}{\text{Futures Contract Value}} \] \[ \text{Futures Contract Value} = \text{Index Level} \times \text{Contract Multiplier} = 4,900 \times 250 = \$1,225,000 \] \[ \text{Hedge Ratio} = -\frac{1 \times 14,000,000}{1,225,000} = -11.43 \] **Interpretation:** **Short 11.43 S&P 500 futures contracts** to fully hedge the market risk. --- ## 2. **Expected Dollar Value of the Portfolio (End of Period)** \[ \text{Expected Portfolio Value} = S \times (1 + \text{Alpha} + \beta \times (\text{Market Return} - \text{Risk-Free Rate}) + \text{Risk-Free Rate}) \] But since the beta is hedged to , only the alpha and risk-free rate remain: \[ \text{Expected Hedged Return} = \text{Alpha} + \text{Risk-Free Rate} = 3.58\% + 2.88\% = 6.46\% \] \[ \text{Expected Portfolio Value} = 14,000,000 \times (1 + .0646) = \$14,904,400 \] --- ## 3. **Expected Dollar Proceeds from the Futures Position** The futures hedge removes exposure to the market return. The **expected gain/loss** on the futures position is: \[ \text{Futures Gain} = - \text{Hedge Ratio} \times \text{Change in Futures Price} \times \text{Contract Multiplier} \] **Expected S&P 500 Return:** $10.49\%$ \[ \text{Expected Index Level (End)} = 4,900 \times (1 + .1049) = 5,414.01 \] \[ \text{Change in Index Level} = 5,414.01 - 4,900 = 514.01 \] \[ \text{Gain per Contract} = 514.01 \times 250 = \$128,502.50 \] \[ \text{Total Gain (Short Position)} = -11.43 \times 128,502.50 = -\$1,469,419 \] **Interpretation:** This is a **loss** from the futures short position, offsetting the market gain in the unhedged portfolio. --- ## 4. **Expected Total Rate of Return of the Hedged Position** After hedging, the portfolio earns only **alpha + risk-free rate**. \[ \text{Expected Hedged Return} = 3.58\% + 2.88\% = \boxed{6.46\%} \] --- ## **Summary Table** | Item | Value | |-------------------------------------- |-------------------------| | Hedge Ratio | -11.43 contracts | | Expected Portfolio Value (End) | $14,904,400 | | Expected Futures Proceeds | -$1,469,419 | | Expected Total Hedged Return | 6.46% | --- ## **Key Points** - **Hedge Ratio**: Short 11.43 contracts. - **Portfolio Return**: Only alpha + risk-free rate remains after hedging ($6.46\%$). - **Futures Position**: Loss on futures cancels out market gain in portfolio. - **Total Hedged Return**: $6.46\%$ or $904,400$ on $14,000,000$. --- **Note:** The expected portfolio value and proceeds are before transaction costs, taxes, and slippage.