• 探讨空头与多头套期保值策略 (Long and Short Hedges)
• 界定基差风险 (Basis Risk)：
  • 探讨影响基差风险的核心因素
  • 基差风险如何影响套期保值的有效性与最终成果？
• 计算并深入解读最小方差套期保值比率 ($h^{\ast }$)
• 确定套期保值中的最优合约数量 ($N^{\ast }$)
• 利用股指期货实现：
  • 对冲股票投资组合的市场系统性风险
  • 灵活调整股票组合的 Beta 系数
• 阐述如何实施“远期滚动对冲” (Stack-and-Roll / 展期对冲)

• Discuss long and short hedges
• Define basis risk:
  • Discuss the factors affecting basis risk
  • How does basis risk affect hedging profitability?
• Calculate and interpret the minimum variance hedge ratio ($h^{\ast }$)
• Determine the optimal number of contracts in a hedge ($N^{\ast }$)
• Use index futures to:
  • Hedge a stock portfolio
  • Change the beta of a stock portfolio
• Explain how to “roll the hedge forward” (stack-and-roll)

• 资产多头风险敞口 (Long Asset Exposure)：
  • 已经拥有现货资产，或已签订协议将在未来生产该资产。
  • 计划在未来特定时间将其出售。
  • 面临风险： 资产价格下跌风险 → 对冲策略：建立期货空头头寸 (Short Futures)。
  • 示例：金矿公司。
• 资产空头风险敞口 (Short Asset Exposure)：
  • 在未来某一时间需要采购该资产。
  • 面临风险： 资产价格上涨风险 → 对冲策略：建立期货多头头寸 (Long Futures)。
  • 示例：珠宝商。
• 套期保值的基本原则：建立一个能够抵消（中和）风险的头寸。

• Long Asset Exposure:
  • Already own the asset or have contracted to produce it.
  • Wish to sell it in the future.
  • Risk: Asset price falls → hedge by taking a Short Futures position.
  • Example: a gold mining company.
• Short Asset Exposure:
  • Need to purchase the asset in the future.
  • Risk: Asset price rises → hedge by taking a Long Futures position.
  • Example: a jeweler.
• Basic principle of hedging: take a position that neutralises the risk.

• 在套期保值中，基差 (Basis) 是被对冲资产的现货价格 $S_t$ 与所用期货合约的价格 $F_t$ 之差：
  $$\text{基差}=b_t=S_t-F_t$$
• 基差可正（当 $S_t>F_t$ 时）可负（当 $S_t<F_t$ 时）。
• 若被对冲的资产与期货合约的基础资产完全一致，则在合约到期时刻 $T$：$b_T=S_T-F_T=0$。

• In a hedging situation the basis is the difference between the spot price $S_t$ of the asset being hedged and the futures price $F_t$ of the contract used for hedging:
  $$\text{Basis}=b_t=S_t-F_t$$
• Basis can be positive (when $S_t>F_t$) or negative (when $S_t<F_t$).
• If the asset being hedged and the asset underlying the futures contract are identical, then at contract maturity $T$: $b_T=S_T-F_T=0$.

• 空头套期保值 (Short Hedge) 是指在期货市场上建立期货空头（卖出）头寸。
• 适用于套期保值者已经拥有某种资产、并预计在未来某个日期将其出售的情形。
• 或者适用于当前尚未拥有该资产、但在未来将生产并销售该资产的企业。
• 价格下跌的风险：价格下跌时，期货空头头寸获利。
• 降低不确定性的方式：
  • 用期货市场的盈利抵消现货市场的亏损（若现货价格上涨，现货出售端的收益则被期货空头的亏损抵消）；
  • 锁定价格。
• 空头套期保值无法保留价格朝有利方向变动时的收益。

• A short hedge involves taking a short position in a futures contract.
• It is appropriate when the hedger already owns an asset and expects to sell it at some future date.
• Alternatively, it is appropriate when an asset is not currently owned but will be produced and sold in the future.
• Risk of a price decrease: the short futures position benefits when the price falls.
• Reduce uncertainty by:
  • offsetting losses in the physical market with gains in the futures market (if the spot price rises, the gain on the spot sale is offset by the loss on the short futures position);
  • locking in the price.
• A short hedge cannot preserve the benefits from favourable price movements.

• 3 月份：农场主预计将在 9 月份出售玉米。
  • 当 12 月合约的期货价格为 $F_1$ 时，建立12 月期货空头头寸进行对冲：即约定在 12 月以 $F_1$ 出售玉米的合约。
• 为什么不使用 9 月份的期货合约？
• 9 月份：农场主准备在现货市场上以 $S_2$ 出售玉米。
  • 当 12 月合约的期货价格为 $F_2$ 时，建立12 月期货多头头寸以平仓：即约定在 12 月以 $F_2$ 买入玉米的合约。

• March: the farmer expects to sell corn in September.
  • Hedges by entering into a short December futures position when the futures price for December contracts is $F_1$: a contract to sell corn in December at $F_1$.
• Why is the September futures contract NOT used?
• September: the farmer is ready to sell corn in the physical (spot) market at $S_2$.
  • Closes out the futures position by entering a long December futures position when the futures price for December contracts is $F_2$: a contract to buy corn in December at $F_2$.

• 假定某空头对冲策略在时刻 $t_1$ 建立，并在时刻 $t_2$ 平仓结束：
  • $F_1$：$t_1$ 时刻建立对冲时的期货价格
  • $S_2$：$t_2$ 时刻出售现货资产时的即期价格
  • $F_2$：$t_2$ 时刻平仓期货头寸时的期货价格
  • $b_2=S_2-F_2$：$t_2$ 时刻平仓时的基差
• 实际收到的价格：现货价格 $S_2$，加上期货交易的现金结算 $F_1-F_2$：
  $$S_2+(F_1-F_2)=F_1+(S_2-F_2)=F_1+b_2$$
• 若基差为 0（$S_2=F_2$），则当前的期货价格 $F_1$ 被锁定。
• 若基差 $\ne 0$，则实际价格 = 期货价格 + 基差。
• 若基差意外走强（$b_2$ 增大），空头套期保值者的状况改善；反之则恶化。

• Suppose a short hedge is initiated at time $t_1$ and closed out at time $t_2$:
  • $F_1$: Futures price when hedge entered at $t_1$
  • $S_2$: Spot price when asset sold at $t_2$
  • $F_2$: Futures price when hedge closed out at $t_2$
  • $b_2=S_2-F_2$: Basis at $t_2$
• Effective price received: the spot price $S_2$ AND the cash settlement from the futures transactions $F_1-F_2$:
  $$S_2+(F_1-F_2)=F_1+(S_2-F_2)=F_1+b_2$$
• If the basis is 0 ($S_2=F_2$), then the current futures price $F_1$ is locked in.
• If the basis $\ne 0$, then the effective price = futures price + basis.
• The short hedger's position improves if the basis strengthens unexpectedly ($b_2$ increases), and vice versa.

• 多头套期保值 (Long Hedge) 适用于套期保值者将在未来买入某种资产、并希望提前锁定采购价格的情形。
• 价格上涨的风险：通过建立期货多头头寸来对冲该风险；价格上涨时期货多头获利。
• 在 $t_1$ 时刻：套期保值者预期未来要买入该资产，建立期货多头头寸（$F_1$）来对冲这笔采购。
• 在 $t_2$ 时刻：套期保值者准备在现货市场上以 $S_2$ 买入该资产，并通过建立期货空头头寸（$F_2$）来平仓。
• 案例分析： 某电气制造厂商已知 5 月份需要采购 100,000 磅铜。1 月份时，5 月到期铜期货价格为 $3.20/磅。
  • 该制造商买入 4 张 5 月铜期货合约（每张 25,000 磅）。
  • 5 月份时，在现货市场上以价格 $S_2$ 买入铜，同时以价格 $F_2$ 平仓期货头寸（即建立期货空头头寸）。

• A long hedge is appropriate when the hedger will purchase the asset in the future and wants to lock in the purchase price.
• Risk of a price increase: hedge this risk by entering into a long position in futures, which benefits when the price rises.
• At $t_1$: the hedger expects to buy the asset in the future and hedges this purchase by entering a long futures position ($F_1$).
• At $t_2$: the hedger is ready to buy the asset in the physical market at $S_2$ and closes out the futures position by entering a short futures position ($F_2$).
• Example: An electrical manufacturer knows it will need 100,000 lbs of copper in May. In January, May copper futures trade at $3.20/lb.
  • Manufacturer buys 4 May copper futures contracts (25,000 lbs each).
  • In May, buys copper in the physical spot market at $S_2$ and closes out the futures at $F_2$ (a short futures position).

• 实际支付的价格：支付的现货价格 $S_2$，减去期货交易收到的现金结算 $F_2-F_1$：
  $$S_2-(F_2-F_1)=F_1+(S_2-F_2)=F_1+b_2$$
• 若基差为 0（$S_2=F_2$），则当前的期货价格 $F_1$ 被锁定。
• 若基差 $\ne 0$，则实际价格 = 期货价格 + 基差，即 $F_1+b_2$，与空头套期保值相同。
• 若基差意外走弱（$b_2$ 减小），多头套期保值者的状况改善；反之则恶化。空头套期保值者则在基差走强时获益。

• Effective price paid: the spot price paid $S_2$ AND the cash settlement received from the futures transactions $F_2-F_1$:
  $$S_2-(F_2-F_1)=F_1+(S_2-F_2)=F_1+b_2$$
• If the basis is 0 ($S_2=F_2$), then the current futures price $F_1$ is locked in.
• If the basis $\ne 0$, then the effective price = futures price + basis: $F_1+b_2$, the same as for a short hedge.
• The long hedger's position improves if the basis weakens unexpectedly ($b_2$ decreases), and vice versa; a short hedger gains when the basis strengthens.

• 套期保值策略能够极大地降低风险，但在现实中极少能够实现完全消除风险。这一残存的风险就是基差风险 (Basis Risk)。
• 基差风险源于平仓时基差 $b_2$ 的不确定性：在 $t_1$ 建立对冲时，它是未知的。
• 实际中的成因：
  • 套期保值者可能无法确定买入或卖出资产的具体日期。
  • 对冲可能需要在期货合约到期日 $T$ 之前很早就平仓：在 $t_1$ 时刻建立对冲，在 $t_2<T$ 时刻平仓。
  • 需要对冲价格的资产与期货合约的基础资产并不完全相同（交叉对冲 Cross-Hedging）。
  • 地点错配：合约规定的交割地点与现货所在地不同。
• 这些问题造成基差风险：
  • 基差风险是商品期货的普遍问题，但在金融期货中同样可能出现，例如公司用股指期货对冲自己的股票组合。

• Hedging strategies minimize risk, but in practice rarely eliminate it completely. This remaining risk is basis risk.
• Causes in practice:
  • The hedger may be uncertain about the exact date when the asset will be bought or sold.
  • The hedge may require the futures contract to be closed out well before its maturity $T$: the hedge is initiated at $t_1$ and closed out at $t_2<T$.
  • The asset whose price is to be hedged is not exactly the same as the asset underlying the futures contract (cross hedging).
  • Location mismatch: delivery is specified at a different geographical point.
• These problems create basis risk:
  • Basis risk arises from the uncertainty about the basis $b_2$ when the hedge is closed out.
  • Basis risk is a generic problem with commodities, but it can also arise with financial futures, for example when a company hedges its own portfolio with index futures.

• 关键在于期货合约的选择：
  • 交割月份：选择交割月份尽可能接近、但晚于套期保值结束日期的合约。尽量避免将期货持有至交割月份：进入交割月后交易流动性往往枯竭，且交割物流会造成价格异常波动。
  • 基础资产：当被对冲资产没有对应的期货合约时，选择其期货价格与该资产价格相关性最高的合约。
• 当被对冲资产（图中 $S$）与期货合约的基础资产（$S^{\ast }$）不同时，基差包含两个部分：品质基差和期限基差：
  $$\text{实际有效价格}=S_2+(F_1-F_2)=F_1+(S_2-S_2^{\ast })+(S_2^{\ast }-F_2)$$

• A key factor is the choice of the futures contract:
  • Delivery month: choose a delivery month that is as close as possible to, but later than, the end of the life of the hedge. Avoid holding futures into the actual delivery month: trading can become illiquid and erratic price spikes occur due to delivery logistics.
  • Underlying asset: when there is no futures contract on the asset being hedged, choose the instrument whose futures price is most highly correlated with the asset price.
• When the asset to be hedged ($S$ in the chart) is different from the asset underlying the futures contract ($S^{\ast }$), there are two components to the basis: grade basis and time basis:
  $$\text{effective price}=S_2+(F_1-F_2)=F_1+(S_2-S_2^{\ast })+(S_2^{\ast }-F_2)$$

• 若期货合约的基础资产与被对冲资产相同，期货头寸的规模应与风险敞口的规模相同。
  • 但这未必能使风险最小。
• 最优的套期保值比率应使头寸的方差最小。
• 套期保值比率 (Hedge Ratio) $h$ 是指持有的期货头寸规模与面临的现货风险敞口规模之比。
• 为使被对冲组合资产价值的方差最小化：
  $$h^{\ast }=\rho \frac{{\sigma }_S}{{\sigma }_F}$$
• 公式参数说明：
  • ${\sigma }_S$：在对冲期间现货价格变动量 $\mathrm{\Delta }S$ 的标准差
  • ${\sigma }_F$：在对冲期间期货价格变动量 $\mathrm{\Delta }F$ 的标准差
  • $\rho$：$\mathrm{\Delta }S$ 与 $\mathrm{\Delta }F$ 之间的线性相关系数
• $h^{\ast }$ 在统计学上恰好等于以 $\mathrm{\Delta }S$ 对 $\mathrm{\Delta }F$ 进行普通最小二乘法 (OLS) 线性回归方程的最佳拟合斜率项。

• If the asset underlying the futures contract is the same as the asset being hedged, the size of the futures position should be the same as the size of the exposure.
  • This may not necessarily minimize risk.
• The optimal hedge ratio should minimize the variance of the position.
• The hedge ratio $h$ is the size of the position taken in futures contracts relative to the size of the exposure.
• To minimize the variance of the hedged position value:
  $$h^{\ast }=\rho \frac{{\sigma }_S}{{\sigma }_F}$$
• where:
  • ${\sigma }_S$: standard deviation of $\mathrm{\Delta }S$ (change in spot price over hedge horizon)
  • ${\sigma }_F$: standard deviation of $\mathrm{\Delta }F$ (change in futures price over hedge horizon)
  • $\rho$: coefficient of correlation between $\mathrm{\Delta }S$ and $\mathrm{\Delta }F$
• $h^{\ast }$ is the slope of the best-fit OLS linear regression of $\mathrm{\Delta }S$ on $\mathrm{\Delta }F$.

• 变量设定：
  • $Q_A$：被对冲资产的头寸规模（物理单位计量）
  • $Q_F$：单张期货合约所代表的资产数量（物理单位计量）
  • $h^{\ast }$：最优最小方差套期保值比率
• 最优合约数量计算公式：
  $$N^{\ast }=h^{\ast }\frac{Q_A}{Q_F}=\rho \frac{{\sigma }_S}{{\sigma }_F}\frac{Q_A}{Q_F}$$
• 在实际操作中，所用合约的数量必须是整数，因此只能近似地实现最优对冲。
• 若考虑每日逐日盯市结算的利率再投资效应（尾随对冲 Tailing the Hedge）： $$N^{\ast }=h^{\ast }\frac{V_A}{V_F}$$ 其中 $V_A=Q_{A}S$ 为现货资产市场总市值，$V_F=Q_{F}F$ 为单张期货合约名义价值。

• Let:
  • $Q_A$: Size of the position being hedged (in units)
  • $Q_F$: Size of one futures contract (in units)
  • $h^{\ast }$: Optimal minimum-variance hedge ratio
• Optimal Number of Contracts Formula:
  $$N^{\ast }=h^{\ast }\frac{Q_A}{Q_F}=\rho \frac{{\sigma }_S}{{\sigma }_F}\frac{Q_A}{Q_F}$$
• In practice the number of contracts used must be an integer, therefore only an approximation to the optimal hedge is possible.
• If daily settlement (tailing the hedge) is taken into account: $$N^{\ast }=h^{\ast }\frac{V_A}{V_F}$$ where $V_A=Q_{A}S$ is the spot value and $V_F=Q_{F}F$ is contract value.

• 某农场主利用期货对冲 18,000 千克 牛肉的销售。每张期货合约的交割规模为 1,500 千克。
  • ${\sigma }_S=15\mathrm{\%}$ • ${\sigma }_F=10\mathrm{\%}$ • $\rho =0.80$ • $S_0=\mathrm{\$}2.50$ • $F_0=\mathrm{\$}2.45$
• 套期保值比率：
  $$h^{\ast }=0.80\times \frac{15\mathrm{\%}}{10\mathrm{\%}}=1.20$$
  做空的期货合约价值应为被对冲头寸价值的 120%。
• 因此应做空：
  $$N^{\ast }=1.20\times \frac{18,000\times \mathrm{\$}2.50}{1,500\times \mathrm{\$}2.45}=14.69\approx 15\text{ 张期货合约}$$

• A farmer hedges the sale of 18,000 kg of beef using futures. Each contract is for the delivery of 1,500 kg.
  • ${\sigma }_S=15\mathrm{\%}$ • ${\sigma }_F=10\mathrm{\%}$ • $\rho =0.80$ • $S_0=\mathrm{\$}2.50$ • $F_0=\mathrm{\$}2.45$
• Hedge ratio:
  $$h^{\ast }=0.80\times \frac{15\mathrm{\%}}{10\mathrm{\%}}=1.20$$
  The futures contracts to be shorted should have 120% of the value of the position being hedged.
• Thus short:
  $$N^{\ast }=1.20\times \frac{18,000\times \mathrm{\$}2.50}{1,500\times \mathrm{\$}2.45}=14.69\approx 15\text{ futures contracts}$$

• 非系统性风险（个股特有风险）可以通过构建充分多元化的投资组合加以分散消除。
• 系统性风险（大盘市场风险）无法通过分散化投资消除，但可以借助股票指数期货完美对冲！
• 资本资产定价模型 (CAPM) 将组合收益率 $R_P$ 与市场收益率 $R_M$、无风险利率 $R_F$、组合 Beta $\beta$ 联系起来：
  $$R_P-R_F=\beta (R_M-R_F)$$
• 若股票组合与指数完全一致，最优套期保值比率为 1.0。
• 若组合并不与指数一致，则最优套期保值比率的估计值是 CAPM 中该组合的 Beta。
• 为彻底消除市场系统性风险（将组合有效 $\beta$ 降至 0）：
  $$N^{\ast }=\beta \frac{P}{F}=\beta \frac{V_P}{V_F}$$
  其中 $P=V_P$ 为当前股票组合的总市值，$F=V_F$ 为单张股指期货合约名义价值。

• Unsystematic (firm-specific) risk is eliminated by diversification.
• Systematic (market) risk cannot be diversified away, but it can be hedged using stock index futures!
• The Capital Asset Pricing Model (CAPM) links the portfolio return $R_P$ to the market return $R_M$, risk-free rate $R_F$ and portfolio beta $\beta$:
  $$R_P-R_F=\beta (R_M-R_F)$$
• If the equity portfolio is identical to the index, the optimal hedge ratio is 1.0.
• When the portfolio does not mirror the index, an estimate of the optimal hedge ratio is the portfolio beta from the CAPM.
• To eliminate market risk (reduce $\beta$ to 0):
  $$N^{\ast }=\beta \frac{P}{F}=\beta \frac{V_P}{V_F}$$
  where $P=V_P$ is current portfolio market value and $F=V_F$ is index futures contract value.

资料来源：cme.com 与 sfe.com.au

Source: cme.com and sfe.com.au

资料来源：sfe.com.au（原第 16 页右半部分）。与所有主要股指期货一样，该合约采用现金结算：无法按指数权重交割整篮成分股。

Source: sfe.com.au (second half of the original slide 16). Like all major stock index futures, the contract is cash-settled: one cannot deliver the whole index basket of shares in exact index proportions.

• 某投资者持有 5000 万美元 的股票组合，Beta 系数 $\beta =0.80$，对未来 4 个月的市场持悲观态度。
• 使用 SFE SPI200 期货合约：
  • $T$ = 6 个月 • 每个指数点价值 $25 • 指数期货价格 = 5,583
• 指数点位 = 5,500 • 标普/澳交所 200 指数股息率 = 每年 3% • 无风险利率 = 每年 6%
• 套期保值所需的期货合约数量为
  $$N^{\ast }=0.80\times \frac{\mathrm{\$}50,000,000}{\mathrm{\$}25\times 5,583}=286.58\approx 287$$
• 套期保值者建立 287 张 6 个月期标普/澳交所 200 指数期货的空头头寸。
• 如果 4 个月后，指数为 5,200、期货价格为 5,226 时对冲平仓，结果会怎样？

• An investor holds a $50 million portfolio with beta $\beta =0.80$ and is pessimistic about the market over the next 4 months.
• Use the SFE SPI200 futures contract:
  • $T$ = 6 months • value = $25 per index point • index futures price = 5,583
• Index level = 5,500 • dividend yield on the S&P/ASX200 index = 3% p.a. • risk-free rate = 6% p.a.
• The number of futures contracts required for hedging is
  $$N^{\ast }=0.80\times \frac{\mathrm{\$}50,000,000}{\mathrm{\$}25\times 5,583}=286.58\approx 287$$
• The hedger enters into a SHORT position in 287 6-month S&P/ASX200 index futures contracts.
• What will happen if in 4 months the hedge is closed out when the index is 5,200 and the futures price is 5,226?

4 个月后：

• 期货空头平仓的盈利：$287\times \mathrm{\$}25\times (5,583-5,226)=\mathrm{\$}2,561,475$
• 股票组合的预期损失：
  • 指数跌幅 $=-300/5,500=-5.45\mathrm{\%}$
  • 指数 4 个月的股息 $=3\mathrm{\%}\times 4/12=1\mathrm{\%}$
  • 指数收益率 $=-5.45\mathrm{\%}+1\mathrm{\%}=-4.45\mathrm{\%}$
  • 4 个月的无风险利率 $=6\mathrm{\%}\times 4/12=2\mathrm{\%}$
  • CAPM：$E(R_p)=r_f+\beta (E(R_m)-r_f)=2\mathrm{\%}+0.8(-4.45\mathrm{\%}-2\mathrm{\%})=-3.16\mathrm{\%}$
  • 4 个月后组合的预期价值 $=\mathrm{\$}50,000,000\times (1-0.0316)=\mathrm{\$}48,420,000$
• 套期保值者头寸的预期总价值 $=\mathrm{\$}2,561,475+\mathrm{\$}48,420,000=\mathrm{\$}50,981,475$

In 4 months:

• Gain from closing out the short futures position: $287\times \mathrm{\$}25\times (5,583-5,226)=\mathrm{\$}2,561,475$
• Expected loss on the portfolio:
  • Loss on index $=-300/5,500=-5.45\mathrm{\%}$
  • Dividend from index in 4 months $=3\mathrm{\%}\times 4/12=1\mathrm{\%}$
  • Return on index $=-5.45\mathrm{\%}+1\mathrm{\%}=-4.45\mathrm{\%}$
  • Risk-free rate in 4 months $=6\mathrm{\%}\times 4/12=2\mathrm{\%}$
  • CAPM: $E(R_p)=r_f+\beta (E(R_m)-r_f)=2\mathrm{\%}+0.8(-4.45\mathrm{\%}-2\mathrm{\%})=-3.16\mathrm{\%}$
  • Expected value of the portfolio in 4 months $=\mathrm{\$}50,000,000\times (1-0.0316)=\mathrm{\$}48,420,000$
• Total expected value of the hedger's position $=\mathrm{\$}2,561,475+\mathrm{\$}48,420,000=\mathrm{\$}50,981,475$

请注意，套期保值者头寸的 Beta 为零。

图：指数对冲的绩效表现

Note that the beta of the hedger's position is zero.

Figure: Performance of index hedge

• 套期保值者头寸 4 个月后的收益：
  • 头寸的预期价值 $\approx$ 5100 万美元
  • 头寸的收益率 $\approx (51-50)/50=2\mathrm{\%}$
  • 4 个月的无风险利率 $=2\mathrm{\%}$
• 对冲使投资者的头寸按无风险利率增长。他/她也可以卖出股票组合，投资于无风险工具（短期国债）。那为什么还要套期保值？
  • 套期保值者计划长期持有该组合，但需要短期的保护。
  • 市场表现的不确定性：套期保值者只承担组合相对于市场的表现风险；套期保值消除了来自市场的风险。
  • 套期保值使不利结果变得不那么不利，也使有利结果变得不那么有利。

• The return on the hedger's position in 4 months:
  • Expected value of the hedger's position $\approx$ $51 million
  • Return of the hedger's position $\approx (51-50)/50=2\mathrm{\%}$
  • 4-month risk-free rate $=2\mathrm{\%}$
• The hedge results in the investor's position growing at the risk-free rate. He/she could sell the portfolio and invest in risk-free instruments (T-bills). So why hedge?
  • The hedger plans to hold the portfolio for a long period, but requires short-term protection.
  • Uncertainty about market performance: the hedger is exposed only to the performance of the portfolio relative to the market; hedging removes the risk arising from the market.
  • The hedge makes unfavourable outcomes less unfavourable and favourable outcomes less favourable.

• 完全对冲的股票头寸，其期望收益率等于无风险利率：
  $$E(R_{\text{对冲}})=R_F$$
  若组合相对于 CAPM 具有正的超额收益（$\alpha >0$），对冲后的收益率为 $R_F+\alpha$。
• 为什么不直接清仓股票、转而持有无风险短期国债？
  • 保留选股的阿尔法收益：基金经理相信所选个股能跑赢基准，即使大盘整体下挫。
  • 避免高额交易成本：一次性抛售 5000 万美元的组合，要承担佣金、买卖价差和市场冲击成本。
  • 避免兑现资本利得税：卖出股票会立即产生纳税义务。
  • 投资契约约束：养老基金和公募基金的契约通常要求保持较高的股票仓位。

• The expected return on a perfectly hedged equity position is the risk-free rate:
  $$E(R_{\text{hedged}})=R_F$$
  If the portfolio earns a positive alpha relative to the CAPM ($\alpha >0$), the hedged return is $R_F+\alpha$.
• Why not just liquidate the stock portfolio and invest in risk-free T-bills?
  • Preserve stock-selection alpha: the manager may be confident that the selected stocks will outperform their benchmarks, even during a market-wide downturn.
  • Avoid large transaction costs: brokerage, bid-ask spread slippage and market impact from dumping a $50M portfolio.
  • Avoid realizing capital gains tax: selling triggers immediate tax liabilities.
  • Mandate and institutional constraints: pension and mutual funds often have strict mandates requiring them to remain fully invested in equities.

• $(\beta -{\beta }^{\ast }){\displaystyle \frac{P}{F}}$ 张期货合约构成的头寸，可将组合的 Beta 从 $\beta$ 调整为 ${\beta }^{\ast }$：
  $$N^{\ast }=(\beta -{\beta }^{\ast })\frac{P}{F}$$
• 若 ${\beta }^{\ast }>\beta$，需要建立期货多头头寸。
  • 示例：要把投资者头寸的 Beta 提高到 1.5，需要什么头寸？ $$\text{多头 }{N}^{\ast }=(0.8-1.5)\frac{50,000,000}{25\times 5,583}=-250.76\approx 251\text{ 张合约}$$
• 若 ${\beta }^{\ast }<\beta$，需要建立期货空头头寸。
  • 示例：要把投资者头寸的 Beta 降低到 0.70，需要什么头寸？ $$\text{空头 }{N}^{\ast }=(0.8-0.7)\frac{50,000,000}{25\times 5,583}=35.82\approx 36\text{ 张合约}$$

• A position in
  $$N^{\ast }=(\beta -{\beta }^{\ast })\frac{P}{F}$$
  futures contracts will change the beta of the position from $\beta$ to ${\beta }^{\ast }$.
• If ${\beta }^{\ast }>\beta$, then a long futures position is required.
  • Example: what position is necessary to increase the beta of the investor's position to 1.5? $$\text{long }{N}^{\ast }=(0.8-1.5)\frac{50,000,000}{25\times 5,583}=-250.76\approx 251\text{ contracts}$$
• If ${\beta }^{\ast }<\beta$, then a short futures position is required.
  • Example: what position is necessary to reduce the beta of the investor's position to 0.70? $$\text{short }{N}^{\ast }=(0.8-0.7)\frac{50,000,000}{25\times 5,583}=35.82\approx 36\text{ contracts}$$

通过建立 251 张期货合约的多头头寸，将投资者头寸的 Beta 提高到 1.5。

图：该策略的绩效表现

Increase the beta of the investor's position to 1.5 by taking a long position in 251 futures contracts.

Figure: Performance of the Strategy

• 面临难题： 套期保值者面临长周期的风险敞口（例如 1 年），而市场上只有短期限的期货合约可用（例如 3 个月）。
• 应对策略：
  • 在近期活跃合约上建立初始对冲头寸。
  • 在临近合约到期日之前，对冲平仓原有合约，并在下一期活跃合约上建立新的对冲头寸（向前展期滚动）。
• 展期基差风险 (Roll-over Basis Risk)：
  • 保值者在每次合约展期平仓和新开仓时，都会承担新的基差波动风险。
  • 最著名的历史案例为 Metallgesellschaft (MG，1993年)：该公司在原油市场处于长期升水 (Contango) 结构下，对巨额长达10年的能源供货合同实施短期期货滚动对冲，展期损失导致数十亿美元灾难性亏损。

• The Problem: Suppose one has a long-dated exposure (e.g., 1 year) and only short-dated futures contracts are available (e.g., 3 months).
• The Strategy:
  • Take a hedge position in short-dated contracts.
  • Just before maturity, close out existing contracts and open a new position in the next contract (rolling forward).
• Roll-over Basis Risk:
  • Each time a contract is rolled over, the hedger incurs basis risk at that rollover point.
  • Made famous by Metallgesellschaft (1993), which suffered massive losses on rolling forward short oil hedges during market contango.