• 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)

• 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.

• 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 $T$: $b_T=S_T-F_T=0$.

• 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.

• 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 $F_2$.

• 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.

• 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).

• 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.

• 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.

• 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:
  $$\text{effective price}=S_2+(F_1-F_2)=F_1+(S_2-S_2^{\ast })+(S_2^{\ast }-F_2)$$

• 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 $\mathrm{\Delta }F$.

• 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.

• 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:
  $$N^{\ast }=1.20\times \frac{18,000\times \mathrm{\$}2.50}{1,500\times \mathrm{\$}2.45}=14.69\approx 15\text{ futures contracts}$$

• 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.

Source: cme.com and sfe.com.au

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.

• 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?

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 $=\mathrm{\$}2,561,475+\mathrm{\$}48,420,000=\mathrm{\$}50,981,475$

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

Figure: Performance of index hedge

• 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.

• 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.

• 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? $$\text{short }{N}^{\ast }=(0.8-0.7)\frac{50,000,000}{25\times 5,583}=35.82\approx 36\text{ contracts}$$

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

• 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.