Chapter 3: Hedging Strategies Using Futures

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Basic Principles

A hedge is a trade designed to reduce risk. When hedging with futures:

A short hedge is appropriate when you will sell an asset in the future and want to lock in the price
A long hedge is appropriate when you will purchase an asset in the future

Perfect hedge: eliminates all risk. Rare in practice due to basis risk, contract size mismatches, and timing mismatches.

Arguments For and Against Hedging

For Hedging

Hedging allows companies to focus on their core business without worrying about commodity prices or exchange rates
Reduces the probability of financial distress
May improve a company's access to debt financing

Against Hedging

Shareholders can diversify away risks themselves
If competitors don't hedge, a hedging company's profits may fluctuate more in absolute terms
Hedging introduces basis risk (an imperfect hedge may be worse than no hedge)

Basis Risk

The basis is defined as:

$used\text{Basis} = \text{Spot price of asset to be hedged} - \text{Futures price of contract used}$

Basis risk arises from uncertainty about the basis at the time the hedge is closed out. For a short hedge:

$price=S2+(F1−F2)=F1+(S2−F2)=F1+basis2\text{Effective price} = S_2 + (F_1 - F_2) = F_1 + (S_2 - F_2) = F_1 + \text{basis}_2$

where $S_2$ is spot price when hedge is closed, $F_1$ is initial futures price, $F_2$ is futures price when closing. The basis $S_2 - F_2$ is unknown at the start.

Key drivers of basis risk:

Asset being hedged differs from the asset underlying the futures contract
Uncertainty about the exact date of purchase/sale
Hedge may need to be closed before delivery month

Cross Hedging

When the asset being hedged is not exactly the same as the asset underlying the futures contract, we perform a cross hedge. The hedge ratio is the ratio of the size of the position in futures to the size of the exposure.

The minimum variance hedge ratio is:

$h∗=ρ⋅σSσFh^* = \rho \cdot \frac{\sigma_S}{\sigma_F}$

where:

$ρ\rho$ = correlation between change in spot price and change in futures price
$σS\sigma_S$ = standard deviation of change in spot price
$σF\sigma_F$ = standard deviation of change in futures price

The optimal number of contracts:

$N∗=h∗⋅QAQFN^* = h^* \cdot \frac{Q_A}{Q_F}$

where $Q_A$ = size of position being hedged, $Q_F$ = size of one futures contract.

Example: An airline wants to hedge 2 million gallons of jet fuel using heating oil futures. If $σS=0.0263\sigma_S = 0.0263$ , $σF=0.0313\sigma_F = 0.0313$ , $ρ=0.928\rho = 0.928$ , then $h∗=0.928×0.0263/0.0313=0.78h^* = 0.928 \times 0.0263/0.0313 = 0.78$ . With heating oil contracts of 42,000 gallons: $N∗=0.78×2,000,000/42,000=37.1N^* = 0.78 \times 2,000,000/42,000 = 37.1$ contracts.

Stock Index Futures

Stock index futures can hedge equity portfolios. The capital asset pricing model (CAPM) provides the theoretical basis:

$r_f + \beta \cdot (r_m - r_f)$

To change the beta of a portfolio from $β\beta$ to $β∗\beta^*$ , the number of futures contracts needed is:

$(\beta^* - \beta) \cdot \frac{V_A}{V_F}$

where $V_A$ is the portfolio value and $V_F$ is the futures contract value (futures price × multiplier).

To eliminate risk completely: Set $β∗=0\beta^* = 0$ , giving $-\beta \cdot V_A/V_F$ (short futures).

Stack and Roll

When hedging horizons exceed the delivery dates of available futures contracts, a stack and roll strategy is used:

Enter into a short maturity futures contract
Close it out and roll into the next contract

This introduces additional basis risk. The Metallgesellschaft case (1993) is a famous example where stack-and-roll went wrong — the company lost $1.3 billion when oil prices fell and margin calls became unsustainable.

Appendix: Capital Asset Pricing Model (CAPM)

The CAPM states that the expected return on an asset is:

$E(ri)=rf+βi⋅[E(rm)−rf]E(r_i) = r_f + \beta_i \cdot [E(r_m) - r_f]$

The beta of a portfolio is the weighted average of the betas of the constituent assets. Stock index futures contracts have a beta of approximately 1.0.