Introduction

• Generally, there is a high correlation in price movements between the futures market and the cash market.
• Futures position acts as a substitute for later cash transaction.
• Hedge is a position established to minimize the exposure to the unwanted risks. Hedge can be either a short hedge or a long hedge.
• Many of the participants in futures markets are hedgers. They generally take equal and opposite position in cash and futures.  Their aim is to use futures markets to reduce a particular risk that they face. This risk might relate to fluctuations in the price of oil, a foreign exchange rate, the level of the stock market, or some other variable. A perfect hedge is one that completely eliminates the risk. Perfect hedges are rare. For the most part, therefore, a study of hedging using futures contracts is a study of the ways in which hedges can be constructed so that they perform as close to perfectly as possible.

Short And Long Hedge

• Short hedge (selling hedge) is a hedge in which an investors takes a short position in a contract.
• EXAMPLE
  • A farmer expects to harvest 1000 bushels of corn in May 2019.
  • Today, the price of corn is $3.65 per bushel.  The farmer faces the risk of price of corn decreasing in 3 months time from now (i.e. in May).  
  • The 3 months futures contract (i.e. expiry date 31^st May 2019) of the same grade of corn trades at $3.35 per bushel.
  • Farmer has a naturally long position in the cash market (i.e. in May the farmer would be having 1000 bushels of corn).
  • To hedge his position, the farmer places a short hedge (selling hedge), to sell corn bushels in at $3.35 per bushel in the month of May 2019.
  • Long hedge (buying hedge) is a hedge in which an investors takes a long position in a contract.
• EXAMPLE
  • A corn flakes producer, MF Cornos, needs 1000 bushels of corn in April 2019, to produce corn flakes for future demand.
  • Today, the price of corn is $3.65 per bushel. MF Cornos, is concerned about the increase in price of corn in two months time.
  • The 2 months futures contract (i.e. expiry date 30^th April 2019) of the same grade of corn trades at $3.70 per bushel.
  • MF Cornos has a naturally short position in the cash market (i.e. in April the producer would be needing 1000 bushels of corn).
  • To hedge its position, MF Cornos thus places a long hedge (buying hedge), to  buy 1000 bushels of corn in April 2019 at $3.70 per bushel.
  • The previous two examples assume that the futures position is closed out in the delivery month. The hedge has the same basic effect if delivery is allowed to happen. However, making or taking delivery can be costly and inconvenient. For this reason, delivery is not usually made even when the hedger keeps the futures contract until the delivery month. As will be discussed later, hedgers with long positions usually avoid any possibility of having to take delivery by closing out their positions before the delivery period, and the calculations related to profit are different.
• We have also assumed in the two examples that there is no daily settlement. In practice, daily settlement does have a small effect on the performance of a hedge.

Arguments For Hedging

• Most nonfinancial companies have no specific skills or expertise in predicting variables such as interest rates, exchange rates, and commodity prices.  Hence, they need to hedge the risks associated with these variables.
• If financial risks are hedged, the companies can focus on their main activities, without worrying about these risks, and without focusing on market forces over which they have no control.
• By hedging, companies can avoid unpleasant surprises such as sharp rises in the price of a commodity that is being purchased.

Arguments Against Hedging

Hedging and shareholders:

• In theory, shareholders can hedge the risk themselves and  companies don’t need to do it for them. But practically, companies carry out high volume transactions hence cost of hedge is lower per dollar of hedging.
• Also shareholders can diversify away the risks. For example, in addition to holding shares in a company that uses copper, a well-diversified shareholder may hold shares in a copper producer, so that there is very little overall exposure to the price of copper.

Hedging and competitors:

• If hedging is not the norm in a certain industry, it may not make sense for one particular company to choose to be different from all others. Competitive pressures within the industry may be such that the prices of the goods and services produced by the industry fluctuate to reflect raw material costs, interest rates, exchange rates, and so on. A company that does not hedge can expect its profit margins to be roughly constant. However, a company that does hedge can expect its profit margins to fluctuate!

Hedging can lead to a worse outcome

• Hedging using futures contracts can result in a decrease or an increase in a company’s profits relative to the position it would be in with no hedging.

Basis Risk

• With a perfect hedge, the loss on a hedged position will be perfectly offset by the gain on the futures position. Perfect hedges are not easy to implement. This is because of one or more of the following issues :
  • The asset whose price is to be hedged may not be exactly the same as the asset underlying the futures contract.
  • The hedger may be uncertain as to the exact date when the asset will be bought or sold.
  • The hedge may require the futures contract to be closed out before its delivery month.

The existence of either one of these conditions gives rise to what is called basis risk.

• Basis = spot price of asset – futures price of contract. Basis is equal to 0 when spot price = futures price.
• Prior to expiration, the basis may be positive or negative.  As time passes, the spot price and the futures price for a particular month do not necessarily change by the same amount. As a result, the basis changes. The change in basis over the hedge horizon is termed basis risk, and it may work in favor of the hedger or against him. An increase in the basis is knows as a strengthening of the basis, whereas a decrease in the basis is known as a weakening of the basis. If the asset to be hedged and the asset underlying the futures contract are the same, the basis should be zero at the expiration of the futures contract.

• To examine the nature of basis risk, let’s use the following notation:

S₁ : Spot price at time t₁

S₂ : Spot price at time t₂

F₁ : Futures price at time t₁

F₂ : Futures price at time t₂

b₁ : Basis at time t₁

b₂ : Basis at time t₂

where t₁ is the time when the hedge is initiated and t₂ is the time when the hedge is closed out.

• Assume that
  • when the time the hedge is initiated (i.e. at t₁) the spot price is S₁ = $4.00 and futures price is F₁= $3.50
  • when the time the hedge is closed out (i.e. at t₂) the spot price is S₂= $4.2 and futures price is F₂= $3.80
• b₁ = S₁ – F₁
• b₂ = S₂ – F₂

• Consider first the situation of a hedger who knows that the asset will be sold at time t₂ and takes a short futures position at time t₁.
  • The price realized for the asset is S₂ (which is the spot price at which it is sold at that time t₂)
  • The profit on the futures position is F₁-F₂ (since it was shorted at F_1 and closed at F_2).
  • The effective price that is obtained for the asset with hedging is

\(S_2+F_1-F_2=F_1+S_2-F_2=F_1+b_2\)

The value of F_1, is known at time t₁. If \(b_2\) were also known at this time, a perfect hedge would result. The hedging risk is the uncertainty associated with \(b_2\) and is known as basis risk.

If the basis strengthens (i.e., increases) unexpectedly, the hedger’s position improves because it will get a higher price for the asset after futures gains or losses are considered; if the basis weakens (i.e., decreases) unexpectedly, the hedger’s position worsens.

• Consider next the situation of a hedger who knows that the asset will be bought at time t₂ and takes a long futures position at time t₁.
  • The price paid for the asset is S₂ (which is the spot price at which it is bought at that time t₂)
  • The profit on the futures position is F_2-F_1 (as long position was entered at F_1 and closed at F_2).
  • The effective price that is paid for the asset with hedging is

\(S_2-(F_2-F_1)=F_1+S_2-F_2=F_1+b_2\)

This is same as before and the hedging risk is the uncertainty associated with b_2 and is known as basis risk.

If the basis strengthens unexpectedly, the hedger’s position worsens because it will pay a higher price for the asset after futures gains or losses are considered; if the basis weakens unexpectedly, the hedger’s position improves.

Cross Hedging And Choice Of Contract

• The asset that gives rise to the hedger’s exposure is sometimes different from the asset underlying the futures contract that is used for hedging. This is known as cross hedging. Hence cross hedging occurs when the two assets are different. For example, an airline company might choose to use heating oil futures contracts to hedge its exposure to a change in jet oil prices, as jet fuel futures are not actively traded.
• The contract which is used for hedging should be chosen considering the following points:
  • The delivery month that is as close as possible to, but later than end of life of the hedge, is chosen.
  • When there is no futures contract on the asset being hedged, the contract whose futures price is most highly correlated with the asset price is chosen.

Cross Hedging And Hedge Ratio

• The hedge ratio is the ratio of the size of the position taken in futures contracts to the size of the exposure.
• When the asset underlying the futures contract is the same as the asset being hedged, it is natural to use a hedge ratio of 1.0.
• When cross hedging is used, setting the hedge ratio equal to 1.0 is not always optimal. The hedger should choose a value for the hedge ratio that minimizes the variance of the value of the hedged position.
• It can be shown that the optimal hedge ratio \(h^∗\) is the slope of the best-fit line from a linear regression of δS (the change in the spot price during the hedging period)  against δF (the change in the futures price during the hedging period). This result is intuitively reasonable. We would expect \(h^∗\) to be the ratio of the average change in S for a particular change in F
• Optimal Hedge Ratio is given by : \(h^∗=ρ_(S,F)×σ_S/σ_F\)
• where \(σ_S\) is the standard deviation of δS, the change in the spot price during the hedging period. \(σ_F\) is the standard deviation of δF, the change in the futures price during the hedging period. \(ρ_(S,F)\) is the coefficient of correlation between δS and δF.

• The hedge effectiveness can be defined as the proportion of the variance that is eliminated by hedging. This is the R^2 from the regression of δS against δF and equals ρ_(S,F)^2.
• This figure shows how the variance of the value of the hedger’s position depends on the hedge ratio chosen.

Optimal Number Of Contracts

• The optimal number of contracts (\(N^∗\)) to hedge a portfolio is given by

\( N^* = h^* \times \frac{Q_A}{Q_F} \)

where
\(Q_A\) is the size of position being hedged (units)

\(Q_F\) is the size of one futures contract (units)

EXAMPLE – An airline expects to purchase 1 million gallons of jet fuel in 2 months and decides to use heating oil futures for hedging.

\(σ_S\)= 3.5%

\(σ_F\) = 4.1%

\(ρ_(S,F)\)= 89%

Tailing The Hedge

• The daily settlement of futures contract means that, when futures contracts are used, there are a series of one-day hedges, not a single hedge. Instead of the quantities (\(Q_A\)) and (\(Q_F\)), the dollar value of the position being hedged (\(V_A\)) and the dollar value of one futures contract (\(V_F\)), is used. So the number of contracts is given by

\( N^* = h^* \times \frac{V_A}{V_F} = h^* \times \frac{Q_A}{Q_F} \times \frac{S}{F} \)

In effect, the hedge ratio is multiplied by the daily spot price to futures price ratio.

EXAMPLE –

• In practice, it is not efficient to adjust the hedge for every daily change in the spot-to-futures ratio.

Stock Index Futures – Optimal Number Of Contracts

• Stock index futures can be used to hedge a well-diversified equity portfolio. While hedging a well diversified equity portfolio using stock index futures, the optimal hedge ratio can be assumed to be 1.0 if the portfolio mirrors (or traces) the index. Hence the number of contracts to be shorted is given by

\( N^* = \frac{V_A}{V_F} \)

where
\(V_A\) = Current value of the portfolio

\(V_F\) = Current value of one futures contract (the futures price times multiplier).

• If you have a portfolio of $500,000 which mirrors S&P 500. Each S&P 500 contract is $250 times the index when the index is at 500. Calculate the number of contracts to be hedged?
• In practical cases investors don’t typically have portfolios that mirror the index. So we can use the parameter beta β from the capital asset pricing model as the optimal hedge ratio.
• β is the slope of the best-fit line obtained when excess return on the portfolio over the risk-free rate is regressed against the excess return of the index over the risk-free rate. Hence, the number of contracts to be shorted is given by

\( N^* = \beta \frac{V_A}{V_F} \)

Changing The Beta Of A Portfolio

• A well diversified $150 million portfolio has a beta of 1.32, with respect to the S&P 500. The March 18 S&P 500 futures are trading at 2,726.50, and the multiplier is 250. Find the number of S&P 500 contracts you need hedge your exposure over the next month.
• The current value of the S&P 500 index is 2,703, and each S&P futures contract is for delivery of 250 times the index. A well diversified equity portfolio with market value of USD 2,500,00,000 has beta of 1.05. Find out how many contracts you need to buy or sell to change the portfolio beta to a) 0.6 b)1.25

Stock Index Futures – Optimal Number Of Contracts

• In order to change the original beta (β) of the portfolio to a target beta (β^∗), we need to long or short the N^∗ number of contracts depending on the sign of N^∗. The number of contracts used for hedging is given by

\( N^* = (\beta^* – \beta) \frac{V_A}{V_F} \)

When \(β^∗<β\), the sign of \(N^∗\) is negative and we need to short the contracts.

When \(β^∗>β\), the sign of \(N^∗\) is positive and we need to go long in the contracts

Stack And Roll

• If the expiration date of the hedge is later than the delivery dates of all the futures contracts that are available, then the hedge is rolled forward by closing out one futures contract and taking the same position in a futures contract with a later delivery date.
• Hedges can be rolled forward many times. The procedure is known as stack and roll.
• When the hedge is rolled forward, hedgers are exposed to the basis risk of a new position each time the hedge is rolled forward. This is known as rollover (basis) risk.