After completing this reading, you should be able to:

- Define and differentiate between short and long hedges and identify their appropriate uses.
- Describe the arguments for and against hedging and the potential impact of hedging on firm profitability.
- Define the basis and explain the various sources of basis risk, and explain how basis risks arise when hedging with futures.
- Define cross hedging, and compute and interpret the minimum variance hedge ratio and hedge effectiveness.
- Compute the optimal number of futures contracts needed to hedge an exposure, and explain and calculate the “tailing the hedge” adjustment.
- Explain how to use stock index futures contracts to change a stock portfolio’s beta.
- Explain the term “rolling the hedge forward” and describe some of the risks that arise from this strategy.

## Short Hedges vs. Long Hedges

A **short hedge** occurs when the trader shorts (sells) a futures contract to hedge against a price decrease in an existing long position. A short hedger already owns the underlying asset or is likely to gain ownership of the asset in the near future, after which they will sell it at some point in future. When the price of the hedged/underlying asset decreases, the short futures position realizes a corresponding positive return that offsets the loss in asset value.

A **long hedge** occurs when the trader buys a futures contract to hedge against a price increase in an existing short position. A long hedger plans to buy the underlying asset in future and fears a rise in price, triggering a loss. When the price of the hedged/underlying asset increases causing a loss, the long futures position realizes a corresponding positive return that offsets the loss in asset value.

## Advantages and Disadvantages of Hedging

### Advantages:

**It helps asset holders to lock in a price for their assets.**By taking a short position, a corn farmer, for example, who is anticipating a bumper harvest in a few months is able to lock in a predetermined price for their corn. By so doing, they eliminate – or at least reduce the risk of a price decrease.**It helps prospective buyers to lock in a price for the goods they intend to purchase.**Instead of a cereal company waiting to buy corn at the prevailing post-harvest price, the company can lock in a predetermined purchase price by getting into a long futures contract. Even if prices rise dramatically between the signing of the contract and the maturity date, the company will benefit from a fixed price.

### Disadvantages:

**Hedging might lock asset holders out of improving market prices.**Although hedging shields asset holders from price declines, it locks them out of increases in value. Even if the asset’s price rises, the short futures contract holder is obliged to honor all the terms of the deal. They must sell the underlying at the contract price.**Hedging is necessarily not beneficial to a company’s shareholders.**Risk-averse shareholders are assumed to hold diversified portfolios that mitigate specific risks and reduce the effect of systematic risks. Thus, hedging at the company level doesn’t necessarily benefit individual shareholders.

## Basis Risk and Its Causes

Basis risk is the risk that the value of a futures contract will not move in normal, steady correlation with the price of the underlying asset. For example, if the current spot price of \(gold\) is \($1,500\), and the six-month futures price of gold is \($1,550\), then the basis, the differential, is \($50\). Basis risk, in this case, is the risk that between now and maturity of the contract in six months, the price of gold will fluctuate by more than \($50\).

The fluctuation in the basis makes hedges less effective than they are meant to be. Between contract initiation and liquidation, the price spread (the difference between the cash price and futures price) may either narrow or widen.

### Sources of basis risk:

- Imperfect matching between the cash asset and the hedge asset, e.g., hedging jet fuel with motor vehicle fuel.
- Changes in the components of the cost of carry, e.g., interest, storage and safekeeping, and insurance.
- Maturity mismatch, e.g., hedging an exposure to physical prices in May with a June futures contract.
- Location mismatch, e.g., hedging crude oil sold in New York with crude oil futures traded on a Mumbai futures exchange.

To minimize basis risk, it’s imperative to choose the hedge tool that’s most correlated with the underlying.

## Cross Hedging

There are instances when it may be impossible to find futures contracts on a particular underlying. In such scenarios, the hedger may turn to futures on securities that exhibit positive correlation with the underlying. This is called cross hedging. The hedger takes opposite positions in the two assets. Since the assets are not entirely identical, there must be enough correlation for the hedge to work.

### The Optimal Hedge Ratio

The optimal hedge ratio, also called the minimum variance ratio, is the degree of correlation between the underlying asset and the futures contract purchased to hedge financial risks. It’s the ratio of the futures position to the spot position.

$$ HR={ \rho }_{ SF }\times \frac { { \sigma }_{ S } }{ { \sigma }_{ F } } $$

Where:

\({ \rho }_{ SF }\)=correlation between spot prices and the future prices

\({ \sigma }_{ S }\)=standard deviation of the spot price

\({ \sigma }_{ F }\)=standard deviation of the futures price

Note:

$$ \rho =\frac { { Cov }_{ SF } }{ { \sigma }_{ S }{ \sigma }_{ F } } $$

And,

$$ \frac { { Cov }_{ SF } }{ { \sigma }_{ S }{ \sigma }_{ F } } \times \frac { { \sigma }_{ S } }{ { \sigma }_{ F } } =\frac { { Cov }_{ SF } }{ { \sigma }_{ F }^{ 2 } } ={ \beta }_{ SF } $$

The effectiveness of a hedge measures the amount of variance that’s reduced by implementing the optimal hedge ratio. It can be evaluated using \({ R }^{ 2 }\), the coefficient of determination, where:

Independent variable = change in futures price, and

Dependent variable = change is spot price

## The Optimal Number of Futures Contracts Needed to Hedge an Exposure

The number of futures contracts required to completely hedge an equity position is given by:

$$ N={ \beta }_{ portfolio }\times \left[ \frac { portfolio\quad value }{ value\quad of\quad futures\quad contract } \right] $$

$$ ={ \beta }_{ portfolio }\times \left[ \frac { portfolio\quad value }{ futures\quad price\times contract\quad multiplier } \right] $$

### Trailing the Hedge

There is always the risk that a hedger will over-hedge the underlying exposure. The trailing the hedge strategy is implemented to avoid this. It involves multiplying the number of futures contracts required by the daily spot price to futures price ratio.

### Adjusting a Stock Portfolio’s Beta using Stock Index Futures

Beta, as defined in the capital asset pricing model, is a measure of a portfolio’s systematic risk. When a trader uses index futures to hedge a position in an equity portfolio, they are effectively trying to reduce the portfolio’s systematic risk. As such, hedging is actually an attempt to reduce a portfolio’s beta.

Let’s define the following variables:

\(\beta\)=portfolio beta

\({ \beta }^{ \ast }\)=target beta after hedging

\(P\)=portfolio value

\(A\)=value of the underlying

Number of contracts required=\(\left( { \beta }^{ \ast }-\beta \right) \left( \frac { P }{ A } \right) \)

If the above result is positive, the trader would have to buy the relevant number of futures. If negative, they would have to sell futures.

### Rolling Hedge

A rolling hedge is a strategy used to replace expiring futures contracts by obtaining new exchange-traded futures. The trader gets a new contract with similar terms, except for the maturity date. Typically, as a maturity date nears, the trader closes out the existing futures position in readiness for a new position.

However, a rolling hedge strategy comes with **roll-over risk**. The trader is exposed to basis risk each time the hedge is rolled forward.

## Questions

### Question 1

Peter McLeish, FRM, is a portfolio manager in charge of a \($200 \quad million\) portfolio that has a beta of \(1.5\) relative to the Nasdaq-100. The Nasdaq-100 futures are trading at \(2,400\), and the multiplier is \(300\). Mr. McLeish intends to hedge his exposure to market risk over the coming months. Determine the type and number of Nasdaq-100 contracts required to implement the hedge.

- Buy 417 contracts
- Sell 417 contracts
- Buy 277 contracts
- Sell 277 contracts

The correct answer is **B**.

The manager is long the Nasdaq-100, so he should construct a short hedge and sell futures contracts. The exact number to sell is given by:

$$ N={ \beta }_{ portfolio }\times \left[ \frac { portfolio\quad value }{ futures\quad price\times contract\quad multiplier } \right] $$

$$ =1.5\times \frac { $200,000,000 }{ 2,400\times 300 } $$

$$ =417 \quad contracts $$

### Question 2

Rachel Zane, FRM, is a portfolio manager in charge of a \($200 \quad million\), well-diversified portfolio that has a beta of \(1.5\) relative to the Nasdaq-100. The current value of the 3-month Nasdaq-100 Index is \(2,500\), and the multiplier is \(300\). Over the next three months, Ms. Zane wants to use the Nasdaq-100 futures to reduce the systematic risk of the portfolio to \(1.0\). To pull that off, which of the following moves is required?

- Sell 100 contracts
- Buy 133 contracts
- Sell 133 contracts
- Buy 100 contracts

The correct answer is **C**.

Number of contracts required=\(\left( { \beta }^{ \ast }-\beta \right) \left( \frac { P }{ A } \right) \)

Where:

\(\beta\)=portfolio beta

\( { \beta }^{ \ast }\)=target beta after hedging

\(P\)=portfolio value

\(A\)=value of futures contract

$$ =1.0-1.5\times \frac { 200,000,000 }{ 2,500\times 300 } $$

$$ =-133 $$

The negative sign implies 133 contracts need to be sold.