Cross Hedging occurs when the asset underlying the hedger’s exposure is different from the asset underlying the
futures contract that is used for hedging, and leads to an increase in basis risk.
Hedge Ratio is the ratio of the size of the position in the futures contract to the size of the exposure. When the
asset underlying the futures contract is the same as the asset being hedged, the hedge ratio = _________.
When employing a cross‐hedge, we must choose a value for the hedge ratio that will minimize the variance of the
value of the hedged position.
Assume no daily settlement in futures. The minimum variance hedge ratio depends on the relation between
changes in the spot price (ΔS) and changes in the futures price (ΔF):
ΔS = change in the spot price (S) during a period of time equal to the life of the hedge
ΔF = change in the futures price (F) during a period of time equal to the life of the hedge
Denote the minimum variance hedge ratio by h*:
In the above, the hedge ratio is based on regressing actual changes in spot prices on actual changes in futures
prices.
σS = standard deviation of ΔS; σF = standard deviation of ΔF
ρ = correlation between the above two changes
The above parameters are typically estimated from historical data on ΔS and ΔF by choosing a number of non‐
overlapping intervals. Ideally, the length of each time interval should be equal to length of the time interval of the
hedge. Practically, this may not be feasible and a shorter time interval is used instead.
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Optimal Number of Contracts (N*) for Hedging:
QA = size of the position being hedged (units)
QF = size of one futures contract (units)
h* = minimum variance hedge ratio
Example: An airline expects to purchase 2 million gallons of jet fuel in 1 month and decides to use heating‐oil
futures for hedging. Size of 1 heating‐oil futures contract = 42,000 gallons. Here, σS = 0.0263; σF = 0.0313; ρ = 0.928:
The above analysis is appropriate for forward contracts. With futures and daily settlement, get a series of one‐day
hedges.
We can derive an alternative hedge ratio by regressing daily percentage changes in spot on daily percentage
changes in futures prices:
VA = SQA is the value of the position being hedged; VF = FQF is the futures price x size of contract
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Fin 662 – Chapter 3
Continuing with the example of Page 9, spot price = 1.10, futures price = 1.30:
Stock Index Futures
Stock Index futures are settled in cash rather than by physical delivery of the underlying asset. We can use stock
index futures to hedge a well‐diversified equity portfolio.
If the portfolio mirrors the index, then the optimal hedge ratio = _____ and
Value of one futures contract = Futures Price x Contract Size
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Fin 662 – Chapter 3
Example: A portfolio with a value of $5,050,000 mirrors a well‐diversified index.
Index futures price = 1,010; contract ‘multiplier’ = 250 (each contract is on $250 times the index):
When the portfolio does not mirror the index, then we use beta (β) from the CAPM to calculate the number of
futures contracts to use. β is based on the regression of excess return of the portfolio ON the excess return of the
index:
The above formula assumes that the maturity of the futures contract is close to the maturity of the hedge.
Example: Suppose that we wish to hedge the value of a portfolio over the next three months with a 4‐month futures
contract:
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Fin 662 – Chapter 3
Index level = 1000, Index futures price = 1010, value of the portfolio = 5,050,000, contract multiplier 250,
Beta (β) of portfolio = 1.5, risk‐free interest rate = 4% per annum, and dividend yield on index = 1% per annum:
In 3 months, Index = 900, Futures price = 902, dividend yield over 3 months = 0.25%
Use CAPM to get E(Rp) over 3 months:
Refer to Table 3.4: Total value of the hedged portfolio is almost independent of the value of the index.
When return = risk‐free rate, then β = 0.
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Fin 662 – Chapter 3
Changing the Beta of a Portfolio
We can use futures contracts to change beta to a value other than zero.
In the above example, we need to short 30 contracts for a complete hedge. To change the beta of a portfolio from
β to β*:
Go back to the example where we need to short 30 contracts for a complete hedge. Now,
i) to reduce the beta of the portfolio from 1.50 to 0.75, we must
ii) to increase the beta of the portfolio from 1.50 to 2, we must
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Fin 662 – Chapter 3
One can use stock index futures to lock in the benefits of stock picking (a single stock or a small portfolio) by hedging
‘market’ risk.
One must short
β = beta of the portfolio; VA = total value of the portfolio to be hedged; VF = current value of one futures contract
Example: It is April. You hold 20,000 shares of a stock whose price = 100 per share and beta = 1.10. You will use the
August futures contract on the Mini S&P 500 to hedge the market return over the next 3 months.
Current futures price = 2100, contract multiplier = $50:
Number of futures contracts to be ______________:
Close out the position in July: Futures price = 1850, Stock price = 90
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Fin 662 – Chapter 3
Stack and Roll
What should a hedger do if the expiration date of the hedge is later than the delivery dates of all usable futures
contracts? Here, the hedger can roll the hedge forward by closing out one futures contract and taking the same
position in a futures contract with a later delivery date – this can be done many times. This process is referred to
as Stack and Roll.
Example: It is April 2021. A company plans to sell 100,000 barrels of oil in June 2022. Current spot price = $49.
Only the first six delivery months have sufficient liquidity. See Table 3.5 in the textbook. Contract size = 1000 barrels:
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