A Note On Currency And Index Futures
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TRADING USING CURRENCY FUTURES Contd..
Optimal Hedge Ratio
A hedger has to determine the number of futures contracts to buy in order
for the best hedge for his/her return profile. A hedge ratio is the ratio of
the size of the future position to the size of the underlying asset
position. In terms of futures, it is be defined as the number of futures
contracts to hold for a given position in an underlying asset.
A hedge ratio of 1.0 is not always optimal and will give a perfect hedge
only when there is no change in the basis. An illustration will explain the
concept better:
Let
D St = Change in the spot price during the period of hedge
D Ft = Change in the futures prices during the period of hedge
σ St = Standard deviation of
D St
σ
Ft = Standard deviation of
D Ft
r = Coefficient of correlation between
D St and D Ft
h = hedge ratio
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 When the hedger has taken a long position on the asset
and is short on futures, the change in the value of the hedger's
position during the period of the hedge would be:
D St – h
D Ft
When the hedger is long on futures it is
h D Ft -
D St
Whatever may be the hedge, the variance, v, in either case would be
given by:
V =
σ2 St+ h2 σ2 Ft –-2hrσStσFt
So that, first derivative with respect to the hedge is
= 2hσ2
Ft - 2rσStσFt………………… (Equation 1)
Setting equation 1 equal to zero and noting that the second derivative
is positive, the value of h, which minimizes the variance, is
σSt
h= r ———
σFt
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Therefore the product of the
coefficient of correlation between St and Ft and the ratio of the
standard deviation of St to the standard deviation of Ft, gives the
optimal hedge ratio. If the spot price and the futures price are perfectly
positively correlated and St = Ft, the optimal hedge ratio would be 1.0.
Example:
A company anticipates that it will be requiring 2 million gallons of heating
oil in six months. The standard deviation of the change in the price per
gallon of heating oil over six months period is calculated as 0.045. The
company decides to buy a six months futures contract on unleaded gasoline to
hedge the risk. The standard deviation of the change in the future price
over a six-month period is 0.054 and the coefficient of correlation between
the two is 0.12. The optimal hedge ratio is therefore
0.12 X (0.045/0.054) = 0.10
One unleaded gasoline futures contract is on 42,000 gallons. The number of
contracts the company should therefore buy is
0.10 X 20,00,000/42,000 = 4.76 contracts
Rounding off the decimal places to the nearest whole number, 5 contracts
have to be purchased.
TRADING USING INDEX FUTURES
CONCLUSION
EXHIBIT I DIFFERENCE BETWEEN FUTURES AND FORWARDS CONTRACTS
EXHIBIT II A NOTE ON ANALYZING FUTURES PRICES
ADDITIONAL READINGS AND REFERENCES
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