A trader owns 100,000 units of a particular asset and decides to hedge the value
ID: 2612273 • Letter: A
Question
A trader owns 100,000 units of a particular asset and decides to hedge the value of her
position with futures contracts on another related asset. Each futures contract is written on 5,000 units. The spot price of the asset that is owned is $76 and the standard deviation of the
change in this price over the life of the hedge is estimated to be $1.33. The futures price of
the related asset is $80 and the standard deviation of the change in this futures price over the life of the hedge is $1.55. The coefficient of correlation between the spot price change and futures
price change is 0.92.
(a) What is the minimum variance hedge ratio?
(b) Should the hedger take a long or short futures position?
(c) What is the optimal number of futures contracts with no tailing of the hedge?
(d) What is the optimal number of futures contracts with tailing of the hedge?
Explanation / Answer
Amount of Asset owned by trader A = 100,000 units
Spot Price of the Asset P1= $76
Standard Deviation of price over life of hedge SD1 = 1.33
Future Price of Related Asset F = $ 80
Standard Deviation of price over life of contract SD2 = 1.55
Futures Contract Size of related Asset S = 5,000 Units
Coefficient of Correlation between Spot price and Future price changes p = 0.92
Answer (a)
Minimum Variance Hedge Ratio - This is a ratio of futures position relative to the spot position that minimizes the variance of the position. This is calculated using the formula
Minimum Variance Hedge Ratio h = p* SD(S)/SD(F) where
p = correlation coefficient
SD(S) = Standard Deviation of Spot Price change
SD(F) = Standard Deviation of Future Price change
Substituting above values
h = 0.92 * (1.33/1.55) ==> h = 0.92 * 0.858
h = 0.789
Answer (b)
As the trader already owns the asset he should take a short position in futures
Answer (C)
No of units of Asset owned A = 100,000
Contract size of related asset underlying futures contract S = 5,000
Minimum Variance Hedge ratio h = 0.789
Optimal number of Futures contracts to be held without tailing can be calculated by using the formula
N = h * A / s = 0.789 * 100,000 / 5,000 = 15.78 or 16 contracts
Answer (d)
Optimal Number of Futures contracts with tailing can be calculated by using the formula
N = h* (A*P1)/(S*F) = 0.789 * (100000*76)/(5000*80)
N = 0.789 * 7,600,000/400,000 = 14.991 or 15 Contracts
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