Chapter 3

Suppose the one-year gold lease rate is 1.5% and the one-year risk-free rate is 5.0%. Both rates are compounded annually. Use the discussion in Business Snapshot 3.1 to calculate the maximum one-year forward price Goldman Sachs should quote for gold when the spot price is $1,200.

Goldman Sachs can borrow 1 ounce of gold and sell it for $1200. It invests the $1,200 at 5% so that it becomes $1,260 at the end of the year. It must pay the lease rate of 1.5% on $1,200. This is $18 and leaves it with $1,242. It follows that if it agrees to buy the gold for less than $1,242 in one year it will make a profit.

It is July 16. A company has a portfolio of stocks worth $100 million. The beta of the portfolio is 1.2. Suppose that the company decides to increase the beta of the portfolio from 1.2 to 1.5 during the period July 16 to November 16. The index is currently 2,000, and each contract is on $250 times the index. What position should the company take?

Number of contracts that should be bought to increase beta to 1.5 is: (1.5-1.2)(100,000,000/(250*2000)=60 contracts

It is July 16. A company has a portfolio of stocks worth $100 million. The beta of the portfolio is 1.2. The company would like to use the December futures contract on a stock index to change beta of the portfolio to 0.5 during the period July 16 to November 16. The index is currently 2,000, and each contract is on $250 times the index. What position should the company take?

Number of contracts that should be shorted to decrease beta to 0.5 is: (1.2-0.5)(100,000,000/(250*2000)=140 contracts

A company has a $20 million portfolio with a beta of 1.2. It would like to use futures contracts on a stock index to hedge its risk. The index futures is currently standing at 1080, and each contract is for delivery of $250 times the index. What should the company do if it wants to reduce the beta of the portfolio to 0.6?

Number of contracts that should be shorted to reduce beta to 0.6 is: (1.2-0.6)(20,000,000/(250*1080)=44 contracts

Sixty futures contracts are used to hedge an exposure to the price of silver. Each futures contract is on 5,000 ounces of silver. At the time the hedge is closed out, the basis is $0.20per ounce. What is the effect of the basis on the hedger's financial position if (a) the trader is hedging the purchase of silver and (b) the trader is hedging the sale of silver?

The excess of the spot over the futures at the time the hedge is closed out is $0.20 per ounce. If the trader is hedging the purchase of silver (long), the price paid is the futures price plus the basis. The trader therefore loses 60×5,000×$0.20=$60,000. If the trader is hedging the sales of silver (short), the price received is the futures price plus the basis. The trader therefore gains $60,000

A company wishes to hedge its exposure to a new fuel whose price changes have a 0.6 correlation with gasoline futures price changes. The company will lose $1 million for each 1 cent increase in the price per gallon of the new fuel over the next three months. The new fuel's price change has a standard deviation that is 50% greater than price changes in gasoline futures prices. If gasoline futures are used to hedge the exposure what should the hedge ratio be? What is the company's exposure measured in gallons of the new fuel? What position measured in gallons should the company take in gasoline futures? How many gasoline futures contracts should be traded? Each contract is on 42,000 gallons.

The hedge ratio should be 0.6 x 1.5 = 0.9. The company has an exposure to the price of $1 million /$0.01 = 100 million gallons of the new fuel. The company should take a position of 90 million gallons in gasoline futures. (Hedge ratio 0.9 x $100 million = $90 million) Each futures contract is on 42,000 gallons. The number of contracts required would be 90,000,000/42,000 = 2142.857 or, rounding to the nearest whole number, 2143.

On March 1 a commodity's spot price is $60 and its August futures price is $59. On July 1 the spot price is $64 and the August futures price is $63.50. A company entered into futures contracts on March 1 to hedge its purchase of the commodity on July 1. It closed out its position on July 1. What is the effective price (after taking account of hedging) paid by the company?

The user of the commodity takes a long futures position. The gain on the futures is 63.50−59 or $4.50. The effective paid realized is therefore 64−4.50 or $59.50. This can also be calculated as the March 1 futures price (=59) plus the basis on July 1 (=0.50).

A company has a $36 million portfolio with a beta of 1.2. The index futures price is 900. Futures contracts on $250 times the index can be traded. What trade is necessary to increase beta to 1.8?

To increase beta by 0.6 we need to go long 0.6×36,000,000/(900×250) or 96 contracts