When the expected return on the market is ?30% the expected return on a portfolio with a beta of 0.2 is
0.05 + 0.2 × (?0.30 ? 0.05) = ?0.02
or –2%. The actual return of –10% is worse than the expected return. The portfolio manager has achieved an alpha of –8%!
Problem 3.27.
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 CME December futures contract on the S&P 500 to change the beta of the portfolio to 0.5 during the period July 16 to November 16. The index is currently 1,000, and each contract is on $250 times the index. a) What position should the company take?
b) Suppose that the company changes its mind and decides to increase the beta of the portfolio from 1.2 to 1.5. What position in futures contracts should it take?
a) The company should short
(1?2?0?5)?100?000?000
1000?250or 280 contracts.
b) The company should take a long position in
(1?5?1?2)?100?000?000
1000?250or 120 contracts.
Problem 3.28. (Excel file)
The following table gives data on monthly changes in the spot price and the futures price for a certain commodity. Use the data to calculate a minimum variance hedge ratio. Spot Price Change ?0?50 ?0?61 ?0?22 ?0?35 ?0?79 Futures Price Change ?0?56 ?0?63 ?0?12 ?0?44 ?0?60 Spot Price Change Futures Price Change ?0?04 ?0?06 ?0?15 ?0?01 ?0?70 ?0?80 ?0?51 ?0?56 ?0?41 ?0?46
Denotexiandyiby the i-th observation on the change in the futures price and the change in the spot price respectively.
?xi?0?96?yi?1?30
?x2i?2?4474?y2i?2?3594
?xyii?2?352
An estimate of ?F is
2?44740?962??0?5116 910?92?35941?302??0?4933 910?910?2?352?0?96?1?3022An estimate of ?S is
An estimate of ? is
(10?2?4474?0?96)(10?2?3594?1?30)The minimum variance hedge ratio is
?0?4933 ?S?0?981??0?946
?F0?5116
Problem 3.29.
It is now October 2013. A company anticipates that it will purchase 1 million pounds of copper in each of February 2014, August 2014, February 2015, and August 2015. The
company has decided to use the futures contracts traded in the COMEX division of the CME Group to hedge its risk. One contract is for the delivery of 25,000 pounds of copper. The initial margin is $2,000 per contract and the maintenance margin is $1,500 per contract. The company?s policy is to hedge 80% of its exposure. Contracts with maturities up to 13 months into the future are considered to have sufficient liquidity to meet the company?s needs. Devise a hedging strategy for the company.
Assume the market prices (in cents per pound) today and at future dates are as follows. What is the impact of the strategy you propose on the price the company pays for copper? What is the initial margin requirement in October 2013? Is the company subject to any margin calls? Date Spot Price Mar 2014 futures price Sept 2014 futures price Mar 2015 futures price Sept 2015 futures price Oct 2013 372.00 372.30 372.80 Feb 2014 369.00 369.10 370.20 370.70 Aug 2014 365.00 364.80 364.30 364.20 Feb 2015 377.00 376.70 376.50 Aug 2015 388.00 388.20 ?0?981
To hedge the February 2014 purchase the company should take a long position in March 2014 contracts for the delivery of 800,000 pounds of copper. The total number of contracts required is 800?000?25?000?32. Similarly a long position in 32 September 2014 contracts is required to hedge the August 2014 purchase. For the February 2015 purchase the company could take a long position in 32 September 2014 contracts and roll them into March 2015 contracts during August 2014. (As an alternative, the company could hedge the February 2015 purchase by taking a long position in 32 March 2014 contracts and rolling them into March 2015 contracts.) For the August 2015 purchase the company could take a long position in 32 September 2014 and roll them into September 2015 contracts during August 2014.
The strategy is therefore as follows
Oct 2013: Enter into long position in 96 Sept. 2014 contracts
Enter into a long position in 32 Mar. 2014 contracts
Feb 2014: Close out 32 Mar. 2014 contracts Aug 2014: Close out 96 Sept. 2014 contracts
Enter into long position in 32 Mar. 2015 contracts Enter into long position in 32 Sept. 2015 contracts
Feb 2015: Close out 32 Mar. 2015 contracts Aug 2015: Close out 32 Sept. 2015 contracts
With the market prices shown the company pays 369?00?0?8?(372?30?369?10)?371?56 for copper in February, 2014. It pays 365?00?0?8?(372?80?364?80)?371?40
for copper in August 2014. As far as the February 2015 purchase is concerned, it loses 372?80?364?80?8?00 on the September 2014 futures and gains 376?70?364?30?12?40 on the February 2015 futures. The net price paid is therefore
377?00?0?8?8?00?0?8?12?40?373?48
As far as the August 2015 purchase is concerned, it loses 372?80?364?80?8?00 on the September 2014 futures and gains 388?20?364?20?24?00 on the September 2015 futures. The net price paid is therefore
388?00?0?8?8?00?0?8?24?00?375?20
The hedging strategy succeeds in keeping the price paid in the range 371.40 to 375.20. In October 2013 the initial margin requirement on the 128 contracts is 128?$2?000 or $256,000. There is a margin call when the futures price drops by more than 2 cents. This happens to the March 2014 contract between October 2013 and February 2014, to the September 2014 contract between October 2013 and February 2014, and to the September 2014 contract between February 2014 and August 2014. (Under the plan above the March 2015 contract is not held between February 2014 and August 2014, but if it were there would be a margin call during this period.)
Problem 3.30. (Excel file)
A fund manager has a portfolio worth $50 million with a beta of 0.87. The manager is concerned about the performance of the market over the next two months and plans to use three-month futures contracts on the S&P 500 to hedge the risk. The current level of the
index is 1250, one contract is on 250 times the index, the risk-free rate is 6% per annum, and the dividend yield on the index is 3% per annum. The current 3 month futures price is 1259. a) What position should the fund manager take to eliminate all exposure to the market over the next two months?
b) Calculate the effect of your strategy on the fund manager?s returns if the level of the market in two months is 1,000, 1,100, 1,200, 1,300, and 1,400. Assume that the one-month futures price is 0.25% higher than the index level at this time.
a) The number of contracts the fund manager should short is
50?000?0000?87??138?20
1259?250Rounding to the nearest whole number, 138 contracts should be shorted.
b) The following table shows that the impact of the strategy. To illustrate the
calculations in the table consider the first column. If the index in two months is 1,000, the futures price is 1000×1.0025. The gain on the short futures position is therefore (1259?1002?50)?250?138?$8?849?250
The return on the index is 3?2?12=0.5% in the form of dividend and
?250?1250??20% in the form of capital gains. The total return on the index is therefore ?19?5%. The risk-free rate is 1% per two months. The return is therefore ?20?5% in excess of the risk-free rate. From the capital asset pricing model we expect the return on the portfolio to be 0?87??20?5%??17?835% in excess of the risk-free rate. The portfolio return is therefore ?16?835%. The loss on the portfolio is 0?16835?50?000?000 or $8,417,500. When this is combined with the gain on the futures the total gain is $431,750. Index now12501250125012501250Index Level in Two Months10001100120013001400Return on Index in Two Months-0.20-0.12-0.040.040.12Return on Index incl divs-0.195-0.115-0.0350.0450.125Excess Return on Index-0.205-0.125-0.0450.0350.115Excess Return on Portfolio-0.178-0.109-0.0390.0300.100Return on Portfolio-0.168-0.099-0.0290.0400.110Portfolio Gain-8,417,500-4,937,500-1,457,5002,022,5005,502,500Futures NowFutures in Two MonthsGain on FuturesNet Gain on Portfolio125912591259125912591002.501102.751203.001303.251403.508,849,2505,390,6251,932,000-1,526,625-4,985,250431,750453,125474,500495,875517,250