Action now cf in action at period end cf in buy one

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Action Now CF in $ Action at period-end CF in $ Buy one TOBEC index futures contract 0 Sell one TOBEC index futures contract $100 × (F 1 F 0 ) Sell spot TOBEC index +$18,500 Buy spot TOBEC index $100 × S 1 Lend $18,500 $18,500 Collect loan repayment $18,500 × 1.03 = +$19,055 Pay transaction costs $15.00 Total 0 Total $100F 0 + $19,040 (Note that F 1 = S 1 at expiration.) The lower bound for F 0 is: 19,040/100 = 190.40 6. a. You should be short the index futures contracts. If the stock value falls, you need futures profits to offset the loss. b. Each contract is for $250 times the index, currently valued at 1,350. Therefore, each contract controls stock worth: $250 × 1,350 = $337,500 In order to hedge a $13.5 million portfolio, you need: 40 500 , 337 $ 000 , 500 , 13 $ = contracts c. Now, your stock swings only 0.6 as much as the market index. Hence, you need 0.6 as many contracts as in (b): 0.6 × 40 = 24 contracts 23-3
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7. If the beta of the portfolio were 1.0, she would sell $1 million of the index. Because beta is 1.25, she should sell $1.25 million of the index. 8. a. 1,200 × 1.01 = 1,212 b. $12 million /($250 × 1,200) = 40 contracts short c. 40 × 250 × (1,212 – S T ) = 12,120,000 – 10,000S T d. The expected return on a stock is: α + r f + β [E(r M ) – r f ] The CAPM predicts that α = 0. In this case, however, if you believe that α = 2% (i.e., 0.02), then you forecast a portfolio return of: r P = 0.02 + 0.01 + 1.0 × (r M – 0.01) + ε = 0.03 + [ 1 × (r M – 0.01) ] + ε where ε is diversifiable risk, with an expected value of zero. e. Because the market is assumed to pay no dividends: r M = (S T – 1,200)/1,200 = (S T /1,200) – 1 The rate of return can also be written as: r P = 0.03 + ( 1 × [(S T /1,200) – 1 – 0.01] ) + ε The dollar value of the stock portfolio as a function of the market index is therefore: $12 million × (1 + r P ) = $12 million × [0.03 + (S T /1,200) – 0.01 + ε ] = $240,000 + 10,000S T + ($12 million × ε ) The dollar value of the short futures position will be (from part c): 12,120,000 – 10,000S T The total value of the portfolio plus the futures proceeds is therefore: [240,000 + 10,000S T + (12 million × ε )] + [12,120,000 – 10,000S T ] = $12,360,000 + ($12 million × ε ) The payoff is independent of the value of the stock index. Systematic risk has been eliminated by hedging (although firm-specific risk remains). f. The portfolio-plus-futures position cost $12 million to establish. The expected end-of-period value is $12,360,000. The rate of return is therefore 3%. 23-4
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g. The beta of the hedged position is 0. The fair return should be r f = 1%. Therefore, the alpha of the position is (3% – 1%) = 2%, the same as the alpha of the portfolio. Now, however, one can take a position on the alpha without incurring systematic risk. 9. You would short $0.50 of the market index contract and $0.75 of the computer industry stock for each dollar held in IBM. 10. a. The strategy would be to sell Japanese stock index futures to hedge the market risk of Japanese stocks, and to sell yen futures to hedge the currency exposure. b. Some possible practical difficulties with this strategy include: • Contract size on futures may not match size of portfolio.
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