bkmsol_ch23 - CHAPTER 23 FUTURES AND SWAPS MARKETS AND...

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CHAPTER 23: FUTURES AND SWAPS: MARKETS AND APPLICATIONS 1. a. S 0 × (1 + r M ) D = (1,425 × 1.06) – 15 = 1,495.50 b. S 0 × (1 + r f ) D = (1,425 × 1.03) – 15 = 1,452.75 c. The futures price is too low. Buy futures, short the index, and invest the proceeds of the short sale in T-bills: CF Now CF in 6 months Buy futures 0 S T 1,422 Short index 1,425 S T 15 Buy T-bills 1,425 1,467.75 Total 0 30.75 2. a. The value of the underlying stock is: $250 × 1,350 = $337,500 25/$337,500 = 0.000074 = 0.0074% of the value of the stock b. $40 × 0.000074 = $0.0030 (less than half of one cent) c. $0.20/$0.0030 = 67 The transaction cost in the stock market is 67 times the transaction cost in the futures market. 3. a. From parity: F 0 = 1,200 × (1 + 0.03) – 15 = 1,221 Actual F 0 is 1,218; so the futures price is 3 below the “proper” level. b. Buy the relatively cheap futures, sell the relatively expensive stock and lend the proceeds of the short sale: CF Now CF in 6 months Buy futures 0 S T 1,218 Sell shares 1,200 S T 15 Lend $1,200 1,200 1,236 Total 0 3 23-1
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c. If you do not receive interest on the proceeds of the short sales, then the $1200 you receive will not be invested but will simply be returned to you. The proceeds from the strategy in part (b) are now negative: an arbitrage opportunity no longer exists. CF Now CF in 6 months Buy futures 0 S T 1,218 Sell shares 1,200 S T 15 Place $1,200 in margin account 1,200 1,200 Total 0 33 d. If we call the original futures price F 0 , then the proceeds from the long- futures, short-stock strategy are: CF Now CF in 6 months Buy futures 0 S T F 0 Sell shares 1,200 S T 15 Place $1,200 in margin account 1,200 1,200 Total 0 1,185 F 0 Therefore, F 0 can be as low as 1,185 without giving rise to an arbitrage opportunity. On the other hand, if F 0 is higher than the parity value (1,221), then an arbitrage opportunity (buy stocks, sell futures) will exist. There is no short-selling cost in this case. Therefore, the no-arbitrage range is: 1,185 F 0 1,221 4. a. Call p the fraction of proceeds from the short sale to which we have access. Ignoring transaction costs, the lower bound on the futures price that precludes arbitrage is the following usual parity value (except for the factor p): S 0 (l + r f p) – D The dividend (D) equals: 0.012 × 1,350 = $16.20 The factor p arises because only this fraction of the proceeds from the short sale can be invested in the risk-free asset. We can solve for p as follows: 1,350 × (1 + 0.022p) – 16.20 = 1,351 p = 0.579 23-2
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b. With p = 0.9, the no-arbitrage lower bound on the futures price is: 1,350 × [1 + (0.022 × 0.9)] – 16.20 = 1,360.53 The actual futures price is 1,351. The departure from the bound is therefore 9.53. This departure also equals the potential profit from an arbitrage strategy. The strategy is to short the stock, which currently sells at 1,350. The investor receives 90% of the proceeds (1,215) and the remainder (135) remains in the margin account until the short position is covered in 6 months. The investor buys futures and lends 1,215: CF Now CF in 6 months Buy futures 0 S T 1,351 Sell shares 1350 135 135 S T 16.20 Lend 1,215 1,215 × 1.022 = 1,241.73 Total 0 9.53 The profit is: 9.53 × $250 per contract = $2,382.50 5. a. By spot-futures parity: F 0 = S 0 × (l + r f ) = 185 × [1 + (0.06/2)] = 190.55 b. The lower bound is based on the reverse cash-and-carry strategy.
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