Homework 8 Solutions Essay
Professor Yates
Homework 8 Solutions
FI 4000
Homework 8 Solutions
1. Selling a contract is a short position. If the price rises, you lose money.
Loss = (1,250 – 1,200) $250 = $12,500
2. Futures price = S0 (1+ rf − d)T = $1,200 (1 + .01 – .02) = $1,188
3. The theoretical futures price = S0 (1+ rf)T = $1,700 (1 + .02) = $1,734. At $1,641, the gold futures contract is underpriced relative to the gold spot price. To benefit from the mispricing, we sell what is overpriced (gold) and buy what is underpriced (futures contract). Specifically, we will short gold, buy 1 futures contract and lend
. The payoff table below shows the proceeds: Short gold
Long gold futures
Invest $1641/1.02
Combined
CF Today
1700
1641/1.02 …show more content…
91.18
CF in 1 Year
1641
1641
0
Since this results in a positive CF today with no future cashflows, it is an arbitrage.
(This answer presumes that that the commodity is available for short sale without fees and with full access to the proceeds of the short sale. In real-world practice, failure to satisfy these conditions may limit the apparent arbitrage opportunity.)
4.
a. The required margin is 1,164.50 $250 .10 = $29,112.50
b. Total Return = (1,200 – 1,164.50) $250 = $8,875
Percentage Return = $8,875/$29,112.5 = .3049 = 30.49%
c. Total Loss = [1,164.5 (1 – .01)] – 1,164.5) $250 = …show more content…
–$2,911.25
Percentage Loss = –$2,911.25/$29,112.5 = – .10 or 10% loss
5.
The ability to buy on margin is one advantage of futures. Another is the ease with which one can alter holdings of the asset. This is especially important if one is dealing in commodities, for which the futures market is far more liquid than the spot market so that transaction costs are lower in the futures market.
Professor Yates
Homework 8 Solutions
FI 4000
6.
a.
Action
Buy stock
Initial Cash Flow
–S0
Cash Flow at Time T
ST + D
Short futures
0
F0 – ST
Borrow
S0
–S0(1 + r)
Total
0
F0 + D – S0(1 + r)
b. The net initial investment is zero, whereas the final cash flow is not zero. Therefore, in order to avoid arbitrage opportunities, the equilibrium futures price will be the final cash flow equated to zero. Accordingly:
F0 = S0 (1 + r) – D
c. Noting that D = (d S0), we substitute and rearrange to find that:
F0 = S0 (1 + r – d)
7.
a. F0 = S0 (1 + rf) = $120 1.06 = $127.20
b. The stock price falls to: $120 (1 – .03) = $116.40
The futures price falls to: $116.40 1.06 = $123.384
The investor loses: ($127.20 – $123.384) 1,000 = $3,816.00
c. The percentage return is: –$3,816/$12,000 =
–31.8%
Homework 8 Solutions
FI 4000
Homework 8 Solutions
1. Selling a contract is a short position. If the price rises, you lose money.
Loss = (1,250 – 1,200) $250 = $12,500
2. Futures price = S0 (1+ rf − d)T = $1,200 (1 + .01 – .02) = $1,188
3. The theoretical futures price = S0 (1+ rf)T = $1,700 (1 + .02) = $1,734. At $1,641, the gold futures contract is underpriced relative to the gold spot price. To benefit from the mispricing, we sell what is overpriced (gold) and buy what is underpriced (futures contract). Specifically, we will short gold, buy 1 futures contract and lend
. The payoff table below shows the proceeds: Short gold
Long gold futures
Invest $1641/1.02
Combined
CF Today
1700
1641/1.02 …show more content…
91.18
CF in 1 Year
1641
1641
0
Since this results in a positive CF today with no future cashflows, it is an arbitrage.
(This answer presumes that that the commodity is available for short sale without fees and with full access to the proceeds of the short sale. In real-world practice, failure to satisfy these conditions may limit the apparent arbitrage opportunity.)
4.
a. The required margin is 1,164.50 $250 .10 = $29,112.50
b. Total Return = (1,200 – 1,164.50) $250 = $8,875
Percentage Return = $8,875/$29,112.5 = .3049 = 30.49%
c. Total Loss = [1,164.5 (1 – .01)] – 1,164.5) $250 = …show more content…
–$2,911.25
Percentage Loss = –$2,911.25/$29,112.5 = – .10 or 10% loss
5.
The ability to buy on margin is one advantage of futures. Another is the ease with which one can alter holdings of the asset. This is especially important if one is dealing in commodities, for which the futures market is far more liquid than the spot market so that transaction costs are lower in the futures market.
Professor Yates
Homework 8 Solutions
FI 4000
6.
a.
Action
Buy stock
Initial Cash Flow
–S0
Cash Flow at Time T
ST + D
Short futures
0
F0 – ST
Borrow
S0
–S0(1 + r)
Total
0
F0 + D – S0(1 + r)
b. The net initial investment is zero, whereas the final cash flow is not zero. Therefore, in order to avoid arbitrage opportunities, the equilibrium futures price will be the final cash flow equated to zero. Accordingly:
F0 = S0 (1 + r) – D
c. Noting that D = (d S0), we substitute and rearrange to find that:
F0 = S0 (1 + r – d)
7.
a. F0 = S0 (1 + rf) = $120 1.06 = $127.20
b. The stock price falls to: $120 (1 – .03) = $116.40
The futures price falls to: $116.40 1.06 = $123.384
The investor loses: ($127.20 – $123.384) 1,000 = $3,816.00
c. The percentage return is: –$3,816/$12,000 =
–31.8%