Question 1

Background: Metal lgesellschaft AG was formerly one of Germany's largest industrial conglomerates based in Frankfurt. It had over 20,000 employees and revenues in excess of 10 billion US dollars. It had over 250 subsidiaries specializing in mining, specialty chemicals (Chemetall), commodity trading, financial services, and engineering (Lurgi). In 1993, the company lost 1.3 billion dollars suffering from flawed long hedge strategy in near term futures contracts that was meant to protect against forward sales commitments. A fall in spot prices forced margin calls for the company and the contracts were closed at a loss. Subsequently, the spot price increased and the company suffered even greater losses covering its customer commitments. It is debated whether the company was speculating after unwinding the long futures hedge since they became essentially exposed or naked against their forward customer commitments.

The following illustrates how important it is to consider tailing factor in taking a correct hedge strategy.

Suppose the current spot price is $3. With 50 percent of the probability, the spot price will increase or decrease by 1 dollar for first year and then remain the same as shown in the graph below.

Please answer the following questions.

1. If the annual discrete compounding risk-free rate is 10%, and the cost of carry offsets the convenience yield exactly, then the future price is equal to the spot price. Please explain.

=Future price has comprised by the shot price.

Future price F=(1+r+s-y)^T-tX S Where: s=Storage Cost

S= Spot Price

y= convenience yield

r= Risk free rate

Therefore, as per the question cost of carry offsets the convenience yield. It means:

r+s=y

Therefore:

F=(1+r+s-y)^T-t.S

Hence:

F=S

Therefore future price is equal to the spot price.

b) Assume hedger takes hedge ratio as h*, i.e, if the risk exposure is a long position of 100 units of spot commodity, to hedge the risk, hedger will short 100h* futures underlying on that commodity. Please answer the questions in the right panel in analogy to the left panel, by filling the blanks in j) r)

NEED TO ANSWERS ONLY THIS BELOW QUESTIONS.

C) To make the hedge portfolio risk-free, the value of the portfolio should not vary no matter stock price rises or falls. Please based on the above fact, solve the optimal hedge ratio h*.

D) Equivalently, to make the hedge portfolio risk-free, the hedge portfolio should not make profit in one state and make loss in the other, where the state means stock price rises or falls. Please use the optimal hedge ratio solved in (c) to verify this fact.

t=0 t=1 t=2 Pu=0.5 F1,2=4 > F2,2=4 FO,2=3 F1,2=2 F2,2=2 Pd=0.5 t=0 t=1 t=2 Pu=0.5 F1,2=4 > F2,2=4 FO,2=3 F1,2=2 F2,2=2 Pd=0.5 If the stock price falls from 3 to 2, then If the stock price rises from 3 to 4, then a) the future price falls from 3 to 2; j) the future price rises from 3 to 4. b) hedger makes profit (3-2)x100h*=100h* from the future, at t=1; k) hedger makes loss (3-4)*100h*=100h* from the future, at t=1; c) by investing the profit from t=1 till t=2, 100h* dollars profit becomes (1+0.1)x100h*=110 h*; 1) Margin call from t=1 till t=2, 100h* dollars loss becomes (-1 + 0.1) X 100h*= -90 loss; d) meanwhile, hedger makes a loss of 100x(2-3)= -100 from the stock at t=1; m) meanwhile, hedger makes a profit of 100 X (4-3) = 100 from the stock at t=1; e) hedger doesn't make further loss on stock from t=1 to t=2; n) hedger doesn't make further profit on stock from t=1 to t=2; f) net profit/loss is [110h*-1000 dollars; o) net profit/loss is 100h*-90; g) stock price is 2 at t=2, hence the value of stock at t=2 is 2x100=200; p) stock price is 4 at t=2, hence the value of stock at t=2 is 4 X 100 = 400; h) the value of futures is its gain and loss, hence, at t=2, its value is 110h*; q) the value of futures is its gain and loss, hence, at t=2, its value is -90; i) the value of the portfolio is the sum of the value of stock and the value of futures, which is [200+110h*). r) the value of the portfolio is the sum of the value of stock and the value of futures, which is 400-90h* t=0 t=1 t=2 Pu=0.5 F1,2=4 > F2,2=4 FO,2=3 F1,2=2 F2,2=2 Pd=0.5 t=0 t=1 t=2 Pu=0.5 F1,2=4 > F2,2=4 FO,2=3 F1,2=2 F2,2=2 Pd=0.5 If the stock price falls from 3 to 2, then If the stock price rises from 3 to 4, then a) the future price falls from 3 to 2; j) the future price rises from 3 to 4. b) hedger makes profit (3-2)x100h*=100h* from the future, at t=1; k) hedger makes loss (3-4)*100h*=100h* from the future, at t=1; c) by investing the profit from t=1 till t=2, 100h* dollars profit becomes (1+0.1)x100h*=110 h*; 1) Margin call from t=1 till t=2, 100h* dollars loss becomes (-1 + 0.1) X 100h*= -90 loss; d) meanwhile, hedger makes a loss of 100x(2-3)= -100 from the stock at t=1; m) meanwhile, hedger makes a profit of 100 X (4-3) = 100 from the stock at t=1; e) hedger doesn't make further loss on stock from t=1 to t=2; n) hedger doesn't make further profit on stock from t=1 to t=2; f) net profit/loss is [110h*-1000 dollars; o) net profit/loss is 100h*-90; g) stock price is 2 at t=2, hence the value of stock at t=2 is 2x100=200; p) stock price is 4 at t=2, hence the value of stock at t=2 is 4 X 100 = 400; h) the value of futures is its gain and loss, hence, at t=2, its value is 110h*; q) the value of futures is its gain and loss, hence, at t=2, its value is -90; i) the value of the portfolio is the sum of the value of stock and the value of futures, which is [200+110h*). r) the value of the portfolio is the sum of the value of stock and the value of futures, which is 400-90h*