Questions 1-3 Are Based On This Slack Partnership

Questionsquestions 1 3 Are Based On Thisslack Is A Partnership Between

Questions questions 1-3 are based on this Slack is a partnership between two traders: Simon and Jack. Simon and Jack each invested money in different commodity futures. Simon’s coffee trades can lose 3.5 million Euros with probability 1/3 or earn a profit of 1 million Euros with probability 2/3. Jack's cereal trades can lose 1 million Euros with probability 1/4 or earn a profit of 1 million Euros with probability 3/4. 1. What is the probability that Slack’s profits will be non-negative (larger than or equal to 0)? 2. What are Slack’s expected profits (in millions)? 3. If the coffee and cereal trades are statistically independent, what is the probability that Slack’s profits will be exactly equal to zero?

Paper For Above instruction

Introduction

Partnerships in trading involve complex risk assessments, especially when diversifying across different commodities. This analysis examines a hypothetical joint venture, Slack, involving two traders, Simon and Jack, who have invested in coffee and cereal futures, respectively. Their investment outcomes are probabilistically modeled to evaluate the joint profit expectations and risks.

Analysis of Profit Probabilities

Firstly, the probability that Slack's profits will be non-negative is determined based on the combined outcomes of Simon's and Jack's trades. Given that Simon's coffee trade can lead to a significant loss of €3.5 million with probability 1/3, or a profit of €1 million with probability 2/3, and Jack's cereal trade can result in a €1 million loss with probability 1/4 or a €1 million gain with probability 3/4, the calculation involves enumerating all outcome combinations where the total profit is non-negative.

Specifically, the combinations where total profit ≥ 0 are as follows:

  • Simon's profit of €1 million (probability 2/3) combined with Jack's profit of €1 million (probability 3/4);
  • Simon's profit of €1 million with Jack’s loss of €1 million (probability 1/4), resulting in total profit €0, which is also non-negative;
  • Simon's loss of €3.5 million (probability 1/3) combined with Jack's profit of €1 million (probability 3/4), resulting in a total of -€2.5 million, which is negative, so it doesn't contribute.
  • Simon's loss of €3.5 million with Jack's loss of €1 million sums to -€4.5 million, also negative.

Thus, the total probability of non-negative profits is the sum of the probabilities of the favorable outcome combinations, i.e., (Simon's profit €1 million, Jack's profit €1 million) and (Simon's profit €1 million, Jack's loss €1 million).

Calculating these:

  • P(Simon profit €1M) = 2/3;
  • P(Jack profit €1M) = 3/4;
  • P(Jack loss €1M) = 1/4;

The probability that Slack's profits are ≥ 0 is therefore:

\[

P = P(Simon profit €1M & Jack profit €1M) + P(Simon profit €1M & Jack loss €1M) = \left(\frac{2}{3}\times\frac{3}{4}\right) + \left(\frac{2}{3}\times\frac{1}{4}\right) = \frac{1}{2} + \frac{1}{6} = \frac{2}{3}

\]

Expected Profits

The expected profit from Simon's trade is:

\[

E(Simon) = (1\, \text{million} \times \frac{2}{3}) + (-3.5\, \text{million} \times \frac{1}{3}) = \frac{2}{3} - \frac{3.5}{3} = -\frac{1.5}{3} = -0.5\, \text{million}

\]

Similarly, Jack's expected profit:

\[

E(Jack) = (1\, \text{million} \times \frac{3}{4}) + (-1\, \text{million} \times \frac{1}{4}) = \frac{3}{4} - \frac{1}{4} = \frac{2}{4} = 0.5\, \text{million}

\]

Therefore, the expected total profit for Slack, assuming independence, is:

\[

E(Slack) = E(Simon) + E(Jack) = -0.5 + 0.5 = 0\, \text{million}

\]

which indicates a zero expected profit despite the risks.

Probability the profits are exactly zero

Assuming independence between Simon and Jack's trades, the probability that Slack's total profit is exactly zero can occur when Simon's profit is €1 million and Jack's loss is €1 million, yielding total zero; or Simon's loss of €3.5 million and Jack's profit of €1 million, totaling -2.5 million, which is not zero, so not relevant. The only relevant outcome is when Simon's profit €1 million and Jack's loss €1 million:

\[

P = P(Simon\ profit = €1M) \times P(Jack\ loss = €1M) = \frac{2}{3} \times \frac{1}{4} = \frac{2}{12} = \frac{1}{6}

\]

In conclusion, the probability that Slack’s profits will be exactly zero is 1/6, with the overall probability of a non-negative profit being 2/3. The expected profits come out to being zero, indicating a balanced expectation with considerable risk in the trade outcomes.

References

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