Question

You have $1000 and a certain commodity presently sells for $2 per ounce. Suppose that after one week the commodity will sell for either $1 or $4 an ounce, with these two possibilities being equally likely. (a) If your objective is to maximize the expected amount of money that you possess at the end of the week, what strategy should you employ? (b) If your objective is to maximize the expected amount of the commodity that you possess at the end of the week, what strategy should you employ?

Answer 1

To maximize the expected amount of money, invest all $1000 in the commodity. To maximize the expected amount of the commodity, there isn't a distinct action as the future price is uncertain with an equal likelihood of increasing or decreasing.

Step-by-step explanation:

To solve both parts of this problem, we must calculate the expected values for each strategy based on the available information. In part (a), the objective is to maximize the expected amount of money by the end of the week.

If you invest all $1000 into the commodity at $2 per ounce, you would have 500 ounces. The possible outcomes at the end of the week are:

• If the commodity falls to $1 per ounce, your investment would then be worth $500.
• If the commodity rises to $4 per ounce, your investment would be worth $2000.

The expected value (EV) of your investment at the end of the week would be:

EV = (0.5 * $500) + (0.5 * $2000) = $250 + $1000 = $1250.

In part (b), the objective is to maximize the expected amount of the commodity you possess. If the price drops to $1 per ounce, you could buy more ounces with your money. However, since the future price is uncertain, and you have no control over it, the best strategy is to simply hold onto the $1000 and see what happens. Ultimately, the expected amount of the commodity does not change based on the actions you take now because either you can afford 500 ounces at the current price or potentially more if the price drops, so there isn't a specific strategy that maximizes the amount of commodity in this scenario.

Therefore, the strategy to maximize the expected amount of money is to invest all $1000 in the commodity, while the strategy to maximize the amount of commodity doesn't offer a distinct action given the equal likelihood of a price increase or decrease.

Answer 2

(a) If your objective is to maximize the expected amount of money that you possess at the end of the week, you should not buy any of the commodity. This is because the price of the commodity is equally likely to either increase or decrease, so there is no guarantee that you will make a profit by buying it. Instead, you should simply keep the $1,000 that you currently have and wait until the end of the week to see what the price of the commodity is. This way, you will have the maximum expected amount of money at the end of the week.

(b) If your objective is to maximize the expected amount of the commodity that you possess at the end of the week, you should buy as much of the commodity as you can with the $1,000 that you currently have. This is because the price of the commodity is equally likely to either increase or decrease, so there is a 50% chance that the price will increase and you will be able to sell the commodity for a profit. By buying as much of the commodity as you can, you will maximize the amount of the commodity that you possess, and therefore maximize your potential profit if the price increases.