Problem 17 - Algebra and Trigonometry

Question

Gambling on Stocks An investor buys 1000 shares of a risky stock for \(\$ 5\) a share. She estimates that the probability that the stock will rise in value to \(\$ 20\) a share is 0.1 and the probability that it will fall to \(\$ 1\) a share is \(0.9 .\) If the only criterion for her decision to buy this stock was the expected value of her profit, did she make a wise investment?

Step-by-Step Solution

Answer

The investment was not wise, as the expected value was negative.

Step 1: Calculate Profit if Stock Rises

If the stock rises to $20 per share, the investor will have 1000 shares * $20 = $20,000 total. Since the initial investment was 1000 shares * $5 = $5000, the profit will be $20,000 - $5,000 = $15,000.

Step 2: Calculate Loss if Stock Falls

If the stock falls to $1 per share, the investor's total will be 1000 shares * $1 = $1,000. Since the initial investment was $5,000, the loss will be $1,000 - $5,000 = -$4,000.

Step 3: Compute Expected Value of Profit

The expected value (EV) is calculated by multiplying each outcome by its probability and adding the results. EV = (Probability of rise * Profit if rises) + (Probability of fall * Loss if falls). So, EV = (0.1 * $15,000) + (0.9 * -$4,000).

Step 4: Simplify the Expected Value Calculation

Calculate each term: 0.1 * $15,000 = $1,500 and 0.9 * -$4,000 = -$3,600. Add these to get the overall expected value: $1,500 - $3,600 = -$2,100.

Step 5: Determine Investment Wisdom Based on EV

Since the expected value of the profit is -$2,100, the investment, based on expected value, is not wise. A negative expected value indicates that the average outcome of this investment is a loss.

Key Concepts

Probability

Probability is at the heart of making predictions in uncertain situations. In this exercise, we consider two possible outcomes for the stock price, each with its own probability. The two key situations are:

• The stock rising to $20 per share with a probability of 0.1.
• The stock falling to $1 per share with a probability of 0.9.

By understanding probability, the investor can estimate how likely each outcome is to occur. This helps in assessing the risks involved in the investment. Remember, probabilities range from 0 (impossible outcome) to 1 (certain outcome). Here, most of the likelihood lies with the stock's price falling. Understanding these probabilities is crucial for calculating expected outcomes in any investment scenario.

Investment Decision

An investment decision involves evaluating potential risks and returns. For our investor, the choice to buy this stock hinged on the expected value of the potential profit. One important thing to consider is that a good investment typically has a positive expected value, which would suggest that over time, the investor could expect to gain more than they lose. However, in this case, because the expected value of this investment turned out to be negative, it suggests that, on average, the investor should expect a loss. This decision highlights the importance of thorough analysis and mathematical evaluation before making investment decisions.

Profit Calculation

Profit calculation is necessary to understand the potential upside of an investment. If the stock reaches $20, our investor calculates the profit from the initial investment. Here's a breakdown:

• Initial cost: 1000 shares x $5/share = $5,000
• Value if stock rises: 1000 shares x $20/share = $20,000
• Profit: $20,000 - $5,000 = $15,000

This potential profit of $15,000 represents the possible gain from making the right investment choice, assuming the stock rises as predicted. The high value achieved would counterbalance the initial expenditure, but only with a low probability of 0.1.

Loss Calculation

Calculating potential loss is equally as important as calculating potential profit. When the probability of falling stock prices is higher, as in this case, loss calculation becomes critical. If the stock price drops to $1, here is how the loss pans out:

• Total valuation if stock falls: 1000 shares x $1/share = $1,000
• Initial cost: $5,000
• Loss: $1,000 - $5,000 = -$4,000

The loss of $4,000 is significant, particularly with the high probability of 0.9. Big risks with a high chance of loss highlight why evaluating expected value is essential before committing to such investments.