Problem 15 - 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 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 unwise based on expected value, as it results in a loss of $2100.

Step 1: Understand the Investment

The investor bought 1000 shares at \(5 each, totaling an investment of \(1000 \times 5 = \\)5000\). The stock could either rise to \(20 or fall to \)1.

Step 2: Define Profit Outcomes

If the stock rises to \(20, the total value of the investment will be \(1000 \times 20 = \\)20000\). If the stock falls to \(1, the total value will be \(1000 \times 1 = \\)1000\).

Step 3: Calculate Profits from Possible Outcomes

If the stock rises to \(20, the profit will be \(\\)20000 - \\(5000 = \\)15000\). If it falls to \(1, the loss will be \(\\)1000 - \\(5000 = -\\)4000\).

Step 4: Assign Probabilities to Each Outcome

The probability of the stock rising to $20 is given as 0.1, and the probability of it falling to $1 is 0.9.

Step 5: Calculate Expected Profit

The expected profit can be calculated using the probability-weighted outcomes: \(E(profit) = (0.1 \times 15000) + (0.9 \times -4000)\).

Step 6: Perform the Calculation

Substitute the values: \(E(profit) = 0.1 \times 15000 + 0.9 \times -4000 = 1500 - 3600 = -2100\).

Step 7: Conclusion

The expected value of the profit is \(-\$2100\). Since this is a negative value, it implies that, on average, the investment decision was not wise based solely on the expected value criterion.

Key Concepts

Probability

Probability is an essential concept that measures how likely an event is to occur. It's expressed as a number between 0 and 1, where 0 means an event will not happen, and 1 indicates certainty.
Understanding probability helps us make informed decisions by evaluating potential outcomes and their likelihoods.
In the given stock scenario, the investor estimates two probability outcomes:

• There's a 0.1 probability that the stock will increase to $20.
• There's a 0.9 probability that it will decrease to $1.

These probabilities guide the investor in understanding the risk involved.
Unlike a guarantee, probability offers a mathematical representation of potential outcomes.

Investment Decision

An investment decision involves choosing where to allocate financial resources, in this case, buying stocks. It's crucial to weigh potential profits against possible risks.
The decision relies on various factors such as expected returns, risk tolerance, market analysis, and personal financial goals.
In our exercise, the only criterion used was the expected value of profit, which considers the probabilities of the stock's rise or fall. This may simplify the decision-making, but it's often not comprehensive enough.
Investors typically evaluate many factors, including company performance and market trends, to ensure a well-rounded decision.

Stock Market

The stock market represents a collection of markets for buying and selling stock shares, offering an avenue for investment.
Stocks are considered risky investments since their values fluctuate based on various factors such as company performance, economic conditions, and investor sentiments.
In this scenario, the stock's price could drastically change to either $20 or $1.
Understanding market volatility is crucial for investors.
They should be prepared for the unpredictability of stock prices.
Knowing this, investors use different strategies, like diversification, to manage risk in the stock market.

Profit and Loss Analysis

Profit and loss analysis helps investors determine the financial viability of an investment by assessing the potential gains or losses.
In our example, there are two potential outcomes:

• If the stock price rises to \(20, the investor gains a profit of \)15,000.
• If it falls to \(1, the investor faces a loss of \)4,000.

These outcomes must be weighted by their probabilities to calculate the expected profit.
The formula used: \( E(profit) = (0.1 \times 15000) + (0.9 \times -4000) = -2100 \).This negative expected value indicates the investment might not be wise.
Understanding potential profits and losses, and their probabilities is critical when evaluating investment risks and returns, helping to make more informed financial decisions.