Question

Eva is risk-averse. Currently she has \(\$ 50,000\) to invest. She faces the following choice: she can invest in the stock of a dot-com company, or she can invest in IBM stock. If she invests in the dot-com company, then with probability 0.5 she will lose \(\$ 30,000\), but with probability 0.5 she will gain \(\$ 50,000\). If she invests in IBM stock, then with probability 0.5 she will lose only \(\$ 10,000,\) but with probability 0.5 she will gain only \(\$ 30,000\). Can you tell which investment she will prefer to make?

Short answer

Answer: Eva is more likely to choose the IBM investment due to its lower potential loss of $10,000 compared to the dot-com investment's potential loss of $30,000, even though both investments have the same expected value of $10,000.

Step-by-step solution

Calculate the expected value of each investment

To calculate the expected value of each investment, multiply the gain/loss amount of each outcome by their respective probabilities, and then add them together. For the dot-com investment: Expected Value = (0.5 * (-\(30,000)) + (0.5 * \)50,000) For the IBM investment: Expected Value = (0.5 * (-\$10,000)) + (0.5 * \$30,000)

Computations

Now let's perform the calculations for each investment: Dot-com investment Expected Value: = (-\$15,000) + (\$25,000) = \$10,000 IBM investment Expected Value: = (-\$5,000) + (\$15,000) = \$10,000

Comparing the expected values

Both the dot-com and IBM investments have the same expected value of \$10,000. However, the dot-com investment has a higher potential loss of \$30,000, while the IBM investment has a smaller potential loss of only \$10,000.

Conclusion

Since Eva is risk-averse, she will prioritize minimizing potential losses over potential gains with the same expected value. Therefore, she will likely prefer investing in the IBM stock as it has a lower potential loss when compared to the dot-com investment.

Key concepts

Risk Aversion

Risk aversion is a key concept in finance and investment decisions. It refers to an investor's preference to avoid uncertainty and potential losses in favor of safer options. In other words, a risk-averse individual, like Eva, will tend to prioritize investments that promise stable returns, even if the returns are lower compared to riskier investments. This tendency stems from the desire to protect one's principal or initial investment from substantial losses.

When faced with two investment choices that have the same expected value, a risk-averse person evaluates the potential downside risks. In the given problem, Eva must choose between investing in a dot-com company or IBM. Both investments have an expected value of \(\\(10,000\), but the potential loss varies - \(\\)30,000\) for the dot-com company and \(\$10,000\) for IBM.

This illustrates the core principle of risk aversion: minimizing the potential for loss, even if it means forgoing the possibility of greater gains. As such, Eva will likely prefer the IBM stock since it has a lower risk associated with loss.

Expected Value

The expected value is a statistical measure used to determine the average outcome of an investment based on all possible scenarios and their probabilities. It helps investors to understand which choice might bring the best result in the long run. Calculating the expected value requires multiplying each possible outcome by its probability and summing these values together.

In the example of Eva's investment decision:

• For the dot-com investment, she faces the possibilities of losing \(\\(30,000\) or gaining \(\\)50,000\), both having a probability of 0.5. The expected value is calculated as \[ (0.5 \times (-30,000)) + (0.5 \times 50,000) = 10,000 \].
• For the IBM investment, the probabilities of losing \(\\(10,000\) or gaining \(\\)30,000\) are also both 0.5. The expected value here is \[ (0.5 \times (-10,000)) + (0.5 \times 30,000) = 10,000 \].

Thus, despite different levels of risk, both investments yield the same expected value. This shows that expected value alone does not fully inform the risk preferences and decisions of risk-averse investors.

Probability

Probability is the measure of the likelihood that a particular event will occur, and it is expressed as a number between 0 and 1. In investment analysis, probability helps investors anticipate the possible outcomes of an investment, allowing them to assess both risks and potential returns.

In Eva's example, the probabilities are straightforward: for both investments, the probabilities of loss and gain are equal. This symmetry simplifies the expected value calculation, but it also highlights the importance of understanding outcomes beyond mere probabilities. For Eva, the probability of each outcome might be the same, but the potential losses are notably different.

Therefore, while probability helps determine the expected value, it must be considered alongside risk assessments. Investors should use these probabilities in combination with risk evaluation to make informed decisions that align with their risk tolerance.