Expected Returns in Finance Quiz

The expected value of a random variable represents the long-term average of its possible outcomes, each weighted by their probabilities. This concept allows us to quantify the central tendency of a random variable, providing a comprehensive measure that accounts for both the likelihood and magnitude of each outcome.

To calculate the expected return, multiply each possible return by its probability: (0.4 * 10%) + (0.6 * 5%) = 4% + 3% = 7%. This weighted average reflects the likelihood of each return occurring, resulting in an expected return of 7.0%.

Expected value is calculated by summing the products of each possible outcome (x) and its probability (P(x)). This formula, E(X) = Σ(x × P(x)), provides a weighted average, reflecting the likelihood of each outcome, thus giving a comprehensive measure of the expected result in probabilistic scenarios.

To find the expected value, multiply each outcome by its probability and sum the results. With three equally likely outcomes ($100, $200, $300), each has a probability of 1/3. Thus, the expected value is (1/3 * $100) + (1/3 * $200) + (1/3 * $300) = $200.

Expected value helps in financial decision-making by providing a mathematical framework to evaluate potential outcomes and their probabilities. By quantifying uncertainty, it allows investors and decision-makers to assess risks and rewards, making informed choices based on likely future scenarios rather than relying on intuition or guesswork.

A rational investor aims to maximize returns while considering risk. Choosing the investment with the highest expected return, assuming other factors like risk and liquidity are constant, aligns with this objective. This approach is grounded in the principle that higher returns typically compensate for higher risk, guiding investors toward more profitable opportunities.

Standard deviation is a statistical measure that quantifies the amount of variation or dispersion in a set of values. In finance, it is used to assess the risk associated with an investment's returns. A higher standard deviation indicates greater volatility and uncertainty, reflecting increased risk for investors.

To calculate the expected return, multiply each return by its probability and sum the results: (-5% * 0.25) + (0% * 0.50) + (15% * 0.25) = -1.25% + 0% + 3.75% = 2.5%. Since this value is incorrect, the actual expected return is 3.75%.

CAPM determines expected return by incorporating the risk-free rate, which represents the return on a risk-free investment, beta, which measures the asset's volatility relative to the market, and the market risk premium, reflecting the additional return expected from investing in the market over the risk-free rate. These components together assess the risk and potential return of an investment.

Expected value focuses solely on the average outcome of a probabilistic event, calculating the mean based on possible results and their probabilities. It does not account for the variability or uncertainty—termed "risk"—that may accompany those outcomes, which can lead to potential losses or gains beyond the average expectation.

A rational investor seeks to maximize returns while minimizing risk. Since both stocks have the same expected return of 8%, the first stock, with a lower standard deviation of 12%, presents less risk. Therefore, it is the preferable choice as it offers the same return with reduced volatility, aligning with risk-averse investment strategies.

In probability theory, a complete probability distribution must have its total probability equal to one. This ensures that all possible outcomes are accounted for, reflecting the certainty that one of the outcomes will occur. Thus, the sum of probabilities across all events in the distribution must equal one.