Expected outcome

What Is Expected Outcome?

Expected outcome, within the realm of portfolio theory and statistical analysis, refers to the average value one anticipates from a random variable over a large number of trials. It is a probabilistic measure that weighs each possible result of an event by its probability of occurring. This concept is fundamental in finance for assessing potential returns and risks of various investments, representing a long-term average rather than a guaranteed singular result.

History and Origin

The concept of expected outcome has roots in the 17th century with the development of probability theory, notably through the work of mathematicians like Blaise Pascal and Pierre de Fermat. However, Daniel Bernoulli formally introduced the idea of expected utility to address the limitations of simply relying on expected monetary value in his 1738 paper, "Exposition of a New Theory on the Measurement of Risk." This was famously in response to the St. Petersburg paradox, a theoretical game where the expected monetary value is infinite, yet most rational individuals would only pay a small amount to play. Bernoulli's insight, discussed in Daniel Bernoulli's work on expected utility, demonstrated that the value of money might not increase linearly with its amount, and that individual preferences for risk and wealth play a crucial role in decision-making⁴. The challenge of the St. Petersburg paradox, where the expected monetary value and the amount a reasonable person would pay vastly diverge, has been a cornerstone in the development of modern economic thought on risk and utility³.

Key Takeaways

The expected outcome is a weighted average of all possible results of an uncertain event, with weights based on their probabilities.
In finance, it helps investors estimate the potential return of an investment over the long term.
It is a theoretical average and does not guarantee any specific actual result in a single instance.
Expected outcome is a core component of modern portfolio theory for optimizing portfolios based on expected returns and risk.

Formula and Calculation

The expected outcome (EV) for a discrete random variable is calculated by multiplying each possible outcome by its respective probability and then summing these products.

For a series of discrete outcomes, the formula is:

E(X) = \sum_{i=1}^{n} (x_i \cdot P(x_i))

Where:

(E(X)) = The expected outcome
(x_i) = The (i)-th possible outcome
(P(x_i)) = The probability of the (i)-th outcome occurring
(n) = The total number of possible outcomes

This formula essentially calculates the mean of the probability distribution.

Interpreting the Expected Outcome

The expected outcome provides a theoretical average, indicating what one would anticipate over many repetitions of an event. For example, if an investment has an expected outcome of a 7% annual return, it means that over a very long period, the average annual return would converge to 7%. It is crucial to understand that this is not a forecast for any single year's performance. Individual outcomes can deviate significantly from the expected outcome in the short term, experiencing both higher and lower results. Therefore, the expected outcome should be considered alongside measures of dispersion, such as variance or standard deviation, to gain a complete understanding of the potential range of results and associated risk.

Hypothetical Example

Consider an investor evaluating a new venture that has three potential outcomes:

Success: A 40% chance of a $1,000,000 gain.
Moderate Success: A 30% chance of a $200,000 gain.
Failure: A 30% chance of a $500,000 loss.

To calculate the expected outcome of this venture:

E(Venture) = (0.40 \cdot \$1,000,000) + (0.30 \cdot \$200,000) + (0.30 \cdot -\$500,000) \\ E(Venture) = \$400,000 + \$60,000 - \$150,000 \\ E(Venture) = \$310,000

The expected outcome for this venture is $310,000. This suggests that, on average, if the investor were to engage in many identical ventures, they would expect a gain of $310,000 per venture. However, for a single instance of this venture, the actual result will be one of the three discrete outcomes, not $310,000. This example highlights how the expected outcome can guide decision-making even when individual outcomes are uncertain.

Practical Applications

The concept of expected outcome is widely applied across various fields within finance:

Investment Analysis: Investors use expected outcome to evaluate potential returns from stocks, bonds, and other securities. It is a critical input in assessing the attractiveness of an investment against its associated risk.
Portfolio Management: In constructing a portfolio, financial professionals often use expected outcomes for individual assets to determine the overall expected return of the portfolio. This feeds into strategies like asset allocation aimed at optimizing risk-adjusted returns.
Corporate Finance: Businesses utilize expected outcomes in capital budgeting to assess the viability of potential projects, weighing anticipated cash flows against the probabilities of different project scenarios. This involves detailed financial modeling and scenario analysis ².
Risk Management: Actuaries and insurers rely heavily on expected outcome to price insurance policies, estimate future claims, and manage reserves. By calculating the expected value of losses, they can determine premiums that cover anticipated payouts and ensure solvency.
Financial Planning: Individuals and financial planners may use expected outcome to project the potential future value of retirement savings or other long-term financial goals, using historical data and probability distributions of returns.

Limitations and Criticisms

While a powerful analytical tool, the expected outcome has several limitations. It represents a long-run average, which means it may not accurately reflect results over shorter time horizons. Actual outcomes can deviate substantially from expected outcomes, especially in periods of market volatility or during "black swan" events that are difficult to predict or model. For instance, an investment with a positive expected outcome might still experience significant losses in the short term due to short-term deviations from long-term expectations ¹.

Critics also point out that expected outcome assumes risk neutrality, meaning all that matters is the average result, not the path taken or the potential for extreme losses. This overlooks the psychological aspects of risk tolerance and the diminishing marginal utility of wealth, as highlighted by Daniel Bernoulli. Furthermore, accurately assigning probabilities to future events, especially rare or unprecedented ones, can be challenging and subjective, potentially leading to inaccurate expected outcome calculations. The reliance on past data to forecast future probabilities assumes that historical patterns will continue, which is not always the case in dynamic financial markets.

Expected Outcome vs. Expected Return

While often used interchangeably in financial contexts, "expected outcome" is a broader statistical term, whereas "expected return" is its specific application in finance.

Feature	Expected Outcome	Expected Return
Category	General statistical concept, probability theory	Financial analysis, investment analysis
What it measures	The average value of any random variable	The anticipated profit or loss on an investment over time
Application	Any scenario with uncertain results	Specifically related to financial assets and portfolios
Output Unit	Can be any unit (e.g., dollars, points, items)	Typically expressed as a percentage or monetary value

Essentially, expected return is a type of expected outcome, specifically the weighted average of possible rates of return on an investment, based on their probabilities. The concepts are closely related, with expected return being the financial application of the more general statistical principle of expected outcome.

FAQs

How is expected outcome used in investment decisions?

In investment decisions, the expected outcome (often referred to as expected return) helps investors estimate the average gain or loss they might anticipate from an investment over a prolonged period. It allows for a quantitative comparison of different investment opportunities, taking into account the various possible results and their likelihoods. For example, an investor might compare the expected outcome of a stock versus a bond.

Can the actual outcome differ significantly from the expected outcome?

Yes, the actual outcome can and often does differ significantly from the expected outcome, particularly over short time frames. The expected outcome is a theoretical average that would only be realized over an extremely large number of identical trials. In a single investment or event, the actual result will be one of the discrete possibilities, which may be far from the calculated average. This divergence highlights the importance of understanding risk and the inherent uncertainty in financial markets.

Is a higher expected outcome always better?

Not necessarily. While a higher expected outcome suggests a greater potential average gain, it often comes with higher risk or volatility. Investors must consider their individual risk tolerance and the potential for negative outcomes (e.g., significant losses) before making decisions based solely on the expected outcome. A comprehensive analysis also includes measures like standard deviation to understand the range of possible results.

Does the expected outcome account for all risks?

The expected outcome accounts for quantifiable risks where probabilities can be assigned to various results. However, it may not fully capture all aspects of risk, such as extreme, unforeseen events (often called "black swans") or liquidity risks. Its accuracy depends on the quality and reliability of the probability estimates used in its calculation, which can be challenging to determine for complex or novel situations.