Expected Value

Expected value is the average outcome of a random event repeated many times, found by multiplying each possible result by its probability and adding the results together. In finance, it helps investors and analysts estimate expected returns or expected losses so they can weigh a decision before an uncertain outcome actually happens.

Probability Theory In Finance

Probability, in its simplest form, measures the likelihood of an event occurring. It's expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. Probability theory is a fundamental aspect of financial markets, providing the groundwork for understanding and managing uncertainty in various financial decisions.

That’s because a lot of finance involves trying to predict the future! Even though the future can never be predicted with absolute certainty, investment professionals rely upon forecasts and prediction models to envision what may happen to account for any event, no matter how unlikely those predictions may be.

First, Some Key Statistical Concepts in Probability

##### Expected Value

The expected value is the average outcome of a random event repeated many times. For instance, we can compute the expected value of a lottery ticket by examining how much you would expect to make if you bought 1,000 lottery tickets. In finance, the expected return on investments or the expected loss in a risky venture can be helpful to account for all possible scenarios in the future.

##### Variance and Standard Deviation

Variance and standard deviation measure the spread of possible outcomes around the expected value. This can basically be thought of how far an event is from the average or mean event outcome. These metrics are used to understand the risk associated with a financial decision, indicating how much the actual outcome might differ from the expected value. If you win 9/10 times with a certain strategy, but the time you lose, you lose all of your money, then it isn’t a very successful strategy!

##### Law of Large Numbers

This law states that as the number of trials or observations increases, the empirical probability (observed in data) will tend to get closer to the theoretical probability. This concept is fundamental in long-term investment strategies or actuarial work in insurance and basically states that if we repeat an event with an uncertain future thousands of times, eventually the average outcome will trend towards the theoretical average.

##### Bayes’ Theorem

Bayes’ Theorem is used for revising probabilities based on new information. It’s particularly useful in finance for updating investment forecasts or risk assessments in light of new data or changing market conditions. With better data, analysts are able to make better predictions.

##### Monte Carlo Simulations

These simulations use probability distributions to simulate various scenarios for portfolio performance by running the same analysis more than 5,000 times! This technique helps in strategic decision-making by providing a range of possible outcomes and their likelihoods, allowing for better risk management and investment planning.

How It Works With Investing & Portfolio Management

Understanding probability is crucial in assessing risks and expected returns in investment decisions. Investors determine the likelihood of a stock’s price increase based on various factors such as market conditions, economic indicators, and the company's performance. This assessment helps in making informed decisions about buying, holding, or selling stocks because they can predict which direction a stock’s price may trend.

For Example: An investor might evaluate the probability of a tech company's stock rising based on the company's quarterly earnings report and trends in the tech industry.

Financial institutions rely on probability to assess various types of risks, including credit risk, market risk, and operational risk. Probability theory also plays a significant role in portfolio management, aiding in diversification and risk optimization. Portfolio managers determine the mix of assets that maximizes the probability of achieving desired returns while managing risk of any asset class declining.

For Example: Banks use probability models to assess the likelihood of loan default based on an applicant's credit history, income level, and current economic conditions.

In the insurance industry, probability theory is fundamental in calculating premiums and managing risk. Insurance companies use probability to estimate the likelihood of events, like accidents or illnesses, and set premiums accordingly so that they are not insuring more than they can afford to cover.

For Example: Health insurance companies analyze factors such as age, lifestyle, and medical history to predict the risk of illness and thereby determine insurance premiums for different individuals.

The Bottom Line

Probability theory is a cornerstone of financial decision-making, providing a framework for understanding and managing risk in investments, insurance, and portfolio management. While it offers valuable insights and aids in strategic planning, acknowledging its limitations is crucial in the dynamic and often unpredictable world of finance.

Financial markets can sometimes exhibit irrational behavior, and unforeseen events like the 2008 financial crisis or the COVID-19 pandemic can significantly disrupt probabilistic predictions. This unpredictability underscores the need for cautious application of probability in financial decisions.

Ultimately, analysts and investors are only as smart as the data they receive and this makes predicting events like COVID-19 exceptionally challenging for any investor. But by having a firm understanding of probability, investors can at least appreciate that the unexpected may happen, and prepare their investments to account for these changes.

Questions

1. How does the article explain the connection between probability and financial decision-making?
2. What key concepts from the article could help someone make better financial decisions regarding risk?
3. In your own words, describe how the principles discussed in the article apply to everyday financial situations.

Doing the Math on Expected Value

The formula is simpler than it sounds: multiply each possible outcome by its probability, then add the results together. A coin flip that pays $10 on heads and nothing on tails has an expected value of $5, even though no single flip actually lands on $5. That's the idea worth sitting with. Expected value describes the average across many repeated events, not a prediction for what happens on any one try. Those probabilities often start out as fractions before they get converted to decimals for the formula, a conversion worth practicing on its own in a fractions to decimals worksheet.

That distinction matters for investing because a single trade can lose money even when it was the right decision going in. A stock with a 70% chance of gaining 10% and a 30% chance of losing 15% still carries a positive expected value, even on the day it happens to drop. The article above walks through how variance and standard deviation measure how far an actual result can stray from that average, which is exactly why a good expected value alone never tells the whole story.

Students can test this directly with the Rapunzl investing simulator, where a simulated $10,000 portfolio lets a class hold several stocks at once and watch how individual results diverge from the average outcome the math predicted. Pulling up live market data alongside those positions turns the abstract math into something visible: a stock can move against the odds on any single day, which is exactly what a probability distribution predicts will happen sometimes. For a closer look at how investors weigh the flip side of this math, see how investment risk factors into the same decision.

This explainer comes from Module 22 of the Rapunzl curriculum, part of the Financial Probabilities unit. Teachers: the accompanying activity and answer key are in the teacher portal.