Expected value (EV) is a fundamental concept in decision-making, particularly in finance. It helps individuals predict the average outcome of uncertain scenarios. EV provides a framework for evaluating a decision’s potential benefits or losses by considering all possible outcomes and their associated probabilities. While it’s a valuable tool, EV is not without its limitations. This article explores the nuances of EV, its calculation, and the often-overlooked risks that can undermine its utility in practical scenarios, particularly when financial losses threaten long-term sustainability.

**What is Expected Value?**

The expected value represents the average result of a decision that could lead to various outcomes, each with its likelihood of occurring. The concept allows decision-makers to account for uncertainty by weighing each potential outcome according to its probability. In doing so, EV offers insight into what can be expected, on average, if a particular choice were repeated many times. It is a cornerstone of probability theory and is widely used in fields ranging from economics to finance.

However, expected value assumes that outcomes balance over time and that individuals or organizations can endure both gains and losses. While it provides a useful starting point for evaluating decisions, it does not consider the potential for catastrophic outcomes—such as bankruptcy—that can remove a player from the game entirely.

**How Expected Value is Calculated**

The expected value calculation involves two key factors: the potential outcomes and the probability of each outcome occurring. To calculate EV, you first identify each possible result of a decision and estimate its probability. Next, you assign a value to each outcome, whether a financial gain, loss, or another quantifiable result. The final step is to multiply the value of each outcome by its respective probability and then sum these weighted outcomes to arrive at the expected value.

In essence, EV tells you what to expect in the long run if the decision is repeated multiple times. While this sounds straightforward, the calculation often oversimplifies complex real-world risks by assuming that long-term averages can be realized without interruption—an assumption that may not hold when dealing with high-stakes decisions.

**Expected Value in the Lottery**

To illustrate, consider the decision to purchase a lottery ticket. In this scenario, there are two possible outcomes: winning the jackpot or losing your money. The probability of winning is extremely low, while the likelihood of losing is very high. To calculate the expected value of buying a lottery ticket, you would first consider the prize money and multiply it by the slim chance of winning. Next, you would weigh the ticket cost by the much higher probability of losing.

Once these probabilities and values are multiplied, the results are combined to calculate the expected value. For most lotteries, this EV is negative, meaning that, on average, you would lose money over time if you repeatedly bought tickets. This example demonstrates the practical limits of expected value in situations where the risk of loss is overwhelming despite the potential for significant gains.

**The Risk of Bankruptcy**

While expected value estimates average outcomes, it does not account for the risk of ruin, particularly in cases where a decision could lead to bankruptcy or the loss of all available capital. Even if a decision has a positive EV, the risk of a catastrophic loss can make it unsustainable.

For example, imagine repeatedly making investments with a favourable expected value but with a slight chance of severe losses. If one of those losses wipes out your capital, you lose the ability to continue investing in future opportunities, making the expected value irrelevant. This is why considering EV without accounting for risk is dangerous; while the long-term outlook may be positive, a single significant loss can prevent further participation in the game, cutting off the potential for future gains.

**The Role of Variance and Compounded Returns**

Another factor that affects decision-making beyond EV is variance, which refers to the unpredictability or volatility of outcomes. Variance plays a crucial role in investments where returns are compounded over time. In theory, an investment with a positive EV should lead to long-term growth, but high variance can erode these gains due to how compounded returns work.

Compounded returns build on previous results, meaning a loss in one period reduces the base for future returns. The more volatile the returns, the greater the risk of losing ground that cannot be quickly recovered, even if subsequent gains are substantial. This effect is better understood through the *geometric mean*, which considers the compounding nature of growth and is often lower than the simple average when variance is high. As a result, high-risk investments may underperform even when their expected value appears favourable on paper.

**Balancing Expected Value and Risk**

Given the limitations of relying solely on expected value, decision-makers must consider EV and the associated risks. While EV can guide long-term strategy, it does not fully account for the potential impact of significant losses, especially when they could lead to bankruptcy or otherwise prevent future participation. Strategies such as diversification, which spreads risk across different investments, can help mitigate the effects of variance and protect against catastrophic outcomes.

By incorporating risk management techniques alongside expected value calculations, individuals and businesses can make more informed, sustainable decisions that not only focus on potential gains but also guard against the losses that could eliminate future opportunities.

**Bottom Line**

Expected value is an essential tool for evaluating decisions in uncertain environments, providing insight into the average outcomes of repeated choices. However, it is critical to recognize that EV does not account for the risks that could prevent individuals or organizations from realizing these long-term benefits. The risk of bankruptcy, high variance in returns, and the compounding effects of losses all undermine the expected value calculations. By understanding and managing these risks, decision-makers can better balance the potential for reward with the need for long-term, fundamentally sound investments.

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