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Expected Value (EV) is a term commonly used in the investment industry and is used to express the expected average value of an investment at a certain point in the future.
Expected Value (EV) is a term commonly used in the investment industry and is used to express the expected average value of an investment at a certain point in the future. Investors use expected value to estimate the value of an investment, usually relative to its risk.
By calculating expected value, investors are able to select scenarios that are most likely to achieve the desired outcome. In statistics and probability analysis, expected value is calculated by multiplying each possible outcome by its probability of occurring and then adding all these values.
Expected value refers to the expected value of an asset in the future. The expected value of a random variable provides a measure of the center of the distribution of that variable. The expected value is essentially the long-term average of that variable.
According to the law of large numbers, when the number of repetitions approaches infinity, the average value of the variable will converge to the expected value. Expected value is also called expectation, mean or first moment.
Expected value can be calculated for a single discrete variable, a single continuous variable, multiple discrete variables, and multiple continuous variables. In the case of continuous variables, integrals need to be used.
Scenario analysis is a technique for calculating the expected value of an investment opportunity. It uses estimated probabilities and multivariate models to examine the likely outcomes of an investment. Scenario analysis also helps investors judge whether the level of risk they are taking is appropriate for the expected outcomes.
**IMPORTANT NOTE:**The difference between expected value and arithmetic mean is that the former involves a probability distribution, while the latter involves the distribution of occurrences of events.
The formula for expected value is:
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Therefore, the expected value of a random variable X is the sum of the products of each value of the random variable and its probability.
Investors need to understand several key factors when constructing an investment or financial portfolio, including how the asset works and its associated risks. Investors should also be clear about their financial situation, investment objectives and investment horizon.
Once these factors are well understood, investors and their financial advisors can use expected values to construct portfolios that maximize returns while minimizing risk.
You can use expected value to determine the potential return on an investment, allowing you to choose which assets to add to your portfolio to match your return preferences.
For example, when calculating expected value, you first multiply the probability of a positive outcome by the potential reward. Assuming that an investment has a 60% chance of increasing in value by 10,000 = $6,000.
Then, multiply the probability of a negative outcome by the potential loss. For example, there is a 40% chance that the investment will lose 5,000 = $2,000.
Finally, subtract the second result from the first: 2,000 = $4,000. This is the expected value of the investment.
Next, you can compare two or more investments of interest, following the same steps, and compare expected returns. This can help you make choices when building your portfolio.
Also remember that different assets may have different expected values. That is, the expected value (and risk characteristics) of stocks versus bonds or exchange-traded funds (ETFs) are different. It is useful to calculate and compare expected values for various assets that interest you.
You can also use expectations to make adjustments after your portfolio is built. For example, comparing expected values to decide whether to sell an asset that is underperforming and has no upside expectations and replace it with another asset that has a higher expected value.
To calculate the expected value of a single discrete random variable, you must multiply each value of the variable by its probability of occurrence.
Take an ordinary six-sided die, for example. When you roll a die, there is a 1 in 6 chance that it will land on a value of 1, 2, 3, 4, 5, or 6. Based on this information, the calculation is:
<!-- @@@ -->If you rolled a six-sided die an infinite number of times, you would find that the average is equal to 3.5.
The expected value of a stock is estimated as the net present value (NPV) of all future dividends from the stock. If you can estimate the growth rate of dividends, you can use a dividend discount model (such as the Gordon Growth Model) to predict how much investors are willing to pay for the stock. However, it is important to note that this is a different formula than the statistical expected value described in this article.
For non-dividend stocks, analysts often use a multiple method to arrive at expected values. For example, the price-to-earnings ratio (P/E) is often used and compared to industry peers. Therefore, if the average P/E ratio in the technology industry is 25 times, a technology stock's expected value is 25 times its earnings per share. Again, this is not the same as the statistical expected value described in this article, but it is another commonly used way of looking at stock value.
Modern portfolio theory and related models use mean-variance optimization to derive optimal portfolio allocations based on risk adjustment. Risk is measured as the standard deviation of the portfolio, while the mean is the expected value (expected return) of the portfolio. This really takes advantage of the concepts described in this article.
Understanding the concept of expected value is critical for investors. It helps them judge the level of return an investment is likely to generate. Expected value and scenario analysis can provide insights into investment risks and returns, helping investors decide whether to include them in their portfolios.
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其中：
因此，随机变量 X 的期望值为随机变量的每个值与其概率的乘积之和。
投资者在构建投资或金融组合时，需要了解几个关键因素，包括资产的运作机制及其相关风险。投资者还应清楚自己的财务状况、投资目标和投资期限。
一旦深入理解这些因素，投资者及其财务顾问便可以利用期望值来构建最大化收益的投资组合，同时尽量降低风险。
你可以利用期望值来确定投资的潜在收益，从而选择哪些资产加入你的投资组合，以符合你的回报偏好。
例如，计算期望值时，首先将正面结果的概率与潜在回报相乘。假设一项投资有60%的可能性增值1万美元，计算为：0.6 x 1万美元 = 6000美元。
然后，将负面结果的概率与潜在损失相乘。例如，该投资还有40%的可能性贬值5000美元，计算为：0.4 x 5000美元 = 2000美元。
最后，将第二个结果从第一个结果中减去：6000美元 - 2000美元 = 4000美元。这便是该项投资的期望值。
接下来，你可以比较两个或多个感兴趣的投资，遵循相同的步骤，并比较期望回报。这可以帮助你在构建投资组合时做出选择。
还要记住，不同资产的期望值可能不同。也就是说，股票与债券或交易所交易基金（ETF）的期望值（和风险特征）是不同的。对你感兴趣的各种资产计算期望值并进行比较是非常有用的。
你还可以在投资组合构建完成后利用期望值进行调整。例如，比较期望值，以决定是否出售一项表现不佳且没有上涨预期的资产，并用另一项期望值更高的资产进行替换。
为了计算单一离散随机变量的期望值，你必须将变量每个值与其发生概率相乘。
以普通的六面骰子为例。当你掷骰子时，它有六分之一的概率落在1、2、3、4、5或6任一值上。基于这一信息，计算为：
期望值（Expected Value，EV）是投资行业中常用的术语，用于表示某项投资在未来某个时点的预期平均价值。投资者利用期望值来估算投资的价值，通常是相对于其风险进行相对分析。
通过计算期望值，投资者能够选择最可能实现预期结果的情境。在统计学和概率分析中，期望值的计算方法是将每个可能结果与其发生的概率相乘，然后将所有这些值相加。
期望值指的是未来资产的预期价值。随机变量的期望值提供了该变量分布中心的度量。期望值实质上是该变量的长期平均值。
根据大数法则，当重复次数趋近于无穷大时，变量的平均值会收敛于期望值。期望值也被称为期望、均值或第一矩。
期望值可以用于单个离散变量、单个连续变量、多重离散变量和多重连续变量的计算。在连续变量的情况下，需要使用积分。
情境分析是计算投资机会期望值的一种技术。它利用估计的概率和多变量模型来检查某项投资的可能结果。情境分析还帮助投资者判断他们所承担的风险水平是否适合预期结果。
**重要提示：**期望值与算术均值的区别在于，前者涉及概率分布，而后者涉及事件发生的分布。
期望值的公式为：
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