什么是变异系数 (CV)？

What is the coefficient of variation (CV)?

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The coefficient of variation (CV) is a statistical measure of the spread of data points in a data set around the mean. The coefficient of variation represents the ratio of the standard deviation to the mean, and is a useful statistic for comparing the degree of variation between different data sets, even if the means are very different.

Main gains

• The coefficient of variation (CV) is a statistical measure of the relative dispersion of data points in a data set around the mean.
• It represents the ratio of standard deviation to mean.
• CV can be used to compare the degree of variation between different data sets, even if the means are very different.
• In finance, the coefficient of variation allows investors to determine how much volatility or risk is being taken compared to the expected return on an investment.
• The lower the ratio of standard deviation to average return, the better the risk-return trade-off.

Understanding the coefficient of variation (CV)

The coefficient of variation shows the degree of variation in the data within a sample, relative to the population mean.

In finance, the coefficient of variation allows investors to determine how much volatility or risk is being taken compared to the expected return on an investment. Ideally, if the coefficient of variation formula should yield a lower ratio of standard deviation to average return, the better the risk-return trade-off. [1]

They are most commonly used to analyze dispersion around the mean, but quartile, quintile or decile CVs can also be used to understand variation around the median or, for example, the 10th percentile.

Important Note: The coefficient of variation formula or calculation can be used to determine the deviation between the historical average price and current price performance of a stock, commodity, or bond, relative to other assets.

Coefficient of variation (CV) formula

Here is the formula for how to calculate the coefficient of variation: [2]

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To calculate the CV of a sample, the formula is:

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Tip: Multiplying the coefficient by 100 is an optional step to get a percentage instead of a decimal.

The coefficient of variation formula can be performed in Excel by first using the standard deviation function of the data set. Next, calculate the average using the provided Excel function. Since the coefficient of variation is the standard deviation divided by the mean, the cells containing the standard deviation are divided by the cells containing the mean.

Coefficient of variation (CV) and standard deviation

Standard deviation is a statistic that measures the spread of a data set relative to its mean. It is used to determine the distribution of values ​​within a single data set rather than comparing different units.

The coefficient of variation is used when we want to compare two or more data sets. CV is the ratio of the standard deviation to the mean. And because it is independent of the unit of measurement, it can be used to compare data sets that have different units or have widely different means.

Simply put, the standard deviation measures the distance from the mean, while the coefficient of variation measures the ratio of the standard deviation to the mean. [2]

Advantages and Disadvantages of Coefficient of Variation (CV)

The coefficient of variation can be useful when comparing data sets that have different units or have widely different means. [3]

This includes when using risk/reward ratio to select investments. For example, risk-averse investors may want to consider assets that have historically been less volatile relative to the overall market or their industry. Conversely, risk-seeking investors may seek to invest in assets that have historically been more volatile.

When the mean is close to zero, CV is very sensitive to small changes in the mean. Using the example above, one notable drawback is if the expected return in the denominator is negative or zero. In this case, the coefficient of variation may be misleading. [3]

If the expected return in the denominator of the coefficient of variation formula is negative or zero, the results can be misleading. [3]

How to use the coefficient of variation (CV)?

The coefficient of variation is used in many different fields, including chemistry, engineering, physics, economics, and neuroscience.

In addition to helping when using risk/return ratio to choose investments, economists also use it to measure economic inequality. Outside the financial field, it is often used to audit the accuracy of a specific process and achieve perfect balance.

Select Coefficient of Variation (CV) Example for Investments

For example, consider a risk-averse investor who wants to invest in an exchange-traded fund (ETF), a basket of securities that tracks a broad market index. Investors choose the SPDR S&P 500 ETF (SPY), the Invesco QQQ ETF (QQQ) and the iShares Russell 2000 ETF (IWM). Investors then analyze the ETF's returns and volatility over the past 15 years and assume that the ETF's returns are likely to be similar to its long-term average.

For ease of illustration, the following 15 years of historical information is used in investor decision-making:

• If the SPDR S&P 500 ETF has an average annual return of 5.47% and a standard deviation of 14.68%, the coefficient of variation for SPY is 2.68.
• If the Invesco QQQ ETF has an average annual return of 6.88% and a standard deviation of 21.31%, the coefficient of variation for QQQ is 3.10.
• If the iShares Russell 2000 ETF has an average annual return of 7.16% and a standard deviation of 19.46%, the IWM's coefficient of variation is 2.72.

Based on the approximate numbers, investors could invest in the SPDR S&P 500 Index ETF or the iShares Russell 2000 ETF, as the risk/reward ratio is roughly the same and suggests a better risk-reward tradeoff than the Invesco QQQ ETF.

What does the coefficient of variation tell us?

The coefficient of variation (CV) indicates the size of the standard deviation relative to its mean. The higher the coefficient of variation, the greater the dispersion around the mean. [2]

What is considered a good coefficient of variation?

It depends on what you are looking at and comparing. There is no set value that can be universally considered "good". In general, however, a lower coefficient of variation is usually preferable because it indicates that the data values ​​are spread over a smaller range relative to the mean.

How to calculate the coefficient of variation?

To calculate the coefficient of variation, first find the mean, then find the sum of squares, and then calculate the standard deviation. Once you have this information, you can calculate the coefficient of variation by dividing the standard deviation by the mean. [2]

Bottom line

The coefficient of variation is a simple way to compare the degree of variation between different data sets. It can be applied to almost anything, including the process of choosing the right investments.

In general, higher CVs indicate that the group is more variable, while lower values ​​indicate the opposite.

References

[1] JoVE. “JoVE Core Statistics; Chapter 4, Measures of Variation; 4.7: Coefficient of Variation.”

[2] Penn State, Eberly College of Science. “STAT 500: Applied Statistics; 1.5.3Measures of Variability.”

[3] UCLA, Advanced Research Computing: Statistical Methods and Data Analytics. “FAQ: What Is the Coefficient of Variation?”

About TheQuantHackers

TheQuantHackers is a cutting-edge community composed of a group of people from the world's top universities and quantitative finance practitioners, dedicated to exploring the infinite possibilities in the fields of artificial intelligence (AI) and quantification (Quant). Our team members come from world-renowned universities such as the University of Cambridge, Oxford University, Harvard University, ETH Zurich, Peking University, and the University of Science and Technology of China. Our external consultants come from first-class companies such as Microsoft, HSBC, Citadel, Man Group, Citi, Jump Trading, and top domestic private equity companies.$\begin{aligned} &\text{CV} = \frac { \sigma }{ \mu } \\ &\textbf{其中：} \\ &\sigma = \text{标准差} \\ &\mu = \text{均值} \\ \end{aligned}$

要计算样本的 CV，公式为：

$CV = s/x * 100$

变异系数 (CV) 是一种统计度量，用于衡量数据集中数据点围绕均值的离散程度。 变异系数表示标准差与均值的比率，它是一种有用的统计量，可以比较不同数据集之间的变异程度，即使均值差异很大。

主要收获

• 变异系数 (CV) 是一种统计度量，用于衡量数据集中数据点围绕均值的相对离散程度。
• 它表示标准差与均值的比率。
• CV 可用于比较不同数据集之间的变异程度，即使均值差异很大。
• 在金融领域，变异系数允许投资者确定与投资预期回报相比，承担了多少波动性或风险。
• 标准差与平均回报的比率越低，风险回报权衡就越好。

理解变异系数 (CV)

变异系数显示了样本中数据的变异程度，相对于总体均值而言。

在金融领域，变异系数允许投资者确定与投资预期回报相比，承担了多少波动性或风险。 理想情况下，如果变异系数公式应得出标准差与平均回报的较低比率，那么风险回报权衡就越好。[1]

它们最常用于分析围绕均值的离散程度，但四分位数、五分位数或十分位数 CV 也可用于理解围绕中位数或例如第 10 个百分位数的变异。

重要提示： 变异系数公式或计算可用于确定股票、商品或债券的历史平均价格与当前价格表现之间的偏差，相对于其他资产而言。

变异系数 (CV) 公式

以下是如何计算变异系数的公式：[2]