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In the finance and investment industries, correlation is a statistic that measures the extent to which two securities move in relation to each other.
# What is correlation?
In the finance and investment industries, correlation is a statistic that measures the extent to which two securities move in relation to each other. Correlation is used in advanced portfolio management and is calculated as the correlation coefficient, whose value must be between -1.0 and +1.0.
Correlation shows the strength of the relationship between two variables and is expressed as a numerical value of the correlation coefficient. The correlation coefficient ranges in value from -1.0 to 1.0.
A perfect positive correlation means that the correlation coefficient is exactly 1. This means that when one security moves up or down, the other security moves simultaneously in the same direction. A perfect negative correlation means the two assets move in opposite directions, while a zero correlation means there is no linear relationship at all.
For example, large-cap mutual funds typically have a highly positive correlation, close to 1, with the Standard & Poor's (S&P) 500 Index. Small-cap stocks tend to be positively correlated with the S&P, but the correlation is modest at about 0.8.
However, put option prices and their underlying stock prices tend to be negatively correlated. A put option gives the owner the right (but not the obligation) to sell a specific amount of the underlying security at a predetermined price within a specified time frame.
Put option contracts become more profitable when the price of the underlying stock falls. In other words, as stock prices rise, put option prices fall, a direct and high-magnitude negative correlation.
There are several ways to calculate correlation. This article will further discuss the most commonly used method, the Pearson product-moment correlation. The Pearson product-moment correlation measures the linear relationship between two variables. It can be used for any data set with a finite covariance matrix. Here are the steps for calculating correlation.
IMPORTANT: To avoid complex manual calculations, consider using the CORREL function in Excel.
Using the Pearson product-moment correlation method, the correlation coefficient r can be found using the following formula:
<!-- @@@ -->Investment managers, traders, and analysts find it important to calculate correlation because the risk-reducing benefits of diversification rely on this statistic. Financial spreadsheets and software can quickly calculate the value of correlation.
As a hypothetical example, suppose an analyst needs to calculate the correlation of the following two data sets:
X: (41, 19, 23, 40, 55, 57, 33)
Y: (94, 60, 74, 71, 82, 76, 61)
Finding correlations involves three steps. The first is to sum all the X values to find SUM(X), sum all the Y values to find SUM(Y), multiply each X value with its corresponding Y value, and then add them together to find SUM(X,Y):
SUM(X) = (41 + 19 + 23 + 40 + 55 + 57 + 33) = 268
SUM(Y) = (94 + 60 + 74 + 71 + 82 + 76 + 61) = 518
SUM(X,Y) = (41 x 94) + (19 x 60) + (23 x 74) + ... (33 x 61) = 20,391
The next step is to take each X value, square it, and then add all those values together to find SUM(x^2). This must also be done for Y values:
SUM(X^2) = (41^2) + (19^2) + (23^2) + ... (33^2) = 11,534
SUM(Y^2) = (94^2) + (60^2) + (74^2) + ... (61^2) = 39,174
Noting that there are seven observations n, the correlation coefficient r can be found using the following formula:
<!-- @@@ -->In this example, the correlation would be:
r = 0.54
In investing, correlation is most important when related to a diversified portfolio. Investors who wish to reduce risk can do so by investing in uncorrelated assets. For example, consider an investor who owns airline stock. If the correlation between the airline industry and the social media industry is found to be low, investors may choose to invest in social media stocks, knowing that a negative impact on one industry may not affect the other.
This is typically the approach when considering investing across asset classes. Stocks, bonds, precious metals, real estate, cryptocurrencies, commodities and other types of investments have different relationships to each other. While some may be highly correlated, if they are not, others may act as hedges to spread risk.
Note: Risk that can be diversified is called unsystematic risk. This type of risk is specific to a company, industry, or asset class. Investing in different assets reduces the correlation of your portfolio and reduces your exposure to unsystematic risk.
Correlation is often determined by and related to other statistical considerations. When variables are analyzed using statistics, correlation is often cited.
In statistics, the p-value is used to indicate whether a study result is statistically significant. It is possible to determine that two variables are related, but there may not be enough supporting evidence to make this a strong claim. A high p-value indicates that there is enough evidence to meaningfully conclude that the overall correlation coefficient is different from zero.
The easiest way to visualize whether two variables are related is to depict them graphically using a scatter plot. Each point on the scatter plot represents a sample item. The x-axis of the scatter plot represents one of the variables being tested, and the y-axis of the scatter plot represents the other variable.
The correlation coefficient of two variables is usually represented graphically as a linear line to show the relationship between the two variables. If two variables are positively correlated, an increasing linear line can be drawn on a scatter plot. If two variables are negatively correlated, a decreasing linear line can be drawn. The stronger the relationship between data points, the closer each data point is to the line.
Scatter plots may be more useful when analyzing more complex data that may have changing relationships. For example, two variables may be positively correlated up to a certain point and then their relationship becomes negative. This nonlinear relationship may be more difficult to identify using a formula, but is easier to spot when plotted on a scatter plot.
Finally, correlations can be easily depicted when scatterplots include density shading. Density shading or density ellipses are shaded areas on a scatter plot that visually display the areas on the scatter plot where data points are most densely packed. If the variables are correlated, the density ellipse usually reflects the direction of the linear correlation line. Otherwise, a more rounded density ellipse with no clear orientation indicates lower correlation.
Another inherent difficulty in statistics is determining whether the relationship between two variables is caused by those variables. Consider the following statements:
"Most basketball players are tall. So if you play basketball, you get taller."
Clearly, the above statement is incorrect. Individuals who are tall and understand this advantage may be drawn to basketball because their natural physical abilities are best suited to the sport. However, because height and basketball performance may be positively correlated, statisticians and data scientists must be aware that a strong relationship between two variables may be caused by either variable.
As with other aspects of statistical analysis, correlations can be misinterpreted. Even if the correlation between two variables appears to be strong, a small sample size can produce unreliable results. Alternatively, a small sample size may produce uncorrelated results when two variables are actually related.
Correlations are often skewed when outliers are present. Correlation only shows how one variable is related to another variable, and it may not be possible to clearly identify how individual instances or results affect the correlation coefficient.
Correlations can also be misinterpreted if the relationship between two variables is non-linear. It is much easier to identify two variables that have a positive or negative correlation. However, two variables may still be related in a more complex relationship.
Correlation is a statistical term that describes the extent to which two variables move in harmony with each other. If two variables move in the same direction, they are said to have a positive correlation. If they move in opposite directions, they have a negative correlation.
Correlations play an important role in finance as they are used to predict future trends and manage risk in investment portfolios. Today, correlations between assets can be easily calculated using a variety of software programs and online services. Correlation and other statistical concepts play an important role in the creation and pricing of derivatives and other complex financial instruments.
Correlation is a widely used concept in modern finance. For example, traders might use historical correlations to predict whether a company's stock will rise or fall due to changes in interest rates or commodity prices. Likewise, a portfolio manager might reduce his or her risk by ensuring that the individual assets in his portfolio are not overly correlated with each other.
Investors may have preferences for the level of correlation in their portfolios. Generally speaking, most investors prefer lower correlations because this reduces the risk that their portfolio of different assets or securities will be affected by similar market conditions. However, risk-seeking investors or investors looking to put their money into a very specific type of industry or company may be willing to have a higher correlation in their portfolio in exchange for greater potential returns.
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投资经理、交易员和分析师发现计算相关性非常重要，因为多元化带来的风险降低收益依赖于此统计量。金融电子表格和软件可以快速计算相关性的值。
作为一个假设的例子，假设分析师需要计算以下两个数据集的相关性：
X: (41, 19, 23, 40, 55, 57, 33)
Y: (94, 60, 74, 71, 82, 76, 61)
找到相关性涉及三个步骤。首先是加总所有 X 值以找到 SUM(X)，加总所有 Y 值以找到 SUM(Y)，并将每个 X 值与其对应的 Y 值相乘，然后将它们相加，以找到 SUM(X,Y)：
SUM(X) = (41 + 19 + 23 + 40 + 55 + 57 + 33) = 268
SUM(Y) = (94 + 60 + 74 + 71 + 82 + 76 + 61) = 518
SUM(X,Y) = (41 x 94) + (19 x 60) + (23 x 74) + ... (33 x 61) = 20,391
下一步是取每个 X 值，将其平方，然后将所有这些值相加，以找到 SUM(x^2)。对于 Y 值也必须这样做：
SUM(X^2) = (41^2) + (19^2) + (23^2) + ... (33^2) = 11,534
SUM(Y^2) = (94^2) + (60^2) + (74^2) + ... (61^2) = 39,174
注意到有七个观察值 n，可以使用以下公式找到相关系数 r：
&r=\text{相关系数}\\ &n=\text{观测值的数量}\end{aligned} $$ 在金融和投资行业中，相关性是一种统计量，用于衡量两种证券彼此相关的变动程度。相关性用于高级投资组合管理，计算为相关系数，其值必须介于 -1.0 和 +1.0 之间。 ### 主要内容 - 相关性是一种统计量，用于衡量两个变量彼此相关的变动程度。 - 在金融领域，相关性可以衡量股票与基准指数（如标准普尔 500 指数）的变动关系。 - 相关性与多元化密切相关，多元化是指通过投资于不相关的资产来减轻某些类型风险的概念。 - 相关性衡量的是关联性，但不能表明 x 是否导致 y，反之亦然，或者关联性是否由第三个因素引起。 - 相关性最容易通过散点图识别，特别是当变量具有非线性但仍然很强的相关性时。 ## 相关性可以告诉你什么 相关性显示了两个变量之间关系的强度，并以相关系数的数值表示。相关系数的值范围在 -1.0 和 1.0 之间。 完全正相关意味着相关系数正好是 1。这意味着当一种证券向上或向下移动时，另一种证券以相同的方向同步移动。完全负相关意味着两种资产向相反的方向移动，而零相关意味着根本没有线性关系。 例如，大盘共同基金通常与标准普尔 (S&P) 500 指数具有高度正相关性，接近于 1。小盘股票往往与标准普尔指数呈正相关，但相关性不高，约为 0.8。 然而，看跌期权价格及其标的股票价格往往呈负相关。看跌期权赋予所有者权利（但没有义务）在指定的时间范围内以预先确定的价格出售特定数量的标的证券。 当标的股票价格下跌时，看跌期权合约会变得更有利可图。换句话说，随着股票价格上涨，看跌期权价格下降，这是一种直接且高幅度的负相关。 ## 如何计算相关性 有几种计算相关性的方法。本文将进一步讨论最常用的方法，即皮尔逊积矩相关。皮尔逊积矩相关衡量两个变量之间的线性关系。它可以用于任何具有有限协方差矩阵的数据集。以下是计算相关性的步骤。 **重要提示：** 为了避免复杂的的手动计算，请考虑使用 Excel 中的 CORREL 函数。 使用皮尔逊积矩相关方法，可以使用以下公式找到相关系数 r：Upgrade to unlock all institutional-grade algorithms, derivative pricing engines, factor backtesting frameworks, and live QuantLab execution.
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