What Is the Coefficient of Determination?
Let me explain the coefficient of determination directly to you—it's also known as r-squared, and it quantifies how much a stock’s price changes are influenced by movements in its associated stock index. As an investor, you can use this measure to see what percentage of a stock’s price movement is explained by the broader index shifts.
Key Takeaways
You should know that the coefficient of determination is a statistical concept focused on data analysis and financial modeling. It explains the relationship between an independent variable, like an index, and a dependent one, like a stock price. Commonly called r-squared or r2, this measure ranges from 0.0 to 1.0, where 1.0 means perfect correlation and a strong model for forecasts, while 0.0 indicates no dependency on the index.
Understanding the Coefficient of Determination
Think of the coefficient of determination as a tool that shows how much variability in one factor comes from its relationship with another. It’s expressed as a value between 0.0 and 1.0, or 0% to 100%. A value of 1.0 means 100% correlation, making it reliable for predictions, whereas 0.0 shows no functional dependency. For instance, a 0.20 value means 20% of the asset’s price movement is explained by the index, and 0.50 means half is explained.
Remember, this is the square of the correlation coefficient 'r' from statistics. While 'r' can be negative, r-squared is always positive because it's 'r' multiplied by itself.
Calculating the Coefficient of Determination
To calculate it, you start by creating a scatter plot of the data with a trend line. Take, for example, closing prices for the S&P 500 and Apple (AAPL) stock from December 21, 2022, to January 20, 2023. You plot these to assess the goodness of fit, which measures how well the data aligns with the regression line by looking at distances from the trend line to the points.
Example Data Table
- Jan. 20: S&P $3,972.61, APPL $137.87
- Jan. 19: S&P $3,898.85, APPL $135.27
- Jan. 18: S&P $3,928.86, APPL $135.21
- Jan. 17: S&P $3,990.97, APPL $135.94
- Jan. 13: S&P $3,999.09, APPL $134.76
- Jan. 12: S&P $3,983.17, APPL $133.41
- Jan. 11: S&P $3,969.61, APPL $133.49
- Jan. 10: S&P $3,919.25, APPL $130.73
- Jan. 9: S&P $3,892.09, APPL $130.15
- Jan. 6: S&P $3,895.08, APPL $129.62
- Jan. 5: S&P $3,808.10, APPL $125.02
- Jan. 4: S&P $3,852.97, APPL $126.36
- Jan. 3: S&P $3,824.14, APPL $125.07
- Dec. 30: S&P $3,839.50, APPL $139.93
- Dec. 29: S&P $3,849.28, APPL $129.61
- Dec. 28: S&P $3,783.22, APPL $126.04
- Dec. 27: S&P $3,829.25, APPL $130.03
- Dec. 23: S&P $3,844.82, APPL $131.86
- Dec. 22: S&P $3,822.39, APPL $132.23
- Dec. 21: S&P $3,878.44, APPL $135.45
Using Spreadsheets
In spreadsheets, you can use a simple formula like =RSQ(A1:A20, B1:B20) for columns with S&P and Apple data. For this example, it gives an r-squared of 0.347, showing the prices are less correlated than if it were 0.5 or higher.
Manual Calculation
If you’re doing it manually, gather the data and compute x (S&P), x², y (APPL), y², and xy for each day, then sum them up. Use the formula r² = [n(∑xy) - (∑x)(∑y)]² / {[n∑x² - (∑x)²] × [n∑y² - (∑y)²]}, where n=20. Plugging in the sums—∑x=77,781.69, ∑x²=302,584,424, ∑y=2,638.05, ∑y²=348,307.23, ∑xy=10,262,772.73—yields r²=0.347. This process is tedious and error-prone with large datasets, so I recommend spreadsheets for accuracy.
Interpreting the Coefficient of Determination
Once you have r-squared, like 0.347 for Apple and S&P, it tells you how closely the asset’s prices match the index. A value of 0.347 means somewhat correlated, not highly so. You decide if this is useful for trend analysis—r-squared doesn’t label it good or bad on its own. Since Apple is on multiple indexes, you could calculate r-squared for others to compare.
Frequently Asked Questions
You might wonder how to interpret it: r-squared shows correlation level between dependent and independent variables, from 0.0 (low) to 1.0 (high). In regression, it reveals dependency strength. It can’t exceed 1.0 or go negative; if it does, there’s a calculation error.
The Bottom Line
In summary, the coefficient of determination is a ratio showing dependency between variables, like a stock and its index. If r-squared is near zero, there’s little dependency; near 1.0, it’s highly dependent. Use it to gauge correlations in your investments.
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