You are given monthly annualized total return data for two mutual funds, A and B. Using linear regression, develop a relationship which gives the annualized return of fund B as a function of the return of fund A. What is the coefficient of determination of this relationship?
Month Total return of A (%) Total return of B (%)
Jan -26.2 30.8
Feb 12.0 30.5
Mar 52.0 -13.4
Apr 39.0 24.6
May -21.4 -27.7
Jun -38.4 -23.3
Jul 15.4 28.4
Aug 15.4 13.3
Sep 17.8 126.1
Oct -52.8 83.4
Nov -10.4 -19.8
Dec -73.4 79.8
To develop a relationship between the annualized return of fund B and the return of fund A using linear regression, we can follow these steps:
1. Calculate the monthly annualized returns for both funds A and B. This can be done by dividing the total return of each month by 12. For example, for fund A in January, the annualized return would be -26.2 / 12 = -2.18%.
2. Create a scatter plot with the annualized return of fund A on the x-axis and the annualized return of fund B on the y-axis. Each data point on the plot represents a month's return for both funds.
3. Use a linear regression model to fit a line to the scatter plot. The line will represent the relationship between the annualized return of fund A and fund B. The equation of the line will be in the form: B = m * A + c, where B represents the annualized return of fund B, A represents the annualized return of fund A, and m and c are the coefficients of the equation.
4. Once the line is fitted, the coefficient of determination (R-squared) can be calculated to measure the goodness of fit for the linear regression model. The coefficient of determination represents the proportion of the variation in the dependent variable (fund B) that can be explained by the independent variable (fund A).
To calculate the coefficient of determination, we need to calculate the sum of squared errors (SSE) and the total sum of squares (SST).
SSE represents the sum of the squared differences between the actual values of fund B and the predicted values from the linear regression model.
SST represents the sum of the squared differences between the actual values of fund B and the mean value of fund B.
The coefficient of determination can be calculated using the formula:
R-squared = 1 - (SSE / SST)
Using the given data, let's calculate the coefficient of determination: