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Principles of Finance

14.6 Use of R Statistical Analysis Tool for Regression Analysis

Principles of Finance14.6 Use of R Statistical Analysis Tool for Regression Analysis

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Table of contents
  1. Preface
  2. 1 Introduction to Finance
    1. Why It Matters
    2. 1.1 What Is Finance?
    3. 1.2 The Role of Finance in an Organization
    4. 1.3 Importance of Data and Technology
    5. 1.4 Careers in Finance
    6. 1.5 Markets and Participants
    7. 1.6 Microeconomic and Macroeconomic Matters
    8. 1.7 Financial Instruments
    9. 1.8 Concepts of Time and Value
    10. Summary
    11. Key Terms
    12. Multiple Choice
    13. Review Questions
    14. Video Activity
  3. 2 Corporate Structure and Governance
    1. Why It Matters
    2. 2.1 Business Structures
    3. 2.2 Relationship between Shareholders and Company Management
    4. 2.3 Role of the Board of Directors
    5. 2.4 Agency Issues: Shareholders and Corporate Boards
    6. 2.5 Interacting with Investors, Intermediaries, and Other Market Participants
    7. 2.6 Companies in Domestic and Global Markets
    8. Summary
    9. Key Terms
    10. CFA Institute
    11. Multiple Choice
    12. Review Questions
    13. Video Activity
  4. 3 Economic Foundations: Money and Rates
    1. Why It Matters
    2. 3.1 Microeconomics
    3. 3.2 Macroeconomics
    4. 3.3 Business Cycles and Economic Activity
    5. 3.4 Interest Rates
    6. 3.5 Foreign Exchange Rates
    7. 3.6 Sources and Characteristics of Economic Data
    8. Summary
    9. Key Terms
    10. CFA Institute
    11. Multiple Choice
    12. Review Questions
    13. Problems
    14. Video Activity
  5. 4 Accrual Accounting Process
    1. Why It Matters
    2. 4.1 Cash versus Accrual Accounting
    3. 4.2 Economic Basis for Accrual Accounting
    4. 4.3 How Does a Company Recognize a Sale and an Expense?
    5. 4.4 When Should a Company Capitalize or Expense an Item?
    6. 4.5 What Is “Profit” versus “Loss” for the Company?
    7. Summary
    8. Key Terms
    9. Multiple Choice
    10. Review Questions
    11. Problems
    12. Video Activity
  6. 5 Financial Statements
    1. Why It Matters
    2. 5.1 The Income Statement
    3. 5.2 The Balance Sheet
    4. 5.3 The Relationship between the Balance Sheet and the Income Statement
    5. 5.4 The Statement of Owner’s Equity
    6. 5.5 The Statement of Cash Flows
    7. 5.6 Operating Cash Flow and Free Cash Flow to the Firm (FCFF)
    8. 5.7 Common-Size Statements
    9. 5.8 Reporting Financial Activity
    10. Summary
    11. Key Terms
    12. CFA Institute
    13. Multiple Choice
    14. Review Questions
    15. Problems
    16. Video Activity
  7. 6 Measures of Financial Health
    1. Why It Matters
    2. 6.1 Ratios: Condensing Information into Smaller Pieces
    3. 6.2 Operating Efficiency Ratios
    4. 6.3 Liquidity Ratios
    5. 6.4 Solvency Ratios
    6. 6.5 Market Value Ratios
    7. 6.6 Profitability Ratios and the DuPont Method
    8. Summary
    9. Key Terms
    10. CFA Institute
    11. Multiple Choice
    12. Review Questions
    13. Problems
    14. Video Activity
  8. 7 Time Value of Money I: Single Payment Value
    1. Why It Matters
    2. 7.1 Now versus Later Concepts
    3. 7.2 Time Value of Money (TVM) Basics
    4. 7.3 Methods for Solving Time Value of Money Problems
    5. 7.4 Applications of TVM in Finance
    6. Summary
    7. Key Terms
    8. CFA Institute
    9. Multiple Choice
    10. Review Questions
    11. Problems
    12. Video Activity
  9. 8 Time Value of Money II: Equal Multiple Payments
    1. Why It Matters
    2. 8.1 Perpetuities
    3. 8.2 Annuities
    4. 8.3 Loan Amortization
    5. 8.4 Stated versus Effective Rates
    6. 8.5 Equal Payments with a Financial Calculator and Excel
    7. Summary
    8. Key Terms
    9. CFA Institute
    10. Multiple Choice
    11. Problems
    12. Video Activity
  10. 9 Time Value of Money III: Unequal Multiple Payment Values
    1. Why It Matters
    2. 9.1 Timing of Cash Flows
    3. 9.2 Unequal Payments Using a Financial Calculator or Microsoft Excel
    4. Summary
    5. Key Terms
    6. CFA Institute
    7. Multiple Choice
    8. Review Questions
    9. Problems
    10. Video Activity
  11. 10 Bonds and Bond Valuation
    1. Why It Matters
    2. 10.1 Characteristics of Bonds
    3. 10.2 Bond Valuation
    4. 10.3 Using the Yield Curve
    5. 10.4 Risks of Interest Rates and Default
    6. 10.5 Using Spreadsheets to Solve Bond Problems
    7. Summary
    8. Key Terms
    9. CFA Institute
    10. Multiple Choice
    11. Review Questions
    12. Problems
    13. Video Activity
  12. 11 Stocks and Stock Valuation
    1. Why It Matters
    2. 11.1 Multiple Approaches to Stock Valuation
    3. 11.2 Dividend Discount Models (DDMs)
    4. 11.3 Discounted Cash Flow (DCF) Model
    5. 11.4 Preferred Stock
    6. 11.5 Efficient Markets
    7. Summary
    8. Key Terms
    9. CFA Institute
    10. Multiple Choice
    11. Review Questions
    12. Problems
    13. Video Activity
  13. 12 Historical Performance of US Markets
    1. Why It Matters
    2. 12.1 Overview of US Financial Markets
    3. 12.2 Historical Picture of Inflation
    4. 12.3 Historical Picture of Returns to Bonds
    5. 12.4 Historical Picture of Returns to Stocks
    6. Summary
    7. Key Terms
    8. Multiple Choice
    9. Review Questions
    10. Video Activity
  14. 13 Statistical Analysis in Finance
    1. Why It Matters
    2. 13.1 Measures of Center
    3. 13.2 Measures of Spread
    4. 13.3 Measures of Position
    5. 13.4 Statistical Distributions
    6. 13.5 Probability Distributions
    7. 13.6 Data Visualization and Graphical Displays
    8. 13.7 The R Statistical Analysis Tool
    9. Summary
    10. Key Terms
    11. CFA Institute
    12. Multiple Choice
    13. Review Questions
    14. Problems
    15. Video Activity
  15. 14 Regression Analysis in Finance
    1. Why It Matters
    2. 14.1 Correlation Analysis
    3. 14.2 Linear Regression Analysis
    4. 14.3 Best-Fit Linear Model
    5. 14.4 Regression Applications in Finance
    6. 14.5 Predictions and Prediction Intervals
    7. 14.6 Use of R Statistical Analysis Tool for Regression Analysis
    8. Summary
    9. Key Terms
    10. Multiple Choice
    11. Review Questions
    12. Problems
    13. Video Activity
  16. 15 How to Think about Investing
    1. Why It Matters
    2. 15.1 Risk and Return to an Individual Asset
    3. 15.2 Risk and Return to Multiple Assets
    4. 15.3 The Capital Asset Pricing Model (CAPM)
    5. 15.4 Applications in Performance Measurement
    6. 15.5 Using Excel to Make Investment Decisions
    7. Summary
    8. Key Terms
    9. CFA Institute
    10. Multiple Choice
    11. Review Questions
    12. Problems
    13. Video Activity
  17. 16 How Companies Think about Investing
    1. Why It Matters
    2. 16.1 Payback Period Method
    3. 16.2 Net Present Value (NPV) Method
    4. 16.3 Internal Rate of Return (IRR) Method
    5. 16.4 Alternative Methods
    6. 16.5 Choosing between Projects
    7. 16.6 Using Excel to Make Company Investment Decisions
    8. Summary
    9. Key Terms
    10. CFA Institute
    11. Multiple Choice
    12. Review Questions
    13. Problems
    14. Video Activity
  18. 17 How Firms Raise Capital
    1. Why It Matters
    2. 17.1 The Concept of Capital Structure
    3. 17.2 The Costs of Debt and Equity Capital
    4. 17.3 Calculating the Weighted Average Cost of Capital
    5. 17.4 Capital Structure Choices
    6. 17.5 Optimal Capital Structure
    7. 17.6 Alternative Sources of Funds
    8. Summary
    9. Key Terms
    10. CFA Institute
    11. Multiple Choice
    12. Review Questions
    13. Problems
    14. Video Activity
  19. 18 Financial Forecasting
    1. Why It Matters
    2. 18.1 The Importance of Forecasting
    3. 18.2 Forecasting Sales
    4. 18.3 Pro Forma Financials
    5. 18.4 Generating the Complete Forecast
    6. 18.5 Forecasting Cash Flow and Assessing the Value of Growth
    7. 18.6 Using Excel to Create the Long-Term Forecast
    8. Summary
    9. Key Terms
    10. Multiple Choice
    11. Review Questions
    12. Problems
    13. Video Activity
  20. 19 The Importance of Trade Credit and Working Capital in Planning
    1. Why It Matters
    2. 19.1 What Is Working Capital?
    3. 19.2 What Is Trade Credit?
    4. 19.3 Cash Management
    5. 19.4 Receivables Management
    6. 19.5 Inventory Management
    7. 19.6 Using Excel to Create the Short-Term Plan
    8. Summary
    9. Key Terms
    10. Multiple Choice
    11. Review Questions
    12. Video Activity
  21. 20 Risk Management and the Financial Manager
    1. Why It Matters
    2. 20.1 The Importance of Risk Management
    3. 20.2 Commodity Price Risk
    4. 20.3 Exchange Rates and Risk
    5. 20.4 Interest Rate Risk
    6. Summary
    7. Key Terms
    8. CFA Institute
    9. Multiple Choice
    10. Review Questions
    11. Problems
    12. Video Activity
  22. Index

Learning Outcomes

By the end of this section, you will be able to:

  • Generate correlation coefficients using the R statistical tool.
  • Generate linear regression models using the R statistical tool.

Generate Correlation Coefficients Using the R Statistical Tool

R is an open-source statistical analysis tool that is widely used in the finance industry. R is available as a free program and provides an integrated suite of functions for data analysis, graphing, and statistical programming. R provides many functions and capabilities for regression analysis.

Recall that most calculations in R are handled via functions.

The typical method for using functions in statistical applications is to first create a vector of data values. There are several ways to create vectors in R. For example, the c function is often used to combine values into a vector. For example, this R command will generate a vector called salaries, containing the data values 40,000, 50,000, 75,000, and 92,000:

      > salaries <- c(40000, 50000, 75000, 92000)
      

To calculate the correlation coefficient r, we use the R command called cor.

As an example, consider the data set in Table 14.8, which tracks the return on the S&P 500 versus return on Coca-Cola stock for a seven-month time period.

Month

S&P 500

Monthly

Return (%)

Coca-Cola

Monthly

Return (%)

Jan 8 6
Feb 1 0
Mar 0 -2
Apr 2 1
May -3 -1
Jun 7 8
Jul 4 2
Table 14.8 Monthly Returns of Coca-Cola Stock versus Monthly Returns for the S&P 500

Create two vectors in R, one vector for the S&P 500 returns and a second vector for Coca-Cola returns:

      > SP500 <- c(8,1,0,2,-3,7,4)

      > CocaCola <- c(6,0,-2,1,-1,8,2)
      

The R command called cor returns the correlation coefficient for the x-data vector and y-data vector:

      > cor(SP500, CocaCola)
      

Generate Linear Regression Models Using the R Statistical Tool

To create a linear model in R, assuming the correlation is significant, the command lm (for linear model) will provide the slope and y-intercept for the linear regression equation.

The format of the R command is

      lm(dependent_variable_vector ~ independent_variable_vector)
      

Notice the use of the tilde symbol as the separator between the dependent variable vector and the independent variable vector.

We use the returns on Coca-Cola stock as the dependent variable and the returns on the S&P 500 as the independent variable, and thus the R command would be

      > lm(CocaCola ~ SP500)

      Call:
      
      lm(formula = CocaCola ~ SP500)
      
      Coefficients:
      
      (Intercept)    SP500
      
       -0.3453    0.8641
      

The R output provides the value of the y-intercept as -0.3453-0.3453 and the value of the slope as 0.8641. Based on this, the linear model would be

y^ = a+bxy^ = -0.3453+0.8641xy^ = a+bxy^ = -0.3453+0.8641x

where x represents the value of S&P 500 return and y represents the value of Coca-Cola stock return.

The results can also be saved as a formula and called “model” using the following R command. To obtain more detailed results for the linear regression, the summary command can be used, as follows:

      > model <- lm(CocaCola ~ SP500)
      
      > summary(model)
      
      Call:
      
      lm(formula = CocaCola ~ SP500)
      
      Residuals:
      
        1    2    3    4    5    6    7
      
      -0.5672 -0.5188 -1.6547 -0.3828 1.9375 2.2969 -1.1109
      
      Coefficients:
      
           Estimate Std. Error t value Pr(>|t|)
      
      (Intercept) -0.3453   0.7836 -0.441 0.67783
      
      SP500     0.8641   0.1734  4.984 0.00416 **
      
      ---
      
      Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
      
      Residual standard error: 1.658 on 5 degrees of freedom
      
      Multiple R-squared: 0.8325,  Adjusted R-squared: 0.7989
      
      F-statistic: 24.84 on 1 and 5 DF, p-value: 0.004161
      

In this output, the y-intercept and slope is given, as well as the residuals for each x-value. The output includes additional statistical details regarding the regression analysis.

Predicted values and prediction intervals can also be generated within R.

First, we can create a structure in R called a data frame to hold the values of the independent variable for which we want to generate a prediction. For example, we would like to generate the predicted return for Coca-Cola stock, given that the return for the S&P 500 is 6.

We use the R command called predict.

To generate a prediction for the linear regression equation called model, using the data frame where the value of the S&P 500 is 6, the R commands will be

      > a <- data.frame(SP500=6)
      
      > predict(model, a)
      
         1
      
      4.839062
      

The output from the predict command indicates that the predicted return for Coca-Cola stock will be 4.8% when the return for the S&P 500 is 6%.

We can extend this analysis to generate a 95% prediction interval for this result by using the following R command, which adds an option to the predict command to generate a prediction interval:

      > predict(model,a, interval="predict")
      
         fit    lwr   upr
      
      1 4.839062 0.05417466 9.62395
      

Thus the 95% prediction interval for Coca-Cola return is (0.05%, 9.62%) when the return for the S&P 500 is 6%.

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