# Regression Analysis in Finance

## Correlation Analysis

### Learning Outcomes

By the end of this section, you will be able to:
1. Calculate a correlation coefficient.
2. Interpret a correlation coefficient.
3. Test for the significance of a correlation coefficient.

### Calculate a Correlation Coefficient

In correlation analysis, we study the relationship between bivariate data, which is data collected on two variables where the data values are paired with one another.
Correlation is the measure of association between two numeric variables. For example, we may be interested to know if there is a correlation between bond prices and interest rates or between the age of a car and the value of the car. To investigate the correlation between two numeric quantities, the first step is to create a scatter plot that will graph the (x, y) ordered pairs. The independent, or explanatory, quantity is labeled as the x-variable, and the dependent, or response, quantity is labeled as the y-variable.
For example, we may be interested to know if the price of Nike stock is correlated with the value of the S&P 500 (Standard & Poor’s 500 stock market index). To investigate this, monthly data can be collected for Nike stock prices and value of the S&P 500 for a period of time, and a scatter plot can be created and examined. A scatter plot, or scatter diagram, is a graphical display intended to show the relationship between two variables. The setup of the scatter plot is that one variable is plotted on the horizontal axis and the other variable is plotted on the vertical axis. Each pair of data values is considered as an (x, y) point, and the various points are plotted on the diagram. A visual inspection of the plot is then made to detect any patterns or trends on the scatter diagram.  shows the relationship between the Nike stock price and its S&P value over a one-year time period.
To assess linear correlation, the graphical trend of the data points is examined on the scatter plot to determine if a straight-line pattern exists. If a linear pattern exists, the correlation may indicate either a positive or a negative correlation. A positive correlation indicates that as the independent variable increases, the dependent variable tends to increase as well, or, as the independent variable decreases, the dependent variable tends to decrease (the two quantities move in the same direction). A negative correlation indicates that as the independent variable increases, the dependent variable decreases, or, as the independent variable decreases, the dependent variable increases (the two quantities move in opposite directions). If there is no relationship or association between the two quantities, where one quantity changing does not affect the other quantity, we conclude that there is no correlation between the two variables.
From the scatter plot in the Nike stock versus S&P 500 example (see ), we note that the trend reflects a positive correlation in that as the value of the S&P 500 increases, the price of Nike stock tends to increase as well.
When inspecting a scatter plot, it may be difficult to assess a correlation based on a visual inspection of the graph alone. A more precise assessment of the correlation between the two quantities can be obtained by calculating the numeric correlation coefficient (referred to using the symbol r).
The correlation coefficient, which was developed by statistician Karl Pearson in the early 1900s, is a measure of the strength and direction of the correlation between the independent variable x and the dependent variable y.
The formula for r is shown below; however, technology, such as Excel or the statistical analysis program R, is typically used to calculate the correlation coefficient.
where n refers to the number of data pairs and the symbol  indicates to sum the x-values.
 provides a step-by-step procedure on how to calculate the correlation coefficient r.
Note that since r is calculated using sample data, r is considered a sample statistic used to measure the strength of the correlation for the two population variables. Sample data indicates data based on a subset of the entire population.
Given the complexity of this calculation, Excel or other software is typically used to calculate the correlation coefficient.
The Excel command to calculate the correlation coefficient uses the following format:
where A1:A10 are the cells containing the x-values and B1:B10 are the cells containing the y-values.

### Interpret a Correlation Coefficient

Once the value of r is calculated, this measurement provides two indicators for the correlation:
1. the strength of the correlation based on the value of r
2. the direction of the correlation based on the sign of r
The value of r gives us this information:
1. The value of r is always between  and : .
2. The size of the correlation r indicates the strength of the linear relationship between the two variables. Values of r close to  or to  indicate a stronger linear relationship.
3. If , there is no linear relationship between the two variables (no linear correlation).
4. If , there is perfect positive correlation.
5. If  there is perfect negative correlation. In both of these cases, all the original data points lie on a straight line.
The sign of r gives us this information:
1. A positive value of r means that when x increases, y tends to increase, and when x decreases, y tends to decrease (positive correlation).
2. A negative value of r means that when x increases, y tends to decrease, and when x decreases, y tends to increase (negative correlation).
The Excel command used to find the value of the correlation coefficient for the Nike stock versus S&P 500 example (refer back to ) is
In this example, the value of  is calculated by Excel to be .
Since this is a positive value close to 1, we conclude that the relationship between Nike stock and the value of the S&P 500 over this time period represents a strong, positive correlation.
The correlation coefficient r can also be determined using the statistical capability on the financial calculator:
1. Step 1 is to enter the data in the calculator (using the [DATA] function that is located above the 7 key).
2. Step 2 is to access the statistical results provided by the calculator (using the [STAT] function that is located above the 8 key) and scroll to the correlation coefficient results.
Follow the steps in  for calculating the correlation data for the data set of Nike stock price and value of the S&P 500 shown previously.
From the statistical results shown on the calculator display, the correlation coefficient r is 0.93, which indicates that the relationship between Nike stock and the value of the S&P 500 over this time period represents a strong, positive correlation.
Note: A strong correlation does not suggest that x causes y or y causes x. We must remember that correlation does not imply causation.

### Test a Correlation Coefficient for Significance

The correlation coefficient, r, tells us about the strength and direction of the linear relationship between x and y. The sample data are used to compute r, the correlation coefficient for the sample. If we had data for the entire population (that is, all measurements of interest), we could find the population correlation coefficient, which is labeled as the Greek letter ρ (pronounced “rho”). But because we have only sample data, we cannot calculate the population correlation coefficient. The sample correlation coefficient, r, is our estimate of the unknown population correlation coefficient.
1. ρ = population correlation coefficient (unknown)
2. r = sample correlation coefficient (known; calculated from sample data)
An important step in the correlation analysis is to determine if the correlation is significant. By this, we are asking if the correlation is strong enough to allow meaningful predictions for y based on values of x. One method to test the significance of the correlation is to employ a hypothesis test. The hypothesis test lets us decide whether the value of the population correlation coefficient ρ is close to zero or significantly different from zero. We decide this based on the sample correlation coefficient r and the sample size n.
If the test concludes that the correlation coefficient is significantly different from zero, we say that the correlation coefficient is significant.
1. Conclusion: There is sufficient evidence to conclude that there is a significant linear relationship between x and y variables because the correlation coefficient is significantly different from zero.
2. What the conclusion means: There is a significant linear relationship between the x and y variables. If the test concludes that the correlation coefficient is not significantly different from zero (it is close to zero), we say that the correlation coefficient is not significant.
A hypothesis test can be performed to test if the correlation is significant. A hypothesis test is a statistical method that uses sample data to test a claim regarding the value of a population parameter. In this case, the hypothesis test will be used to test the claim that the population correlation coefficient ρ is equal to zero.
Use these hypotheses when performing the hypothesis test:
1. Null hypothesis:
2. Alternate hypothesis:
The hypotheses can be stated in words as follows:
1. Null hypothesis : The population correlation coefficient is not significantly different from zero. There is not a significant linear relationship (correlation) between x and y in the population.
2. Alternate hypothesis : The population correlation coefficient is significantly different from zero. There is a significant linear relationship (correlation) between x and y in the population.
A quick shorthand way to test correlations is the relationship between the sample size and the correlation. If  then this implies that the correlation between the two variables demonstrates that a linear relationship exists and is statistically significant at approximately the 0.05 level of significance. As the formula indicates, there is an inverse relationship between the sample size and the required correlation for significance of a linear relationship. With only 10 observations, the required correlation for significance is 0.6325; for 30 observations, the required correlation for significance decreases to 0.3651; and at 100 observations, the required level is only 0.2000.
NOTE:
1. If r is significant and the scatter plot shows a linear trend, the line can be used to predict the value of y for values of x that are within the domain of observed x-values.
2. If r is not significant OR if the scatter plot does not show a linear trend, the line should not be used for prediction.
3. If r is significant and the scatter plot shows a linear trend, the line may not be appropriate or reliable for prediction outside the domain of observed x-values in the data.
Correlations may be helpful in visualizing the data, but they are not appropriately used to explain a relationship between two variables. Perhaps no single statistic is more misused than the correlation coefficient. Citing correlations between health conditions and everything from place of residence to eye color have the effect of implying a cause-and-effect relationship. This simply cannot be accomplished with a correlation coefficient. The correlation coefficient is, of course, innocent of this misinterpretation. It is the duty of analysts to use a statistic that is designed to test for cause-and-effect relationships and to report only those results, if they are intending to make such a claim. The problem is that passing this more rigorous test is difficult, therefore lazy and/or unscrupulous researchers fall back on correlations when they cannot make their case legitimately.

###

Correlation is the measure of association between two numeric variables. A correlation coefficient called r is used to assess the strength and direction of the correlation. The value of r is always between  and . The size of the correlation r indicates the strength of the linear relationship between the two variables. Values of r close to  or to  indicate a stronger linear relationship. A positive value of r means that when x increases, y tends to increase and when x decreases, y tends to decrease (positive correlation). A negative value of r means that when x increases, y tends to decrease and when x decreases, y tends to increase (negative correlation).


# Regression Analysis in Finance

## Linear Regression Analysis

### Learning Outcomes

By the end of this section, you will be able to:
1. Analyze a regression using the method of least squares and residuals.
2. Test the assumptions for linear regression.

### Method of Least Squares and Residuals

Once the correlation coefficient has been calculated and a determination has been made that the correlation is significant, typically a regression model is then developed. In this discussion we will focus on linear regression, where a straight line is used to model the relationship between the two variables. Once a straight-line model is developed, this model can then be used to predict the value of the dependent variable for a specific value of the independent variable.
Recall from algebra that the equation of a straight line is given by
where m is the slope of the line and b is the y-intercept of the line.
The slope measures the steepness of the line, and the y-intercept is that point on the y-axis where the graph crosses, or intercepts, the y-axis.
In linear regression analysis, the equation of the straight line is written in a slightly different way using the model
In this format, b is the slope of the line, and a is the y-intercept. The notation  is called y-hat and is used to indicate a predicted value of the dependent variable y for a certain value of the independent variable x.
If a line extends uphill from left to right, the slope is a positive value, and if the line extends downhill from left to right, the slope is a negative value. Refer to .
When generating the equation of a line in algebra using , two (x, y) points were required to generate the equation. However, in regression analysis, all (x, y) points in the data set will be utilized to develop the linear regression model.
The first step in any regression analysis is to create the scatter plot. Then proceed to calculate the correlation coefficient r, and check this value for significance. If we think that the points show a linear relationship, we would like to draw a line on the scatter plot. This line can be calculated through a process called linear regression. However, we only calculate a regression line if one of the variables helps to explain or predict the other variable. If x is the independent variable and y the dependent variable, then we can use a regression line to predict y for a given value of x.
As an example of a regression equation, assume that a correlation exists between the monthly amount spent on advertising and the monthly revenue for a Fortune 500 company. After collecting (x, y) data for a certain time period, the company determines the regression equation is of the form
where x represents the monthly amount spent on advertising (in thousands of dollars) and  represents the monthly revenues for the company (in thousands of dollars).
A scatter plot of the (x, y) data is shown in .
The Fortune 500 company would like to predict the monthly revenue if its executives decide to spend $150,000 in advertising next month. To determine the estimate of monthly revenue, let  in the regression equation and calculate a corresponding value for :
This predicted value of y indicates that the anticipated revenue would be $18,646,700, given the advertising spend of $150,000.
Notice that from past data, there may have been a month where the company actually did spend $150,000 on advertising, and thus the company may have an actual result for the monthly revenue. This actual, or observed, amount can be compared to the prediction from the linear regression model to calculate a residual.
A residual is the difference between an observed y-value and the predicted y-value obtained from the linear regression equation. As an example, assume that in a previous month, the actual monthly revenue for an advertising spend of $150,000 was $19,200,000, and thus . The residual for this data point can be calculated as follows:
Notice that residuals can be positive, negative, or zero. If the observed y-value exactly matches the predicted y-value, then the residual will be zero. If the observed y-value is greater than the predicted y-value, then the residual will be a positive value. If the observed y-value is less than the predicted y-value, then the residual will be a negative value.
When formulating the linear regression line of best fit to the points on the scatter plot, the mathematical analysis generates a linear equation where the sum of the squared residuals is minimized. This analysis is referred to as the method of least squares. The result is that the analysis generates a linear equation that is the “best fit” to the points on the scatter plot, in the sense that the line minimizes the differences between the predicted values and observed values for y.
The goal in the regression analysis is to determine the coefficients a and b in the following regression equation:
Once the (x, y) has been collected, the slope (b) and y-intercept (a) can be calculated using the following formulas:
where n refers to the number of data pairs and  indicates sum of the x-values.
Notice that the formula for the y-intercept requires the use of the slope result (b), and thus the slope should be calculated first and the y-intercept should be calculated second.
When making predictions for y, it is always important to plot a scatter diagram first. If the scatter plot indicates that there is a linear relationship between the variables, then it is reasonable to use a best-fit line to make predictions for y, given x within the domain of x-values in the sample data, but not necessarily for x-values outside that domain.
Note: Computer spreadsheets, statistical software, and many calculators can quickly calculate the best-fit line and create the graphs. The calculations tend to be tedious if done by hand.

### Assumptions for Linear Regression

Testing the significance of the correlation coefficient requires that certain assumptions about the data are satisfied. The premise of this test is that the data are a sample of observed points taken from a larger population. We have not examined the entire population because it is not possible or feasible to do so. We are examining the sample to draw a conclusion about whether the linear relationship that we see between x and y in the sample data provides strong enough evidence that we can conclude that there is a linear relationship between x and y in the population.
The regression line equation that we calculate from the sample data gives the best-fit line for our particular sample. We want to use this best-fit line for the sample as an estimate of the best-fit line for the population (). Examining the scatter plot and testing the significance of the correlation coefficient helps us determine if it is appropriate to do this.
These are the assumptions underlying the test of significance:
1. There is a linear relationship in the population that models the average value of y for varying values of x. In other words, the expected value of y for each particular value lies on a straight line in the population. (We do not know the equation for the line for the population. Our regression line from the sample is our best estimate of this line in the population.)
2. The y-values for any particular x-value are normally distributed about the line. This implies that there are more y-values scattered closer to the line than are scattered farther away. Assumption (1) implies that these normal distributions are centered on the line: the means of these normal distributions of y-values lie on the line.
3. The standard deviations of the population y-values about the line are equal for each value of x. In other words, each of these normal distributions of y-values has the same shape and spread about the line.
4. The residual errors are mutually independent (no pattern).
5. The data are produced from a well-designed, random sample or randomized experiment.

###

Linear regression analysis uses a straight-line fit to model the relationship between the two variables. Once a straight-line model is developed, this model can then be used to predict the value of the dependent variable for a specific value of the independent variable. Two parameters are calculated for the linear model, the slope of the best-fit line and the y-intercept of the best-fit line. The method of least squares is used to generate these parameters; this method is based on minimizing the squared differences between the predicted values and observed values for y.


# Regression Analysis in Finance

## Best-Fit Linear Model

### Learning Outcomes

By the end of this section, you will be able to:
1. Calculate the slope and y-intercept for a linear regression model using technology.
2. Interpret and apply the slope and y-intercepts.

### Calculate the Slope and y-Intercept for a Linear Regression Model Using Technology

Once a correlation has been deemed as significant, a best-fit linear regression model is developed. The goal in the regression analysis is to determine the coefficients a and b in the following regression equation:
The slope (b) and y-intercept (a) can be calculated using the following formulas:
These formulas can be quite cumbersome, especially for a significant number of data pairs, and thus technology is often used (such as Excel, a calculator, R statistical software, etc.).
Using Excel: To calculate the slope and y-intercept of the linear model, start by entering the (x, y) data in two columns in Excel. Then the Excel commands =SLOPE and =INTERCEPT can be used to calculate the slope and intercept, respectively.
The following data set will be used as an example: the monthly amount spent on advertising and the monthly revenue for a Fortune 500 company for 12 months (data is shown in ).
To calculate the slope of the regression model, use the Excel command
It’s important to note that this Excel command expects that the y-data range is entered first and the x-data range is entered second. Since revenue depends on amount spent on advertising, revenue is considered the y-variable and amount spent on advertising is considered the x-variable. Notice the y-data is contained in cells C2 through C13 and the x-data is contained in cells B2 through B13. Thus the Excel command for slope would be entered as
In the same way, the Excel command to calculate the y-intercept of the regression model is
For the data set shown in the above table, the Excel command would be
The results are shown in , where
Based on this, the regression equation can be written as
where x represents the amount spent on advertising (in thousands of dollars) and y represents the amount of revenue (in thousands of dollars).

### Using a Financial Calculator

The financial calculator provides the slope and y-intercept for the linear regression model once the (x, y) data is inputted into the calculator.
Follow the steps in  for calculating the slope and y-intercept for the data set of amounts spent on advertising and revenue shown previously.
From the statistical results shown on the calculator display, the slope b is 61.8 and the y-intercept a is 9,367.7.
Based on this, the regression equation can be written as

### Interpret and Apply the Slope and y-Intercept

The slope of the line, b, describes how changes in the variables are related. It is important to interpret the slope of the line in the context of the situation represented by the data. You should be able to write a sentence interpreting the slope in plain English.

### Interpretation of the Slope

The slope of the best-fit line tells us how the dependent variable (y) changes for every one unit increase in the independent (x) variable, on average.
In the previous example, the linear regression model for the monthly amount spent on advertising and the monthly revenue for a Fortune 500 company for 12 months was generated as follows:
Since the slope was determined to be 61.8, the company can interpret this to mean that for every $1,000 dollars spent on advertising, on average, this will result in an increase in revenues of $61,800.
The intercept of the regression equation is the corresponding y-value when.

### Interpretation of the Intercept

The intercept of the best-fit line tells us the expected mean value of y in the case where the x-variable is equal to zero.
However, in many scenarios it may not make sense to have the x-variable equal zero, and in these cases, the intercept does not have any meaning in the context of the problem. In other examples, the x-value of zero is outside the range of the x-data that was collected. In this case, we should not assign any interpretation to the y-intercept.
In the previous example, the range of data collected for the x-variable was from $49 to $153 spent per month on advertising. Since this interval does not include an x-value of zero, we would not provide any interpretation for the intercept.

###

Once a correlation has been deemed significant, a linear regression model is developed. The goal in the regression analysis is to determine the coefficients a and b in the following regression equation: . Typically some technology, such as Excel, R statistical tool, or a calculator, is used to generate the coefficients a and b since manual calculations are cumbersome.


# Regression Analysis in Finance

## Regression Applications in Finance

### Learning Outcomes

By the end of this section, you will be able to:
1. Calculate the regression model for a single independent variable as applied to financial forecasting.
2. Extract measures of slope and intercept from regression analysis in financial applications.

### Regression Model for a Single Independent Variable

Regression analysis is used extensively in finance-related applications. Many typical applications involve determining if there is a correlation between various stock market indices such as the S&P 500, the Dow Jones Industrial Average (DJIA), and the Russell 2000 index.
As an example, suppose we would like to determine if there is a correlation between the Russell 2000 index and the DJIA. Does the value of the Russell 2000 index depend on the value of the DJIA? Is it possible to predict the value of the Russell 2000 index for a certain value of the DJIA? We can explore these questions using regression analysis.
 shows a summary of monthly closing prices of the DJIA and the Russell 2000 for a 12-month time period. We consider the DJIA to be the independent variable and the Russell 2000 index to be the dependent variable.
The first step is to create a scatter plot to determine if the data points appear to follow a linear pattern. The scatter plot is shown in . The scatter plot clearly shows a linear pattern; the next step is to calculate the correlation coefficient and determine if the correlation is significant.
1. Using the Excel command =CORREL, the correlation coefficient is calculated to be 0.947. This value of the correlation coefficient is significant using the test for significance referenced earlier in Correlation Analysis.
2. Using the Excel commands =SLOPE and =INTERCEPT, the value of the slope and y-intercept are calculated as 0.11 and , respectively, when rounded to two decimal places.
The Excel output is shown below:
Based on these results, the corresponding linear regression model is
Assume the DJIA has reached a value of 32,000. Predict the corresponding value of the Russell 2000 index. To determine this, substitute the value of the independent variable,  (this is the given value of the DJIA), and calculate the corresponding value for the dependent variable, which is the predicted value for the Russell 2000 index:
Thus the predicted value for the Russell 2000 index is approximately 2,024 when the DJIA reached a value of 32,000.

### Measures of Slope and Intercept from Regression Analysis

An important application of regression analysis is to determine the systematic risk for a particular stock, which is referred to as beta. A stock’s beta is a measure of the volatility of the stock compared to a benchmark such as the S&P 500 index. If a stock has more volatility compared to the benchmark, then the stock will have a beta greater than 1.0. If a stock has less volatility compared to the benchmark, then the stock will have a beta less than 1.0.
Beta can be determined as the slope of the regression line when the stock returns are plotted versus the returns for the benchmark, such as the S&P 500. As an example, consider the calculation for beta of Nike stock based on monthly returns of Nike stock versus monthly returns for the S&P 500 over the time period from May 2020 to March 2021. The monthly return data is shown in .
The scatter plot that graphs S&P monthly return versus Nike monthly return is shown in .
The slope of the regression line is 0.83, obtained by using the =SLOPE command in Excel.
This indicates the value of beta for Nike stock is 0.83, which indicates that Nike stock had lower volatility versus the S&P 500 for the time period of interest.

###

Regression analysis is used extensively in finance-related applications. Many typical applications involve determining if there is a correlation between various stock market indices such as the S&P 500, the DJIA, and the Russell 2000 index. The procedure is to first generate a scatter plot to determine if a visual trend is observed, then calculate a correlation coefficient and check for significance. If the correlation coefficient is significant, a linear model can then be generated and used for predictions.


# Regression Analysis in Finance

## Predictions and Prediction Intervals

### Learning Outcomes

By the end of this section, you will be able to:
1. Calculate predictions for the dependent variable using the regression model.
2. Generate prediction intervals based on a prediction for the dependent variable.

### Predicting the Dependent Variable Using the Regression Model

A key aspect of generating the linear regression model is to use the model for predictions, provided the correlation is significant. To generate predictions or forecasts using the linear regression model, substitute the value of the independent variable (x) in the regression equation and solve the equation for the dependent variable (y).
In a previous example, the linear regression equation was generated to relate the amount of monthly revenue for a Fortune 500 company to the amount of monthly advertising spend. From the previous example, it was determined that the regression equation can be written as
where x represents the amount spent on advertising (in thousands of dollars) and y represents the amount of revenue (in thousands of dollars).
Let’s assume the Fortune 500 company would like to predict the monthly revenue for a month where it plans to spend $80,000 for advertising. To determine the estimate of monthly revenue, let  in the regression equation and calculate a corresponding value for ŷ:
This predicted value of y indicates that the forecasted revenue would be $14,320,700, assuming an advertising spend of $80,000.
1. Excel can provide this forecasted value directly using the =FORECAST command.
2. To use this command, enter the value of the independent variable
3. Using this Excel command, the forecasted value for the revenue is $14,320.52 when the advertising spend is $80 (in thousands of dollars) (see ). (Note: The discrepancy in the more precise Excel result and the formula result is due to rounding in interim calculations.)
A word of caution when predicting values for y: it is generally recommended to only predict values for y using values of x that are in the original range of the data collection.
As an example, assume we have developed a linear model to predict the height of male children based on their age. We have collected data for the age range from  years old to  years old, and we have confirmed that the scatter plot shows a linear trend and that the correlation is significant.
It would be erroneous to use this model to predict the height of a 25-year-old male since  is outside the range of the x-data, which was from 3 to 10 years old. The reason this is not recommended is that a linear pattern cannot be assumed to continue beyond the x-value of 10 years old unless some data collection has occurred at ages greater than 10 to confirm that the linear pattern is consistent for x-values beyond 10 years old.

### Generating Prediction Intervals

One important value of an estimated regression equation is its ability to predict the effects on y of a change in one or more values of the independent variables. The value of this is obvious. Careful policy cannot be made without estimates of the effects that may result. Indeed, it is the desire for particular results that drive the formation of most policy. Regression models can be, and have been, invaluable aids in forming such policies.
Remember that point estimates do not carry a particular level of probability, or level of confidence, because points have no “width” above which there is an area to measure. There are actually two different approaches to the issue of developing estimates of changes in the independent variable (or variables) on the dependent variable. The first approach wishes to measure the expected mean value of y from a specific change in the value of x.
The second approach to estimate the effect of a specific value of x on y treats the event as a single experiment: you choose x and multiply it times the coefficient, and that provides a single estimate of y. Because this approach acts as if there were a single experiment, the variance that exists in the parameter estimate is larger than the variance associated with the expected value approach.
The conclusion is that we have two different ways to predict the effect of values of the independent variable(s) on the dependent variable, and thus we have two different intervals. Both are correct answers to the question being asked, but there are two different questions. To avoid confusion, the first case where we are asking for the expected value of the mean of the estimated y is called a confidence interval. The second case, where we are asking for the estimate of the impact on the dependent variable y of a single experiment using a value of x, is called the prediction interval.
The prediction interval for an individual y for  can be calculated as
where s is the standard deviation of the error term, s is the standard deviation of the x-variable, and  is the critical value of the t-distribution at the  confidence level.
Tabulated values of the t-distribution are available in online references such as the Engineering Statistics Handbook. The mathematical computations for prediction intervals are complex, and usually the calculations are performed using software. The formula above can be implemented in Excel to create a 95% prediction interval for the forecast for monthly revenue when  is spent on monthly advertising.  shows the detailed calculations in Excel to arrive at a 95% prediction interval of (13,270.95, 15,370.09) for the monthly revenue. (The commands refer to the Excel data table shown in .)
This prediction interval can be interpreted as follows: there is 95% confidence that when the amount spent on monthly advertising is $80,000, the corresponding monthly revenue will be between $13,270.95 and $15,370.09.
Various computer regression software packages provide programs within the regression functions to provide answers to inquiries of estimated predicted values of y given various values chosen for the x-variable(s). For example, the statistical program R provides these prediction intervals directly. It is important to know just which interval is being tested in the computer package because the difference in the size of the standard deviations will change the size of the interval estimated. This is shown in .
 shows visually the difference the standard deviation makes in the size of the estimated intervals. The confidence interval, measuring the expected value of the dependent variable, is smaller than the prediction interval for the same level of confidence. The expected value method assumes that the experiment is conducted multiple times rather than just once, as in the other method. The logic here is similar, although not identical, to that discussed when developing the relationship between the sample size and the confidence interval using the central limit theorem. There, as the number of experiments increased, the distribution narrowed, and the confidence interval became tighter around the expected value of the mean.
It is also important to note that the intervals around a point estimate are highly dependent upon the range of data used to estimate the equation, regardless of which approach is being used for prediction. Remember that all regression equations go through the point of means—that is, the mean value of y and the mean values of all independent variables in the equation. As the value of x gets further and further from the (x, y) point corresponding to the mean value of x and the mean value of y, the width of the estimated interval around the point estimate increases. Choosing values of x beyond the range of the data used to estimate the equation poses an even greater danger of creating estimates with little use, very large intervals, and risk of error.  shows this relationship.
 demonstrates the concern for the quality of the estimated interval, whether it is a prediction interval or a confidence interval. As the value chosen to predict y,  in the graph, is further from the central weight of the data, , we see the interval expand in width even while holding constant the level of confidence. This shows that the precision of any estimate will diminish as one tries to predict beyond the largest weight of the data and most certainly will degrade rapidly for predictions beyond the range of the data. Unfortunately, this is just where most predictions are desired. They can be made, but the width of the confidence interval may be so large as to render the prediction useless.

###

A key aspect of generating the linear regression model is to then use the model for predictions, provided that the correlation is significant. To generate predictions or forecasts using the linear regression model, substitute the value of the independent variable (x) in the regression equation and solve the equation for the dependent variable (y). When making predictions using the linear model, it is generally recommended to only predict values for y using values of x that are in the original range of the data collection.


# Regression Analysis in Finance

## Use of R Statistical Analysis Tool for Regression Analysis

### Learning Outcomes

By the end of this section, you will be able to:
1. Generate correlation coefficients using the R statistical tool.
2. 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:
To calculate the correlation coefficient r, we use the R command called cor.
As an example, consider the data set in , which tracks the return on the S&P 500 versus return on Coca-Cola stock for a seven-month time period.
Create two vectors in R, one vector for the S&P 500 returns and a second vector for Coca-Cola returns:
The R command called cor returns the correlation coefficient for the x-data vector and y-data vector:

### 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
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
The R output provides the value of the y-intercept as  and the value of the slope as 0.8641. Based on this, the linear model would be
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:
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
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:
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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R is an open-source statistical analysis tool that is widely used in the finance industry and can be found online. R provides an integrated suite of functions for data analysis, graphing, and correlation and regression analysis. R is increasingly being used as a data analysis and statistical tool because it is an open-source language and additional features are constantly being added by the user community. The tool can be used on many different computing platforms.

### Multiple Choice

### Questions

### Problems

###

### Simple Linear Regression

### How to Calculate Correlation for Stocks, Bonds, and Funds


# How to Think about Investing

## Why It Matters

Having finished her college degree and embarked on her career, Maria is now contemplating her financial future. She is considering how she might invest some of her hard-earned money. As a short-term goal, she wants to build an emergency fund so that she could cover her expenses for six months if she became ill or injured and had to take time off of work. She would also like to save money for a down payment on a home and to purchase new furniture. Although she is not yet 30 years old, Maria also knows that it is prudent to begin saving for retirement.
What should she do with her savings? Maria has some friends who have told her how successful they have been investing in stocks. Bart bragged about doubling his money in just over a year when he purchased Facebook stock, and Tiffany quickly tripled her money when she purchased shares in Netflix. But Maria also knows that her uncle lost a significant amount of money when his Boeing stock dropped from over $300 per share to under $150 within a couple of months at the beginning of 2020. Just how risky would it be to invest in stocks? What type of return might Maria expect? Are there strategies she could follow that would allow her to avoid her uncle’s fate?


# How to Think about Investing

## Risk and Return to an Individual Asset

### Learning Outcomes

By the end of this section, you will be able to:
1. Compute the realized return from an individual investment.
2. Compute the average return and volatility of returns from historical data.
3. Describe firm-specific risk.

### Measuring Historical Returns

Risk and return are often referred to as the two Rs of finance. Investors are interested in both risk and return because understanding one without the other is really meaningless. In terms of investment, the concept of return is fairly straightforward; return is the benefit, or profit, the investor expects from an expenditure. It is the reward for investing—the reason an investment is made in the first place. However, no investment is a sure thing. The return may not be what the investor was expecting. This uncertainty about what the return will be is referred to as risk.
We begin by looking at how to measure both risk and return when considering an individual asset, such as one stock. If your grandparents bought 100 shares of Apple, Inc. stock for you when you were born, you are interested in knowing how well that investment has done. You may even want to compare how that investment has fared to how an investment in a different stock, perhaps Disney, would have done. You are interested in measuring the historical return.

### Individual Investment Realized Return

The realized return of an investment is the total return that occurs over a particular time period. Suppose that you purchased a share of Target (TGT) at the beginning of January 2020 for $128.74. At the end of the year, you sold the stock for $176.53, which was $47.79 more than you paid for it. This increase in value is known as a capital gain. As the owner of the stock, you also received $2.68 in dividends during 2020. The total dollar return from your investment is calculated as
It is common to express investment returns in percentage terms rather than dollar terms. This allows you to answer the question “How much do I receive for each dollar invested?” so that you can compare investments of different sizes. The total percent return from your investment is
The dividend yield is calculated by dividing the dividends you received by the initial stock price. This calculation says that for each dollar invested in TGT in 2020, you received $0.0208 in dividends. The capital gain yield is the change in the stock price divided by the initial stock price. This calculation says that for each dollar invested in TGT in 2020, you received $0.3712 in capital gains. Your total percent return of 39.20% means that you made $0.392 for every dollar invested when your gains from both dividends and stock price appreciation are totaled together.
Of course, investors seldom purchase a stock and then sell it exactly one year later. Assume that you purchased shares of Facebook (FB) on June 1, 2020, for $228.50 per share and sold the shares three months later for $261.90. You received no dividends. In this case, your holding period percentage return is calculated as
This 14.62% is your return for a three-month holding period. To compare them to other investment opportunities, you need to express returns on a per-year, or annualized, basis. The holding period returned is converted to an effective annual rate (EAR) using the formula
where m is the number of holding periods in a year.
There are four three-month periods in a year. So, the EAR for this investment is
What happens if you own a stock for more than one year? Your holding period return would have occurred over a period longer than a year, but the process to calculate the EAR is the same. Suppose you purchased shares of FB in May 2015, when it was selling for $79.30 per share. You held the stock until May 2020, when you sold it for $224.59. Your holding period percentage return would be . You more than tripled your money, but it took you five years to do so. Your EAR, which will be smaller than this five-year holding period return rate, is calculated as

### Average Annual Returns

Suppose that you purchased shares of Delta Airlines (DAL) at the beginning of 2011 for $11.19 and held the stock for 10 years before selling it for $40.21. You made  on your investment over a 10-year period. This is a 259.34% holding period return. The EAR for this investment is
To calculate the EAR using the above formula, the holding period return must first be calculated. The holding period return represents the percentage return earned over the entire time the investment is held. Then the holding period return is converted to an annual percentage rate using the formula.
You can also use the basic time value of money formula to calculate the EAR on an investment. In time value of money language, the initial price paid for the investment, $11.19, is the present value. The price the stock is sold for, $40.21, is the future value. It takes 10 years for the $11.19 to grow to $40.21. Using the time value of money will result in a calculation of
The EAR formula and the time value of money both result in a 13.65% annual return. Mathematically, the two formulas are the same; one is simply an algebraic rearrangement of the other.
If you earned 13.65% each year, compounded for 10 years, you would have converted your $11.19 per share investment to $40.21 per share. Of course, DAL stock did not increase by exactly 13.65% each year. The returns for DAL for each year are shown in . Some years, the return was much higher than 13.65%. In 2013, the return was almost 133%! Other years, the return was much lower than 13.65%; in fact, in the return was negative in four of the years.
Although an investment in DAL of $11.19 at the beginning of 2011 grew to $40.20 by the end of 2020, this growth was not consistent each year. The amount that the stock was worth at the end of each year is also shown in . During 2011, the return for DAL was −35.79%, resulting in the value of the investment falling to . The following year, 2012, the return for DAL was 46.72%. Therefore, the value of the investment was  at the end of 2012. This process continues each year that the stock is held.
The compounded annual return derived from the EAR and time value of money formulas is also known as a geometric average return. A geometric average return is calculated using the formula
where R is the return for each year in the time period for which the average is calculated.
The calculation of the geometric average return for DAL is shown in the right column of . (The slight difference in the geometric average return of 13.64% from the 13.65% derived from the EAR and time value of money calculations is due to rounding errors.)
Looking at , you will notice that the geometric average return differs from the mean return. Adding each of the annual returns and dividing the sum by 10 results in a 22.4% average annual return. This 22.4% is called the arithmetic average return.
The geometric average return will be smaller than the arithmetic average return (unless the returns for all years are identical). This is due to the basic arithmetic of compounding. Think of a very simple example in which you invest $100 for two years. If you have a positive return of 50% the first year and a negative 50% return the second year, you will have an arithmetic average return of , but you will have a geometric average return of . With a 50% positive return the first year, you ended the year with $150. The second year, you lost 50% of that balance and were left with only $75.
Another important fact when studying average returns is that the order in which you earn the returns is not important. Consider what would have occurred if the returns in the two years were reversed, so that you faced a loss of 50% in the first year of your investment and a gain of 50% in the second year of your investment. With a −50% return in the first year, you would have ended that year with only $50. Then, if that $50 earned a positive 50% return the second year, you would have a $75 balance at the end of the two-year period. A negative return of 50% followed by a positive return of 50% still results in an arithmetic average return of 0% and a geometric average return of .
Both the arithmetic average return and the geometric average return are “correct” calculations. They simply answer different questions. The geometric average tells you what you actually earned per year on average, compounded annually. It is useful for calculating how much a particular investment grows over a period of time. The arithmetic average tells you what you earned in a typical year. When we are looking at the historical description of the distribution of returns and want to predict what to expect in a particular year, the arithmetic average is the relevant calculation.

### Measuring Risk

Although the arithmetic average return for Delta Airlines (DAL) for 2011–2020 was 22.4%, there is not a year in which the return was exactly 22.4%. In fact, in some years, the return was much higher than the average, such as in 2013, when it was 132.61%. In other years, the return was negative, such as 2011, when it was −35.79%. Looking at the yearly returns in , the return for DAL varies widely from year to year. In finance, this volatility of returns is considered risk.

### Volatility of Returns

The most commonly used measure of volatility of returns in finance is the standard deviation of the returns. The standard deviation of returns for DAL for the sample period 2011–2020 is 51.9%. Remember that if the normal distribution (a bell−shaped curve) describes returns, then 68% (or about two-thirds) of the time, the return in a particular year will be within one standard deviation above and one standard deviation below the arithmetic average return. Given DAL’s average return of 22.4%, the actual yearly return will be somewhere between −29.5% and 74.29% in two out of three years. A very high return of greater than 74.29% would occur 16% of the time; a very large loss of more than 29.5% would also occur 16% of the time.
As you can see, there is a wide range of what can be considered a “typical” year for DAL. Although we can calculate an average return, the return in any particular year is likely to vary from that average. The larger the standard deviation, the greater this range of returns is. Thus, a larger standard deviation indicates a greater volatility of returns and, hence, more risk.

### Firm-Specific Risk

Investors purchase a share of stock hoping that the stock will increase in value and they will receive a positive return. You can see, however, that even with well-established companies such as ExxonMobil and CVS, returns are highly volatile. Investors can never perfectly predict what the return on a stock will be, or even if it will be positive.
The yearly returns for four companies—Delta Airlines (DAL), Southwest Airlines (LUV), ExxonMobil (XOM), and CVS Health Corp. (CVS)—are shown in . Each of these stocks had years in which the performance was much better or much worse than the arithmetic average. In fact, none of the stocks appear to have a typical return that occurs year after year.
 contains a graph of the returns for each of these four stocks by year. In this graph, it is easy to see that DAL and LUV both have more volatility, or returns that vary more from year to year, than do XOM or CVS. This higher volatility leads to DAL and LUV having higher standard deviations of returns than XOM or CVS.
Standard deviation is considered a measure of the risk of owning a stock. The larger the standard deviation of a stock’s annual returns, the further from the average that stock’s return is likely to be in any given year. In other words, the return for the stock is highly unpredictable. Although the return for CVS varies from year to year, it is not subject to the wide swings of the returns for DAL or LUV.
Why are stock returns so volatile? The value of the stock of a company changes as the expectations of the future revenues and expenses of the company change. These expectations may change due to a number of events and new information. Good news about a company will tend to result in an increase in the stock price. For example, DAL announcing that it will be opening new routes and flying to cities it has not previously serviced suggests that DAL will have more customers and more revenue in future years. Or if CVS announces that it has negotiated lower rent for many of its locations, investors will expect the expenses of the company to fall, leading to more profits. Those types of announcements will often be associated with a higher stock price. Conversely, if the pilots and flight attendants for DAL negotiate higher salaries, the expenses for DAL will increase, putting downward pressure on profits and the stock price.

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Investors are interested in both the return they can expect to receive when making an investment and the risk associated with that investment. In finance, risk is considered the volatility of the return from time period to time period. Historical returns are measured by the arithmetic average, and the risk is measured by the standard deviation of returns.

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### How to Double Your Money in Seven Years

In this video, Jim Cramer explains how compounding can help investors build and preserve wealth. He provides suggestions for how young people can use the stock market to build financial independence.


# How to Think about Investing

## Risk and Return to Multiple Assets

### Learning Outcomes

By the end of this section, you will be able to:
1. Explain the benefits of diversification.
2. Describe the relationship between risk and return for large portfolios.
3. Compare firm-specific and systematic risk.
4. Discuss how portfolio size impacts risk.

### Diversification

So far, we have looked at the return and the volatility of an individual stock. Most investors, however, own shares of stock in multiple companies. This collection of stocks is known as a portfolio. Let’s explore why it is wise for investors to hold a portfolio of stocks rather than to pick just one favorite stock to own.
We saw that investors who owned DAL experienced an average annual return of 20.87% but also a large standard deviation of 51.16%. Investors who used all their funds to purchase DAL stock did exceptionally well during 2012–2014. But in 2020, those investors lost almost one-third of their money as COVID-19 caused a sharp reduction in air travel worldwide. To protect against these extreme outcomes, investors practice what is called diversification, or owning a variety of stocks in their portfolios.
Suppose, for example, you have saved $50,000 that you want to invest. If you purchased $50,000 of DAL stock, you would not be diversified. Your return would depend solely on the return on DAL stock. If, instead, you used $5,000 to purchase DAL stock and used the remaining $45,000 to purchase nine other stocks, you would be diversifying. Your return would depend not only on DAL’s return but also on the returns of the other nine stocks in your portfolio. Investors practice diversification to manage risk.
It is akin to the saying “Don’t put all of your eggs in one basket.” If you place all of your eggs in one basket and that basket breaks, all of your eggs will fall and crack. If you spread your eggs out across a number of baskets, it is unlikely that all of the baskets will break and all of your eggs will crack. One basket may break, and you will lose the eggs in that basket, but you will still have your other eggs. The same idea holds true for investing. If you own stock in a company that does poorly, perhaps even goes out of business, you will lose the money you placed in that particular investment. However, with a diversified portfolio, you do not lose all your money because your money is spread out across a number of different companies.
 shows the returns of investors who placed 50% of their money in DAL and the remaining 50% in LUV, XOM, or CVS. Notice that the standard deviation of returns is lower for the two-stock portfolios than for DAL as an individual investment.
As investors diversify their portfolios, the volatility of one particular stock becomes less important. XOM has good years with above-average returns and bad years with below-average (and even negative) returns, just like DAL. But the years in which those above-average and below-average returns occur are not always the same for the two companies. In 2014, for example, the return for DAL was greater than 80%, while the return for XOM was negative. On the other hand, in 2011, when DAL had a return of −35.15%, XOM had a positive return. When more than one stock is held, the gains in one stock can offset the losses in another stock, washing away some of the volatility.
When an investor holds only one stock, that one stock’s volatility contributes 100% to the portfolio’s volatility. When two stocks are held, the volatility of each stock contributes to the volatility of the portfolio. However, the volatility of the portfolio is not simply the average of the volatility of each stock held independently. How correlated the two stocks are, or how much they move together, will impact the volatility of the portfolio.
You will recall from our study of correlation in Regression Analysis in Finance that a correlation coefficient describes how two variables move relative to each other. A correlation coefficient of 1 means that there is a perfect, positive correlation between the two variables, while a correlation coefficient of −1 means that the two variables move exactly opposite of each other. Stocks that are in the same industry will tend to be more strongly correlated than stocks that are in much different industries. During the 2011–2020 time period, the correlation coefficient for DAL and LUV was 0.87, the correlation coefficient for DAL and XOM was 0.35, and the correlation coefficient for DAL and CVS was 0.79. Combining stocks that are not perfectly positively correlated in a portfolio decreases risk.
Notice that investors who owned DAL and LUV from 2011 to 2020 would have had a lower portfolio standard deviation, but not much lower, than investors who just owned DAL. Because the correlation coefficient is less than one, the standard deviation fell. However, because the two stocks are in the same industry and exposed to many of the same economic issues, the correlation coefficient is relatively high, and combining those two stocks provides only a small decrease in risk.
This is because, as airlines, DAL and LUV face many of the same market conditions. In years when the economy is strong, the weather is good, fuel prices are low, and people are traveling a lot, both companies will do well. When something such as bad weather conditions reduces the amount of air travel for several weeks, both companies are harmed. By holding LUV in addition to DAL, investors can reduce exposure to risk that is specific to DAL (perhaps a problem that DAL has with its reservation system), but they do not reduce exposure to the risk associated with the airline industry (perhaps rising jet fuel prices). DAL and LUV tend to experience positive returns in the same years and negative returns in the same years.
On the other hand, investors who added XOM to their portfolio saw a significantly lower standard deviation than those who held just DAL. In years when jet fuel prices rise, harming the profits of both DAL and LUV, XOM is likely to see high profits. Diversifying a portfolio across firms that are less correlated will reduce the standard deviation of the portfolio more.

### Portfolio Size and Risk

As you add more stocks to a portfolio, the volatility, or standard deviation, of the portfolio decreases. The volatility of individual assets becomes less and less important. As we discussed earlier, the risk that is associated with events related to a particular company is called firm-specific risk, or unsystematic, risk. Examples of unsystematic risk would include a company facing a product liability lawsuit, a company inventing a new product, or accounting irregularities being detected. Holding a portfolio of stocks means that if one company you have invested in goes out of business because of poor management, you do not lose all your savings because some of your money is invested in other companies. Portfolio diversification protects you from being significantly impacted by unsystematic risk.
However, there is a level below which the portfolio risk does not drop, no matter how diversified the portfolio becomes. The risk that never goes away is known as systematic risk. Systematic risk is the risk of holding the market portfolio.
We have talked about reasons why a firm’s returns might be volatile; for example, the firm discovering a new technology or having a product liability lawsuit brought against it will impact that firm specifically. There are also events that broadly impact the stock market. Changes in the Federal Reserve Bank’s monetary policy and interest rates impact all companies. Geopolitical events, major storms, and pandemics can also impact the entire market. Investors in stocks cannot avoid this type of risk. This unavoidable risk is the systematic risk that investors in stocks have. This systematic risk cannot be eliminated through diversification.
In addition, as per research conducted by Meir Statman,Meir Statman. “How Many Stocks Make a Diversified Portfolio?” Journal of Financial and Quantitative Analysis 22, no. 3 (September 1987): 353–363. https://doi.org/10.2307/2330969 the standard deviation of a portfolio drops quickly as the number of stocks in the portfolio increases from one to two or three (see Figure 2 illustration in this subsequent article by Statman for context). Increasing the size of the portfolio decreases the standard deviation, and thus the risk, of the portfolio. However, as the portfolio increases in size, the amount of risk reduced by adding one more stock to the portfolio will decrease. How many stocks does an investor need for a portfolio to be well-diversified? There is not an exact number that all financial managers agree on. A portfolio of 15 highly correlated stocks will offer less benefits of diversification than a portfolio of 10 stocks with lower correlation coefficients. A portfolio that consists of American Airlines, Spirit Airlines, United Airlines, Southwest Airlines, Delta Airlines, and Jet Blue, along with a few other stocks, is not very diversified because of the heavy concentration in the airline industry. The term diversified portfolio is a relative concept, but the average investor can create a reasonably diversified portfolio with approximately a dozen stocks.

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As investors hold multiple assets in a portfolio, they are able to eliminate firm-specific risk. However, systematic or market risk remains, even if an investor holds the market portfolio. The return to a portfolio is measured by the arithmetic average, and the risk is measured by the standard deviation of the returns of the portfolio. The risk of the portfolio will be lower than the weighted average of the risk of the individual securities because the returns of the securities are not perfectly correlated. A low or negative return for one stock in a period can be offset by a high return for another stock in the same period.

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### John Bogle and the Buy-and-Hold Strategy

In this video, the legendary investor Jack Bogle, founder and former CEO of Vanguard, discusses strategies for investors.


# How to Think about Investing

## The Capital Asset Pricing Model (CAPM)

### Learning Outcomes

By the end of this section, you will be able to:
1. Define risk premium.
2. Explain the concept of beta.
3. Compute the required return of a security using the CAPM.

### Risk-Free Rate

The capital asset pricing model (CAPM) is a financial theory based on the idea that investors who are willing to hold stocks that have higher systematic risk should be rewarded more for taking on this market risk. The CAPM focuses on systematic risk, rather than a stock’s individual risk, because firm-specific risk can be eliminated through diversification.
Suppose that your grandparents have given you a gift of $20,000. After you graduate from college, you plan to work for a few years and then apply to law school. You want to use the $20,000 your grandparents gave you to pay for part of your law school tuition. It will be several years before you are ready to spend the money, and you want to keep the money safe. At the same time, you would like to invest the money and have it grow until you are ready to start law school.
Although you would like to earn a return on the money so that you have more than $20,000 by the time you start law school, your primary objective is to keep the money safe. You are looking for a risk-free investment. Lending money to the US government is considered the lowest-risk investment that you can make. You can purchase a US Treasury security. The chances of the US government not paying its debts is close to zero. Although, in theory, no investment is 100% risk-free, investing in US government securities is generally considered a risk-free investment because the risk is so miniscule.
The rate that you can earn by purchasing US Treasury securities is a proxy for the risk-free rate. It is used as an investing benchmark. The average rate of return for the three-month US Treasury security from 1928 to 2020 is 3.36%.“Historical Return on Stocks, Bonds and Bills: 1928–2020.” Damodaran Online. Stern School of Business, New York University, January 2021. http://pages.stern.nyu.edu/~adamodar/New_Home_Page/datafile/histretSP.html You can see that you will not become immensely wealthy by investing in US Treasury bills. Another characteristic of US Treasury securities, however, is that their volatility tends to be much lower than that of stocks. In fact, the standard deviation of returns for the US Treasury bills is 3.0%. Unlike the returns for stocks, the return on US Treasury bills has never been negative. The lowest annual return was 0.03%, which occurred in 2014.Ibid.

### Risk Premium

You know that if you use your $20,000 to invest in stock rather than in US Treasury bills, the outcome of the investment will be uncertain. Your investments may do well, but there is also a risk of losing money. You will only be willing to take on this risk if you are rewarded for doing so. In other words, you will only be willing to take the risk of investing in stocks if you think that doing so will make you more than you would make investing in US Treasury securities.
From 1928 to 2020, the average return for the S&P 500 stock index has been 11.64%, which is much higher than the 3.36% average return for US Treasury bills.Ibid. Stock returns, with a standard deviation of 19.49%, however, have also been much more volatile. In fact, there were 25 years in which the return for the S&P 500 index was negative.
You may not be willing to take the risk of losing some of the money your grandparents gave you because you have been setting it aside for law school. If that’s the case, you will want to invest in US Treasury securities. You may have money that you are saving for other long-term goals, such as retirement, with which you are willing to take some risk. The extra return that you will earn for taking on risk is known as the risk premium. The risk premium can be thought of as your reward for being willing to bear risk.
The risk premium is calculated as the difference between the return you receive for taking on risk and what you would have returned if you did not take on risk. Using the average return of the S&P 500 (to measure what investors who bear the risk earn) and the US Treasury bill rate (to measure what investors who do not bear risk earn), the risk premium is calculated as

### Beta

The risk premium represents how much an investor who takes on the market portfolio is rewarded for risk. Investors who purchase one stock—DAL, for example—experience volatility, which is measured by the standard deviation of that stock’s returns. Remember that some of that volatility, the volatility caused by firm-specific risk, can be diversified away. Because investors can eliminate firm-specific risk through diversification, they will not be rewarded for that risk. Investors are rewarded for the amount of systematic risk they incur.

### Interpreting Beta

The relevant risk for investors is the systematic risk they incur. The systematic risk of a particular stock is measured by how much the stock moves with the market. The measure of how much a stock moves with the market is known as its beta. A stock that tends to move in sync with the market will have a beta of 1. For these stocks, if the market goes up 10%, the stock generally also goes up 10%; if the market goes down 5%, stocks with a beta of 1 also tend to go down 5%.
If a company has a beta greater than 1, then the stock tends to have a more pronounced move in the same direction as a market move. For example, if a stock has a beta of 2, the stock will tend to increase by 20% when the market goes up by 10%. If the market falls by 5%, that same stock will tend to fall by twice as much, or 10%. Thus, stocks with a beta greater than 1 experience greater swings than the overall market and are considered to be riskier than the average stock.
On the other hand, stocks with a beta less than 1 experience smaller swings than the overall market. A beta of 0.5, for example, means that a stock tends to experience moves that are only 50% of overall market moves. So, if the market increases by 10%, a stock with a beta of 0.5 would tend to rise by only 5%. A market decline of 5% would tend to be associated with a 2.5% decrease in the stock.

### Calculating Betas

The calculation of beta for DAL is demonstrated in . Monthly returns for DAL and for the S&P 500 are plotted in the diagram. Each dot in the scatter plot corresponds to a month from 2018 to 2020; for example, the dot that lies furthest in the upper right-hand corner represents November 2020. The return for the S&P 500 was 10.88% that month; this return is plotted along the horizontal axis. The return for DAL during November 2020 was 31.36%; this return is plotted along the vertical axis.
You can see that generally, when the overall stock market as measured by the S&P 500 is positive, the return for DAL is also positive. Likewise, in months in which the return for the S&P 500 is negative, the return for DAL is also usually negative. Drawing a line that best fits the data, also known as a regression line, summarizes the relationship between the returns for DAL and the S&P 500. The slope of this line, 1.39, is DAL’s beta. Beta measures the amount of systematic risk that DAL has.

### CAPM Equation

Because DAL’s beta of 1.39 is greater than 1, DAL is riskier than the average stock in the market. Finance theory suggests that investors who purchase DAL will expect a higher rate of return to compensate them for this risk. DAL has 139% of the average stock’s systematic risk; therefore, investors in the stock should receive 139% of the market risk premium.
The capital asset pricing model (CAPM) equation is
where Re is the expected return of the asset, Rf is the risk-free rate of return, and Rm is the expected return of the market. Given the average S&P 500 return of 11.64% and the average US Treasury bill return of 3.36%, the expected return of DAL would be calculated as

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The capital asset pricing model (CAPM) relates the expected return of an asset to the systematic risk of that asset. Investors will be rewarded for taking on systematic risk. They will not be rewarded for taking on firm-specific risk, however, because that risk can be diversified away.

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# How to Think about Investing

## Applications in Performance Measurement

### Learning Outcomes

By the end of this section, you will be able to:
1. Interpret a Sharpe ratio.
2. Interpret a Treynor measurement.
3. Interpret Jensen’s alpha.

### Sharpe Ratio

Investors want a measure of how good a professional money manager is before they entrust their hard-earned funds to that professional for investing. Suppose that you see an advertisement in which McKinley Investment Management claims that the portfolios of its clients have an average return of 20% per year. You know that this average annual return is meaningless without also knowing something about the riskiness of the firm’s strategy. In this section, we consider some ways to evaluate the riskiness of an investment strategy.
A basic measure of investment performance that includes an adjustment for risk is the Sharpe ratio. The Sharpe ratio is computed as a portfolio’s risk premium divided by the standard deviation of the portfolio’s return, using the formula
The portfolio risk premium is the portfolio return RP minus the risk-free return Rf; this is the basic reward for bearing risk. If the risk-free return is 3%, McKinley Investment Management’s clients who are earning 20% on their portfolios have an excess return of 17%.
The standard deviation of the portfolio’s return, , is a measure of risk. Although you see that McKinley’s clients earn a nice 20% return on average, you find that the returns are highly volatile. In some years, the clients earn much more than 20%, and in other years, the return is much lower, even negative. That volatility leads to a standard deviation of returns of 26%. The Sharpe ratio would be , or 0.65.
Thus, the Sharpe ratio can be thought of as a reward-to-risk ratio. The standard deviation in the denominator can be thought of as the units of risk the investor has. The numerator is the reward the investor is receiving for taking on that risk.

### Treynor Measurement of Performance

Another reward-to-risk ratio measurement of investment performance is the Treynor ratio. The Treynor ratio is calculated as
Just as with the Sharpe ratio, the numerator of the Treynor ratio is a portfolio’s risk premium; the difference is that the Treynor ratio focus focuses on systematic risk, using the beta of the portfolio in the denominator, while the Shape ratio focuses on total risk, using the standard deviation of the portfolio’s returns in the denominator.
If McKinley Investment Management has a portfolio with a 20% return over the past five years, with a beta of 1.2 and a risk-free rate of 3%, the Treynor ratio would be
Both the Sharpe and Treynor ratios are relative measures of investment performance, meaning that there is not an absolute number that indicates whether an investment performance is good or bad. An investment manager’s performance must be considered in relation to that of other managers or to a benchmark index.

### Jensen’s Alpha

Jensen’s alpha is another common measure of investment performance. It is computed as the raw portfolio return minus the expected portfolio return predicted by the CAPM:
Suppose that the average market return has been 12%. What would Jensen’s alpha be for McKinley Investment Management’s portfolio with a 20% average return and a beta of 1.2?
Unlike the Sharpe and Treynor ratios, which are meaningful in a relative sense, Jensen’s alpha is meaningful in an absolute sense. An alpha of 0.062 indicates that the McKinley Investment Management portfolio provided a return that was 6.2% higher than would be expected given the riskiness of the portfolio. A positive alpha indicates that the portfolio had an abnormal return. If Jensen’s alpha equals zero, the portfolio return was exactly what was expected given the riskiness of the portfolio as measured by beta.

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Because investors are not simply interested in returns but are also interested in risk, the success of a portfolio cannot be measured simply by considering the portfolio’s return. In order to compare investment portfolios, risk and return must both be taken into consideration. The Sharpe ratio and the Treynor ratio are two measures that provide a reward-to-risk measure of a portfolio. Jensen’s alpha provides a measure of the abnormal return of a portfolio, considering the portfolio’s risk level.

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# How to Think about Investing

## Using Excel to Make Investment Decisions

### Learning Outcomes

By the end of this section, you will be able to:
1. Calculate the average return and standard deviation for a stock.
2. Calculate the average return and standard deviation for a portfolio.
3. Calculate the beta of a stock.

### Average Return and Standard Deviation for a Single Stock

Excel can be used to calculate the average returns and the standard deviation of returns for both a single stock and a portfolio of stocks. It can also be used to calculate the beta for a stock. Historic stock price data for stocks you are interested in analyzing can easily be downloaded from sites such as Yahoo! Finance into Excel. The examples in this section use monthly stock data from December 2017 to December 2020 from Yahoo! Finance.
Monthly price data for AMZN (Amazon) is shown in column B of . To begin, monthly returns must be calculated from the price data using the formula
The ending prices shown in  are the last price the stock traded for each month. Each month, the return is calculated under the assumption that you purchased the stock at the last trading price of the previous month and sold at the last price of the current month. Thus, the return for January 2018 is calculated as
This is accomplished in Excel by placing the formula =(B3-B2)/B2 in cell C3. This formula can then be copied down the spreadsheet through row C38. Now that each monthly return is in column C, you can calculate the average of the monthly returns in cell C39 and the standard deviation of returns in cell C40.
Over the three-year period, the average monthly return for AMZN was 3.3%. However, this return was highly volatile, with a standard deviation of 9.33%. Remember that this means that approximately two-thirds of the time, the monthly return from AMZN was between −6.03% and 12.63%.

### Average Return and Standard Deviation for a Portfolio

The Excel screenshot in  shows the return and standard deviation calculation for a portfolio. This sample four-stock portfolio contains AMZN, CVS, AAPL (Apple), and NFLX (Netflix). This portfolio is constructed as an equally weighted portfolio; because there are four stocks in this portfolio, each has a weight of 25%.
The monthly returns for each stock are recorded in their respective columns. The portfolio return for each month is calculated as the weighted average of the four monthly individual stock returns. The formula for the portfolio return is
The formula =$B$1*B3+$C$1*C3+$D$1*D3+$E$1*E3 is placed in cell F3. The formula is then copied down column F to calculate the portfolio return for each month. After the monthly portfolio return is calculated, then the average monthly portfolio return is calculated in cell F39. The average monthly portfolio return is 2.69%.
Because this is an equally weighted portfolio, with each of the four stocks impacting the portfolio return in the same way, the average monthly portfolio return of 2.69% is the same as the sum of the average monthly returns of the four stocks divided by four, or .
The standard deviation of the monthly portfolio returns is calculated in cell F40. This four-stock portfolio has a standard deviation of 7.10%. Unlike the average return, this standard deviation is not equal to the average of the standard deviations of returns of the four stocks. In fact, the standard deviation for the portfolio is less than the standard deviation for any one of the four stocks. Remember that this occurs because the stock returns are not perfectly positively correlated. The high return of one of the stocks in one month is dampened by a lower return in another stock during the same month. Likewise, a negative return in one stock during a month might be offset by a positive return in one of the other three stocks during the same month. This is the risk reduction benefit of holding a portfolio of stocks.

### Calculating Beta

The standard deviation of a stock’s returns indicates the stock’s volatility. Remember that the volatility is caused by both firm-specific and systematic risk. Investors will not be rewarded for firm-specific risk because they can diversify away from it. Investors are, however, rewarded for systematic risk. To determine how much of a firm’s risk is due to systematic risk, you can use Excel to calculate the stock’s beta.
To calculate a stock’s beta, you need the monthly return for the market in addition to the monthly market return for the stock. In column B in , the monthly return for SPY, the SPDR S&P 500 Trust, is recorded. SPY is an ETF that was created to mimic the performance of the S&P 500 index by State Street Global Advisors and is often used as a proxy for the overall market performance. The monthly returns for AMZN are visible in column C. It is important that these returns be lined up so that the returns for a particular month for both securities appear in the same row number. Also, you want to place the returns for SPY in the column to the left of the returns for AMZN so that when you create your graph, SPY will automatically appear on the horizontal axis.
You will use a scatter plot to create a graph. In Excel, go to the Insert tab, and then from the Chart menu, choose the first scatter plot option.
Selecting the scatter plot option will result in a chart being inserted that looks like the chart in . Each dot represents one month’s combination of returns, with the return for SPY measured on the horizontal axis and the return for AMZN measured on the vertical axis. Consider, for example, the dot in the furthest upper right-hand section of the figure. This dot is the plot of returns for the month of April 2020, when the return for SPY was 13.36% (measured on the horizontal axis) and the return for AMZN was 26.89% (measured on the vertical axis).
Hover your mouse over one of the dots, and right-click the dot to pull up a chart formatting menu. This menu will allow you to add labels to your axis and polish your chart in additional ways if you would like. Select the option Add Trendline.
When the trendline is inserted, a formatting box will appear on the right of your screen (see ). If it is not already selected, choose the Linear trendline option. Scroll down and select the “Display Equation on chart” option. You will see the equation  appear on the screen. This is the equation for the best-fit line that shows how AMZN moves when the market moves. The slope of this line, 1.1477, is the beta for AMZN. This tells you that for every 10% move the overall market makes, AMZN tends to move 11.477%. Because AMZN tends to move a little more than the broader market, it has a little more systematic risk than the average stock in the market.

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Using Excel to manipulate publicly available stock data makes calculating the average return of a stock and the standard deviation of returns easy. The average return for a portfolio and the standard deviation of the portfolio returns can also be calculated easily. By comparing the returns of a stock with the returns of the overall market using Excel charting tools, the beta for a stock, which measures systematic risk, can be determined.

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### CFA Institute

This chapter supports some of the Learning Outcome Statements (LOS) in this CFA® Level I Study Session. Reference with permission of CFA Institute.


# How Companies Think about Investing

## Why It Matters

One of the most important decisions a company faces is choosing which investments it should make. Should an automobile manufacturer purchase a new robot for its assembly line? Should an airline purchase a new plane to add to its fleet? Should a hotel chain build a new hotel in Atlanta? Should a bakery purchase tables and chairs to provide places for customers to eat? Should a pharmaceutical company spend money on research for a new vaccine? All of these questions involve spending money today to make money in the future.
The process of making these decisions is often referred to as capital budgeting. In order to grow and remain competitive, a firm relies on developing new products, improving existing products, and entering new markets. These new ventures require investments in fixed assets. The company must decide whether the project will generate enough cash to cover the costs of these initial expenditures once the project is up and running.
For example, Sam’s Sporting Goods sells sporting equipment and uniforms to players on local recreational and school teams. Customers have been inquiring about customizing items such as baseball caps and equipment bags with logos and other designs. Sam’s is considering purchasing an embroidery machine so that it can provide these customized items in-house. The machine will cost $16,000. Purchasing the embroidery machine would be an investment in a fixed asset. If it purchases the machine, Sam’s will be able to charge customers for customization.
The managers think that selling customized items will allow the company to increase its cash flow by $2,000 next year. They predict that as customers become more aware of this service, the ability to customize products in-house will increase the company’s cash flow by $4,000 the following year. The managers expect the machine will be used for five years, with the embroidery products increasing cash flows by $5,000 during each of the last three years the machine is used. Should Sam’s Sporting Goods invest in the embroidery machine? In this chapter, we consider the main capital budgeting techniques Sam’s and other companies can use to evaluate these types of decisions.


# How Companies Think about Investing

## Payback Period Method

### Learning Outcomes

By the end of this section, you will be able to:
1. Define payback period.
2. Calculate payback period.
3. List the advantages and disadvantages of using the payback period method.
The payback period method provides a simple calculation that the managers at Sam’s Sporting Goods can use to evaluate whether to invest in the embroidery machine. The payback period calculation focuses on how long it will take for a company to make enough free cash flow from the investment to recover the initial cost of the investment.

### Payback Period Calculation

In order to purchase the embroidery machine, Sam’s Sporting Goods must spend $16,000. During the first year, Sam’s expects to see a $2,000 benefit from purchasing the machine, but this means that after one year, the company will have spent $14,000 more than it has made from the project. During the second year that it uses the machine, Sam’s expects that its cash inflow will be $4,000 greater than it would have been if it had not had the machine. Thus, after two years, the company will have spent $10,000 more than it has benefited from the machine. This process is continued year after year until the accumulated increase in cash flow is $16,000, or equal to the original investment. The process is summarized in .
Sam’s Sporting Goods is expecting its cash inflow to increase by $16,000 over the first four years of using the embroidery machine. Thus, the payback period for the embroidery machine is four years. In other words, it takes four years to accumulate $16,000 in cash inflow from the embroidery machine and recover the cost of the machine.

### Advantages

The principal advantage of the payback period method is its simplicity. It can be calculated quickly and easily. It is easy for managers who have little finance training to understand. The payback measure provides information about how long funds will be tied up in a project. The shorter the payback period of a project, the greater the project’s liquidity.

### Disadvantages

Although it is simple to calculate, the payback period method has several shortcomings. First, the payback period calculation ignores the time value of money. Suppose that in addition to the embroidery machine, Sam’s is considering several other projects. The cash flows from these projects are shown in . Both Project B and Project C have a payback period of five years. For both of these projects, Sam’s estimates that it will take five years for cash inflows to add up to $16,000. The payback period method does not differentiate between these two projects.
However, we know that money has a time value, and receiving $6,000 in year 1 (as occurs in Project C) is preferable to receiving $6,000 in year 5 (as in Projects B and D). From what we learned about the time value of money, Projects B and C are not identical projects. The payback period method breaks the important finance rule of not adding or comparing cash flows that occur in different time periods.
A second disadvantage of using the payback period method is that there is not a clearly defined acceptance or rejection criterion. When the payback period method is used, a company will set a length of time in which a project must recover the initial investment for the project to be accepted. Projects with longer payback periods than the length of time the company has chosen will be rejected. If Sam’s were to set a payback period of four years, Project A would be accepted, but Projects B, C, and D have payback periods of five years and so would be rejected. Sam’s choice of a payback period of four years would be arbitrary; it is not grounded in any financial reasoning or theory. No argument exists for a company to use a payback period of three, four, five, or any other number of years as its criterion for accepting projects.
A third drawback of this method is that cash flows after the payback period are ignored. Projects B, C, and D all have payback periods of five years. However, Projects B and C end after year 5, while Project D has a large cash flow that occurs in year 6, which is excluded from the analysis. The payback method is shortsighted in that it favors projects that generate cash flows quickly while possibly rejecting projects that create much larger cash flows after the arbitrary payback time criterion.
Fourth, no risk adjustment is made for uncertain cash flows. No matter how careful the planning and analysis, a business is seldom sure what future cash flows will be. Some projects are riskier than others, with less certain cash flows, but the payback period method treats high-risk cash flows the same way as low-risk cash flows.

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The payback period is the simplest project evaluation method. It is the time it takes the company to recoup its initial investment. Its usefulness is limited, however, because it ignores the time value of money.

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