Problem 49
Question
Earnings per Share The earnings per share z (in dollars) for Starbucks Corporation from 1998 through 2006 can be modeled by z = 0.106 \(x-0.036 y-0.005,\) where \(x\) is sales (in billions of dollars) and \(y\) is the shareholder's equity (in billions of dollars). (a) Find the earnings per share when \(x=8\) and \(y=5 .\) (b) Which of the two variables in this model has the greater influence on the earnings per share? Explain.
Step-by-Step Solution
Verified Answer
The earnings per share when \(x = 8\) and \(y = 5\) is calculated using the provided model equation to be approximately \(0.588\) dollars. The variable \(x\) (sales) has a greater influence on earnings per share, as it has a larger absolute coefficient in the equation.
1Step 1: Substitution
For part (a), substitute \(x = 8\) and \(y = 5\) into the model equation: \(z = 0.106(8) - 0.036(5) - 0.005\).
2Step 2: Calculation
Solve the equation to determine the earnings per share. It's a simple arithmetic operation where you should calculate the value of \(z\).
3Step 3: Comparison of Variable Influence
For part (b), the variable that corresponds to the larger coefficient in absolute value has a greater impact on the earnings per share. In this case, we compare \(0.106\) (the coefficient of \(x\)) and \(0.036\) (the coefficient of \(y\)).
Key Concepts
Financial ModelingVariable Influence AnalysisMathematical Substitution
Financial Modeling
Financial modeling is a quantitative analysis that is used to forecast a company's financial performance. The model built in the exercise for Starbucks Corporation is an example of how financial modeling helps in understanding the impact of different financial metrics on a company's profitability.
In our case, earnings per share (EPS) is the dependent variable that the model is trying to predict, represented by the letter 'z'. The independent variables affecting EPS in this model are 'x', the sales in billions of dollars, and 'y', the shareholders' equity in billions of dollars. The coefficients within the model (0.106, -0.036, -0.005) are the weights assigned to each of these variables, indicating how much each unit increase in these variables will impact the EPS.
The exercise provided simplifies the concept of financial modeling by using a linear equation to represent the relationship between sales, shareholder's equity, and earnings per share. Such models are critical tools that can help investors, financial analysts, and company managers in making informed business decisions based on projected financial outcomes.
In our case, earnings per share (EPS) is the dependent variable that the model is trying to predict, represented by the letter 'z'. The independent variables affecting EPS in this model are 'x', the sales in billions of dollars, and 'y', the shareholders' equity in billions of dollars. The coefficients within the model (0.106, -0.036, -0.005) are the weights assigned to each of these variables, indicating how much each unit increase in these variables will impact the EPS.
The exercise provided simplifies the concept of financial modeling by using a linear equation to represent the relationship between sales, shareholder's equity, and earnings per share. Such models are critical tools that can help investors, financial analysts, and company managers in making informed business decisions based on projected financial outcomes.
Variable Influence Analysis
Variable influence analysis is the examination of how the different independent variables in a model affect the dependent variable.
In the exercise, we assess which variable between sales ('x') and shareholders' equity ('y') has a more significant influence on the earnings per share ('z'). To do this, we look at the coefficients of those variables in the equation. A larger absolute value of the coefficient signifies a greater impact on the dependent variable, which, in this context, is the EPS.
The sales variable 'x' has a coefficient of 0.106, while the shareholder's equity variable 'y' has a lower coefficient of -0.036. Thus, it is clear that sales have a more substantial influence on the EPS for Starbucks Corporation. Understanding variable influence helps in focusing on the most significant factors that can drive the financial performance of a company. For example, a company may decide to implement strategies to boost sales if the sales have a more pronounced effect on profitability, as indicated by the financial model.
In the exercise, we assess which variable between sales ('x') and shareholders' equity ('y') has a more significant influence on the earnings per share ('z'). To do this, we look at the coefficients of those variables in the equation. A larger absolute value of the coefficient signifies a greater impact on the dependent variable, which, in this context, is the EPS.
The sales variable 'x' has a coefficient of 0.106, while the shareholder's equity variable 'y' has a lower coefficient of -0.036. Thus, it is clear that sales have a more substantial influence on the EPS for Starbucks Corporation. Understanding variable influence helps in focusing on the most significant factors that can drive the financial performance of a company. For example, a company may decide to implement strategies to boost sales if the sales have a more pronounced effect on profitability, as indicated by the financial model.
Mathematical Substitution
Mathematical substitution is a method used to replace variables with their respective values to solve equations or evaluate expressions. This method is demonstrated in step 1 of the provided exercise where the values for sales ('x = 8') and shareholder's equity ('y = 5') are substituted into the earnings per share model.
By performing the substitution \(z = 0.106(8) - 0.036(5) - 0.005\), we can compute the value of 'z', which represents Starbucks' EPS. This approach simplifies the solution process, especially in more complex financial models.
Mathematical substitution is not only essential for solving problems but also for variable sensitivity analysis which tells us how changes in input values affect the output. The method is widely used across various fields such as finance, economics, engineering, and physics, due to its fundamental role in evaluating and simplifying expressions.
By performing the substitution \(z = 0.106(8) - 0.036(5) - 0.005\), we can compute the value of 'z', which represents Starbucks' EPS. This approach simplifies the solution process, especially in more complex financial models.
Mathematical substitution is not only essential for solving problems but also for variable sensitivity analysis which tells us how changes in input values affect the output. The method is widely used across various fields such as finance, economics, engineering, and physics, due to its fundamental role in evaluating and simplifying expressions.
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