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Principles of Finance

13.1 Measures of Center

Principles of Finance13.1 Measures of Center

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

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

  • Calculate various measures of the average of a data set, such as mean, median, mode, and geometric mean.
  • Recognize when a certain measure of center is more appropriate to use, such as weighted mean.
  • Distinguish among arithmetic mean, geometric mean, and weighted mean.

Arithmetic Mean

The average of a data set is a way of describing location. The most widely used measures of the center of a data set are the mean (average), median, and mode. The arithmetic mean is the most common measure of the average. We will discuss the geometric mean later.

Note that the words mean and average are often used interchangeably. The substitution of one word for the other is common practice. The technical term is arithmetic mean, and average technically refers only to a center location. Formally, the arithmetic mean is called the first moment of the distribution by mathematicians. However, in practice among non-statisticians, average is commonly accepted as a synonym for arithmetic mean.

To calculate the arithmetic mean value of 50 stock portfolios, add the 50 portfolio dollar values together and divide the sum by 50. To calculate the arithmetic mean for a set of numbers, add the numbers together and then divide by the number of data values.

In statistical analysis, you will encounter two types of data sets: sample data and population data. Population data represents all the outcomes or measurements that are of interest. Sample data represents outcomes or measurements collected from a subset, or part, of the population of interest.

The notation x¯x¯ is used to indicate the sample mean, where the arithmetic mean is calculated based on data taken from a sample. The notation xx is used to denote the sum of the data values, and nn is used to indicate the number of data values in the sample, also known as the sample size.

The sample mean can be calculated using the following formula:

x¯ = xnx¯ = xn

Finance professionals often rely on averages of Treasury bill auction amounts to determine their value. Table 13.1 lists the Treasury bill auction amounts for a sample of auctions from December 2020.

Maturity Amount ($Billions)
4-week T-bills $32.9
8-week T-bills 38.4
13-week T-bills 63.1
26-week T-bills 59.6
52-week T-bills 39.7
Total $233.7
Table 13.1 United States Treasury Bill Auctions, December 22 and 24, 2020 (source: Treasury Direct)

To calculate the arithmetic mean of the amount paid for Treasury bills at auction, in billions of dollars, we use the following formula:

x¯=xn=$233.75=$46.74x¯=xn=$233.75=$46.74

Median

To determine the median of a data set, order the data from smallest to largest, and then find the middle value in the ordered data set. For example, to find the median value of 50 portfolios, find the number that splits the data into two equal parts. The portfolio values owned by 25 people will be below the median, and 25 people will have portfolio values above the median. The median is generally a better measure of the average when there are extreme values or outliers in the data set.

An outlier or extreme value is a data value that is significantly different from the other data values in a data set. The median is preferred when outliers are present because the median is not affected by the numerical values of the outliers.

The ordered data set from Table 13.1 appears as follows:

32.9, 38.4, 39.7, 59.6, 63.132.9, 38.4, 39.7, 59.6, 63.1

The middle value in this ordered data set is the third data value, which is 39.7. Thus, the median is $39.7 billion.

You can quickly find the location of the median by using the expression n + 12n + 12. The variable n represents the total number of data values in the sample. If n is an odd number, the median is the middle value of the data values when ordered from smallest to largest. If n is an even number, the median is equal to the two middle values of the ordered data values added together and divided by 2. In the example from Table 13.1, there are five data values, so n = 5. To identify the position of the median, calculate n + 12n + 12, which is 5 + 125 + 12, or 3. This indicates that the median is located in the third data position, which corresponds to the value 39.7.

As mentioned earlier, when outliers are present in a data set, the mean can be nonrepresentative of the center of the data set, and the median will provide a better measure of center. The following Think It Through example illustrates this point.

Think It Through

Finding the Measure of Center

Suppose that in a small village of 50 people, one person earns a salary of $5 million per year, and the other 49 individuals each earn $30,000. Which is the better measure of center: the mean or the median?

Mode

Another measure of center is the mode. The mode is the most frequent value. There can be more than one mode in a data set as long as those values have the same frequency and that frequency is the highest. A data set with two modes is called bimodal. For example, assume that the weekly closing stock price for a technology stock, in dollars, is recorded for 20 consecutive weeks as follows:

50, 53, 59, 59, 63, 63, 72, 72, 72, 72, 72, 76, 78, 81, 83, 84, 84, 84, 90, 9350, 53, 59, 59, 63, 63, 72, 72, 72, 72, 72, 76, 78, 81, 83, 84, 84, 84, 90, 93

To find the mode, determine the most frequent score, which is 72. It occurs five times. Thus, the mode of this data set is 72. It is helpful to know that the most common closing price of this particular stock over the past 20 weeks has been $72.00.

Geometric Mean

The arithmetic mean, median, and mode are all measures of the center of a data set, or the average. They are all, in their own way, trying to measure the common point within the data—that which is “normal.” In the case of the arithmetic mean, this is accomplished by finding the value from which all points are equal linear distances. We can imagine that all the data values are combined through addition and then distributed back to each data point in equal amounts.

The geometric mean redistributes not the sum of the values but their product. It is calculated by multiplying all the individual values and then redistributing them in equal portions such that the total product remains the same. This can be seen from the formula for the geometric mean, x̃ (pronounced x-tilde):

x~=x1·x2xnnx~=x1·x2xnn

The geometric mean is relevant in economics and finance for dealing with growth—of markets, in investments, and so on. For an example of a finance application, assume we would like to know the equivalent percentage growth rate over a five-year period, given the yearly growth rates for the investment.

For a five-year period, the annual rate of return for a certificate of deposit (CD) investment is as follows:
3.21%, 2.79%, 1.88%, 1.42%, 1.17%. Find the single percentage growth rate that is equivalent to these five annual consecutive rates of return. The geometric mean of these five rates of return will provide the solution. To calculate the geometric mean for these values (which must all be positive), first multiply1 the rates of return together—after adding 1 to the decimal equivalent of each interest rate—and then take the nth root of the product. We are interested in calculating the equivalent overall rate of return for the yearly rates of return, which can be expressed as 1.0321, 1.0279, 1.0188, 1.0142, and 1.0117:

x~=x1·x2xnn = 1.0321·1.0279·1.0188·1.0142·1.01175 = 1.0209x~=x1·x2xnn = 1.0321·1.0279·1.0188·1.0142·1.01175 = 1.0209

Based on the geometric mean, the equivalent annual rate of return for this time period is 2.09%.

Weighted Mean

A weighted mean is a measure of the center, or average, of a data set where each data value is assigned a corresponding weight. A common financial application of a weighted mean is in determining the average price per share for a certain stock when the stock has been purchased at different points in time and at different share prices.

To calculate a weighted mean, create a table with the data values in one column and the weights in a second column. Then create a third column in which each data value is multiplied by each weight on a row-by-row basis. Then, the weighted mean is calculated as the sum of the results from the third column divided by the sum of the weights.

Think It Through

Calculating the Weighted Mean

Assume your portfolio contains 1,000 shares of XYZ Corporation, purchased on three different dates, as shown in Table 13.2. Calculate the weighted mean of the purchase price for the 1,000 shares.

Date Purchased Purchase Price ($) Number of Shares Purchased Price ($) Times
Number of Shares
January 17 78 200 15,600
February 10 122 300 36,600
March 23 131 500 65,500
Total NA 1,000 117,700
Table 13.2 1,000 Shares of XYZ Corporation

Footnotes

  • 1In this chapter, the interpunct dot will be used to indicate the multiplication operation in formulas.
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