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Principles of Finance

13.5 Probability Distributions

Principles of Finance13.5 Probability Distributions

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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 portfolio weights in an investment.
  • Calculate and interpret the expected values.
  • Apply the normal distribution to characterize average and standard deviation in financial contexts.

Calculate Portfolio Weights

In many financial analyses, the weightings by asset category in a portfolio are a key index used to assess if the portfolio is meeting allocation metrics. For example, an investor approaching retirement age may wish to shift assets in a portfolio to more conservative and lower-volatility investments. Weightings can be calculated in several different ways—for example, based on individual stocks in a portfolio or on various sectors in a portfolio. Weightings can also be calculated based on number of shares or the value of shares of a stock.

To calculate a weighting in a portfolio based on value, take the value of the particular investment and divide it by the total value of the overall portfolio. As an example, consider an individual’s retirement account for which the desired portfolio weighting is determined to be 40% stocks, 50% bonds, and 10% cash equivalents. Table 13.7 shows the current assets in the individual’s portfolio, broken out according to stocks, bonds, and cash equivalents.

Asset Value ($)
Stock A 134,000
Stock B 172,000
Bond C 38,000
Bond D 102,000
Bond E 96,000
Cash in CDs 35,700
Cash in savings 22,500
Total Value 600,200
Table 13.7 Portfolio Assets in Stocks, Bonds, and Cash Equivalents

To determine the weighting in this portfolio for stocks, bonds, and cash, take the total value for each category and divide it by the total value of the entire portfolio. These results are summarized in Table 13.8. Notice that the portfolio weightings shown in the table do not match the target, or desired, allocation weightings of 40% stocks, 50% bonds, and 10% cash equivalents.

Asset Category Category Value ($) Portfolio Weighting
Stocks 306,000306,000 306,000600,200=0.51306,000600,200=0.51
Bonds 236,000236,000 236,000600,200=0.39236,000600,200=0.39
Cash 58,20058,200 58,200600,200=0.1058,200600,200=0.10
Total Value 600,200600,200  
Table 13.8 Portfolio Weightings for Stocks, Bonds, and Cash Equivalents

Portfolio rebalancing is a process whereby the investor buys or sells assets to achieve the desired portfolio weightings. In this example, the investor could sell approximately 10% of the stock assets and purchase bonds with the proceeds to align the asset categories to the desired portfolio weightings.

Calculate and Interpret Expected Values

A probability distribution is a mathematical function that assigns probabilities to various outcomes. For example, we can assign a probability to the outcome of a certain stock increasing in value or decreasing in value. One application of a probability distribution function is determining expected value.

In many financial situations, we are interested in determining the expected value of the return on a particular investment or the expected return on a portfolio of multiple investments. To calculate expected returns, we formulate a probability distribution and then use the following formula to calculate expected value:

ExpectedValue=P1· R1+P2· R2+P3· R3++Pn· RnExpectedValue=P1· R1+P2· R2+P3· R3++Pn· Rn

where P1, P2, P3, ⋯ Pn are the probabilities of the various returns and R1, R2, R3, ⋯ Rn are the various rates of return.

In essence, expected value is a weighted mean where the probabilities form the weights. Typically, these values for Pn and Rn are derived from historical data. As an example, consider a probability distribution for potential returns for United Airlines common stock. Assume that from historical data gathered over a certain time period, there is a 15% probability of generating a 12% return on investment for this stock, a 35% probability of generating a 5% return, a 25% probability of generating a 2% return, a 14% probability of generating a 5% loss, and an 11% probability of resulting in a 10% loss. This data can be organized into a probability distribution table as seen in Table 13.9.

Using the expected value formula, the expected return of United Airlines stock over an extended period of time follows:

Expected Value=P1·R1+P2·R2+P3·R3++Pn·RnExpected Value=P1·R1+P2·R2+P3·R3++Pn·Rn
ExpectedValue = 0.150.12 + 0.350.05 + 0.250.02 + 0.14-0.05 + 0.11-0.10 = 0.0225ExpectedValue = 0.150.12 + 0.350.05 + 0.250.02 + 0.14-0.05 + 0.11-0.10 = 0.0225

Based on the probability distribution, the expected value of the rate of return for United Airlines common stock over an extended period of time is 2.25%.

Historical Return (%) Associated Probability (%)
12 15
5 35
2 25
-5 14
-10 11
Table 13.9 Probability Distribution for Historical Returns on United Airlines Stock

We can extend this analysis to evaluate the expected return for an investment portfolio consisting of various asset categories, such as stocks, bonds, and cash equivalents, where the probabilities are associated with the weighting of each category relative to the total value of the portfolio. Using historical return data for each of the asset categories, the expected return of the overall portfolio can be calculated using the expected value formula.

Assume an investor has assets in stocks, bonds, and cash equivalents as shown in Table 13.10.

Asset Category Value ($) Portfolio Weighting Historical Return (%)
Stocks 306,000306,000 306,000600,200=0.51306,000600,200=0.51 13.0
Bonds 236,000236,000 236,000600,200=0.39236,000600,200=0.39 4.0
Cash 58,20058,200 58,200600,200=0.1058,200600,200=0.10 2.5
Total Value $600,200$600,200    
Table 13.10 Portfolio Weightings and Historical Returns for Various Asset Categories
ExpectedValue=P1·R1+P2·R2+P3·R3++Pn·RnExpectedValue=P1·R1+P2·R2+P3·R3++Pn·Rn
ExpectedValue=0.510.130 + 0.390.040 + 0.100.025 = 0.0844ExpectedValue=0.510.130 + 0.390.040 + 0.100.025 = 0.0844

Based on the probability distribution, the expected value of the rate of return for this portfolio over an extended period of time is 8.44%.

Apply the Normal Distribution in Financial Contexts

The normal, or bell-shaped, distribution can be utilized in many applications, including financial contexts. Remember that the normal distribution has two parameters: the mean, which is the center of the distribution, and the standard deviation, which measures the spread of the distribution. Here are several examples of applications of the normal distribution:

  • IQ scores follow a normal distribution, with a mean IQ score of 100 and a standard deviation of 15.
  • Salaries at a certain company follow a normal distribution, with a mean salary of $52,000 and a standard deviation of $4,800.
  • Grade point averages (GPAs) at a certain college follow a normal distribution, with a mean GPA of 3.27 and a standard deviation of 0.24.
  • The average annual gain of the Dow Jones Industrial Average (DJIA) over a 40-year time period follows a normal distribution, with a mean gain of 485 points and a standard deviation of 1,065 points.
  • The average annual return on the S&P 500 over a 50-year time period follows a normal distribution, with a mean rate of return of 10.5% and a standard deviation of 14.3%.
  • The average annual return on mid-cap stock funds over the five-year period from 2010 to 2015 follows a normal distribution, with a mean rate of return of 8.9% and a standard deviation of 3.7%.

When analyzing data sets that follow a normal distribution, probabilities can be calculated by finding areas under the normal curve. To find the probability that a measurement is within a specific interval, we can compute the area under the normal curve corresponding to the interval of interest.

Areas under the normal curve are available in tables, and Excel also provides a method to find these areas. The empirical rule is one method for determining areas under the normal curve that fall within a certain number of standard deviations of the mean (see Figure 13.4).

Graph of a normal distribution showing mean and increments of standard deviation. It is symmetrical about a vertical line drawn through the mean. The standard deviation up to three deviations are displayed on both sides of the mean.
Figure 13.4 Normal Distribution Showing Mean and Increments of Standard Deviation

If x is a random variable and has a normal distribution with mean µ and standard deviation σσ, then the empirical rule states the following:

  • About 68% of the x-values lie between -1σ-1σ and +1σ+1σ units from the mean µµ (within one standard deviation of the mean).
  • About 95% of the x-values lie between -2σ-2σ and +2σ+2σ units from the mean µµ (within two standard deviations of the mean).
  • About 99.7% of the x-values lie between -3σ-3σ and +3σ+3σ units from the mean µµ (within three standard deviations of the mean). Notice that almost all the x-values lie within three standard deviations of the mean.
  • The z-scores for +1σ+1σ and -1σ-1σ are +1+1 and -1-1, respectively.
  • The z-scores for +2σ+2σ and -2σ-2σ are +2+2 and -2-2, respectively.
  • The z-scores for +3σ+3σ and -3σ-3σ are +3+3 and -3-3, respectively.

As an example of using the empirical rule, suppose we know that the average annual return for mid-cap stock funds over the five-year period from 2010 to 2015 follows a normal distribution, with a mean rate of return of 8.9% and a standard deviation of 3.7%. We are interested in knowing the likelihood that a randomly selected mid-cap stock fund provides a rate of return that falls within one standard deviation of the mean, which implies a rate of return between 5.2% and 12.6%. Using the empirical rule, the area under the normal curve within one standard deviation of the mean is 68%. Thus, there is a probability, or likelihood, of 0.68 that a mid-cap stock fund will provide a rate of return between 5.2% and 12.6%.

If the interval of interest is extended to two standard deviations from the mean (a rate of return between 1.5% and 16.3%), using the empirical rule, we can determine that the area under the normal curve within two standard deviations of the mean is 95%. Thus, there is a probability, or likelihood, of 0.95 that a mid-cap stock fund will provide a rate of return between 1.5% and 16.3%.

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