Investments Chapter 7: Fundamentals of Portfolio Analysis
Investment Problems <ul><li>Problems of  constrained optimization  under uncertainty </li></ul>
Three Parts to this Problem <ul><ul><li>Optimization </li></ul></ul><ul><ul><li>Concept: utility function. </li></ul></ul>...
Portfolio Possibilities: Set I <ul><li>An investor chooses a portfolio that combines the assets in a variety of proportion...
Portfolio Possibilities: Set II <ul><li>Example of a portfolio possibilities set  for two restrictions: </li></ul><ul><li>...
The Probability Distribution: I <ul><li>Objective Probabilities </li></ul><ul><li>Probabilities that are known with certai...
The Probability Distribution: II <ul><li>Subjective probabilities can be described by a probability distribution. </li></u...
Population Statistics: I <ul><li>Sample Statistics   </li></ul><ul><li>Ex-post statistics; summarize historical returns. <...
Population Statistics: II <ul><li>1. Population mean. </li></ul><ul><li>2. Population variance. </li></ul><ul><li>3. Popul...
The Normal Distribution: I <ul><li>Discrete Distribution </li></ul><ul><li>Describes a countable number of  states-of-the-...
The Normal Distribution: II <ul><li>Characteristics of the  two-parameter normal distribution : </li></ul><ul><ul><li>The ...
The Normal Distribution: III <ul><li>Probability Density Function </li></ul><ul><li>Describes the outcomes of a continuous...
The Normal Distribution: IV <ul><li>Cumulative Normal Distribution Function </li></ul><ul><li>Represents the area below th...
The Normal Distribution: V – Illustrations A Normal Probabilty Density Function and its corresponding Cumulative Normal Di...
Normal Distribution Source: ©  Stocks, Bonds, Bills, and Inflation   2000 Yearbook™ , Ibbotson Associates, Inc., Chicago (...
Normal Distribution <ul><li>A large enough sample drawn from a normal distribution looks like a bell-shaped curve. </li></...
Normal Distribution <ul><li>The 20.1-percent standard deviation we found for stock returns from 1926 through 1999 can now ...
The Utility Function <ul><li>Four properties: </li></ul><ul><li>1. Utility is increasing </li></ul><ul><li>Marginal utilit...
Expected Utility Criterion <ul><li>Combines the three key elements of investment (probabilities set, probability distribut...
Risk Premiums <ul><li>The equity premium is the difference in the expected rate of return between stocks and treasury bill...
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  1. 1. Investments Chapter 7: Fundamentals of Portfolio Analysis
  2. 2. Investment Problems <ul><li>Problems of constrained optimization under uncertainty </li></ul>
  3. 3. Three Parts to this Problem <ul><ul><li>Optimization </li></ul></ul><ul><ul><li>Concept: utility function. </li></ul></ul><ul><ul><li>2. Uncertainty </li></ul></ul><ul><ul><li>Concept: probability distribution. </li></ul></ul><ul><ul><li>3. Constraints </li></ul></ul><ul><ul><li>Concept: portfolio possibilities set. </li></ul></ul>
  4. 4. Portfolio Possibilities: Set I <ul><li>An investor chooses a portfolio that combines the assets in a variety of proportions (known as portfolio weights ). </li></ul><ul><li>Generally, there are constrictions on these proportions. </li></ul>
  5. 5. Portfolio Possibilities: Set II <ul><li>Example of a portfolio possibilities set for two restrictions: </li></ul><ul><li>1. All portfolio weights add up to one. </li></ul><ul><li>2. No short sales allowed. </li></ul>
  6. 6. The Probability Distribution: I <ul><li>Objective Probabilities </li></ul><ul><li>Probabilities that are known with certainty (example: coin-flipping experiment). </li></ul><ul><li>Subjective Probabilities </li></ul><ul><li>Probabilities that are uncertain and can only be estimated (example: future values of assets). </li></ul>
  7. 7. The Probability Distribution: II <ul><li>Subjective probabilities can be described by a probability distribution. </li></ul><ul><li>A probability distribution: </li></ul><ul><li>1. Is a mathematical function. </li></ul><ul><li>2. Considers possible outcomes of a random variable or a set of random variables. </li></ul><ul><li>3. Assigns probabilities to these outcomes. </li></ul>
  8. 8. Population Statistics: I <ul><li>Sample Statistics </li></ul><ul><li>Ex-post statistics; summarize historical returns. </li></ul><ul><li>Population Statistics </li></ul><ul><li>Ex-ante statistics; summarize a future return distribution. </li></ul>
  9. 9. Population Statistics: II <ul><li>1. Population mean. </li></ul><ul><li>2. Population variance. </li></ul><ul><li>3. Population covariance. </li></ul><ul><li>4. Population correlation (coefficient). </li></ul>
  10. 10. The Normal Distribution: I <ul><li>Discrete Distribution </li></ul><ul><li>Describes a countable number of states-of-the-world. </li></ul><ul><li>Continuous Distribution </li></ul><ul><li>Describes an infinite number of states-of-the-world. </li></ul>
  11. 11. The Normal Distribution: II <ul><li>Characteristics of the two-parameter normal distribution : </li></ul><ul><ul><li>The distribution is completely characterized by its mean and variance. </li></ul></ul><ul><ul><li>The possible outcomes range from minus infinity to plus infinity. </li></ul></ul><ul><ul><li>The distribution is symmetric around the mean. </li></ul></ul><ul><ul><li>The larger the distance of an outcome from the mean, the lower the probability density assigned to that outcome. </li></ul></ul>
  12. 12. The Normal Distribution: III <ul><li>Probability Density Function </li></ul><ul><li>Describes the outcomes of a continuous distribution: </li></ul><ul><li>1. Rescales the area under the continuous distribution function in such a manner that it equals 1. </li></ul><ul><li>2 Measures the relative probability of outcomes, instead of their absolute probabilities. </li></ul>
  13. 13. The Normal Distribution: IV <ul><li>Cumulative Normal Distribution Function </li></ul><ul><li>Represents the area below the probability density function from minus infinity to a specified value and thereby: </li></ul><ul><li>Represents the probability that an outcome takes a value that is smaller than, or equal, to that specified value. </li></ul>
  14. 14. The Normal Distribution: V – Illustrations A Normal Probabilty Density Function and its corresponding Cumulative Normal Distribution Function:
  15. 15. Normal Distribution Source: © Stocks, Bonds, Bills, and Inflation 2000 Yearbook™ , Ibbotson Associates, Inc., Chicago (annually updates work by Roger G. Ibbotson and Rex A. Sinquefield). All rights reserved.
  16. 16. Normal Distribution <ul><li>A large enough sample drawn from a normal distribution looks like a bell-shaped curve. </li></ul>Probability Return on large company common stocks 68% 95% > 99% – 3 – 47.9% – 2 – 27.6% – 1 – 7.3% 0 13.0% + 1 33.3% + 2 53.6% + 3 73.9% the probability that a yearly return will fall within 20.1 percent of the mean of 13.3 percent will be approximately 2/3.
  17. 17. Normal Distribution <ul><li>The 20.1-percent standard deviation we found for stock returns from 1926 through 1999 can now be interpreted in the following way: if stock returns are approximately normally distributed, the probability that a yearly return will fall within 20.1 percent of the mean of 13.3 percent will be approximately 2/3. </li></ul>
  18. 18. The Utility Function <ul><li>Four properties: </li></ul><ul><li>1. Utility is increasing </li></ul><ul><li>Marginal utility is always positive. </li></ul><ul><li>2. Utility functions are concave </li></ul><ul><li>Utility functions curve to the return axis. </li></ul><ul><li>3. Different investors have different utility functions </li></ul><ul><li>4. Utility functions are subjective </li></ul>
  19. 19. Expected Utility Criterion <ul><li>Combines the three key elements of investment (probabilities set, probability distribution and utility function) into one decision rule : </li></ul><ul><li>‘ An investor will select the portfolio that yields the highest possible expected value for his or her utility function .’ </li></ul>
  20. 20. Risk Premiums <ul><li>The equity premium is the difference in the expected rate of return between stocks and treasury bills. </li></ul><ul><li>Equity premium puzzle: the size of historical equity premiums cannot be justified by the risk of stocks and the risk aversion of investors. </li></ul>
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