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  • 1. Investments Chapter 7: Fundamentals of Portfolio Analysis
  • 2. Investment Problems
    • Problems of constrained optimization under uncertainty
  • 3. Three Parts to this Problem
      • Optimization
      • Concept: utility function.
      • 2. Uncertainty
      • Concept: probability distribution.
      • 3. Constraints
      • Concept: portfolio possibilities set.
  • 4. Portfolio Possibilities: Set I
    • An investor chooses a portfolio that combines the assets in a variety of proportions (known as portfolio weights ).
    • Generally, there are constrictions on these proportions.
  • 5. Portfolio Possibilities: Set II
    • Example of a portfolio possibilities set for two restrictions:
    • 1. All portfolio weights add up to one.
    • 2. No short sales allowed.
  • 6. The Probability Distribution: I
    • Objective Probabilities
    • Probabilities that are known with certainty (example: coin-flipping experiment).
    • Subjective Probabilities
    • Probabilities that are uncertain and can only be estimated (example: future values of assets).
  • 7. The Probability Distribution: II
    • Subjective probabilities can be described by a probability distribution.
    • A probability distribution:
    • 1. Is a mathematical function.
    • 2. Considers possible outcomes of a random variable or a set of random variables.
    • 3. Assigns probabilities to these outcomes.
  • 8. Population Statistics: I
    • Sample Statistics
    • Ex-post statistics; summarize historical returns.
    • Population Statistics
    • Ex-ante statistics; summarize a future return distribution.
  • 9. Population Statistics: II
    • 1. Population mean.
    • 2. Population variance.
    • 3. Population covariance.
    • 4. Population correlation (coefficient).
  • 10. The Normal Distribution: I
    • Discrete Distribution
    • Describes a countable number of states-of-the-world.
    • Continuous Distribution
    • Describes an infinite number of states-of-the-world.
  • 11. The Normal Distribution: II
    • Characteristics of the two-parameter normal distribution :
      • The distribution is completely characterized by its mean and variance.
      • The possible outcomes range from minus infinity to plus infinity.
      • The distribution is symmetric around the mean.
      • The larger the distance of an outcome from the mean, the lower the probability density assigned to that outcome.
  • 12. The Normal Distribution: III
    • Probability Density Function
    • Describes the outcomes of a continuous distribution:
    • 1. Rescales the area under the continuous distribution function in such a manner that it equals 1.
    • 2 Measures the relative probability of outcomes, instead of their absolute probabilities.
  • 13. The Normal Distribution: IV
    • Cumulative Normal Distribution Function
    • Represents the area below the probability density function from minus infinity to a specified value and thereby:
    • Represents the probability that an outcome takes a value that is smaller than, or equal, to that specified value.
  • 14. The Normal Distribution: V – Illustrations A Normal Probabilty Density Function and its corresponding Cumulative Normal Distribution Function:
  • 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. Normal Distribution
    • A large enough sample drawn from a normal distribution looks like a bell-shaped curve.
    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. Normal Distribution
    • 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.
  • 18. The Utility Function
    • Four properties:
    • 1. Utility is increasing
    • Marginal utility is always positive.
    • 2. Utility functions are concave
    • Utility functions curve to the return axis.
    • 3. Different investors have different utility functions
    • 4. Utility functions are subjective
  • 19. Expected Utility Criterion
    • Combines the three key elements of investment (probabilities set, probability distribution and utility function) into one decision rule :
    • ‘ An investor will select the portfolio that yields the highest possible expected value for his or her utility function .’
  • 20. Risk Premiums
    • The equity premium is the difference in the expected rate of return between stocks and treasury bills.
    • Equity premium puzzle: the size of historical equity premiums cannot be justified by the risk of stocks and the risk aversion of investors.