Levy and Post, Investments © Pearson Education Limited 2005

Slide 7.1

Investments

Chapter 7: Fundamentals of Portfolio Analysis

Levy and Post, Investments © Pearson Education Limited 2005

Slide 7.2

Investment Problems

Problems of constrained optimization under uncertainty

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Slide 7.3

Three Parts to this Problem

1. OptimizationConcept: utility function.

2. UncertaintyConcept: probability distribution.

3. ConstraintsConcept: portfolio possibilities set.

Levy and Post, Investments © Pearson Education Limited 2005

Slide 7.4

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.

Levy and Post, Investments © Pearson Education Limited 2005

Slide 7.5

Portfolio Possibilities: Set II

Example of a portfolio possibilities set for two restrictions:

1. All portfolio weightsadd up to one.

2. No short salesallowed.

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Slide 7.6

The Probability Distribution: I

• Objective ProbabilitiesProbabilities that are known with certainty (example: coin-flipping experiment).

• Subjective ProbabilitiesProbabilities that are uncertain and can only be estimated (example: future values of assets).

Levy and Post, Investments © Pearson Education Limited 2005

Slide 7.7

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.

Levy and Post, Investments © Pearson Education Limited 2005

Slide 7.8

Population Statistics: I

• Sample Statistics

Ex-post statistics; summarize historical returns.

• Population Statistics

Ex-ante statistics; summarize a future return distribution.

Levy and Post, Investments © Pearson Education Limited 2005

Slide 7.9

Population Statistics: II

1. Population mean.

2. Population variance.

3. Population covariance.

4. Population correlation (coefficient).

Levy and Post, Investments © Pearson Education Limited 2005

Slide 7.10

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.

Levy and Post, Investments © Pearson Education Limited 2005

Slide 7.11

The Normal Distribution: II

Characteristics of the two-parameter normal distribution:

1. The distribution is completely characterized by its mean and variance.

2. The possible outcomes range from minus infinity to plus infinity.

3. The distribution is symmetric around the mean.4. The larger the distance of an outcome from the

mean, the lower the probability density assigned to that outcome.

Levy and Post, Investments © Pearson Education Limited 2005

Slide 7.12

The Normal Distribution: III

Probability Density FunctionDescribes 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.

Levy and Post, Investments © Pearson Education Limited 2005

Slide 7.13

The Normal Distribution: IV

Cumulative Normal Distribution FunctionRepresents 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.

Levy and Post, Investments © Pearson Education Limited 2005

Slide 7.14

The Normal Distribution: V –Illustrations

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A Normal Probabilty Density Function and its corresponding Cumulative Normal Distribution Function:

Levy and Post, Investments © Pearson Education Limited 2005

Slide 7.15

Normal Distribution S&P 500 Return Frequencies

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Normal approximationMean = 12.8%Std. Dev. = 20.4%

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.

Levy and Post, Investments © Pearson Education Limited 2005

Slide 7.16

Normal Distribution

• A large enough sample drawn from a normal distribution looks like a bell-shaped curve.

Probability

Return onlarge company commonstocks

68%

95%

> 99%

– 3 – 47.9%

– 2 – 27.6%

– 1 – 7.3%

013.0%

+ 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.

Levy and Post, Investments © Pearson Education Limited 2005

Slide 7.17

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.

Levy and Post, Investments © Pearson Education Limited 2005

Slide 7.18

The Utility Function

Four properties:

1. Utility is increasingMarginal utility is always positive.

2. Utility functions are concaveUtility functions curve to the return axis.

3. Different investors have different utility functions

4. Utility functions are subjective

Levy and Post, Investments © Pearson Education Limited 2005

Slide 7.19

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.’

Levy and Post, Investments © Pearson Education Limited 2005

Slide 7.20

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.