1. Portfolio Theory Chapter 7
Charles P. Jones, Investments: Analysis and Management,
Twelfth Edition, John Wiley & Sons

2. Investment Decisions Involve uncertainty Focus on expected returns
Estimates of future returns needed to consider and manage risk
Investors often overly optimistic about expected returns
Goal is to reduce risk without affecting returns
Accomplished by building a portfolio
Diversification is key

3. Dealing With Uncertainty
Risk that an expected return will not be realized
Investors must think about return distributions
Probabilities weight outcomes
Assigned to each possible outcome to create a distribution
History provides guide but must be modified for expected future changes
Distributions can be discrete or continuous

4. Calculating Expected Return
Expected value
The weighted average of all possible return outcomes included in the probability distribution
Each outcome weighted by probability of occurrence
Referred to as expected return

5. Calculating Risk
Variance and standard deviation used to quantify and measure risk
Measure the spread or dispersion around the mean of the probability distribution
Variance of returns: σ² = (Ri - E(R))²pri
Standard deviation of returns:
σ =(σ²)1/2
σ is expected (ex ante)
Actual (ex post) σ useful but not perfect estimate of future

6. Modern Portfolio Theory
Provides framework for selection of portfolios based on expected return and risk
Used, to varying degrees, by financial managers
Shows benefits of diversification
The risk of a portfolio does not equal the sum of the risks of its individual securities
Must account for correlations among individual risks

7. Portfolio Expected Return
Weighted average of the individual security expected returns
Each portfolio asset has a weight, w, which represents the percent of the total portfolio value

8. Portfolio Risk
Portfolio risk not simply the sum or the weighted average of individual security risks
Emphasis on the risk of the entire portfolio and not on risk of individual securities in the portfolio
Diversification almost always lowers risk
Measured by the variance or standard deviation of the portfolio’s return

9. Risk Reduction in Portfolios
Assume all risk sources for a portfolio of securities are independent
This assumption is unrealistic when investing
Market risk affects all firms, cannot be diversified away
If risks independent, the larger the number of securities the smaller the exposure to any particular risk
“Insurance principle”
Only issue is how many securities to hold

10. Risk Reduction in Portfolios
Random (or naïve) diversification
Diversifying without looking at how security returns are related to each other
Marginal risk reduction gets smaller and smaller as more securities are added
Beneficial but not optimal
Risk reduction kicks in as soon as additional securities added
Research suggests it takes a large number of securities for significant risk reduction

11. Measuring Portfolio Risk
Needed to calculate risk of a portfolio:
Weighted individual security risks
Calculated by a weighted variance using the proportion of funds in each security
For security i: (wi × i)2
Weighted co-movements between returns
Return covariances are weighted using the proportion of funds in each security
For securities i, j: 2wiwj × ij

12. Correlation Coefficient
Statistical measure of relative association
ij = correlation coefficient between securities i and j
ij = +1.0 = perfect positive correlation
ij = -1.0 = perfect negative (inverse) correlation
ij = 0.0 = zero correlation

13. Correlation Coefficient
When does diversification pay?
With perfectly positive correlation, risk is a weighted average, therefore there is no risk reduction
With perfectly negative correlation, diversification assures the expected return
With zero correlation
If many securities, provides significant risk reduction
Cannot eliminate risk
Negative correlation or low positive correlation ideal but unlikely

14. Covariance Absolute, not relative, measure of association
Not limited to values between -1 and +1
Sign interpreted the same as correlation
Size depends on units of variables
Related to correlation coefficient

15. Calculating Portfolio Risk
Encompasses three factors
Variance (risk) of each security
Covariance between each pair of securities
Portfolio weights for each security
Goal: select weights to determine the minimum variance combination for a given level of expected return

16. Calculating Portfolio Risk
Generalizations
the smaller the positive correlation between securities, the better
Covariance calculations grow quickly
As the number of securities increases:
The importance of covariance relationships increases
The importance of each individual security’s risk decreases

17. Simplifying Markowitz Calculations
Markowitz full-covariance model
Requires a covariance between the returns of all securities in order to calculate portfolio variance
[n(n-1)]/2 set of covariances for n securities
Markowitz suggests using an index to which all securities are related to simplify

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