1. AN INTRODUCTION TO PORTFOLIO MANAGEMENT
Chapter 8
AN INTRODUCTION TO PORTFOLIO MANAGEMENT

2. Chapter 8 Questions
What do we mean be risk aversion, and what evidence indicates that investors are generally averse to risk?
What are the basic assumptions behind the Markowitz portfolio theory?
What do we mean by risk, and what are some of the measures of risk used in investments?
How does one compute the expected rate of return for an individual risky asset or a portfolio of assets?

3. Chapter 8 Questions
How does one compute the standard deviation of rates of return for an individual risky asset?
What do we mean by the covariance between rates of return, and how is it computed?
What is the relationship between covariance and correlation?

4. Chapter 8 Questions
What is the formula for the standard deviation for a portfolio of risky assets, and how does it differ from the standard deviation of an individual risky asset?
Given the formula for the standard deviation of a portfolio, why and how does one diversify a portfolio?
What happens to the standard deviation of a portfolio when we change the correlation between the assets in the portfolio?

5. Chapter 8 Questions
What is the risk-return efficient frontier of risky assets?
Is it reasonable for alternative investors to select different portfolios from the portfolios on the efficient frontier?
What determines which portfolio on the efficient frontier is selected by an individual investor?

6. Background Assumptions
As an investor you want to maximize the returns for a given level of risk.
Your portfolio includes all of your assets, not just financial assets
The relationship between the returns for assets in the portfolio is important.
A good portfolio is not simply a collection of individually good investments.

7. Risk Aversion
Portfolio theory assumes that investors are averse to risk
Given a choice between two assets with equal expected rates of return, risk averse investors will select the asset with the lower level of risk
It also means that a riskier investment has to offer a higher expected return or else nobody will buy it

8. Are investors risk averse?
The popularity of insurance of various types attests to risk aversion
Yield on bonds increase with risk classifications from AAA to AA to A…., indicating that investors require risk premiums as compensation
Experimental psychology also confirms that humans tend to be risk averse

9. Are investors always risk averse?
Risk preference may have to do with amount of money involved - risking only small amounts.
Trips to the casino might seem to refute risk aversion, but realize that gaming is best thought of as entertainment, not investing

10. Definition of Risk One definition: Uncertainty of future outcomes
Alternative definition: The probability of an adverse outcome
We will discuss several measures of risk that are used in developing portfolio theory

11. Markowitz Portfolio Theory
Derives the expected rate of return for a portfolio of assets and an expected risk measure
Markowitz demonstrated that the variance of the rate of return is a meaningful measure of portfolio risk under reasonable assumptions
The portfolio variance formula shows how to effectively diversify a portfolio

12. Markowitz Portfolio Theory
Assumptions
Investors consider each investment alternative as being presented by a probability distribution of expected returns over some holding period.
Investors minimize one-period expected utility, and their utility curves demonstrate diminishing marginal utility of wealth.
Investors estimate the risk of the portfolio on the basis of the variability of expected returns.

13. Markowitz Portfolio Theory
Assumptions
Investors base decisions solely on expected return and risk, so their utility curves are a function of expected return and the expected variance (or standard deviation) of returns only.
For a given risk level, investors prefer higher returns to lower returns. Similarly, for a given level of expected returns, investors prefer less risk to more risk.

14. Markowitz Portfolio Theory
Under these five assumptions, a single asset or portfolio of assets is efficient if no other asset or portfolio of assets offers higher expected return with the same (or lower) risk, or lower risk with the same (or higher) expected return.

15. Alternative Measures of Risk
Variance or standard deviation of expected return (Main focus)
Based on deviations from the mean return
Larger values indicate greater risk
Other measures
Range of returns
Returns below expectations
Semivariance – measures deviations only below the mean

16. Expected Rates of Return
Individual risky asset (Chapter 2)
Weighted average of all possible returns
Probabilities serve as the weights
Portfolio
Weighted average of expected returns (Ri) for the individual investments in the portfolio
Percentages invested in each asset (wi) serve as the weights
E(Rport) = S wi Ri

17. Variance & Standard Deviation of Returns
Individual Investment (Chapter 2)
Standard deviation is the positive square root of the variance
Both measures are based on deviations of each possible return (Ri) from the expected return (E(R))
Variance:
s2 = SPi(Ri-E(R))2

18. Variance & Standard Deviation of Returns
Before calculating the portfolio variance and standard deviation, several other measures need to be understood
Covariance
Measures the extent to which two variables move together
For two assets, i and j, the covariance of rates of return is defined as:
Covij = E{[Ri,t - E(Ri)][Rj,t - E(Rj)]}

19. Variance & Standard Deviation of Returns
Correlation coefficient
Values of the correlation coefficient (r) go from -1 to +1
Standardized measure of the linear relationship between two variables
rij = Covij/(sisj)
Covij= covariance of returns for securities i and j
si= standard deviation of returns for security i
sj= standard deviation of returns for security j

20. Portfolio Standard Deviation Formula

21. Portfolio Standard Deviation Calculation
The portfolio standard deviation is a function of:
The variances of the individual assets that make up the portfolio
The covariances between all of the assets in the portfolio
The larger the portfolio, the more the impact of covariance and the lower the impact of the individual security variance

22. Implications for Portfolio Formation
Assets differ in terms of expected rates of return, standard deviations, and correlations with one another
While portfolios give average returns, they give lower risk
Diversification works!
Even for assets that are positively correlated, the portfolio standard deviation tends to fall as assets are added to the portfolio

23. Implications for Portfolio Formation
Combining assets together with low correlations reduces portfolio risk more
The lower the correlation, the lower the portfolio standard deviation
Negative correlation reduces portfolio risk greatly
Combining two assets with perfect negative correlation reduces the portfolio standard deviation to nearly zero

24. Estimation Issues
Results of portfolio analysis depend on accurate statistical inputs
Estimates of
Expected returns
Standard deviations
Correlation coefficients
With 100 assets, 4,950 correlation estimates
Estimation risk refers to potential errors

25. Estimation Issues
With assumption that stock returns can be described by a single market model, the number of correlations required reduces to the number of assets
Single index market model:
bi = the slope coefficient that relates the returns for security i to the returns for the aggregate stock market
Rm = the returns for the aggregate stock market

26. The Efficient Frontier
The efficient frontier represents that set of portfolios with the maximum rate of return for every given level of risk, or the minimum risk for every level of return
Frontier will be portfolios of investments rather than individual securities
Exceptions being the asset with the highest return and the asset with the lowest risk

27. Efficient Frontier and Alternative PortfoliosB
E(R) A C
Standard Deviation of Return

28. The Efficient Frontier and Portfolio Selection
Any portfolio that plots “inside” the efficient frontier (such as point C) is dominated by other portfolios
For example, Portfolio A gives the same expected return with lower risk, and Portfolio B gives greater expected return with the same risk
Would we expect all investors to choose the same efficient portfolio?
No, individual choices would depend on relative appetites return as opposed to risk

29. The Efficient Frontier and Investor Utility
An individual investor’s utility curve specifies the trade-offs she is willing to make between expected return and risk
Each utility curve represent equal utility; curves higher and to the left represent greater utility (more return with lower risk)
The interaction of the individual’s utility and the efficient frontier should jointly determine portfolio selection

30. The Efficient Frontier and Investor Utility
The optimal portfolio has the highest utility for a given investor
It lies at the point of tangency between the efficient frontier and the utility curve with the highest possible utility

31. Selecting an Optimal Risky Portfolio
U3’
U2’
U1’ Y U3 X U2 U1

32. Investor Differences and Portfolio Selection
A relatively more conservative investor would perhaps choose Portfolio X
On the efficient frontier and on the highest attainable utility curve
A relatively more aggressive investor would perhaps choose Portfolio Y