NOTE: The figures and tables in this chapter are either the standard figures typically seen in portfolio theory or illustrate calculations and examples. As such, they can be referred to directly or instructors can substitute their own figures and examples without any loss of continuity.
Figure 7-1 illustrates a discrete and a continuous probability distribution.
Figure 7-2 illustrates the concept of risk reduction when returns are independent. Risk continues to decline as the number of observations increase.
Figures 7-3, 7-4 and 7-5 illustrate, respectively: the case of perfect positive correlation, the case of perfect negative correlation,
the case of partial positive correlations between the returns for two securities based on the average correlation for NYSE stocks of approximately +0.55.
Table 7-1 illustrates the calculation of standard deviation when probabilities are involved. Table 7-2 shows the expected standard deviation of annual portfolio returns for various numbers of stocks in a portfolio.
Table 7-3 illustrates the variance-covariance matrix involved in calculating the standard deviation of a portfolio of two securities and of four securities. The point illustrated is
that the number of covariances involved increases quickly as more securities are considered.
Box Inserts
Box 7-1 is an interesting discussion of risk, and how best to understand it. It was written by Peter Bernstein, a well-known investments professional.
ANSWERS TO END-OF-CHAPTER QUESTIONS
7-1. Historical returns are realized returns, such as those reported by Ibbotson Associates and
Wilson and Jones in Chapter 6 (Table 6-6).
Expected returns are ex ante returns--they are the most likely returns for the future,
although they may not actually be realized because of risk.
7-2. The expected return for one security is determined from a probability distribution consisting of the likely outcomes, and their associated probabilities, for the security. The expected return for a portfolio is calculated as a weighted average of the individual securities’ expected returns. The weights used are the percentages of total investable funds invested in each security.
7-3. The basis of portfolio theory is that the whole is not equal to the sum of its parts, at least with respect to risk. Portfolio risk, as measured by the standard deviation, is not equal to the weighted sum of the individual security standard deviations. The reason, of course, is that the covariances must be accounted for.
7-4. In the Markowitz model, three factors determine portfolio risk: individual variances, the covariances between securities, and the weights (percentage of investable funds) given to each security.
7-5. The Markowitz approach is built around return and risk. The return is, in effect, the mean of the probability distributions, and variance is a proxy for risk. Efficient portfolios, a key concept, are defined on the basis of return and risk--that is, mean and variance.
7-6. A stock with a large risk (standard deviation) could be desirable if it has high negative correlation with other stocks. This will lead to large negative covariances which help to reduce the portfolio risk.
7-7. The correlation coefficient is a relative measure of risk ranging from -1 to +1. The covariance is an absolute measure of risk. Since COVAB = ρAB σA σB,
COVAB ρAB = ───── σA σB
7-8. Markowitz was the first to formally develop the concept of portfolio diversification. He showed quantitatively why, and how, portfolio diversification works to reduce the risk of a portfolio to an investor. In effect, he showed that diversification involves the
7-9. The expected return for a portfolio of 500 securities is calculated exactly as the expected return for a portfolio of 2 securities--namely, as a weighted average of the individual security returns. With 500 securities, the weights for each of the securities would be very small.
7-10. Each security in a portfolio, in terms of dollar amounts invested, is a percentage of the
total dollar amount invested in the portfolio. This percentage is a weight, and the general assumption is that these weights sum to 1.0, accounting for all of the portfolio funds.
7-11. The expected return for a portfolio must be between the lowest expected return for a
security in the portfolio and the highest expected return for a security in the portfolio. The exact position depends upon the weights of each of the securities.
7-12. Naive or random diversification refers to the act of randomly diversifying without regard
to relevant investment characteristics such as expected return and industry classification.
7-13. For 10 securities, there would be n (n-1) covariances, or 90. Divide by 2 to obtain unique
covariances; that is, [n(n-1)] / 2, or in this case, 45.
7-14. With 30 securities, there would be 900 terms in the variance-covariance matrix. Of these
900 terms, 30 would be variances, and n (n - 1), or 870, would be covariances. Of the 870 covariances, 435 are unique.
7-15. This statement is CORRECT. As the number of securities in a portfolio increases, the
importance of the covariance relationships increases while the importance of each individual security’s risk decreases.
7-16. The correlation coefficient is more useful in explaining diversification concepts because
it is a relative measure of association between security returns--we always know the boundaries of the association.
7-17. Investors should typically expect stock and bond returns to be positively related, as well
as bond and bill returns. Note, however, that correlations can change depending upon the time period used to measure the correlation. Stocks and gold have been negatively related, and stocks and real estate are typically negatively related.
7-18. The number of unique covariances needed for 500 securities using the Markowitz model
is:
n(n-1) 500(499) 249,500
────── = ─────── = ────── = 124,750 2 2 2
The total pieces of information needed: [n(n+3)]/2 = [500(503)]/2 = 251,500
CFA 7-19. c
7-20. No—their systematic risk differs, and they should priced in relation to their systematic
risk
7-21. c
7-22. d (note: for answer b, expected return is always a weighted average) 7-23. c (30 securities would have 30 x 30 = 900 terms)
7-24. a, b, and d 7-25. b
ANSWERS TO END-OF-CHAPTER PROBLEMS