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0:!!!0"77:! : Chapter 7
CHAPTER OVERVIEW
Chapter 7 is a complement to Chapter 6 in that it is a discussion of expected return and risk, whereas Chapter 6 focuses exclusively on realized return and risk. This organization allows the reader to focus on expected return and risk in Chapter 7 where portfolio theory, which is based on expected returns, is developed.
Chapter 7 covers basic portfolio theory, allowing students to be exposed to the most important, basic concepts of diversification, Markowitz portfolio theory, and capital market theory relatively early in the semester. They can then use these concepts throughout the remaining chapters. For example, it is very useful to know the implications of saying that stock A is very highly correlated with stock C, or with the market.
Chapter 7 serves as an introduction to portfolio theory, centering on the important building blocks of the Markowitz model. Students learn about such well known concepts as diversification, efficient portfolios, the risk of the portfolio, covariances, and so forth.
The first part of the chapter discusses the estimation of individual security return and risk, which provides the basis for considering portfolio return and risk in the next section. It begins with a discussion of uncertainty, and develops the concept of a probability distribution. The important calculation of expected value, or, as used here, expected return, is presented, as is the equation for standard deviation.
The next part of the chapter presents the Markowitz model along the standard dimensions of efficient portfolios, the inputs needed, and so forth. The discussion first examines expected portfolio return and risk. The portfolio risk discussion shows why portfolio risk is not a weighted average of individual security risks, which leads directly into a discussion of analyzing portfolio risk. The concept of risk reduction is illustrated for the cases of independent returns (the insurance principle), random diversification, and Markowitz diversification.
Correlation coefficients and covariances are explained in detail. This is a very standard discussion.
The calculation of portfolio risk is explained in two stages, starting with the two-security case and progressing to the n-security case. Sufficient detail is provided in order for students to really understand the concept of calculating portfolio risk using the Markowitz model, and why the problem of a large number of covariances is significant.
Efficient portfolios are explained and illustrated in brief fashion, which sets the stage for a more thorough discussion in Chapter 8.
CHAPTER OBJECTIVES
To explain the meaning and calculation of expected return and risk for individual securities using probabilities.
To fully explain the concepts of expected return and risk for portfolios based on correlations and covariances.
To present the basics of Markowitz portfolio theory, with an emphasis on portfolio risk.
MAJOR CHAPTER HEADINGS [Contents]
Dealing With Uncertainty
Using Probabilities
[random variable; point estimates]
Probability Distributions
[discrete vs. continuous; the normal distribution]
Calculating Expected Return for a Security
[expected value = expected return; formula]
Calculating Risk for a Security
[variance and standard deviation using probabilities; realized and expected standard deviations]
Introduction to Modern Portfolio Theory (MPT)
[Markowitzs contribution; concept of diversification]
Portfolio Return And Risk
Portfolio Expected Return
[portfolio weights; portfolio expected return is a weighted average of individual security returns; calculation and example]
Portfolio Risk
[portfolio risk is not a weighted average of individual security risks]
Analyzing Portfolio Risk
Risk ReductionThe Insurance Principle
[insurance principlerisk sources are independent]
Diversification
[random diversification; benefits of diversification kick in immediately]
The Components Of Portfolio Risk
The Correlation Coefficient
[description; graphs of perfect positive correlation, perfect negative correlation, 0.55 positive correlation]
Covariance
[description; relation with correlation coefficient]
Relating the Correlation Coefficient and the Covariance
[formula showing linkage between the two]
The Importance of Covariance
[the importance of covariance relationships increases as the number of securities increases]
Calculating Portfolio Risk
The Two-Security Case
[detailed example and explanation; the impact of the correlation coefficient; the impact of portfolio weights]
The n-Security Case
[formula; explanation; the importance of covariance]
Obtaining The Data
Simplifying the Markowitz Calculations
[need for estimates; the variancecovariance matrix illustrated; the problem with the Markowitz model]
POINTS TO NOTE ABOUT CHAPTER 7
Exhibits, Figures and Tables
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.
Table 7-1 illustrates the calculation of standard deviation when probabilities are involved.
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.
Table 7-2 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 the late 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 Markowitz model is based on the calculations for the expected return and risk of a portfolio. Another name associated with expected return is simply mean, and another name associated with the risk of a portfolio is the variance. Hence, the model is sometimes referred to as the mean-variance approach.
7-4. 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.
7-5. 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.
7-6. 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.
7-7. 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 relationships among securities.
7-8. With regard to risk, the whole is not equal to the sum of the parts. We cannot simply add up the individual (weighted) standard deviations of the securities in the portfolio, and obtain portfolio risk. If we could, the whole would be equal to the sum of the parts.
]
7-9. 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.
7-10. The correlation coefficient is a relative measure of risk ranging from -1 to +1.
The covariance is an absolute measure of risk.
Since COVAB = AB A B,
COVAB
AB = %%%%%
A B
7-11. 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.
7-12. 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.
7-13. 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.
7-14. 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 securitys risk decreases.
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.
NOTE: It is important to remember that these correlations can change depending upon the time periods examined, and the indexes used (for example, DJIA, Nasdaq, etc.).
7-16. The inputs for the Markowitz model, supplied by an investor, are expected returns and standard deviations for each security, and the correlation coefficient between each pair of securities.
7-17. A correlation of -1.0 does not guarantee a risk of zero for a portfolio of two securities. Optimal weights must be chosen for each security for this to occur.
7-18. Disagree. The variance of a portfolio is the weighted sum of the variances and covariances of the stocks in the portfolio.
CFA
7-19. A covariance matrix for five assets has 5 x 5 = 25 entries. Subtracting the five diagonal variance terms, we have 25-5 = 20 off-diagonal entries. Because the covariance matrix is symmetric, only 10 entries are unique (10 = 20/2). Hence, you must use 10 unique covariances in your five-stock portfolio variance calculation.
CFA
7-20. The expected return is 0.75E(return on stocks) + 0.25E (return on bonds)
=0.75(15) + 0.25(5)
=12.5 percent
The standard deviation is
= [w2stocks 2stocks + w2bonds 2bonds + 2wstockswbonds
Corr(Rstocks,Rbonds) stocks bonds]1/2
= [0.752 (225) + 0.252 (100) + 2(0.75)(0.25)(0.5)(15)(10)]1/2
= (126.5625 + 6.25 + 28.125)1/2
= (160.9375)1/2
= 12.69%
CFA
7-21. Define
Rp = return on the portfolio
R1 = return on the risk-free asset
R2 = return on the risky asset
w1 = fraction of the portfolio invested in the risk-free asset
w2 = fraction of the portfolio invested in the risky asset
Then the expected return on the portfolio is
E(Rp) = w1E(R1) + w2E(R2)
= 0.10(5%) + 0.9 (13%) = 0.5+11.7 +12.2%
To calculate standard deviation of return, we calculate variance of return and take the square root of variance:
2 (Rp) = w21 2 (R1) + w22 2 (R2) + 2w1w2Cov(R1,R2)
= 0.12(02) + 0.92 (232) + 2(0.1)(0.9)(0)
= 0.92 (232)
= 428.49
Thus the portfolio standard deviation of return is (Rp) = (428.49)1/2 = 20.7 percent.
7-22. No their systematic risk differs, and they should priced in relation to their systematic risk. This will be discussed in Chapter 9.
7-23. c
7-24. d (note: for answer b, expected return is always a weighted average)
7-25. c (30 securities would have 30 x 30 = 900 terms)
7-26. a, b, d
7-27. b
ANSWERS TO END-OF-CHAPTER PROBLEMS
7-1. (.10)(.20) = .020
(.20)(.16) = .032
(.40)(.12) = .048
(.15)(.05) = .0075
(.15)(-.05) = -.0075
.10 or 10% = expected return
To calculate the standard deviation for General Foods, use the formula
n
VARi = [PRi-ERi]2Pi
i=1
VARGF = [(.20-.10)2.10] + [(.16-.10)2.20] +
[(.12-.10)2.40] + [(.05-.10)2.15]
+ [(-.05-.10)2.15]
= .0056
Since i = (VAR)1/2
the for GF = (.0056)1/2 = .0750 = 7.5%
7-2. (a) (.25)(12) + (.25)(15) + (.25)(22) + (.25)(30) = 19.75%
(b) (.10)(12) + (.30)(15) + (.30)(22) + (.30)(30) = 21.30%
(c) (.20)(12) + (.20)(15) + (.30)(22) + (.30)(30) = 21%
7-3. (a) (1) {3 decimal places} (1/3)2(10)2 = 11.089
+ (1/3)2( 8)2 = 7.097
+ (1/3)2(20)2 = 44.360
+ (2)(1/3)(1/3)(.6)( 8)(10) = 10.645
+ (2)(1/3)(1/3)(.2)(20)(10) = 8.871
+ (2)(1/3)(1/3)(-1)(20)( 8) = -35.485
46.577
variance = 46.577; = 6.82%
(2) variance = (.5)2(8)2 + (.5)2(20)2 + 2(.5)(.5)
(-1)(20)(8)
= 16 + 100 - 80
= 36
= 6%
(3) variance = (.5)2(8)2 + (.5)2(16)2 +
2(.5)(.5)(.3)(8)(16)
= 16 + 64 + 19.2
= 99.2
= 9.96%
(4) variance = (.5)2(20)2 + (.5)2(16)2 +
2(.5)(.5)(8)(20)(16)
= 100 + 64 + 128
= 292
= 17.09%
(b) (1) variance = (.3)2(8)2 + (.7)2(20
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(c) In part (a), the minimum risk portfolio is 50% of the portfolio in B and 50% in C. But this may not be the highest return. For the combinations in (a) above, the return/risk combinations are:
Portfolio ER SD
(1) A, B, C 19% 6.82%
(2) B&C 21% 6.00%
(3) B&D 17% 9.96%
(4) C&D 26% 17.09%
Combination (BC) is clearly preferable over (ABC) and (BD), because there is a higher ER at lower risk. The choice between (BC) and (CD) would depend on the investor's risk-return tradeoff.
7-4. We will confirm the expected return for the third case shown in the table-- 0.6 weight on EG&G and 0.4 weight on GF. Each of the other expected returns in column 1 are calculated exactly the same way.
ERp = 0.6 (25) + 0.4 (23) = 24.2
7-5. We will confirm the portfolio variance for the third case, 0.6 weight on EG&G and 0.4 weight on GF. Each of the other portfolio variances in column 2 are calculated exactly the same way.
variancep = (.6)2(30)2 + (.4)2(25)2 + 2(.6)(.4)(112.5)
= 324 + 100 + 54
= 478
7-6. Knowing the variance for any combination of portfolio weights, the standard deviation is, of course, simply the square root. Thus, for the case of 0.6 and 0.4 weights, respectively, using the variance calculated in Problem 7-5, we confirm the standard deviation as
(478)1/2 = 21.86 or 21.9 as per column 3.
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7-8. (a) ER = (.6)(4) + (.4)(15) = 8.4%; SD = .4(20) = 8%
(b) ER = (-.5)(4) + (1.5)(15) = 20.5%; SD = 1.5(20) = 30%
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