Chapter 11 Optimal Portfolio Choice and the Capital

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Chapter 11 Optimal Portfolio Choice and the Capital Asset Pricing Model Copyright © 2011

Chapter 11 Optimal Portfolio Choice and the Capital Asset Pricing Model Copyright © 2011 Pearson Prentice Hall. All rights reserved.

Lesson Outline From Text 11. 1 The Expected Return of a Portfolio 11. 2

Lesson Outline From Text 11. 1 The Expected Return of a Portfolio 11. 2 The Volatility of a Two-Stock Portfolio 11. 3 The Volatility of a Large Portfolio 11. 4 Risk Versus Return: Choosing an Efficient Portfolio From Notes Theory of Choice 6 -2

11. 1 The Expected Return of a Portfolio • Portfolio Weights § The fraction

11. 1 The Expected Return of a Portfolio • Portfolio Weights § The fraction of the total investment in the portfolio held in each individual investment in the portfolio • The portfolio weights must add up to 1. 00 or 100%. 6 -3

11. 1 The Expected Return of a Portfolio (cont'd) • Then the return on

11. 1 The Expected Return of a Portfolio (cont'd) • Then the return on the portfolio, Rp , is the weighted average of the returns on the investments in the portfolio, where the weights correspond to portfolio weights. 6 -4

Chapter 11, problem 1 You are considering how to invest part of your retirement

Chapter 11, problem 1 You are considering how to invest part of your retirement savings. You have decided to put $200, 000 into three stocks: 50% of the money in Gold. Finger (currently $25/share), 25% of the money in Moosehead (currently $80/share), and the remainder in Venture Associates (currently $2/share). If Gold. Finger stock goes up to $30/share. Moosehead stock drops to $60 share, and Venture Associates stock rises to $3 per share, a. What is the new value of the portfolio b. What return did the portfolio earn c. If you don’t buy or sell shares after the price change, what are you new portfolio weights. 6 -5

11. 1 The Expected Return of a Portfolio (cont'd) • The expected return of

11. 1 The Expected Return of a Portfolio (cont'd) • The expected return of a portfolio is the weighted average of the expected returns of the investments within it. 6 -6

 • Problem § Assume your portfolio consists of $25, 000 of Intel stock

• Problem § Assume your portfolio consists of $25, 000 of Intel stock and $35, 000 of ATP Oil and Gas. § Your expected return is 18% for Intel and 25% for ATP Oil and Gas. § What is the expected return for your portfolio? 6 -7

11. 2 The Volatility of a Two-Stock Portfolio • Combining Risks 6 -8

11. 2 The Volatility of a Two-Stock Portfolio • Combining Risks 6 -8

Determining Covariance and Correlation (cont'd) • Covariance § The expected product of the deviations

Determining Covariance and Correlation (cont'd) • Covariance § The expected product of the deviations of two returns from their means § Covariance between Returns Ri and Rj § Estimate of the Covariance from Historical Data • If the covariance is positive, the two returns tend to move together. • If the covariance is negative, the two returns tend to move in opposite directions. 6 -9

Determining Covariance and Correlation (cont'd) • Correlation § A measure of the common risk

Determining Covariance and Correlation (cont'd) • Correlation § A measure of the common risk shared by stocks that does not depend on their volatility • The correlation between two stocks will always be between – 1 and +1. 6 -10

Figure 11. 1 Correlation 6 -11

Figure 11. 1 Correlation 6 -11

Table 11. 2 6 -12

Table 11. 2 6 -12

Question Using the data in the following table, estimate (a) the average return and

Question Using the data in the following table, estimate (a) the average return and volatility for each stock, (b) the covariance between the stocks, and (c) the correlation between these two stocks. Realized Returns Year Stock A Stock B 1998 -10% 21% 1999 20% 30% 2000 5% 7% 2001 -5% -3% 2002 2% -8% 2003 9% 25% Solution 6 -13

Table 11. 3 Historical Annual Volatilities and Correlations for Selected Stocks 6 -14

Table 11. 3 Historical Annual Volatilities and Correlations for Selected Stocks 6 -14

Computing a Portfolio’s Variance and Volatility • For a two security portfolio: § The

Computing a Portfolio’s Variance and Volatility • For a two security portfolio: § The Variance of a Two-Stock Portfolio 6 -15

Alternative Example 11. 6 • Problem § Assume your portfolio consists of $25, 000

Alternative Example 11. 6 • Problem § Assume your portfolio consists of $25, 000 of Intel stock and $35, 000 of ATP Oil and Gas. § Assume the annual standard deviation of returns is 43% for Intel and 68% for ATP Oil and Gas. § If the correlation between Intel and ATP is. 49, what is the standard deviation of your portfolio? 6 -16

11. 3 The Volatility of a Large Portfolio • The variance of a portfolio

11. 3 The Volatility of a Large Portfolio • The variance of a portfolio is equal to the weighted average covariance of each stock with the portfolio: § which reduces to: 6 -17

Diversification with an Equally Weighted Portfolio of Many Stocks • Equally Weighted Portfolio §

Diversification with an Equally Weighted Portfolio of Many Stocks • Equally Weighted Portfolio § A portfolio in which the same amount is invested in each stock • Variance of an Equally Weighted Portfolio of n Stocks 6 -18

Figure 11. 2 Volatility of an Equally Weighted Portfolio Versus the Number of Stocks

Figure 11. 2 Volatility of an Equally Weighted Portfolio Versus the Number of Stocks 6 -19

Chapter 11, problem 16 What is the volatility (standard deviation) of a very large

Chapter 11, problem 16 What is the volatility (standard deviation) of a very large portfolio of equally weighted stocks within an industry in which all the stocks have a volatility of 50% and all pairwise correlations are 40%? 6 -20

Diversification with General Portfolios • For a portfolio with arbitrary weights, the standard deviation

Diversification with General Portfolios • For a portfolio with arbitrary weights, the standard deviation is calculated as: § Volatility of a Portfolio with Arbitrary Weights Security I’s contribution to the volatility of portfolio Amount of i held Total risk of i Fraction of i’s risk that is common to P • Unless all of the stocks in a portfolio have a perfect positive correlation of +1 with one another, the risk of the portfolio will be lower than the weighted average volatility of the individual stocks: 6 -21

11. 4 Risk Versus Return: Choosing an Efficient Portfolio • Efficient Portfolios with Two

11. 4 Risk Versus Return: Choosing an Efficient Portfolio • Efficient Portfolios with Two Stocks § Recall from Chapter 10, in an efficient portfolio there is no way to reduce the volatility of the portfolio without lowering its expected return. § In an inefficient portfolio, it is possible to find another portfolio that is better in terms of both expected return and volatility. 6 -22

11. 4 Risk Versus Return: Choosing an Efficient Portfolio (cont'd) • Efficient Portfolios with

11. 4 Risk Versus Return: Choosing an Efficient Portfolio (cont'd) • Efficient Portfolios with Two Stocks § Consider a portfolio of Intel and Coca-Cola 6 -23

Figure 11. 3 Volatility Versus Expected Return for Portfolios of Intel and Coca-Cola Stock

Figure 11. 3 Volatility Versus Expected Return for Portfolios of Intel and Coca-Cola Stock 6 -24

Example 11. 9 6 -25

Example 11. 9 6 -25

Figure 11. 4 Effect on Volatility and Expected Return of Changing the Correlation between

Figure 11. 4 Effect on Volatility and Expected Return of Changing the Correlation between Intel and Coca-Cola Stock 6 -26

Short Sales • Long Position § A positive investment in a security • Short

Short Sales • Long Position § A positive investment in a security • Short Position § A negative investment in a security § In a short sale, you sell a stock that you do not own and then buy that stock back in the future. § Short selling is an advantageous strategy if you expect a stock price to decline in the future. 6 -27

Example 11. 10 6 -28

Example 11. 10 6 -28

Example 6 -29

Example 6 -29

Figure 11. 5 Portfolios of Intel and Coca-Cola Allowing for Short Sales 6 -30

Figure 11. 5 Portfolios of Intel and Coca-Cola Allowing for Short Sales 6 -30

Risk Versus Return: Many Stocks • Consider adding Bore Industries to the two stock

Risk Versus Return: Many Stocks • Consider adding Bore Industries to the two stock portfolio: • Although Bore has a lower return and the same volatility as Coca-Cola, it still may be beneficial to add Bore to the portfolio for the diversification benefits. 6 -31

Figure 11. 6 Expected Return and Volatility for Selected Portfolios of Intel, Coca-Cola, and

Figure 11. 6 Expected Return and Volatility for Selected Portfolios of Intel, Coca-Cola, and Bore Industries Stocks 6 -32

Figure 11. 7 The Volatility and Expected Return for All Portfolios of Intel, Coca-Cola,

Figure 11. 7 The Volatility and Expected Return for All Portfolios of Intel, Coca-Cola, and Bore Stock 6 -33

Risk Versus Return: Many Stocks (cont'd) • The efficient portfolios, those offering the highest

Risk Versus Return: Many Stocks (cont'd) • The efficient portfolios, those offering the highest possible expected return for a given level of volatility, are those on the northwest edge of the shaded region, which is called the efficient frontier for these three stocks. § In this case none of the stocks, on its own, is on the efficient frontier, so it would not be efficient to put all our money in a single stock. 6 -34

Figure 11. 8 Efficient Frontier with Ten Stocks Versus Three Stocks 6 -35

Figure 11. 8 Efficient Frontier with Ten Stocks Versus Three Stocks 6 -35

Problem Using the data from slide 11 -31 but assuming that Coca-Cola and Intel

Problem Using the data from slide 11 -31 but assuming that Coca-Cola and Intel stocks are perfectly negatively correlated (their correlation is -1). a. Calculate the portfolio weights that remove all risk (note that Intel’s volatility is twice that of Coca-Cola). b. What is the risk-free rate of interest in this economy? 6 -36

Problem Suppose Johnson & Johnson and the Walgreen Company have expected returns and volatilities

Problem Suppose Johnson & Johnson and the Walgreen Company have expected returns and volatilities shown below, with a correlation of 22%. E(R) SD(R) Johnson & Johnson 7% 16% Walgreen Company 10% 20% Calculate (a) the expected return and (b) the volatility (standard deviation) of a portfolio that consists of a long position of $3, 500 in Johnson & Johnson and a short position of $1, 000 in Walgreen’s. 6 -37