Fin432 1 Chapter 5

Chapter 5

Modern Portfolio Concepts

Asset Allocation between risky and risk-free assets

Asset Allocation with Two Risky Assets

Covariance and Correlation

Diversification and Efficient Portfolio

Beta and CAPM

Traditional vs. Modern Asset Allocation Approach

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Asset Allocation - 1 Asset Allocation: Portfolio choice among broad investment classes. Decision on asset allocation has been shown to account for over 90% of the total return on the portfolios. Therefore, we start by exploring the basic asset allocation choice: how much do we put into a risk-free asset and how much into a portfolio of risky assets. Example Suppose you can invest a proportion y of your money in a portfolio (P) of risky stocks, which has an expected return of E[rP] and standard deviation of Pσ . The remaining proportion (1-y) of your money goes into risk-free assets, which has an expected return of rF. What is the expected return and standard deviation of the complete portfolio (C)? The answer is:

E[rC] = yE[rP] + (1-y)rF

PC y σσ ×=

or, E[rC] – rF = y(E[rP] - rF)

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Asset Allocation - 2 Capital Allocation Line (CAL): Plot of risk-return combinations available by varying portfolio allocation between a risk-free asset and a risky asset. Usually plotted in mean-standard deviation space. Reward-to-variability ratio: ratio of risk premium to standard deviation.

P

FP rrESσ−

=)(

• S is the slope of the CAL.

• S shows extra return per extra risk.

• Same reward-to-variability ratio along the same CAL.

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Asset Allocation - 3 Example Suppose there are only two assets in the world: a risky asset Y with E(rY) = 12% and σY = 20%, and a risk-free asset with rF = 4%. Both investor A and B have $6,000 to invest. A puts $3,000 into Y and the rest into risk-free asset. B instead of investing in risk-free asset, he borrows $3,000 and invests all $9,000 into the risky assets. 1) What are the expected return and volatility of A and B’s portfolio? 2) What is the reward-to-variability ratio for A and B’s portfolio?

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Portfolio of Two Stocks - 1 Suppose that you can invest a proportion x of your money in stock A, which has an expected return of E[rA] and standard deviation of

Aσ . The remaining proportion (1-x) of your money goes into company B, which has an expected return of E[rB] and standard deviation of Bσ . Let the correlation coefficient between the returns of stocks A and B be given by ABρ . What is the expected return E[rP] and standard deviation Pσ of your portfolio? The answer is: E[rP] = xE[rA] + (1-x)E[rB]

BAABBAP σσx)ρ2x(12)x)σ((12)(xσσ −+−+= Yes, it does look really complicated.

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Covariance and Correlation - 1 Assuming the covariance between stock A and B is covA,B, and correlation is ρAB. Then we have:

BAAB σσ

ρ BA,cov=

How do we measure covariance and correlation using historical stock return data? EXCEL applications:

Covariance: Function COVAR(array 1, array 2) Correlation: Function CORREL(array 1, array 2)

We can also use “Tool/Data Analysis” and choose the covariance and correlation. Correlation

Always between -1 and +1 Perfectly positively correlated: Corr = +1 Perfectly negatively correlated: Corr = -1 Uncorrelated: Corr = 0

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Covariance and Correlation - 2 The following is the covariance and correlation matrix of five asset classes over 1926—2002. Covariance Matrix:

small stocks large stocks LT bonds intermediate T-Billssmall stocks 0.15241large stocks 0.06226 0.04168LT bonds -0.00061 0.00220 0.00669intermediate -0.00087 0.00139 0.00396 0.00392T-Bills -0.00178 -0.00004 0.00065 0.00083 0.00100 Correlation Matrix:

small stocks large stocks LT bonds intermediate T-Billssmall stocks 1.00large stocks 0.78 1.00LT bonds -0.02 0.13 1.00intermediate -0.04 0.11 0.77 1.00T-Bills -0.14 -0.01 0.25 0.42 1.00

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Covariance and Correlation - 3 The Correlation between Series M, N and P

Combining Negatively Correlated Assets to Diversify Risk

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Portfolio of Two Stocks - 2 Example: Stock A expected return E[rA] = 12%, standard deviation

Aσ = 10%. Stock B expected return is E[rB] = 15%, standard deviation Bσ = 11%. What is the expected return and standard deviation of equally weighted portfolio (x=0.5) of these stocks? The correlation coefficient is 0.62. The answer is: E[rP] = 0.5*0.12 + 0.5*0.15 = 0.135 or 13.5%.

0945.011.010.062.05.05.0220.11).50(2)10.0(0.5σP =∗∗∗∗∗+∗+∗=or 9.45%. Notice how much lower the risk of this portfolio is compared with either of the individual stocks. So, by combining stocks, we can reduce risk without having a negative effect on the expected return. This is one of the most important lessons in the whole of finance.

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Portfolio of Two Stocks - 3 We can now consider the portfolios that we can construct by combining stock A and stock B. Let x be the proportion of our wealth that we place in Stock A, (1-x) the proportion in Stock B

The first thing we should do is to plot these portfolios in mean-standard deviation space. The higher “up” we are on this chart the better (higher expected return). The further we are to the left on this chart the better (the lower the risk). Therefore, we are aiming towards the top left-hand corner.

x 1-x Mean SD0 1 15.00% 11.00%

0.1 0.9 14.70% 10.55%0.2 0.8 14.40% 10.16%0.3 0.7 14.10% 9.85%0.4 0.6 13.80% 9.61%0.5 0.5 13.50% 9.45%0.6 0.4 13.20% 9.39%0.7 0.3 12.90% 9.41%0.8 0.2 12.60% 9.52%0.9 0.1 12.30% 9.72%

1 0 12.00% 10.00%

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Graphing These Portfolios

10.50%

11.50%

12.50%

13.50%

14.50%

15.50%

9.00% 9.50% 10.00%

10.50%

11.00%

11.50%

Standard Deviation

Exp

ecte

d R

etur

n

Stock A

Stock B

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The Correlation - 1 The key thing determining the benefit of diversification is the correlation of the two stocks. The lower the correlation, the greater the benefit:

12.00%

14.00%

16.00%

18.00%

0.00% 2.00% 4.00% 6.00% 8.00% 10.00% 12.00% 14.00%

Standard deviation

Expe

cted

retu

rn

1 0.5 0 0.5 1- ρ −=

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The Correlation - 2 If correlation is +1, there is no diversification benefit.

BAP xx σσσ )1( −+= If correlation is -1, it is able to form a risk-free portfolio by combining the two risky assets.

BAP xx σσσ )1( −−= Let Pσ be zero, we can solve for X:

BA

Bxσσ

σ+

=

Going back to the example on page 9: If correlation between A and B is -1, describe how you can construct a risk-free portfolio: What is the weight on A and B? What is the risk-free rate?

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Combining multiple shares to make portfolios With multiple stocks, although the math is more complex, the idea is just the same. The aim is to gain the highest possible expected return for a given level of risk (standard deviation). This is driven by the expected return of each share, the standard deviations of those returns and the correlation coefficient between the various stock returns.

SEARS GM MCGRAW DELTA PEPSICO GE IBM

SEARS 1.000

GM 0.245 1.000

MCGRAW 0.282 0.325 1.000

DELTA 0.246 0.176 0.343 1.000

PEPSICO 0.263 0.215 0.425 0.360 1.000

GE 0.143 0.258 0.537 0.328 0.501 1.000

IBM 0.134 0.225 0.021 0.096 0.149 0.185 1.000

Company Mean: SDSears 1.36% 8.59%GM 1.76% 8.11%

Delta 0.67% 7.74%Pepsico 1.21% 7.16%

GE 2.43% 5.93%IBM 2.10% 8.99%

McGraw 1.81% 5.82%

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The efficient frontier This graph gives the Markowitz “efficient frontier” using the seven stocks monthly returns from January 1992 till January 2000. The advantages of diversification are clear. Notice that, when dealing with multiple stocks, none of the individual shares lies on the frontier. That is, portfolios (almost) always dominate individual stock holdings.

IBM GE

PEPSICO

DELTA

MCGRAW GM

SEARS

0%

1%

2%

3%

4%

5%

0% 2% 4% 6% 8% 10% 12% 14% 16% 18% 20%

Standard Deviation

Exp

ecte

d R

etur

n

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Diversification - 1 Figure 5.7 The Feasible or Attainable Set and the Efficient Frontier

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Market vs. Firm Specific Risk We identify two predominant types of risk in the economy: Diversifiable (firm specific, unique) Risk

Examples: oil field exploration by a firm, replacement of the senior management, lawsuits against a firm, discovery of a new drug by a firm, takeover by another firm.

Undiversifiable (Economy Wide, Market, Systematic) Risk

Examples: business cycles, interest rate uncertainty, inflation uncertainty, a war

The TOTAL RISK of any stock (portfolio) can be divided into its two parts – unique risk and market risk. That is: Total risk = Nondiversifiable risk + Diversifiable risk

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Diversification - 2 Figure 5.8 Portfolio Risk and Diversification

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Diversification - 3 International Diversification

Offers more diverse investment alternatives than U.S.-only based investing

Foreign economic cycles may move independently from U.S. economic cycle

Foreign markets may not be as “efficient” as U.S. markets, allowing true gains from superior research

Study done between 1984 and 1994 suggests that portfolio 70% S&P 500 and 30% EAFE would reduce risk 5% and increase return 7% over a 100% S&P 500 portfolio

Advantages of International Diversification

Broader investment choices Potentially greater returns than in U.S. Reduction of overall portfolio risk

Disadvantages of International Diversification

Currency exchange risk Less convenient to invest than U.S. stocks More expensive to invest Riskier than investing in U.S.

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Systematic Risk and Beta

Because investors are trying to diversify as much as possible, they evaluate projects on the basis of how it helps them in the diversification process. For example, investors are prepared to discount countercyclical projects at a lower discount rate than procyclical projects with the same standard deviation of cash flows. This is because projects that do well when the rest of the economy is doing badly offer a very good diversification opportunity. For this reason, we need to distinguish between the total risk of a project’s future cash flows (measured as a standard deviation) and the systematic (or undiversifiable) risk – which is the risk that investors consider when determining the appropriate discount rate. The unit of measurement of systematic risk is the “beta”.

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The Beta - 1 If we use βi to be the beta of asset i, ri and rM respectively to be the returns to asset i and the market portfolio then:

)(

)(),(

)(

),(

MrirMrir

MrVarMrirCov

i σ

σρβ ==

Notice that this depends both on the total risk of the asset i (the standard deviation) and how correlated this is with the market portfolio. Dividing by the risk of the market portfolio “normalizes” this measure so that the beta of the market portfolio = 1 by definition.

For example, if returns to the market portfolio have standard deviation of 20% per year, returns to the stock of company A have a correlation coefficient of 0.5 with the returns to the market and standard deviation of 35%, then

875.02.0

35.0*5.0 ==Aβ

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The Beta - 2 Beta: A measure of nondiversifiable risk

Indicates how the price of a security responds to market forces Compares historical return of an investment to the market return (the S&P 500 Index) The beta for the market is 1.00 Stocks may have positive or negative betas. Nearly all are positive. Stocks with betas greater than 1.00 are more risky than the overall market. Stocks with betas less than 1.00 are less risky than the overall market.

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The Beta - 3 The Capital Asset Pricing Model (or CAPM – pronounced “cap-em”) uses the beta to tell us the discount rate (here denoted by E[ri]) that we should use with each project.

][][ frMrEifrirE −+= β

Notice that this depends on three things 1) The rate of return offered by the Treasury bill. 2) The equity premium 3) The systematic risk (not total risk) of the project.

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The Beta - 4 Interpreting Beta

Higher stock betas should result in higher expected returns due to greater risk

If the market is expected to increase 10%, a stock with a beta of 1.50 is expected to increase 15%

If the market went down 8%, then a stock with a beta of 0.50 should only decrease by about 4%

Beta values for specific stocks can be obtained from Value Line reports or online websites such as yahoo.com

Stock Beta Stock Beta Amazon.com 1.60 Int’l Business

Machines 1.10

Anheuser Busch 0.60 Merrill Lynch & Co. 1.60 Bank of America Corp.

1.25 Microsoft 1.15

Dow Jones & Co. 1.00 Nike, Inc. 0.90 Disney 1.25 PepsiCo, Inc. 0.65 eBay 1.55 Qualcomm 1.20 ExxonMobil Corp. 0.80 Sempra Energy 0.85 Gap (The), Inc. 1.35 Wal-Mart Stores 1.00 General Motors Corp. 1.20 Xerox 1.45 Intel 1.35 Yahoo! Inc. 1.85

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The Beta - 5 Graphical Representations of Beta

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Security Market Line - 1 Security Market Line (SML) plots the relationship between expected return and beta.

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Security Market Line - 2 SML has the following features: • A straight line in the expected return-beta space. • Intercept of the line gives the risk-free rate; Slope gives the risk

premium of the market portfolio • SML is valid for both portfolios and individual assets. • “Fairly priced” assets plot exactly on the SML. • Underpriced stocks plot above the SML • Overpriced stocks plot below the SML. • The difference between the fair and actually expected return on a

stock is called the stock’s alpha.

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Security Market Line - 2 Example: Suppose the return on the market is expected to be 14%. A stock has a beta of 1.2, and the T-bill rate is 6%. The SML would predict an expected return on the stock of E(r) = 6% + 1.2 * (14%-6%) = 15.6%. If one believes that the stock will provide instead a return of 17%, it implies that: • The stock is underpriced.

• Its implied alpha is: 17% - 15.6% = 1.4%.

Alpha: α = estimated expected return – “fair return” calculated by CAPM

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Portfolio Beta Beta of a portfolio is simply the weighted average of betas of the stocks in the portfolio. Example: An equally-weighted portfolio consists of 4 stocks. The betas of the four stocks are 0.5, 1, 1.5, and 2 respectively. What is the beta of the portfolio? Recalculate the beta if the portfolio is value-weighted, with the market capitalization given in the following table:

Stock market cap beta W $400m 0.5 X $300m 1.0 Y $200m 1.5 Z $100m 2.0

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Constructing Portfolios Traditional Approaches

Emphasizes “balancing” the portfolio using a wide variety of stocks and/or bonds

Uses a broad range of industries to diversify the portfolio

Tends to focus on well-known companies • Perceived as less risky • Stocks are more more liquid and available • Familiarity provides higher “comfort” levels

for investors Modern Portfolio Approaches

Emphasizes statistical measures to develop a portfolio plan

Focus is on: • Expected returns • Standard deviation of returns • Correlation between returns Combines securities that have negative (or low-positive) correlations between each other’s rates of return

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Approaches to Asset Allocation

Fixed-Weightings Approach: asset allocation plan in which a fixed percentage of the portfolio is allocated to each asset category Flexible-Weightings Approach: asset allocation plan in which weights for each asset category are adjusted periodically based on market analysis Tactical Approach: asset allocation plan that uses stock-index futures and bond futures to change a portfolio’s asset allocation based on market behavior

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Readings and Homework

Readings: Chapter 5

Exercises: Chapter 5 (will NOT be collected)

Text Website: Self-assessment quiz

End-of-Chapter CFA questions (Page 218)

Problems: P5.3, 5.5, 5.13, 5.17, 5.23

Homework (will be collected and graded, due date to be

announced)

Chapter 5: Excel with Spreadsheet (Page 230)