7007 lecture 4

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FINM7007 Applied Corporate FinanceLecture 4

Capital Asset Pricing Model(CAPM)

BD Chapter 11.4-11.8, 12.1-12.3

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11.4 Risk Versus Return: Choosing an Efficient Portfolio

• Efficient Portfolios with Two Stocks– Identifying Inefficient Portfolios

• In an inefficient portfolio, it is possible to find another portfolio that is better in terms of both expected return and volatility.

– Identifying Efficient Portfolios• Recall from Chapter 10, in an efficient portfolio there is

no way to reduce the volatility of the portfolio without lowering its expected return.

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Risk Versus Return: Choosing an Efficient Portfolio (cont'd)

• Efficient Portfolios with Two Stocks– Consider a portfolio of Intel and Coca-Cola

Table 11.4 Expected Returns and Volatility for DifferentPortfolios of Two Stocks

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Figure 11.3 Volatility Versus Expected Return for Portfolios of Intel and Coca-Cola Stock

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Risk Versus Return: Choosing an Efficient Portfolio (cont'd)

• Efficient Portfolios with Two Stocks– Consider investing 100% in Coca-Cola stock. As

shown in on the previous slide, other portfolios—such as the portfolio with 20% in Intel stock and 80% in Coca-Cola stock—make the investor better off in two ways: It has a higher expected return, and it has lower volatility. As a result, investing solely in Coca-Cola stock is inefficient.

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The Effect of Correlation• Correlation has no effect on the expected return of a

portfolio. However, the volatility of the portfolio will differ depending on the correlation.

• The lower the correlation, the lower the volatility we can obtain. As the correlation decreases, the volatility of the portfolio falls.

• The curve showing the portfolios will bend to the left to a greater degree as shown on the next slide.

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Figure 11.4 Effect on Volatility and Expected Return of Changing the Correlation between Intel and Coca-Cola Stock

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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.

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Figure 11.5 Portfolios of Intel and Coca-Cola Allowing for Short Sales

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Efficient Portfolios with 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.

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Figure 11.6 Expected Return and Volatility for Selected Portfolios of Intel, Coca-Cola, and Bore Industries Stocks

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Figure 11.7 The Volatility and Expected Return for All Portfolios of Intel, Coca-Cola, and Bore Stock

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Risk Versus Return: Many Stocks

• 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.

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Figure 11.8 Efficient Frontier with Ten Stocks Versus Three Stocks

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Risk-Free Saving and Borrowing• Risk can also be reduced by investing a portion

of a portfolio in a risk-free investment, like T-Bills. However, doing so will likely reduce the expected return.

• On the other hand, an aggressive investor who is seeking high expected returns might decide to borrow money to invest even more in the stock market.

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Figure 11.9 The Risk–Return Combinations from Combining a Risk-Free Investment and a Risky Portfolio

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Borrowing and Buying Stocks on Margin

• Buying Stocks on Margin– Borrowing money to invest in a stock.

– A portfolio that consists of a short position in the risk-free investment is known as a levered portfolio. Margin investing is a risky investment strategy.

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Identifying the Tangent Portfolio

• To earn the highest possible expected return for any level of volatility we must find the portfolio that generates the steepest possible line when combined with the risk-free investment.

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Identifying the Tangent Portfolio (cont'd)

• Sharpe Ratio– Measures the ratio of reward-to-volatility provided

by a portfolio

• The portfolio with the highest Sharpe ratio is the portfolio where the line with the risk-free investment is tangent to the efficient frontier of risky investments. The portfolio that generates this tangent line is known as the tangent portfolio.

[ ] Portfolio Excess ReturnSharpe Ratio Portfolio Volatility ( )

P f

P

E R rSD R

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Figure 11.10 The Tangent or Efficient Portfolio

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• Combinations of the risk-free asset and the tangent portfolio provide the best risk and return tradeoff available to an investor. – This means that the tangent portfolio is efficient

and that all efficient portfolios are combinations of the risk-free investment and the tangent portfolio. Every investor should invest in the tangent portfolio independent of his or her taste for risk.

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• An investor’s preferences will determine only how much to invest in the tangent portfolio versus the risk-free investment. – Conservative investors will invest a small amount

in the tangent portfolio. – Aggressive investors will invest more in the

tangent portfolio.– Both types of investors will choose to hold the same portfolio of risky

assets, the tangent portfolio, which is the efficient portfolio.

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Expected Returns and the Efficient Portfolio

• Expected Return of a Security

– A portfolio is efficient if and only if the expected return of every available security equals its required return.

[ ] ( [ ] ) effi i f i eff fE R r r E R r

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11.7 The Capital Asset Pricing Model

• The Capital Asset Pricing Model (CAPM) allows us to identify the efficient portfolio of risky assets without having any knowledge of the expected return of each security.

• Instead, the CAPM uses the optimal choices investors make to identify the efficient portfolio as the market portfolio, the portfolio of all stocks and securities in the market.

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The CAPM Assumptions

• Three Main Assumptions– Assumption 1

• Investors can buy and sell all securities at competitive market prices (without incurring taxes or transactions costs) and can borrow and lend at the risk-free interest rate.

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The CAPM Assumptions (cont'd)


• Investors hold only efficient portfolios of traded securities—portfolios that yield the maximum expected return for a given level of volatility.

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The CAPM Assumptions (cont'd)


• Investors have homogeneous expectations regarding the volatilities, correlations, and expected returns of securities.

• Homogeneous Expectations– All investors have the same estimates concerning future

investments and returns.

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Supply, Demand, and the Efficiency of the Market Portfolio

• Given homogeneous expectations, all investors will demand the same efficient portfolio of risky securities.

• The combined portfolio of risky securities of all investors must equal the efficient portfolio.

• Thus, if all investors demand the efficient portfolio, and the supply of securities is the market portfolio, the demand for market portfolio must equal the supply of the market portfolio.

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Example

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Example (cont'd)

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Optimal Investing: The Capital Market Line

• When the CAPM assumptions hold, an optimal portfolio is a combination of the risk-free investment and the market portfolio. – When the tangent line goes through the market

portfolio, it is called the capital market line (CML).

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Optimal Investing: The Capital Market Line (cont'd)

• The expected return and volatility of a capital market line portfolio are: [ ] (1 ) [ ] ( [ ] ) xCML f Mkt f Mkt fE R x r xE R r x E R r

( ) ( )xCML MktSD R xSD R

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Figure 11.11 The Capital Market Line

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Determining the Risk Premium

• Market Risk and Beta– Given an efficient market portfolio, the expected

return of an investment is:

– The beta is defined as:Risk premium for security

[ ] ( [ ] ) Mkt

i i f i Mkt f

i

E R r r E R r

Volatility of that is common with the market

i( ) ( , ) ( , )

( ) ( )

i

Mkt i i Mkt i Mkti

Mkt Mkt

SD R Corr R R Cov R RSD R Var R

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Example

• Problem– Assume the risk-free return is 5% and the

market portfolio has an expected return of 12% and a standard deviation of 44%.

– ATP Oil and Gas has a standard deviation of 68% and a correlation with the market of 0.91.

– What is ATP’s beta with the market?– Under the CAPM assumptions, what is its

expected return?

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Example (cont’d)

• Solution

i( ) ( , ) (.68)(.91) 1.41

( ) .44

i i Mkt

Mkt

SD R Corr R RSD R

[ ] ( [ ] ) 5% 1.41(12% 5%) 14.87% Mkti f i Mkt fE R r E R r

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The Security Market Line

• There is a linear relationship between a stock’s beta and its expected return (See figure on next slide). The security market line (SML) is graphed as the line through the risk-free investment and the market.– According to the CAPM, if the expected return and

beta for individual securities are plotted, they should all fall along the SML.

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Figure 11.12 The Capital Market Line and the Security Market Line

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Figure 11.12 The Capital Market Line and the Security Market Line, panel (a)

(a) The CML depicts portfolios combining the risk-free investment and the efficient portfolio, and shows the highest expected return that we can attain for each level of volatility. According to the CAPM, the market portfolio is on the CML and all other stocks and portfolios contain diversifiable risk and lie to the right of the CML, as illustrated for Exxon Mobil (XOM).

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Figure 11.12 The Capital Market Line and the Security Market Line, panel (b)

(b) The SML shows the expected return for each security as a function of its beta with the market. According to the CAPM, the market portfolio is efficient, so all stocks and portfolios should lie on the SML.

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The Security Market Line (cont'd)

• Beta of a Portfolio– The beta of a portfolio is the weighted average

beta of the securities in the portfolio.

, ( , ) ( , ) ( ) ( ) ( )

i i MktiP Mkt i Mkt

P i i ii iMkt Mkt Mkt

Cov x R RCov R R Cov R Rx xVar R Var R Var R

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Example

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Example (cont’d)

Alpha (cont'd)

• When the market portfolio is efficient, all stocks are on the security market line and have an alpha of zero.

• Investors can improve the performance of their portfolios by buying stocks with positive alphas and by selling stocks with negative alphas.

Figure An Inefficient Market Portfolio

Figure Deviations from the Security Market Line

Summary of the Capital Asset Pricing Model

• The market portfolio is the efficient portfolio.

• The risk premium for any security is proportional to its beta with the market.

Determining Beta

• Estimating Beta from Historical Returns– Recall, beta is the expected percent

change in the excess return of the security for a 1% change in the excess return of the market portfolio.

– Consider Cisco Systems stock and how it changes with the market portfolio.

Monthly Returns for Cisco Stock and for the S&P 500, 1996–2005

Scatterplot of Monthly Excess Returns for Cisco Versus

the S&P 500, 1996–2005

Determining Beta (cont'd)

• Estimating Beta from Historical Returns– As the scatterplot on the previous slide shows,

Cisco tends to be up when the market is up, and vice versa.

– We can see that a 10% change in the market’s return corresponds to about a 20% change in Cisco’s return.

• Thus, Cisco’s return moves about two for one with the overall market, so Cisco’s beta is about 2.

Determining Beta (cont'd)

• Estimating Beta from Historical Returns– Beta corresponds to the slope of the best-fitting

line in the plot of the security’s excess returns versus the market excess return.

Using Linear Regression

• Linear Regression– The statistical technique that identifies the best-

fitting line through a set of points.

• αi is the intercept term of the regression. • Βi(RMkt – rf) represents the sensitivity of the stock to

market risk. • εi is the error term and represents the deviation from the

best-fitting line and is zero on average.

( ) ( ) i f i i Mkt f iR r R r

Using Linear Regression (cont'd)

• Linear Regression– Since E[εi] = 0:

• αi represents a risk-adjusted performance measure for the historical returns.

– If αi is positive, the stock has performed better than predicted by the CAPM.

– If αi is negative, the stock’s historical return is below the SML.

Distance above / below the SMLExpected return for from the SML

[ ] ( [ ] ) i f i Mkt f i

i

E R r E R r

Using Linear Regression (cont'd)

• Linear Regression– Given data for rf , Ri , and RMkt , statistical packages

for linear regression can estimate βi. • A regression for Cisco using the monthly returns for

1996–2004 indicates the estimated beta is 1.94.• The estimate of Cisco’s alpha from the regression is

1.2%.

Investor Information and Rational Expectations

• In the CAPM framework, investors should hold the market portfolio combined with risk-free investments– This investment advice does not depend on the

quality of an investor’s information.

Investor Information and Rational Expectations (cont'd)

• Rational Expectations– Investors may have different information regarding

expected returns, correlations, and volatilities, but they correctly interpret that information and the information contained in market prices and adjust their estimates of expected returns in a rational way.

Example

Example (cont'd)


• Regardless of how much information an investor has access to, he can guarantee himself an alpha of zero by holding the market portfolio.


• Because the average portfolio of all investors is the market portfolio, the average alpha of all investors is zero.

• If no investor earns a negative alpha, then no investor can earn a positive alpha, and the market portfolio must be efficient.


• The market portfolio can be inefficient only if a significant number of investors either:– Misinterpret information and believe they are

earning a positive alpha when they are actually earning a negative alpha, or

– Care about aspects of their portfolios other than expected return and volatility, and so are willing to hold inefficient portfolios of securities.

The CAPM in Practice

• Forecasting Beta– Time Horizon

• For stocks, common practice is to use at least two years of weekly return data or five years of monthly return data.

– The Market Proxy• In practice the S&P 500 is used as the market proxy.

Other proxies include the NYSE Composite Index and the Wilshire 5000.

The CAPM in Practice (cont'd)

• Forecasting Beta– Beta Extrapolation

• Many practitioners prefer to use average industry betas rather than individual stock betas.

• In addition, evidence suggests that betas tend to regress toward the average beta of 1.0 over time.


• Forecasting Beta– Outliers

• The beta estimates obtained from linear regression can be very sensitive to outliers.

Beta Estimation with and without Outliers for Genentech Using Monthly Returns for 2002–2004


• Forecasting Beta– Other Considerations

• Historical betas may not be a good measure if a firm were to change industries.

Evidence Regarding the CAPM

• Historically, researchers have found that expected returns were related to betas, as predicted by the CAPM, rather than to other measures of risk such as the security’s volatility. – However, they did find the empirically

estimated security market line is somewhat flatter than that predicted by the CAPM, as shown on the next slide.

Empirical SML Versus SML Predicted by CAPM (Black, Jensen,

and Scholes, 1972)

Evidence Regarding the CAPM (cont'd)

• More recently, researchers have found problems with the CAPM:– Betas are not observed.

• If betas change over time, evidence against the CAPM may be the result of mismeasuring betas.


• More recently, researchers have found problems with the CAPM:– Expected returns are not observed.

• Even if beta is a perfect measure of risk, average returns need not match expected returns. The realized average return need not match investors’ expectations.


• More recently, researchers have found problems with the CAPM:– The market proxy is not correct.

• Although the S&P 500 is a reasonable proxy for the U.S. stock market, investors hold many other assets. For example, the U.S. stock market represents only about 50% of world equity markets.

• Any failure of the CAPM may simply be the result of our failure to find a good measure of the market portfolio.

The Bottom Line on the CAPM

• The CAPM remains the predominant model used in practice to determine the equity cost of capital.– Although the CAPM is not perfect, it is unlikely

that a truly perfect model will be found in the foreseeable future.

• The imperfections of the CAPM may not be critical in the context of capital budgeting.

– Errors in estimating project cash flows are likely to be far more important than small discrepancies in the cost of capital.

7007 lecture 4

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