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CHAPTER 11 Optimal Portfolio Choice and the Capital Asset Pricing Model

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CHAPTER 11 Optimal Portfolio Choice and the Capital Asset Pricing Model CHAPTER 11 Optimal Portfolio Choice and the Capital Asset Pricing Model Chapter Synopsis 11.1 The Expected Return of a Portfolio The expected return on an n-asset portfolio is simply the weighted-average of the expected returns of the portfolio’s components: nn RxRxRPnniiPii=+++=⇒= 11 22 L xR ∑∑ xRERxER [] []. ii==11 where xi is the value of asset i divided by the portfolio’s total value. 11.2 The Volatility of a Two-Stock Portfolio The standard deviation of an n-asset portfolio is generally less than the weighted-average of the standard deviations of the portfolio’s components. When two or more stocks are combined in a portfolio, some of their risk will generally be eliminated through diversification. The amount of risk that will remain depends on the degree to which the stocks share systematic risk. Thus the risk of a portfolio depends on more than the risk and return of the component stocks, and the degree to which the stocks’ returns move together is important; their covariance or correlation must be considered: Covariance is the expected product of the deviation of each return from its mean, which can be measured from historical data as: 1 T Cov(RRij , ) =−− ∑( R it,, Rij)( R jt R) . T − 1t=1 Correlation is the covariance of the returns divided by the standard deviation of each return: ©2011 Pearson Education 122 Berk/DeMarzo • Corporate Finance, Second Edition Cov(RRij , ) Corr(RRij , ) = . SD(RRij )× SD( ) Correlation is generally easier to interpret because it always lies between –1 and 1. The closer the correlation is to 1, the more the returns tend to move together. Assets with zero correlation move independently; assets with –1.0 correlation, move in opposite directions. The variance of a two-asset portfolio can now be calculated as: 22 Var(RxRxRP ) =++ 1122Var( ) Var( ) 2 xxRR 1212 Cov( , ), or 22 =++ xRxR1122Var( ) Var( ) 2 xxRRRR 12121 Cor( , )SD( )SD( 2 ). As can be seen from the equation, only when the correlation between the two stocks is exactly 1.0 is the Var(P) equal to the weighted average of the stocks’ variances—otherwise it is less. The portfolio’s standard deviation equals Var(P ) and is often referred to as volatility. 11.3 The Volatility of a Large Portfolio The variance of an equally-weighted n-asset portfolio is: 11⎛⎞ Var(RAverageP ) = ( asset variance) + ⎜⎟1− ( Average covariance between assets) . nn⎝⎠ Thus, as the number of stocks, n, grows large, the variance of the portfolio is determined 1 primarily by the average covariance among the stocks (because 1– n →1 ) while the average 1 variance becomes unimportant (because n → 0 ). Therefore, the risk of a stock in a diversified portfolio depends on its contribution to the portfolio’s average covariance. 11.4 Risk Versus Return: Choosing an Efficient Portfolio The set of efficient portfolios, which offer investors the highest possible expected return for a given level of risk, is called the efficient frontier. Investors must choose among the efficient portfolios based on their own preferences for return versus risk. A positive investment in a security is referred to as a long position. It is also possible to invest a negative amount in a stock, called a short position, by engaging in a short sale, a transaction in which you sell a stock that you do not own and then buy that stock back in the future at hopefully a lower price. When investors can use short sales in their portfolios, the portfolio weights on those stocks are negative, and the set of possible portfolios is extended. 11.5 Risk-Free Saving and Borrowing Portfolios can be formed by combining borrowing and lending at the risk-free rate and by investing in a portfolio of risky assets. The expected return for this type of portfolio is the weighted average of the expected returns of the risk-free asset and the risky portfolio, or: ER[] =− (1) xr + xER[] = r + xER([]) − r xPPPfff ©2011 Pearson Education Berk/DeMarzo • Corporate Finance, Second Edition 123 Since the standard deviation of the risk-free investment is zero, the covariance between the risk-free investment and the portfolio is also zero, and the standard deviation for this type of portfolio equals simply xSD(RP ) since Var(rf) and Cov(rRfP , ) are zero: 22 SD[RxrxRxxrRRxPfPfPP ] =++−= (1 - ) Var( ) Var( ) 2(1 )( )Cov( , ) xSD( ) . On a graph, the risk–return combinations of the risk-free investment and a risky portfolio lie on a straight line connecting the two investments. As the fraction x invested in the risk portfolio increases from 0% to 100%, you move along the line from 100% in the risk-free investment, to 100% in the risky portfolio. If you increase x beyond 100%, you reach points beyond the risky portfolio in the graph. In this case, you are borrowing at the risk-free rate and buying stocks on margin. To earn the highest possible expected return for any level of volatility, you must find the portfolio that generates the steepest possible line when combined with the risk-free investment. The slope of the line through a given portfolio P is often referred to as the Sharpe ratio of the portfolio: Portfolio Excess Return ER[]− r Sharpe Ratio = = Pf. Portfolio Volatility SD(RP ) The portfolio with the highest Sharpe ratio is called the tangent portfolio. You can draw a straight line from the risk-free rate through the tangent portfolio that is tangent to the efficient frontier of portfolios. When the risk-free asset is included, all efficient portfolios are combinations of the risk-free investment and the tangent portfolio—no other portfolio that consists of only risky assets is efficient. Therefore, the optimal portfolio of risky investments no longer depends on how conservative or aggressive the investor is. All investors will choose to hold the same portfolio of risky assets, the tangent portfolio, combined with borrowing or lending at the risk-free rate. 11.6 The Efficient Portfolio and Required Returns The beta of an investment with a portfolio is defined as: P Cov(RRiP , ) SD( R i ) × SD( R P ) ×Corr(RRiP , ) SD(RRRiiP ) × Corr( , ) βi ≡= = . Var(RP ) SD(RRPP )× SD( ) SD(RP ) Beta measures the sensitivity of the investment’s return to fluctuations in the portfolio’s return. Buying shares of security i improves the performance of a portfolio if its expected return exceeds the required return: Required Return ≡=rr + β P ([ ERr ] − ). iiPff A portfolio is efficient when E[Ri] = ri for all securities and the following relation holds: ER[] =≡ r r + β efficient portfolio ([ ER ] − r). iif i efficient portfolio f If investors hold portfolio P, this equation yields the return that investors will require to add the asset to their portfolio—i.e. the investment’s cost of capital. Buying shares of a security improves the performance of a portfolio if its expected return exceeds the required return based on its beta with the portfolio. ©2011 Pearson Education 124 Berk/DeMarzo • Corporate Finance, Second Edition 11.7 The Capital Asset Pricing Model The Capital Asset Pricing Model (CAPM) makes three assumptions. Investors can trade all securities at competitive market prices without incurring taxes or transactions costs, and can borrow and lend at the same risk-free interest rate. Investors hold only efficient portfolios of traded securities—portfolios that yield the maximum expected return for a given level of volatility. Investors have homogeneous expectations regarding the volatilities, correlations, and expected returns of securities. It follows that each investor will identify the same efficient portfolio of risky securities, which is the portfolio that has the highest Sharpe ratio in the economy. Because every security is owned by someone, the holdings of all investors’ portfolios must equal the portfolio of all risky securities available in the market, which is known as the market portfolio. Thus, when the CAPM assumptions hold, choosing an optimal portfolio is relatively straightforward for individual investors—they just choose to hold the same portfolio of risky assets, the market portfolio, combined with borrowing or lending at the risk-free rate. The capital market line (CML) is the set of portfolios with the highest possible expected return for a given level of standard deviation, or volatility. Under the CAPM assumptions, on a graph in which expected return is on the y-axis and standard deviation is on the x-axis, the CML is the line through the risk-free security and the market portfolio. 11.8 Determining the Risk Premium When the CAPM holds, a security’s expected return is given by: Mkt ER[]iif==+ r rβiMkt([ ER ] − r f). Thus, there is a linear relation between a stock’s beta and its expected return. On a graph in which expected return is on the y-axis and beta is on the x-axis, the line that goes through the risk-free investment (with a beta of 0) and the market (with a beta of 1) is called the security market line (SML). The security market line applies to portfolios as well. For example, the market portfolio is on the SML and, according to the CAPM, other portfolios such as mutual funds must also be on the SML, or else they are mispriced. Therefore, the expected return of a portfolio should correspond to the portfolio’s beta, which equals the weighted average of the betas of the securities in the portfolio: Cov(RRiMkt , ) βPi== ∑∑xx iiβ . iiVar(RMkt ) While the CAPM is based on rather strong assumptions, financial economists find the intuition underlying the model compelling, so it is still the most commonly used to measure systematic risk. It is also the most popular method used by financial managers to estimate their firm’s equity cost of capital.
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