La Frontera Eficiente de Inversión Dra. Gabriela Siller Pagaza Departamento de Economía masiller@itesm.mx Factores relevantes • A los inversionistas les interesa conocer sobre los activos: – Rendimiento – Riesgo E R n R i pi i 1 Var R n 2 p R i i 2 i 1 • Además, invierten en una canasta de activos, para obtener beneficios por diversificación. Expected Return, Variance, and Covariance Rate of Return Scenario Probability Stock fund Bond fund Recession 33.3% -7% 17% Normal 33.3% 12% 7% Boom 33.3% 28% -3% Consider the following two risky asset world. There is a 1/3 chance of each state of the economy and the only assets are a stock fund and a bond fund. Expected Return, Variance, and Covariance Scenario Recession Normal Boom Expected return Variance Standard Deviation Stock fund Rate of Squared Return Deviation -7% 3.24% 12% 0.01% 28% 2.89% 11.00% 0.0205 14.3% Bond Fund Rate of Squared Return Deviation 17% 1.00% 7% 0.00% -3% 1.00% 7.00% 0.0067 8.2% Expected Return, Variance, and Covariance Scenario Recession Normal Boom Expected return Variance Standard Deviation Stock fund Rate of Squared Return Deviation -7% 3.24% 12% 0.01% 28% 2.89% 11.00% 0.0205 14.3% Bond Fund Rate of Squared Return Deviation 17% 1.00% 7% 0.00% -3% 1.00% 7.00% 0.0067 8.2% E (rS ) 1 (7%) 1 (12%) 1 (28%) 3 3 3 E (rS ) 11% Expected Return, Variance, and Covariance Scenario Recession Normal Boom Expected return Variance Standard Deviation Stock fund Rate of Squared Return Deviation -7% 3.24% 12% 0.01% 28% 2.89% 11.00% 0.0205 14.3% Bond Fund Rate of Squared Return Deviation 17% 1.00% 7% 0.00% -3% 1.00% 7.00% 0.0067 8.2% E (rB ) 1 (17%) 1 (7%) 1 (3%) 3 3 3 E (rB ) 7% Expected Return, Variance, and Covariance Scenario Recession Normal Boom Expected return Variance Standard Deviation Stock fund Rate of Squared Return Deviation -7% 3.24% 12% 0.01% 28% 2.89% 11.00% 0.0205 14.3% Bond Fund Rate of Squared Return Deviation 17% 1.00% 7% 0.00% -3% 1.00% 7.00% 0.0067 8.2% (11% 7%) 3.24% 2 Expected Return, Variance, and Covariance Scenario Recession Normal Boom Expected return Variance Standard Deviation Stock fund Rate of Squared Return Deviation -7% 3.24% 12% 0.01% 28% 2.89% 11.00% 0.0205 14.3% Bond Fund Rate of Squared Return Deviation 17% 1.00% 7% 0.00% -3% 1.00% 7.00% 0.0067 8.2% (11% 12%) .01% 2 Expected Return, Variance, and Covariance Scenario Recession Normal Boom Expected return Variance Standard Deviation Stock fund Rate of Squared Return Deviation -7% 3.24% 12% 0.01% 28% 2.89% 11.00% 0.0205 14.3% Bond Fund Rate of Squared Return Deviation 17% 1.00% 7% 0.00% -3% 1.00% 7.00% 0.0067 8.2% (11% 28%) 2.89% 2 Expected Return, Variance, and Covariance Scenario Recession Normal Boom Expected return Variance Standard Deviation Stock fund Rate of Squared Return Deviation -7% 3.24% 12% 0.01% 28% 2.89% 11.00% 0.0205 14.3% Bond Fund Rate of Squared Return Deviation 17% 1.00% 7% 0.00% -3% 1.00% 7.00% 0.0067 8.2% 1 .0205 (3.24% 0.01% 2.89%) 3 Expected Return, Variance, and Covariance Scenario Recession Normal Boom Expected return Variance Standard Deviation Stock fund Rate of Squared Return Deviation -7% 3.24% 12% 0.01% 28% 2.89% 11.00% 0.0205 14.3% Bond Fund Rate of Squared Return Deviation 17% 1.00% 7% 0.00% -3% 1.00% 7.00% 0.0067 8.2% 14.3% 0.0205 The Return and Risk for Portfolios Scenario Recession Normal Boom Expected return Variance Standard Deviation Stock fund Rate of Squared Return Deviation -7% 3.24% 12% 0.01% 28% 2.89% 11.00% 0.0205 14.3% Bond Fund Rate of Squared Return Deviation 17% 1.00% 7% 0.00% -3% 1.00% 7.00% 0.0067 8.2% Note that stocks have a higher expected return than bonds and higher risk. Let us turn now to the risk-return tradeoff of a portfolio that is 50% invested in bonds and 50% invested in stocks. The Return and Risk for Portfolios Rate of Return Stock fund Bond fund Portfolio -7% 17% 5.0% 12% 7% 9.5% 28% -3% 12.5% Scenario Recession Normal Boom Expected return Variance Standard Deviation 11.00% 0.0205 14.31% 7.00% 0.0067 8.16% squared deviation 0.160% 0.003% 0.123% 9.0% 0.0010 3.08% The expected rate of return on the portfolio is a weighted average of the expected returns on the securities in the portfolio. E (rP ) wB E (rB ) wS E (rS ) 9% 50% (11%) 50% (7%) The Return and Risk for Portfolios Scenario Recession Normal Boom Expected return Variance Standard Deviation Rate of Return Stock fund Bond fund Portfolio -7% 17% 5.0% 12% 7% 9.5% 28% -3% 12.5% 11.00% 0.0205 14.31% 7.00% 0.0067 8.16% squared deviation 0.160% 0.003% 0.123% 9.0% 0.0010 3.08% The variance of the rate of return on the two risky assets portfolio is σ P2 (wB σ B )2 (wS σ S )2 2(wB σ B )(wS σ S )ρBS σ P2 (wB σ B )2 (wS σ S )2 wB wS 2Cov where BS is the correlation coefficient between the returns on the stock and bond funds. The Return and Risk for Portfolios Scenario Recession Normal Boom Expected return Variance Standard Deviation Rate of Return Stock fund Bond fund Portfolio -7% 17% 5.0% 12% 7% 9.5% 28% -3% 12.5% 11.00% 0.0205 14.31% 7.00% 0.0067 8.16% squared deviation 0.160% 0.003% 0.123% 9.0% 0.0010 3.08% Observe the decrease in risk that diversification offers. An equally weighted portfolio (50% in stocks and 50% in bonds) has less risk than stocks or bonds held in isolation. % in stocks Risk Return 0% 5% 10% 15% 20% 25% 30% 35% 40% 45% 50.00% 55% 60% 65% 70% 75% 80% 85% 90% 95% 100% 8.2% 7.0% 5.9% 4.8% 3.7% 2.6% 1.4% 0.4% 0.9% 2.0% 3.08% 4.2% 5.3% 6.4% 7.6% 8.7% 9.8% 10.9% 12.1% 13.2% 14.3% 7.0% 7.2% 7.4% 7.6% 7.8% 8.0% 8.2% 8.4% 8.6% 8.8% 9.00% 9.2% 9.4% 9.6% 9.8% 10.0% 10.2% 10.4% 10.6% 10.8% 11.0% Portfolio Return The Efficient Set for Two Assets Portfolo Risk and Return Combinations 12.0% 11.0% 100% stocks 10.0% 9.0% 8.0% 7.0% 6.0% 100% bonds 5.0% 0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% 16.0% Portfolio Risk (standard deviation) We can consider other portfolio weights besides 50% in stocks and 50% in bonds … % in stocks Risk Return 0% 0% 5% 5% 10% 10% 15% 15% 20% 20% 25% 25% 30% 30% 35% 35% 40% 40% 45% 45% 50% 50% 55% 55% 60% 60% 65% 65% 70% 70% 75% 75% 80% 80% 85% 85% 90% 90% 95% 95% 100% 100% 8.2% 8.2% 7.0% 7.0% 5.9% 5.9% 4.8% 4.8% 3.7% 3.7% 2.6% 2.6% 1.4% 1.4% 0.4% 0.4% 0.9% 0.9% 2.0% 2.0% 3.1% 3.1% 4.2% 4.2% 5.3% 5.3% 6.4% 6.4% 7.6% 7.6% 8.7% 8.7% 9.8% 9.8% 10.9% 10.9% 12.1% 12.1% 13.2% 13.2% 14.3% 14.3% 7.0% 7.0% 7.2% 7.2% 7.4% 7.4% 7.6% 7.6% 7.8% 7.8% 8.0% 8.0% 8.2% 8.2% 8.4% 8.4% 8.6% 8.6% 8.8% 8.8% 9.0% 9.0% 9.2% 9.2% 9.4% 9.4% 9.6% 9.6% 9.8% 9.8% 10.0% 10.0% 10.2% 10.2% 10.4% 10.4% 10.6% 10.6% 10.8% 10.8% 11.0% 11.0% Portfolio Return The Efficient Set for Two Assets Portfolo Risk and Return Combinations 12.0% 11.0% 100% stocks 10.0% 9.0% 8.0% 7.0% 6.0% 100% bonds 5.0% 0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% 16.0% Portfolio Risk (standard deviation) We can consider other portfolio weights besides 50% in stocks and 50% in bonds … The Efficient Set for Two Assets % in stocks Risk Return 0% 5% 10% 15% 20% 25% 30% 35% 40% 45% 50% 55% 60% 65% 70% 75% 80% 85% 90% 95% 100% 8.2% 7.0% 5.9% 4.8% 3.7% 2.6% 1.4% 0.4% 0.9% 2.0% 3.1% 4.2% 5.3% 6.4% 7.6% 8.7% 9.8% 10.9% 12.1% 13.2% 14.3% 7.0% 7.2% 7.4% 7.6% 7.8% 8.0% 8.2% 8.4% 8.6% 8.8% 9.0% 9.2% 9.4% 9.6% 9.8% 10.0% 10.2% 10.4% 10.6% 10.8% 11.0% 100% activo A 100% Activo B Note that some portfolios are “better” than others. They have higher returns for the same level of risk or less. These compromise the efficient frontier. Minimum Variance Portfolio σ P2 (wB σ B )2 ( 1 wB σ S )2 wB 1 wB 2Cov σ 2 wB B2 2 wB S2 2 S2 2Cov 4 wB Cov 0 wB 2 P WB Cov BS 2 S 2 Cov BS 2 S 2 B Portfolio Risk/Return Two Securities: Correlation Effects • Relationship depends on correlation coefficient • -1.0 < < +1.0 • The smaller the correlation, the greater the risk reduction potential • If = +1.0, no risk reduction is possible Portfolio Risk as a Function of the Number of Stocks in the Portfolio In a large portfolio the variance terms are effectively diversified away, but the covariance terms are not. Diversifiable Risk; Nonsystematic Risk; Firm Specific Risk; Unique Risk Portfolio risk Nondiversifiable risk; Systematic Risk; Market Risk n Thus diversification can eliminate some, but not all of the risk of individual securities. return The Efficient Set for Many Securities Individual Assets P Consider a world with many risky assets; we can still identify the opportunity set of risk-return combinations of various portfolios. return The Efficient Set for Many Securities minimum variance portfolio Individual Assets P Given the opportunity set we can identify the minimum variance portfolio. return The Efficient Set for Many Securities minimum variance portfolio Individual Assets P The section of the opportunity set above the minimum variance portfolio is the efficient frontier. return Optimal Risky Portfolio with a Risk-Free Asset 100% stocks rf 100% bonds In addition to stocks and bonds, consider a world that also has risk-free securities like T-bills return Riskless Borrowing and Lending 100% stocks Balanced fund rf 100% bonds Now investors can allocate their money across the T-bills and a balanced mutual fund return Riskless Borrowing and Lending rf P With a risk-free asset available and the efficient frontier identified, we choose the capital allocation line with the steepest slope return Market Equilibrium M rf P With the capital allocation line identified, all investors choose a point along the line—some combination of the risk-free asset and the market portfolio M. In a world with homogeneous expectations, M is the same for all investors. return The Separation Property M rf P The Separation Property states that the market portfolio, M, is the same for all investors—they can separate their risk aversion from their choice of the market portfolio. return The Separation Property M rf P Investor risk aversion is revealed in their choice of where to stay along the capital allocation line—not in their choice of the line. return Market Equilibrium 100% stocks Balanced fund rf 100% bonds Just where the investor chooses along the Capital Asset Line depends on his risk tolerance. The big point though is that all investors have the same CML. return Market Equilibrium 100% stocks Optimal Risky Porfolio rf 100% bonds All investors have the same CML because they all have the same optimal risky portfolio given the risk-free rate. return The Separation Property 100% stocks Optimal Risky Porfolio rf 100% bonds The separation property implies that portfolio choice can be separated into two tasks: (1) determine the optimal risky portfolio, and (2) selecting a point on the CML. return Optimal Risky Portfolio with a Risk-Free Asset r 1 f 0 rf 100% stocks First Optimal Risky Portfolio Second Optimal Risky Portfolio 100% bonds By the way, the optimal risky portfolio depends on the risk-free rate as well as the risky assets. Definition of Risk When Investors Hold the Market Portfolio • Researchers have shown that the best measure of the risk of a security in a large portfolio is the beta (b)of the security. • Beta measures the responsiveness of a security to movements in the market portfolio. bi Cov( Ri , RM ) ( RM ) 2 Security Returns Estimating b with regression Slope = bi Return on market % Ri = a i + biRm + ei The Formula for Beta bi Cov( Ri , RM ) ( RM ) 2 Clearly, your estimate of beta will depend upon your choice of a proxy for the market portfolio. Relationship between Risk and Expected Return (CAPM) • Expected Return on the Market: R M RF Market Risk Premium • Expected return on an individual security: R i RF β i ( R M RF ) Market Risk Premium This applies to individual securities held within welldiversified portfolios. Expected Return on an Individual Security • This formula is called the Capital Asset Pricing Model (CAPM) Ri RF βi ( R M RF ) Expected return on a security = Risk-free rate + Beta of the security × Market risk premium • Assume bi = 0, then the expected return is RF. • Assume bi = 1, then R i R M Expected return Relationship Between Risk & Expected Return Ri RF βi ( R M RF ) RM RF 1.0 Ri RF βi ( R M RF ) b Expected return Relationship Between Risk & Expected Return 13.5% 3% 1.5 β i 1.5 RF 3% b R M 10% R i 3% 1.5 (10% 3%) 13.5%

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# La Frontera Eficiente de Inversión