FIN 614 Risk and Return 2: Portfolios Professor Robert B.H. Hauswald Kogod School of Business, AU Risk and Return: From Securities to Portfolios From securities individual risk and return characteristics to portfolios: risk diversification to security markets: systematic/unsystematic risk, beta This lecture: generalize risk-return to portfolios WYSIWYG: portfolio returns WYSINWYG: interaction of risks and portfolio variance Two fundamental concepts covariance and correlation risk diversification 1/25/2011 Risk and Return: Portfolios Robert B.H. Hauswald 2
Return Statistics The history of capital market returns can be summarized by describing the average return: what is better?? ( R R R 1 + L+ T ) = T the standard deviation of those returns: measures what? 2 2 ( R1 R) + ( R2 R) + L( RT R) SD = VAR = T 1 the frequency distribution of the returns: related to? 2 1/25/2011 Risk and Return: Portfolios Robert B.H. Hauswald 3 Expected Returns Assessing prospects: given probabilities of outcomes find expected returns and risk (variance) of future events Expected return on (stock) investments: let S denote the total number of states ot he world (possible outcomes), r is the return in state s, for stock i, and p s the probability of state s the expected return is given by: [ ] E ri = ps ris Interpretation: probability weighted average s= 1 return return in each state ( scenario ) weighted by likelihood generalizes to all financial instruments Challenge: making definition operational difficult to get probabilities and returns by state by state note the difference to historical returns 1/25/2011 Risk and Return: Portfolios Robert B.H. Hauswald 4 S
Covariance Covariance: measures returns co-movement can also be expressed as the product of individual asset standard deviations and their correlation where: standard deviation = root of variance correlation coefficient What determines the sign of the covariance? interpretation? 1/25/2011 Risk and Return: Portfolios Robert B.H. Hauswald 5 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. 1/25/2011 Risk and Return: Portfolios Robert B.H. Hauswald 6
Expected Return, Variance, and Covariance Stock fund Bond Fund Rate of Squared Rate of Squared Scenario Return Deviation Return Deviation Recession -7% 3.24% 17% 1.00% Normal 12% 0.01% 7% 0.00% Boom 28% 2.89% -3% 1.00% Expected return 11.00% 7.00% Variance 0.0205 0.0067 Standard Deviation 14.3% 8.2% 1/25/2011 Risk and Return: Portfolios Robert B.H. Hauswald 7 Expected Return, Variance, and Covariance Stock fund Bond Fund Rate of Squared Rate of Squared Scenario Return Deviation Return Deviation Recession -7% 3.24% 17% 1.00% Normal 12% 0.01% 7% 0.00% Boom 28% 2.89% -3% 1.00% Expected return 11.00% 7.00% Variance 0.0205 0.0067 Standard Deviation 14.3% 8.2% E( r S E( r S ) = 1 ( 7%) + 3 ) = 11% 1 3 (12%) + 1 3 (28%) 1/25/2011 Risk and Return: Portfolios Robert B.H. Hauswald 8
Expected Return, Variance, and Covariance Stock fund Bond Fund Rate of Squared Rate of Squared Scenario Return Deviation Return Deviation Recession -7% 3.24% 17% 1.00% Normal 12% 0.01% 7% 0.00% Boom 28% 2.89% -3% 1.00% Expected return 11.00% 7.00% Variance 0.0205 0.0067 Standard Deviation 14.3% 8.2% E( r B E( r B ) = 1 (17%) + 3 ) = 7% 1 3 (7%) + 1 3 ( 3%) 1/25/2011 Risk and Return: Portfolios Robert B.H. Hauswald 9 Expected Return, Variance, and Covariance Stock fund Bond Fund Rate of Squared Rate of Squared Scenario Return Deviation Return Deviation Recession -7% 3.24% 17% 1.00% Normal 12% 0.01% 7% 0.00% Boom 28% 2.89% -3% 1.00% Expected return 11.00% 7.00% Variance 0.0205 0.0067 Standard Deviation 14.3% 8.2% (11% 7%) 2 = 3.24% 1/25/2011 Risk and Return: Portfolios Robert B.H. Hauswald 10
Expected Return, Variance, and Covariance Stock fund Bond Fund Rate of Squared Rate of Squared Scenario Return Deviation Return Deviation Recession -7% 3.24% 17% 1.00% Normal 12% 0.01% 7% 0.00% Boom 28% 2.89% -3% 1.00% Expected return 11.00% 7.00% Variance 0.0205 0.0067 Standard Deviation 14.3% 8.2% (11% 12%) 2 =.01% 1/25/2011 Risk and Return: Portfolios Robert B.H. Hauswald 11 Expected Return, Variance, and Covariance Stock fund Bond Fund Rate of Squared Rate of Squared Scenario Return Deviation Return Deviation Recession -7% 3.24% 17% 1.00% Normal 12% 0.01% 7% 0.00% Boom 28% 2.89% -3% 1.00% Expected return 11.00% 7.00% Variance 0.0205 0.0067 Standard Deviation 14.3% 8.2% (11% 28%) 2 = 2.89% 1/25/2011 Risk and Return: Portfolios Robert B.H. Hauswald 12
Expected Return, Variance, and Covariance Stock fund Bond Fund Rate of Squared Rate of Squared Scenario Return Deviation Return Deviation Recession -7% 3.24% 17% 1.00% Normal 12% 0.01% 7% 0.00% Boom 28% 2.89% -3% 1.00% Expected return 11.00% 7.00% Variance 0.0205 0.0067 Standard Deviation 14.3% 8.2% 1 2.05% = (3.24% + 0.01% + 2.89%) 3 1/25/2011 Risk and Return: Portfolios Robert B.H. Hauswald 13 Expected Return, Variance, and Covariance Stock fund Bond Fund Rate of Squared Rate of Squared Scenario Return Deviation Return Deviation Recession -7% 3.24% 17% 1.00% Normal 12% 0.01% 7% 0.00% Boom 28% 2.89% -3% 1.00% Expected return 11.00% 7.00% Variance 0.0205 0.0067 Standard Deviation 14.3% 8.2% 14.3% = 0.0205 1/25/2011 Risk and Return: Portfolios Robert B.H. Hauswald 14
Market Exposure Investment Market-Beta Correlation w/ Market TB 3M 0.04 23% TB 10Y 0.10 15% S&P 500 1.00 100% KO 1.02 65% PEP 0.86 65% SONY 1.47 41% (UAL) (1.65) (56%) 15 Comovement of Stock and Market 16
Risk and Diversification Risk consists of surprises: unanticipated changes realization of uncertain events Surprises come in two flavors and turn into risk: systematic or market risk: a surprise that affects a large number of assets to varying degrees common shock unsystematic (unique) risk: a surprise that affects at most a small number of assets idiosyncratic shock Examples: three levels economy, industry, firm market risk: idiosyncratic risk: industry risk: 1/25/2011 Risk and Return: Portfolios Robert B.H. Hauswald 17 The Diversification Principle Two approaches to egg preservation: Your parents: not all eggs in one basket Mark Twain: all eggs in one basket and watch that basket!! Diversification principle: portfolios reduce total risk variability of multiple assets held together (more, less, equal?) than variability of typical stock. diversifiable or unique risk: the portion of an asset s variability not present in a large group of assets (portfolio) held together undiversifiable risk: the level of variance that is present in collections of assets (portfolio riks) A typical single stock on NYSE: σ(annual) = 49.24% 100 or more NYSE stock portfolio: σ(annual) < 20% 1/25/2011 Risk and Return: Portfolios Robert B.H. Hauswald 18
Portfolio Risk (%) 0.44 0.27 Examples of Diversification The diversification success depends on the market: compare the portfolio risk of the 50 most liquid stocks in three different cases Swiss stocks U.S. stocks 0.12 International stocks 1 10 20 30 40 50 Number of Stocks 1/25/2011 Risk and Return: Portfolios Robert B.H. Hauswald 19 Spreading Your Investments does not always mean spreading your risks 1/25/2011 Risk and Return: Portfolios Robert B.H. Hauswald 20
Systematic and Unsystematic Return Components Total return = Expected return + Unexpected return: Unexpected return = (investor) surprises that come in two flavors systematic (market, economy) returns: unsystematic, firm-specific returns: Market (systematic) return and firm (idiosyncratic) return imply systematic and unsystematic return risk 1/25/2011 Risk and Return: Portfolios Robert B.H. Hauswald 21 Diversification and Unsystematic Risk Key insight: combining securities into portfolios, their unsystematic risks tend to cancel each other out what is left? only the variability that affects all securities to a greater or lesser degree (called what?) diversifiable risk is interchangeable with unsystematic risk large portfolios have little or no unsystematic risk Diversification cannot eliminate systematic risk Why not? For individual stocks? Total risk = systematic risk + unsystematic risk Total risk = undiversifiable risk + diversifiable risk What is the relative magnitude for portfolios? 1/25/2011 Risk and Return: Portfolios Robert B.H. Hauswald 22
Diversification Effects σ 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. NYSE: N > 100 1/25/2011 Risk and Return: Portfolios Robert B.H. Hauswald 23 Portfolios Definition: a portfolio is a collection of assets or securities (stocks and bonds) held by an investor folio (feuille): sheets of papers representing securities portare (porter): to carry around Portfolio weights: fraction of individual asset percentage of the portfolio's total value invested in each security, i.e., by the security s portfolio weights, α i. Portfolio s Expected Returns: weighted average sum of the product of the individual security's expected returns and their portfolio weights: N [ r ] = w E[ r ] p i i i= 1 1/25/2011 Risk and Return: Portfolios Robert B.H. Hauswald 24 E
Portfolio Variance For a 2 stock (asset) portfolio: σ For an N-stock (asset) portfolio: ( r ) 2 2 2 2 2 p = wi σ i + w jσ j + 2wi w jcov i,r j NB: unlike expected return, the variance of a portfolio is NOT the weighted sum of the individual variances Combining securities into portfolios can reduce the variability of returns: why? 1/25/2011 Risk and Return: Portfolios Robert B.H. Hauswald 25 As N gets large As N gets large, the average covariance of the securities with the portfolio dominates any individual security s measure of risk weights become insignificant when squared: Left with Cov(i,p): measure of how much risk any one security contributes to portfolio covariance: statistical measure of co-movement of variables Proportion of risk any one asset contributes to overall portfolio risk is: Cov( i, p) σ 2 1/25/2011 Risk and Return: Portfolios p Robert B.H. Hauswald 26
Portfolio Return and Risk Stock fund Bond Fund Rate of Squared Rate of Squared Scenario Return Deviation Return Deviation Recession -7% 3.24% 17% 1.00% Normal 12% 0.01% 7% 0.00% Boom 28% 2.89% -3% 1.00% Expected return 11.00% 7.00% Variance 0.0205 0.0067 Standard Deviation 14.3% 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. 1/25/2011 Risk and Return: Portfolios Robert B.H. Hauswald 27 Portfolio Return and Risk Rate of Return Scenario Stock fund Bond fund Portfolio squared deviation Recession -7% 17% 5.0% 0.160% Normal 12% 7% 9.5% 0.003% Boom 28% -3% 12.5% 0.123% Expected return 11.00% 7.00% 9.0% Variance 0.0205 0.0067 0.0010 Standard Deviation 14.31% 8.16% 3.08% The rate of return on the portfolio is a weighted average of the returns on the stocks and bonds in the portfolio: r = w r + w P 9.5% = 50% (12%) + 50% (7%) 1/25/2011 Risk and Return: Portfolios Robert B.H. Hauswald 28 B B S r S
Portfolio Return and Risk Rate of Return Scenario Stock fund Bond fund Portfolio squared deviation Recession -7% 17% 5.0% 0.160% Normal 12% 7% 9.5% 0.003% Boom 28% -3% 12.5% 0.123% Expected return 11.00% 7.00% 9.0% Variance 0.0205 0.0067 0.0010 Standard Deviation 14.31% 8.16% 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 r ) = w E( r ) + w E( r ) ( P B B S S 9 % = 50% (11%) + 50% (7%) 1/25/2011 Risk and Return: Portfolios Robert B.H. Hauswald 29 Portfolio Return and Risk Rate of Return Scenario Stock fund Bond fund Portfolio squared deviation Recession -7% 17% 5.0% 0.160% Normal 12% 7% 9.5% 0.003% Boom 28% -3% 12.5% 0.123% Expected return 11.00% 7.00% 9.0% Variance 0.0205 0.0067 0.0010 Standard Deviation 14.31% 8.16% 3.08% The variance of the rate of return on the two risky assets portfolio is σ (w σ σ + σ σ ρ 2 2 2 P = B B ) + (ws S ) 2(wB B )(ws S ) where ρ BS is the correlation coefficient between the returns on the stock and bond funds. 1/25/2011 Risk and Return: Portfolios Robert B.H. Hauswald 30 BS
Portfolio Return and Risk Rate of Return Scenario Stock fund Bond fund Portfolio squared deviation Recession -7% 17% 5.0% 0.160% Normal 12% 7% 9.5% 0.003% Boom 28% -3% 12.5% 0.123% Expected return 11.00% 7.00% 9.0% Variance 0.0205 0.0067 0.0010 Standard Deviation 14.31% 8.16% 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. 1/25/2011 Risk and Return: Portfolios Robert B.H. Hauswald 31 Efficient Set for Two Assets % in stocks Risk Return 0% 8.2% 7.0% 5% 7.0% 7.2% 10% 5.9% 7.4% 15% 4.8% 7.6% 20% 3.7% 7.8% 25% 2.6% 8.0% 30% 1.4% 8.2% 35% 0.4% 8.4% 40% 0.9% 8.6% 45% 2.0% 8.8% 50.00% 3.08% 9.00% 55% 4.2% 9.2% 60% 5.3% 9.4% 65% 6.4% 9.6% 70% 7.6% 9.8% 75% 8.7% 10.0% 80% 9.8% 10.2% 85% 10.9% 10.4% 90% 12.1% 10.6% 95% 13.2% 10.8% 100% 14.3% 11.0% P o rtf o lio R e t u rn Portfolo Risk and Return Combinations 7.0% 100% 6.0% bonds 5.0% 0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% 16.0% 1/25/2011 Risk and Return: Portfolios Robert B.H. Hauswald 32 12.0% 11.0% 10.0% 9.0% 8.0% Portfolio Risk (standard deviation) 100% stocks We can consider other portfolio weights besides 50% in stocks and 50% in bonds
Efficient Set for Two Assets Portfolo Risk and Return Combinations % in stocks Risk Return 0% 8.2% 7.0% 5% 7.0% 7.2% 12.0% 10% 5.9% 7.4% 11.0% 15% 4.8% 7.6% 10.0% 100% 20% 3.7% 7.8% 9.0% stocks 25% 2.6% 8.0% 8.0% 30% 1.4% 8.2% 7.0% 100% 35% 0.4% 8.4% 6.0% bonds 40% 0.9% 8.6% 5.0% 45% 2.0% 8.8% 0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% 16.0% 50% 3.1% 9.0% 55% 4.2% 9.2% Portfolio Risk (standard deviation) 60% 5.3% 9.4% 65% 6.4% 9.6% 70% 7.6% 9.8% We can consider other 75% 8.7% 10.0% 80% 9.8% 10.2% portfolio weights besides 85% 10.9% 10.4% 50% in stocks and 50% in 90% 12.1% 10.6% 95% 13.2% 10.8% bonds 1/25/2011 100% Risk 14.3% and Return: 11.0% Portfolios Robert B.H. Hauswald 33 P o rt f o lio R e t u rn Efficient Set for Two Assets P o rt f o lio R e t u rn Portfolo Risk and Return Combinations % in stocks Risk Return 0% 8.2% 7.0% 5% 7.0% 7.2% 12.0% 10% 5.9% 7.4% 11.0% 15% 4.8% 7.6% 10.0% 100% 20% 3.7% 7.8% 9.0% stocks 25% 2.6% 8.0% 8.0% 30% 1.4% 8.2% 7.0% 100% 35% 0.4% 8.4% 6.0% bonds 40% 0.9% 8.6% 5.0% 45% 2.0% 8.8% 0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% 16.0% 50% 3.1% 9.0% 55% 4.2% 9.2% Portfolio Risk (standard deviation) 60% 5.3% 9.4% 65% 6.4% 9.6% 70% 7.6% 9.8% 75% 8.7% 10.0% 80% 9.8% 10.2% 85% 10.9% 10.4% 90% 12.1% 10.6% 95% 13.2% 10.8% 1/25/2011 100% Risk 14.3% and Return: 11.0% Portfolios Robert B.H. Hauswald 34 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.
Two-Security Portfolios with Various Correlations return ρ = -1.0 100% stocks 100% bonds ρ = 0.2 ρ = 1.0 σ 1/25/2011 Risk and Return: Portfolios Robert B.H. Hauswald 35 Summary and Outlook Risk and return of portfolios: pitfalls unlike expected return, the variance of a portfolio is not the weighted sum of the individual security variances combining securities into portfolios can reduce the variability of returns - by reducing unsystematic (unique) risk. From individual assets to portfolios: risk aggregation unsystematic vs. systematic risk unsystematic risk diversified away: power of correlation From portfolios to markets via systematic and unsystematic risk to security market line Capital Asset Pricing Model: risk and return in equilibrium fundamental concept for pricing risk 1/25/2011 Risk and Return: Portfolios Robert B.H. Hauswald 36