Chapter 6 Why Diversification Is a Good Idea 1 Lessons from Evans and Archer Introduction Methodology Results Implications Words of caution 2 Introduction Evans and Archer’s 1968 Journal of Finance article • Very consequential research regarding portfolio construction • Shows how naïve diversification reduces the dispersion of returns in a stock portfolio – Naïve diversification refers to the selection of portfolio components randomly 3 Methodology Used computer simulations: • Measured the average variance of portfolios of different sizes, up to portfolios with dozens of components • Purpose was to investigate the effects of portfolio size on portfolio risk when securities are randomly selected 4 Results Definitions General results Strength in numbers Biggest benefits come first Superfluous diversification 5 Definitions Systematic risk is the risk that remains after no further diversification benefits can be achieved Unsystematic risk is the part of total risk that is unrelated to overall market movements and can be diversified • Research indicates up to 75 percent of total risk is diversifiable 6 Definitions (cont’d) Investors are rewarded only for systematic risk • Rational investors should always diversify • Explains why beta (a measure of systematic risk) is important – Securities are priced on the basis of their beta coefficients 7 General Results Portfolio Variance Number of Securities 8 Strength in Numbers Portfolio variance (total risk) declines as the number of securities included in the portfolio increases • On average, a randomly selected ten-security portfolio will have less risk than a randomly selected three-security portfolio • Risk-averse investors should always diversify to eliminate as much risk as possible 9 Biggest Benefits Come First Increasing the number of portfolio components provides diminishing benefits as the number of components increases • Adding a security to a one-security portfolio provides substantial risk reduction • Adding a security to a twenty-security portfolio provides only modest additional benefits 10 Superfluous Diversification Superfluous diversification refers to the addition of unnecessary components to an already well-diversified portfolio • Deals with the diminishing marginal benefits of additional portfolio components • The benefits of additional diversification in large portfolio may be outweighed by the transaction costs 11 Implications Very effective diversification occurs when the investor owns only a small fraction of the total number of available securities • Institutional investors may not be able to avoid superfluous diversification due to the dollar size of their portfolios – Mutual funds are prohibited from holding more than 5 percent of a firm’s equity shares 12 Implications (cont’d) Owning all possible securities would require high commission costs It is difficult to follow every stock 13 Words of Caution Selecting securities at random usually gives good diversification, but not always Industry effects may prevent proper diversification Although naïve diversification reduces risk, it can also reduce return • Unlike Markowitz’s efficient diversification 14 Markowitz’s Contribution Harry Markowitz’s “Portfolio Selection” Journal of Finance article (1952) set the stage for modern portfolio theory • The first major publication indicating the important of security return correlation in the construction of stock portfolios • Markowitz showed that for a given level of expected return and for a given security universe, knowledge of the covariance and correlation matrices are required 15 Quadratic Programming The Markowitz algorithm is an application of quadratic programming • The objective function involves portfolio variance • Quadratic programming is very similar to linear programming 16 Portfolio Programming in a Nutshell Various portfolio combinations may result in a given return The investor wants to choose the portfolio combination that provides the least amount of variance 17 Markowitz Quadratic Programming Problem 18 Concept of Dominance Dominance is a situation in which investors universally prefer one alternative over another • All rational investors will clearly prefer one alternative 19 Concept of Dominance (cont’d) A portfolio dominates all others if: • For its level of expected return, there is no other portfolio with less risk • For its level of risk, there is no other portfolio with a higher expected return 20 Concept of Dominance (cont’d) Example (cont’d) In the previous example, the B/C combination dominates the A/C combination: 0.14 Expected Return 0.12 0.1 B/C combination dominates A/C 0.08 0.06 0.04 0.02 0 0 0.005 0.01 0.015 Risk 0.02 0.025 0.03 21 Terminology Security Universe Efficient frontier Capital market line and the market portfolio Security market line Expansion of the SML to four quadrants Corner portfolio 22 Security Universe The security universe is the collection of all possible investments • For some institutions, only certain investments may be eligible – E.g., the manager of a small cap stock mutual fund would not include large cap stocks 23 Efficient Frontier Construct a risk/return plot of all possible portfolios • Those portfolios that are not dominated constitute the efficient frontier 24 Efficient Frontier (cont’d) Expected Return No points plot above the line All portfolios on the line are efficient 100% investment in security with highest E(R) Points below the efficient frontier are dominated 100% investment in minimum variance portfolio Standard Deviation 25 Efficient Frontier (cont’d) When a risk-free investment is available, the shape of the efficient frontier changes • The expected return and variance of a risk-free rate/stock return combination are simply a weighted average of the two expected returns and variance The risk-free rate has a variance of zero 26 Efficient Frontier (cont’d) Expected Return C B Rf A Standard Deviation 27 Efficient Frontier (cont’d) The efficient frontier with a risk-free rate: • Extends from the risk-free rate to point B – The line is tangent to the risky securities efficient frontier • Follows the curve from point B to point C 28 Theorem For any constant Rf on the returns axis, the weights of the tangency portfolio B are: z n1 , zj j 1 zn , n z j j 1 where z is given by: z V 1[ R f ] In other words, the weights of tangency portfolio B are the normalized weights of portfolio z. 29 Example with Rf=0 and Rf=6.5% A 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 B C Variance-covariance matrix 0.400 0.030 0.020 0.030 0.200 0.001 0.020 0.001 0.300 0.000 -0.060 0.030 Rf= 0.00 z 0.1019 0.5657 0.1141 1.1052 D E 0.000 -0.060 0.030 0.100 Rf= B 0.0540 0.2998 0.0605 0.5857 0.065 z -0.0101 -0.0353 0.0047 0.1274 F G Mean returns 0.06 0.05 0.07 0.08 Mean minus constant -0.005 -0.015 0.005 0.015 B -0.1163 -0.4067 0.0544 1.4687 H I J <-- =F6-$E$11 <-- =F7-$E$11 <-- =F8-$E$11 <-- =F9-$E$11 Cells E13:E16 contain the array function =MMULT(MINVERSE(A6:D9),G6:G9). Cell F13 contains the function =E13/SUM($E$13:$E$16). This function is copied to cells F14:F16. 30 Finding Envelope Portfolios Graphically: 12% 10% Portfolio mean return B, the tangency portfolio given Rf 8% 6% Rf Zero beta portfolio 4% 2% 0% 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% Portfolio standard deviation 31 What is the zero-beta portfolio? The zero beta portfolio P0 is the portfolio determined by the intersection of the frontier with a horizontal line originating from the constant Rf selected. Property: whatever Rf we choose, we always have Cov(B,P0)=0 (Notice, however, that the location of B and P0 will depend on the value selected for Rf) 32 Note that the last proposition is true even if the risk-free rate (i.e. a riskless security) doesn’t exist in the economy. The way the tangency portfolio B was determined also remains valid even if there is no riskless rate in the economy. All one has to do is replace Rf by a chosen constant c. The mathematics of the last propositions will remain valid. 33 Fisher Black zero beta CAPM (1972) For a chosen constant c on the vertical axis of returns, the tangency portfolio B can be computed, and for ANY portfolio x we have a linear relationship if we regress the returns of x on the returns of B: E ( Rx ) c x [ E ( RB ) c] where x Cov( x, B) / B2 Moreover, c is the expected rate of return of a portfolio P0 whose covariance with B is zero. c E ( RP0 ) and Cov( P0 , B) 0 34 Fisher Black zero beta CAPM (Cont’d) The name “zero beta” stems from the fact that the covariance between P0 and B is zero, since a zero covariance implies that the beta of P0 with respect to B is zero too. If a riskless asset DOES exist in the economy, however, we can replace the constant c in Black’s zero beta CAPM by Rf and the portfolio B is the market portfolio. 35 Capital Market Line and the Market Portfolio The tangent line passing from the risk-free rate through point B is the capital market line (CML) • When the security universe includes all possible investments, point B is the market portfolio – It contains every risky assets in the proportion of its market value to the aggregate market value of all assets – It is the only risky assets risk-averse investors will hold 36 Capital Market Line and the Market Portfolio (cont’d) Implication for investors: • Regardless of the level of risk-aversion, all investors should hold only two securities: – The market portfolio – The risk-free rate • Conservative investors will choose a point near the lower left of the CML • Growth-oriented investors will stay near the market portfolio 37 Capital Market Line and the Market Portfolio (cont’d) Any risky portfolio that is partially invested in the risk-free asset is a lending portfolio Investors can achieve portfolio returns greater than the market portfolio by constructing a borrowing portfolio 38 Capital Market Line and the Market Portfolio (cont’d) Expected Return C B Rf A Standard Deviation 39 Security Market Line The graphical relationship between expected return and beta is the security market line (SML) • The slope of the SML is the market price of risk • The slope of the SML changes periodically as the risk-free rate and the market’s expected return change 40 Security Market Line (cont’d) Expected Return E(R) Market Portfolio Rf 1.0 Beta 41 THE SECURITY MARKET LINE--A SIMPLE EXAMPLE 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 Mean Beta AMR -0.3505 0.7083 0.7329 -0.2034 0.1663 -0.2659 0.0124 -0.0264 1.0642 0.1942 BS -0.1154 0.2472 0.3665 -0.4271 -0.0452 0.0158 0.4751 -0.2042 -0.1493 0.3680 GE -0.4246 0.3719 0.2550 -0.0490 -0.0573 0.0898 0.3350 -0.0275 0.6968 0.3110 HR -0.2107 0.2227 0.5815 -0.0938 0.2751 0.0793 -0.1894 -0.7427 -0.2615 1.8682 MO -0.0758 0.0213 0.1276 0.0712 0.1372 0.0215 0.2002 0.0913 0.2243 0.2066 UK 0.2331 0.3569 0.0781 -0.2721 -0.1346 0.2254 0.3657 0.0479 0.0456 0.2640 Market -0.2647 0.3720 0.2384 -0.0718 0.0656 0.1844 0.3242 -0.0491 0.2141 0.2251 0.2032 1.4820 0.0531 1.0840 0.1501 1.3107 0.1529 1.2991 0.1025 0.2622 0.1210 0.4939 0.1238 1.0000 =SLOPE(B4:B13,$H$4:$H$13) =COVAR(B4:B13,$H$4:$H$13)/VARP($H$4:$H$13) Regressing the means on the betas: Intercept 0.0766 <-- =INTERCEPT(B15:G15,B16:G16) Slope 0.0545 <-- =SLOPE(B15:G15,B16:G16) R-squared 0.2793 <-- =RSQ(B15:G15,B16:G16) 42 Notice that we obtained very poor results. The R-squared is only 27.93% ! However, the math of the CAPM is undoubtedly true. How then can CAPM fail in the real world? Possible explanations are that true asset returns distributions are unobservable, individuals have non-homogenous expectations, the market portfolio is unobservable, the riskless rate is ambiguous, markets are not friction-free. 43 Using “Artificial Market Portfolio” 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 ANNUAL RETURNS ON SIX ASSETS AND "MARKET" PORTFOLIO "Market" AMR BS GE HR MO UK portfolio -0.3505 -0.1154 -0.4246 -0.2107 -0.0758 0.2331 0.3560 0.7083 0.2472 0.3719 0.2227 0.0213 0.3569 0.3845 0.7329 0.3665 0.2550 0.5815 0.1276 0.0781 0.0189 -0.2034 -0.4271 -0.0490 -0.0938 0.0712 -0.2721 0.1558 0.1663 -0.0452 -0.0573 0.2751 0.1372 -0.1346 0.0872 -0.2659 0.0158 0.0898 0.0793 0.0215 0.2254 0.2070 0.0124 0.4751 0.3350 -0.1894 0.2002 0.3657 -0.1121 -0.0264 -0.2042 -0.0275 -0.7427 0.0913 0.0479 0.2340 1.0642 -0.1493 0.6968 -0.2615 0.2243 0.0456 0.5550 0.1942 0.3680 0.3110 1.8682 0.2066 0.2640 0.3918 Mean 0.2032 0.0531 Beta with respect to "market" 0.7938 -0.4647 0.1501 0.1529 0.1025 0.1210 0.2278 0.3484 0.3716 -0.0504 0.1043 1.0000 =SLOPE(E50:E59,$L$50:$L$59) Regressing the means on the betas Intercept 0.1086 <-- =INTERCEPT(E61:J61,E63:J63) Slope 0.1193 <-- =SLOPE(E61:J61,E63:J63) R-squared 1.0000 <-- =RSQ(E61:J61,E63:J63) 44 We obtained a perfect 100% R-squared this time ! The reason is that when portfolio returns are regressed on their betas with respect to an efficient portfolio, an exact linear relationship holds. 45 Expansion of the SML to Four Quadrants There are securities with negative betas and negative expected returns • A reason for purchasing these securities is their risk-reduction potential – E.g., buy car insurance without expecting an accident – E.g., buy fire insurance without expecting a fire 46 Security Market Line (cont’d) Expected Return Securities with Negative Expected Returns Beta 47 Diversification and Beta Beta measures systematic risk • Diversification does not mean to reduce beta • Investors differ in the extent to which they will take risk, so they choose securities with different betas – E.g., an aggressive investor could choose a portfolio with a beta of 2.0 – E.g., a conservative investor could choose a portfolio with a beta of 0.5 48 Capital Asset Pricing Model Introduction Systematic and unsystematic risk Fundamental risk/return relationship revisited 49 Introduction The Capital Asset Pricing Model (CAPM) is a theoretical description of the way in which the market prices investment assets • The CAPM is a positive theory 50 Systematic and Unsystematic Risk Unsystematic risk can be diversified and is irrelevant Systematic risk cannot be diversified and is relevant • Measured by beta – Beta determines the level of expected return on a security or portfolio (SML) 51 CAPM The more risk you carry, the greater the expected return: E ( Ri ) R f i E ( Rm ) R f where E ( Ri ) expected return on security i R f risk-free rate of interest i beta of Security i E ( Rm ) expected return on the market 52 CAPM (cont’d) The CAPM deals with expectations about the future Excess returns on a particular stock are directly related to: • The beta of the stock • The expected excess return on the market 53 CAPM (cont’d) CAPM assumptions: • Variance of return and mean return are all investors care about • Investors are price takers – They cannot influence the market individually • All investors have equal and costless access to information • There are no taxes or commission costs 54 CAPM (cont’d) CAPM assumptions (cont’d): • Investors look only one period ahead • Everyone is equally adept at analyzing securities and interpreting the news 55 SML and CAPM If you show the security market line with excess returns on the vertical axis, the equation of the SML is the CAPM • The intercept is zero • The slope of the line is beta 56 Note on the CAPM Assumptions Several assumptions are unrealistic: • People pay taxes and commissions • Many people look ahead more than one period • Not all investors forecast the same distribution Theory is useful to the extent that it helps us learn more about the way the world acts • Empirical testing shows that the CAPM works reasonably well 57 Stationarity of Beta Beta is not stationary • Evidence that weekly betas are less than monthly betas, especially for high-beta stocks • Evidence that the stationarity of beta increases as the estimation period increases The informed investment manager knows that betas change 58 Equity Risk Premium Equity risk premium refers to the difference in the average return between stocks and some measure of the risk-free rate • The equity risk premium in the CAPM is the excess expected return on the market • Some researchers are proposing that the size of the equity risk premium is shrinking 59 Using A Scatter Diagram to Measure Beta Correlation of returns Linear regression and beta Importance of logarithms Statistical significance 60 Correlation of Returns Much of the daily news is of a general economic nature and affects all securities • Stock prices often move as a group • Some stock routinely move more than the others regardless of whether the market advances or declines – Some stocks are more sensitive to changes in economic conditions 61 Linear Regression and Beta To obtain beta with a linear regression: • Plot a stock’s return against the market return • Use Excel to run a linear regression and obtain the coefficients – The coefficient for the market return is the beta statistic – The intercept is the trend in the security price returns that is inexplicable by finance theory 62 Importance of Logarithms Taking the logarithm of returns reduces the impact of outliers • Outliers distort the general relationship • Using logarithms will have more effect the more outliers there are 63 Statistical Significance Published betas are not always useful numbers • Individual securities have substantial unsystematic risk and will behave differently than beta predicts • Portfolio betas are more useful since some unsystematic risk is diversified away 64 Arbitrage Pricing Theory APT background The APT model Comparison of the CAPM and the APT 65 APT Background Arbitrage pricing theory (APT) states that a number of distinct factors determine the market return • Roll and Ross state that a security’s long-run return is a function of changes in: – Inflation – Industrial production – Risk premiums – The slope of the term structure of interest rates 66 APT Background (cont’d) Not all analysts are concerned with the same set of economic information • A single market measure such as beta does not capture all the information relevant to the price of a stock 67 The APT Model General representation of the APT model: RA E ( RA ) b1 A F1 b2 A F2 b3 A F3 b4 A F4 where RA actual return on Security A E ( RA ) expected return on Security A biA sensitivity of Security A to factor i Fi unanticipated change in factor i 68 APT R E ( R ) 1 F1 2 F2 3 F3 R E ( R ) 1[ R1 E ( R1 )] 2 [ R2 E ( R2 )] 3[ R3 E ( R3 )] R E ( R ) 1 E ( R1 ) 2 E ( R2 ) 3 E ( R3 ) 1R1 2 R2 3 R3 Fixed Random (Notice that the security index "A" has been ignored for clarity purposes) 69 Replicating the Randomness Let’s try to replicate the random component of security A by forming a portfolio with the following weights: 1 on R1 , 2 on R2 , 3 on R3 , and finally 1-1 2 3 on R f We get the following return (for this portfolio of factors): R (1-1 2 3 )R f 1 R1 2 R2 3 R3 Fixed Random 70 Key Point in Reasoning Since we were able to match the random components exactly, the only terms that differ at this point are the fixed components. But if one fixed component is larger than the other, arbitrage profits are possible by investing in the highest yielding security (either A or the portfolio of factors) while short-selling the other (being “long” in one and “short” in the other will assure an exact cancellation of the random terms). 71 Therefore the fixed components MUST BE THE SAME for security A and the portfolio of factors created, otherwise unlimited profits would be possible. So we have: E ( R) 1E ( R1 ) 2 E ( R2 ) 3 E ( R3 ) (1-1 2 3 )R f Rearranging terms yields: E ( R) R f 1[ E ( R1 ) R f ] 2 [ E ( R2 ) R f ] 2 [ E ( R3 ) R f ] 72 Comparison of the CAPM and the APT The CAPM’s market portfolio is difficult to construct: • Theoretically all assets should be included (real estate, gold, etc.) • Practically, a proxy like the S&P 500 index is used APT requires specification of the relevant macroeconomic factors 73 Comparison of the CAPM and the APT (cont’d) The CAPM and APT complement each other rather than compete • Both models predict that positive returns will result from factor sensitivities that move with the market and vice versa 74