Chapter 6 Efficient Diversification Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 6.1 Diversification and Portfolio Risk 6.2 Asset Allocation With Two Risky Assets 6-* = W1 + W2 W1 = Proportion of funds in Security 1 W2 = Proportion of funds in Security 2 = Expected return on Security 1 = Expected return on Security 2 Two-Security Portfolio: Return r1 E( ) rp r2 r1 r2 6-* E(rp) = W1r1 + W2r2 W1 = W2 = = = Two-Security Portfolio Return E(rp) = 0.6(9.28%) + 0.4(11.97%) = 10.36% Wi = % of total money invested in security i 0.6 0.4 9.28% 11.97% r1 r2 6-* Combinations of risky assets When we put stocks in a portfolio, p < Why? When Stock 1 has a return E[r1] it is likely that Stock 2 has a return E[r2] so that rp that contains stocks 1 and 2 remains close to What statistics measure the tendency for r1 to be above expected when r2 is below expected? Covariance and Correlation (Wii) E[rp] < > n = # securities in the portfolio Averaging principle 6-* Portfolio Variance and Standard Deviation Variance of a Two Stock Portfolio: 6-* Expost Covariance Calculations If when r1 > E[r1], r2 > E[r2], and when r1 < E[r1], r2 < E[r2], then COV will be _______. If when r1 > E[r1], r2 < E[r2], and when r1 < E[r1], r2 > E[r2], then COV will be _______. positive negative Which will “average away” more risk? 6-* Covariance and Correlation The problem with covariance Covariance does not tell us the intensity of the comovement of the stock returns, only the direction. We can standardize the covariance however and calculate the correlation coefficient which will tell us not only the direction but provides a scale to estimate the degree to which the stocks move together. 6-* Measuring the Correlation Coefficient Standardized covariance is called the _____________________ For Stock 1 and Stock 2 correlation coefficient or 6-* r and Diversification in a 2 Stock Portfolio is always in the range __________ inclusive. What does (1,2) = +1.0 imply? What does (1,2) = -1.0 imply? The two are perfectly positively correlated. Means? If (1,2) = +1, then (1,2) = W11 + W22 The two are perfectly negatively correlated. Means? If (1,2) = -1, then (1,2) = ±(W11 – W22) It is possible to choose W1 and W2 such that (1,2) = 0. -1.0 to +1.0 There are very large diversification benefits from combining 1 and 2. Are there any diversification benefits from combining 1 and 2? 6-* What does -1 < (1,2) < 1 imply? If -1 < (1,2) < 1 then sp2 = W12s12 + W22s22 + 2W1W2 Cov(r1r2) And since Cov(r1r2) = r1,2s1s2 sp2 = W12s12 + W22s22 + 2W1W2 r1,2s1s2 There are some diversification benefits from combining stocks 1 and 2 into a portfolio. 6-* r and Diversification in a 2 Stock Portfolio Typically r is greater than ____________________ r(1,2) = r(2,1) and the same is true for the COV The covariance between any stock such as Stock 1 and itself is simply the variance of Stock 1, r(1,1) = +1.0 by definition We have no measure for how three or more stocks move together. zero and less than 1.0 6-* r and Diversification in a 2 Stock Portfolio The Effects of Correlation & Covariance on Diversification Asset A Asset B Portfolio AB 6-* 6-* The Effects of Correlation & Covariance on Diversification Asset C Asset C Portfolio CD Naïve Diversification 6-* Most of the diversifiable risk eliminated at 25 or so stocks The power of diversification sp2 = W12s12 + W22s22 + 2W1W2 Cov(r1r2) s12 = Variance of Security 1 s22 = Variance of Security 2 Cov(r1r2) = Covariance of returns for Security 1 and Security 2 Two-Security Portfolio: Risk 6-* 6-* 2ABC = ABC = 2XYZ = XYZ = 1.37387 / (10-1) = 0.15265 39.07% 1.57885 / (10-1) = 0.17543 41.88% Calculating Variance and Covariance Ex post 6-* COV(ABC,XYZ) = rABC,XYZ = rABC,XYZ = 0.533973 / (10-1) = 0.059330 COV / (sABCXYZ) = 0.3626 0.059330 / (0.3907 x 0.4188) Ex ante Covariance Calculation Using scenario analysis with probabilities the covariance can be calculated with the following formula: 6-* sp2 = sp2 = sp2 = sp = sp < Two-Security Portfolio Risk W12s12 + 2W1W2 Cov(r1r2) + W22s22 0.36(0.15265) + 0.1115019 = variance of the portfolio 33.39% Let W1 = 60% and W2 = 40% Stock 1 = ABC; Stock 2 = XYZ 40.20% 2(.6)(.4)(0.05933) + 0.16(0.17543) 33.39% < [0.60(0.3907) + 0.40(0.4188)] = W1s1 + W2s2 6-* s2p = W12s12 + W22s22 + W32s32 + 2W1W2 Cov(r1r2) Cov(r1r3) + 2W1W3 Cov(r2r3) + 2W2W3 Three-Security Portfolio n or Q = 3 6-* For an n security portfolio there would be _ variances and _____ covariance terms. The ___________ are the dominant effect on n n(n-1) covariances s2p r = +1 r = .3 E(r) 13% 8% 12% 20% St. Dev TWO-SECURITY PORTFOLIOS WITH DIFFERENT CORRELATIONS Stock A Stock B WA = 0% WB = 100% WA = 100% WB = 0% r = 0 r = -1 50%A 50%B 6-* Summary: Portfolio Risk/Return Two Security Portfolio Amount of risk reduction depends critically on _________________________. Adding securities with correlations _____ will result in risk reduction. If risk is reduced by more than expected return, what happens to the return per unit of risk (the Sharpe ratio)? correlations or covariances < 1 6-* Minimum Variance Combinations -1< r < +1 1 2 - Cov(r1r2) W1 = + - 2Cov(r1r2) 2 W2 = (1 - W1) s 2 s 2 s 2 Choosing weights to minimize the portfolio variance 6-* 1 Minimum Variance Combinations -1< r < +1 2 E(r2) = .14 = .20 Stk 2 12 = .2 E(r1) = .10 = .15 Stk 1 s s r Cov(r1r2) = r1,2s1s2 6-* E[rp] = Minimum Variance: Return and Risk with r = .2 sp2 = 1 .6733(.10) + .3267(.14) = .1131 or 11.31% W12s12 + W22s22 + 2W1W2 r1,2s1s2 6-* Minimum Variance Combination with r = -.3 -.3 1 Cov(r1r2) = r1,2s1s2 6-* Minimum Variance Combination with r = -.3 -.3 E[rp] = sp2 = 0.6087(.10) + 0.3913(.14) = .1157 = 11.57% W12s12 + W22s22 + 2W1W2 r1,2s1s2 1 Notice lower portfolio standard deviation but higher expected return with smaller 12 = .2 E(rp) = 11.31% p = 13.08% 6-* Extending Concepts to All Securities Consider all possible combinations of securities, with all possible different weightings and keep track of combinations that provide more return for less risk or the least risk for a given level of return and graph the result. The set of portfolios that provide the optimal trade-offs are described as the efficient frontier. The efficient frontier portfolios are dominant or the best diversified possible combinations. All investors should want a portfolio on the efficient frontier. … Until we add the riskless asset 6-* E(r) The minimum-variance frontier of risky assets Global minimum variance portfolio Efficient frontier Individual assets Minimum variance frontier St. Dev. Efficient Frontier is the best diversified set of investments with the highest returns Found by forming portfolios of securities with the lowest covariances at a given E(r) level. 6-* E(r) The EF and asset allocation Efficient frontier St. Dev. 20% Stocks 80% Bonds 100% Stocks EF including international & alternative investments Ex-Post 2000-2002 80% Stocks 20% Bonds 60% Stocks 40% Bonds 40% Stocks 60% Bonds 100% Stocks 6-* Efficient frontier for international diversification Text Table 6.1 6-* Efficient frontier for international diversification Text Figure 6.11 6-* 6.3 The Optimal Risky Portfolio With A Risk-Free Asset 6.4 Efficient Diversification With Many Risky Assets 6-* Including Riskless Investments The optimal combination becomes linear A single combination of risky and riskless assets will dominate 6-* E(r) s ALTERNATIVE CALS P E(rA) A CAL (A) F Risk Free A Efficient Frontier CAL (Global minimum variance) G E(rP) CAL (P) P P&F E(rP&F) 6-* E(r) s The Capital Market Line or CML P E(rP&F) F Risk Free P&F Efficient Frontier E(rP) P P&F E(rP&F) CAL (P) = CML The optimal CAL is called the Capital Market Line or CML The CML dominates the EF 6-* Dominant CAL with a Risk-Free Investment (F) CAL(P) = Capital Market Line or CML dominates other lines because it has the the largest slope Slope = (E(rp) - rf) / sp (CML maximizes the slope or the return per unit of risk or it equivalently maximizes the Sharpe ratio) Regardless of risk preferences some combinations of P & F dominate 6-* E(r) s The Capital Market Line or CML P E(rP&F) F Risk Free P&F Efficient Frontier E(rP) P P&F E(rP&F) CML A=2 A=4 Both investors choose the same well diversified risky portfolio P and the risk free asset F, but they choose different proportions of each. 6-* Practical Implications The analyst or planner should identify what they believe will be the best performing well diversified portfolio, call it P. P may include funds, stocks, bonds, international and other alternative investments. This portfolio will serve as the starting point for all their clients. The planner will then change the asset allocation between the risky portfolio and “near cash” investments according to risk tolerance of client. The risky portfolio P may have to be adjusted for individual clients for tax and liquidity concerns if relevant and for the client’s opinions. o o o o 6-* 6.5 A Single Index Asset Market 6-* Individual Securities We have learned that investors should diversify. Individual securities will be held in a portfolio. What do we call the risk that cannot be diversified away, i.e., the risk that remains when the stock is put into a portfolio? How do we measure a stock’s systematic risk? Systematic risk Consequently, the relevant risk of an individual security is the risk that remains when the security is placed in a portfolio. 6-* Systematic risk Systematic risk arises from events that effect the entire economy such as a change in interest rates or GDP or a financial crisis such as occurred in 2007and 2008. If a well diversified portfolio has no unsystematic risk then any risk that remains must be systematic. That is, the variation in returns of a well diversified portfolio must be due to changes in systematic factors. 6-* Individual Securities How do we measure a stock’s systematic risk? Returns Stock A Returns well diversified portfolio Δ interest rates, Δ GDP, Δ consumer spending, etc. Systematic Factors 6-* Single Factor Model 6-* Ri = E(Ri) + ßiM + ei Ri = Actual excess return = ri – rf E(Ri) = expected excess return Two sources of Uncertainty M ßi ei = some systematic factor or proxy; in this case M is unanticipated movement in a well diversified broad market index like the S&P500 = sensitivity of a securities’ particular return to the factor = unanticipated firm specific events Single Index Model Parameter Estimation Risk Prem Market Risk Prem or Index Risk Prem = the stock’s expected excess return if the market’s excess return is zero, i.e., (rm - rf) = 0 ßi(rm - rf) = the component of excess return due to movements in the market index ei = firm specific component of excess return that is not due to market movements αi 6-* Let: Ri = (ri - rf) Rm = (rm - rf) Risk premium format Ri = ai + ßi(Rm) + ei Risk Premium Format The Model: 6-* Estimating the Index Model Excess Returns (i) Security Characteristic Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Excess returns on market index Ri = a i + ßiRm + ei Slope of SCL = beta y-intercept = alpha Scatter Plot 6-* Excess Returns (i) Security Characteristic Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Excess returns on market index Variation in Ri explained by the line is the stock’s _____________ Variation in Ri unrelated to the market (the line) is ________________ Scatter Plot unsystematic risk Ri = a i + ßiRm + ei systematic risk 6-* Estimating the Index Model Components of Risk Market or systematic risk: Unsystematic or firm specific risk: Total risk = Systematic + Unsystematic risk related to the systematic or macro economic factor in this case the market index risk not related to the macro factor or market index ßiM + ei i2 = Systematic risk + Unsystematic Risk 6-* Comparing Security Characteristic Lines Describe e for each. 6-* Measuring Components of Risk si2 = where; bi2 sm2 + s2(ei) si2 = total variance bi2 sm2 = systematic variance s2(ei) = unsystematic variance 6-* Total Risk = Systematic Risk / Total Risk = Examining Percentage of Variance Systematic Risk + Unsystematic Risk ßi2 s m2 / si2 = r2 bi2 sm2 / (bi2 sm2 + s2(ei)) = r2 r2 6-* Advantages of the Single Index Model Reduces the number of inputs needed to account for diversification benefits If you want to know the risk of a 25 stock portfolio you would have to calculate 25 variances and (25x24) = 600 covariance terms With the index model you need only 25 betas Easy reference point for understanding stock risk. βM = 1, so if βi > 1 what do we know? If βi < 1? 6-* Sharpe Ratios and Alphas When ranking portfolios and security performance we must consider both return & risk “Well performing” diversified portfolios provide high Sharpe ratios: Sharpe = (rp – rf) / p You can also use the Sharpe ratio to evaluate an individual stock if the investor does not diversify 6-* Sharpe Ratios and Alphas “Well performing” individual stocks held in diversified portfolios can be evaluated by the stock’s alpha in relation to the stock’s unsystematic risk. Skip Treynor-Black Model 6-* Suppose an investor holds a passive portfolio M but believes that an individual security has a positive alpha. A positive alpha implies the security is undervalued. Suppose it is Google. Adding Google moves the overall portfolio away from the diversified optimum but it might be worth it to earn the positive alpha. What is the optimal portfolio including Google? What is the resulting improvement in the Sharpe ratio? 6-* The Treynor-Black Model The Treynor-Black Model Weight of Google in the optimal portfolio O: The improvement in the Sharpe ratio (S) over the Sharpe of the passive portfolio M can be found as: Notice that the improvement in the Sharpe ratio is a function of This ratio is called the “information ratio” 6-* For multiple stocks in the active portfolio: The optimal weight of each security in the active portfolio is found as: A larger alpha increases the weight of stock i and larger residual variance reduces the weight of stock i. 6-* The Treynor-Black Model If A stands for the “active portfolio,” the active portfolio’s alpha, beta and residual risk can be found from: & 6-* The Treynor-Black Model 6-* M A P E(r) s CML CAL Treynor-Black Allocation Rf 6-* 6.6 Risk of Long-Term Investments 6-* Are Stock Returns Less Risky in the Long Run? Consider the variance of a 2-year investment with serially independent returns r1 and r2: The variance of the 2-year return is double that of the one-year return and σ is higher by a multiple of the square root of 2 ) 6-* Are Stock Returns Less Risky in the Long Run? Generalizing to an investment horizon of n years and then annualizing: For a portfolio: 6-* The Fly in the ‘Time Diversification’ Ointment The annualized standard deviation is only appropriate for short-term portfolios The variance grows linearly with the number of years Standard deviation grows in proportion to 6-* The Fly in the ‘Time Diversification’ Ointment To compare investments in two different time periods: Examine risk of the total rate of return rather than average sub-period returns Must account for both magnitudes of total returns and probabilities of such returns occurring 6-* Selected Problems 6-* Problem 1 E(r) = E(r) = (0.5 x 15%) + (0.4 x 10%) + (0.1 x 6%) 12.1% 6-* Problem 2 Criteria 1: Eliminate Fund B Criteria 2: Choose Fund D Lowest correlation, best chance of improving return per unit of risk ratio. 6-* Problem 3 Subscript OP refers to the original portfolio, ABC to the new stock, and NP to the new portfolio. i. E(rNP) = wOP E(rOP ) + wABC E(rABC ) = ii Cov = OP ABC = iii. NP = [wOP2 OP2 + wABC2 ABC2 + 2 wOP wABC (CovOP , ABC)]1/2 = [(0.92 .02372) + (0.12 .02952) + (2 0.9 0.1 .00028)]1/2 = 2.2673% 2.27% (0.9 0.67) + (0.1 1.25) = 0.728% 0.40 .0237 .0295 = .00027966 0.00028 6-* Problem 3 Subscript OP refers to the original portfolio, GS to government securities, and NP to the new portfolio. i. E(rNP) = wOP E(rOP ) + wGS E(rGS ) = ii. Cov = OP GS = iii. NP = [wOP2 OP2 + wGS2 GS2 + 2 wOP wGS (CovOP , GS)]1/2 = [(0.92 0.02372) + (0.12 0) + (2 0.9 0.1 0)]1/2 = 0.9 x 0.0237 = 2.133% 2.13% (0.9 0.67%) + (0.1 0.42%) = 0.645% 0 .0237 0 = 0 6-* Problem 3 βGS = 0, so adding the risk-free government securities would result in a lower beta for the new portfolio. 6-* Problem 3 The comment is not correct. Although the respective standard deviations and expected returns for the two securities under consideration are equal, the covariances and correlations between each security and the original portfolio are unknown, making it impossible to draw the conclusion stated. 6-* Problem 3 Returns above expected contribute to risk as measured by the standard deviation but her statement indicates she is only concerned about returns sufficiently below expected to generate losses. However, as long as returns are normally distributed, usage of should be fine. 6-* Problem 4 a. Although it appears that gold is dominated by stocks, gold can still be an attractive diversification asset. If the correlation between gold and stocks is sufficiently low, gold will be held as a component in the optimal portfolio. If gold had a perfectly positive correlation with stocks, gold would not be a part of efficient portfolios. The set of risk/return combinations of stocks and gold would plot as a straight line with a negative slope. (See the following graph.) 6-* Problem 4 The graph shows that the stock-only portfolio dominates any portfolio containing gold. This cannot be an equilibrium; the price of gold must fall and its expected return must rise. 6-* Problem 5 No, it is not possible to get such a diagram. Even if the correlation between A and B were 1.0, the frontier would be a straight line connecting A and B. 6-* Problem 6 The expected rate of return on the stock will change by beta times the unanticipated change in the market return: 1.2 (8% – 10%) = – 2.4% Therefore, the expected rate of return on the stock should be revised to: 12% – 2.4% = 9.6% 6-* Problem 7 The undiversified investor is exposed to both firm-specific and systematic risk. Stock A has higher firm-specific risk because the deviations of the observations from the SCL are larger for Stock A than for Stock B. Stock A may therefore be riskier to the undiversified investor. The risk of the diversified portfolio consists primarily of systematic risk. Beta measures systematic risk, which is the slope of the security characteristic line (SCL). The two figures depict the stocks' SCLs. Stock B's SCL is steeper, and hence Stock B's systematic risk is greater. The slope of the SCL, and hence the systematic risk, of Stock A is lower. Thus, for this investor, stock B is the riskiest. 6-*