Chapter 6 Efficient Diversification 6.1 Diversification and Portfolio Risk 6-2  Market risk - The risk that has to do with general economic conditions. - The risk that remains even after diversification. - Systematic risk or non-diversifiable risk.  Firm-specific risk - Diversifying into many more securities reduce exposure to firm-specific factors. - Unique risk, nonsystematic risk, or diversifiable risk. 6-3 6-4 Figure 6.1 Portfolio risk as a function of the number of stocks in the portfolio 6-5 Figure 6.2 Portfolio risk decreases as diversification increases. 6.2 Asset Allocation With Two Risky Assets 6-6 - Need to understand how the uncertainties of asset returns interact when we form a risky portfolio. - The key determinant of portfolio risk is the extent to which the returns on the two assets tend to vary either in tandem or in opposition. - Portfolio risk depends on the covariance between the returns of the assets in the portfolio. 6-7 Asset Allocation With Two Risky Assets = 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 6-8 portfolio the in securities # n ;rW)rE( n 1i i i p == ∑ = 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 6-9 Combinations of risky assets 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 E[rp] What statistics measure the tendency for r1 to be below expected when r2 is above expected? Covariance and Correlation n = # securities in the portfolio 6-10 [...]... (0 .67 33 2 ) (0.15 2 ) + (0.3 267 2 ) (0.2 2 ) + 2 (0 .67 33) (0.3 267 ) (0.2) (0.15) (0.2)      1/2 σ p = 0.01711 / 2 = 13.08 % 6- 26 Minimum Variance Combination with ρ = -.3 1 -.3 Cov(r1r2) = ρ1,2σ1σ2 6- 27 Minimum Variance Combination with ρ = -.3 1 -.3 E[rp] = 0 .60 87(.10) + 0.3913(.14) = 1157 = 11.57% W12σ12 + W22σ22 + 2W1W2 ρ1,2σ1σ2 σp2 = σ p = (0 .60 872 ) (0.15 2 ) + (0.3913 2 ) (0.2 2 ) + 2 (0 .60 87)... 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- 31 6. 3 The Optimal Risky Portfolio With A Risk-Free Asse t 6. 4 Efficient Diversification With Many Risky Assets 6- 32  The optimal combination becomes... i= 1 6- 20 Two-Security Portfolio Risk σp 2 Q Q = ∑ ∑ [WI WJ Cov(I, J)] I =1J =1 σp2 W12σ12 + 2W1W2 Cov(r1r2) + W22σ22 = Let W1 = 60 % and W2 = 40% Stock 1 = ABC; Stock 2 = XYZ σp2 0. 36( 0.15 265 ) + 2( .6) (.4)(0.05933) + 0. 16( 0.17543) = σp2 0.1115019 = variance of the portfolio = σp = 33.39% σp < W1σ1 + W2σ2 (Linear combination: ρ12=1) 33.39% < [0 .60 (0.3907) + 0.40(0.4188)] = 40.20% The benefits of diversification. .. large diversification benefits from combining 1 and 2 6- 15 ρ and diversification in a 2 stock portfolio What does -1 < ρ12 < 1 imply? If -1 < ρ12 < 1, then There are some diversification benefits from combining stocks 1 and 2 into a portfolio σp2 = W12σ12 + W22σ22 + 2W1W2 Cov(r1r2) And since Cov(r1r2) = ρ12σ1σ2 σp2 = W12σ12 + W22σ22 + 2W1W2 ρ12σ1σ2 6- 16 The effects of correlation & covariance on diversification. .. 13.08% 6- 28 The minimum-variance frontier of risky assets Expected Return Efficient Frontier is the best diversified set of investments with the highest returns Efficient frontier Global minimum variance portfolio Found by forming portfolios of securities with the lowest covariances at a given E(r) level Individual assets Minimum variance frontier St Dev Find the mean-variance efficient portfolios! 6- 29... 2W1W2 ρ12σ1σ2 6- 16 The effects of correlation & covariance on diversification Asset A Asset B Portfolio AB 6- 17 The effects of correlation & covariance on diversification Asset C Asset C Portfolio CD 6- 18 The power of diversification Most of the diversifiable risk eliminated at 25 or so stocks 6- 19 Two-Risky Assets Portfolios rp = W1r1 +W2r2 E(rp) = W1E(r1) + W2E(r2) Linear Function σp2 = W12σ12 + W22σ22... “average away” more risk? 6- 12 - 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- 13 Cov(r1 , r2 ) ρ12 = σ1 × σ 2 6- 14 ρ and diversification in a... the mean-variance efficient portfolios! 6- 29 The EF and asset allocation Expected Return EF including international & alternative investments 80% Stocks 20% Bonds 60 % Stocks 40% Bonds 40% Stocks 60 % Bonds 100% Stocks Efficient frontier 20% Stocks 80% Bonds 100% Stocks St Dev 6- 30 Extending Concepts to All Securities  Consider all possible combinations of securities, with all possible different weightings... (the Sharpe ratio)? . Chapter 6 Efficient Diversification 6. 1 Diversification and Portfolio Risk 6- 2  Market risk - The risk that has to do with general. diversifiable risk. 6- 3 6- 4 Figure 6. 1 Portfolio risk as a function of the number of stocks in the portfolio 6- 5 Figure 6. 2 Portfolio risk decreases as diversification increases. 6. 2 Asset Allocation. are some diversification benefits from combining stocks 1 and 2 into a portfolio. 6- 16 The effects of correlation & covariance on diversification Asset A Asset B Portfolio AB 6- 17 6- 18 The