复旦大学：《投资学讲义》（英文版） Chapter 4 asset diversification

INVESTMENTS Chapter 4 Optim Risky porfolios

INVESTMENTS Optimal Optimal Risky Portfolios Risky Portfolios Chapter 4 Chapter 4

INVESTMENTS Risk reduction with diversification St deviation Unique risk Market risk Number of Securities

INVESTMENTS Risk Reduction with Diversification Risk Reduction with Diversification Number of Securities St. Deviation Market Risk Unique Risk

INVESTMENTS T'Wo-Security portfolio: Return r=Wr+war WI=Proportion of funds in Security 1 W2=Proportion of funds in Security 2 rI =Expected return on Security 1 r2=Expected return on Security 2 ∑w

INVESTMENTS r p = W 1 r1 + W 2 r 2 W 1 = Proportion of funds in Security 1 W 2 = Proportion of funds in Security 2 r 1 = Expected return on Security 1 r 2 = Expected return on Security 2 ∑ = 1 = n i 1 wi Two -Security Portfolio: Return Security Portfolio: Return

INVESTMENTS T'wo-Security portfolio Risk 0=W10+W202+2W1W2 Cov(rr2) 01=Variance of Security I 02=Variance of Security 2 Cov(rr2)=covariance of returns for Security I and Security 2

INVESTMENTS σp2 = w12σ12 + w22σ22 + 2W1W2 Cov(r1r2) σ12 = Variance of Security 1 σ22 = Variance of Security 2 Cov(r1r2) = Covariance of returns for Security 1 and Security 2 Two-Security Portfolio: Risk Security Portfolio: Risk

INVESTMENTS Covariance Cov(rr2)=p2 012 P12= Correlation coefficient of returns 0=Standard deviation of returns for Security 1 o2-Standard deviation of returns for Security 2

INVESTMENTS ρ1,2 = Correlation coefficient of returns Cov(r 1 r 2) = ρ 2 σ 1 σ 2 σ 1 = Standard deviation of returns for Security 1 σ 2 = Standard deviation of returns for Security 2 Covariance Covariance

INVESTMENTS Correlation coefficients Possible val Range of values for +1.0≥p,>-10 Ifp. =1.0, the securities would be perfectly positively correlated 1. 0. the securities would be perfectly negatively correlated

INVESTMENTS Range of values for ρ1,2 + 1.0 > ρ  > -1.0 If ρ = 1.0, the securities would be perfectly positively correlated If ρ = - 1.0, the securities would be perfectly negatively correlated Correlation Coefficients: Correlation Coefficients: Possible Values Possible Values

INVESTMENTS Three- sSecurity portfolio rp=W+W22+ W33 0.2 W +Wo2 +w 303 +2W1W2 Cov(r r2) +2W,W3 Cov( r3) +2W,W3 Cov(r2r3)

INVESTMENTS σ 2 p = W1 σ 1 2 + 2W 1 W 2 r p = W 1 r1 + W 2 r2 + W 3 r 3 Cov(r 1 r 2 ) + W 3 2 σ 3 2 Cov(r 1 r 3 + 2W ) 1 W 3 Cov(r 2 r 3 + 2W ) 2 W 3 Three -Security Portfolio Security Portfolio 2 + W2 σ 2

INVESTMENTS In general for an orokolo rp=Weighted average of the n securities o=( Consider all pairwise covariance measures

INVESTMENTS r p = Weighted average of the n securities σ p 2 = (Consider all pairwise covariance measures) In General, For an In General, For an n -Security Portfolio: Security Portfolio:

INVESTMENTS T'Wo-Security portfolio E(D)=Wr+w, 0 2=W1202+W202+2W,W2 Cov(rr2) o=[w101+w202+2W,W2, Cov(rr2)ll/2

INVESTMENTS E(rp) = W1r1 + W2r2 σ p2 = w12σ 12 + w22σ 22 + 2W1W2 Cov(r1r2) σ p = [w12 σ 12 + w22 σ 22 + 2W1W2 Cov(r1r2)]1/2 Two-Security Portfolio Security Portfolio

INVESTMENTS Two- Security Portfolios with Different correlationis 13% 120 20% St Dev

INVESTMENTS Two -Security Portfolios with Security Portfolios with Different Correlations Different Correlations ρ = 1 13% E(r) 12% 20% St. Dev ρ = .3 ρ = -1 8% ρ = -1