Efficient Diversification Chapter 7 Efficient Diversification

Two-Security Portfolio: Return rp = W1r1 + W2r2 W1 = Proportion of funds in Security 1 W2 = Proportion of funds in Security 2 r1 = Expected return on Security 1 r2 = Expected return on Security 2 W i S =1 n = 1

Two-Security Portfolio: Risk 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

Covariance Cov(r1r2) = r1,2s1s2 r1,2 = Correlation coefficient of returns s1 = Standard deviation of returns for Security 1 s2 = Standard deviation of returns for Security 2

Correlation Coefficients: Possible Values Range of values for r 1,2 -1.0 < r < 1.0 If r = 1.0, the securities would be perfectly positively correlated If r = - 1.0, the securities would be perfectly negatively correlated

Three-Security Portfolio rp = W1r1 + W2r2 + W3r3 s2p = W12s12 + W22s22 + W32s32 + 2W1W2 Cov(r1r2) + 2W1W3 Cov(r1r3) + 2W2W3 Cov(r2r3)

In General, For an n-Security Portfolio: rp = Weighted average of the n securities sp2 = (Consider all pair-wise covariance measures)

Two-Security Portfolio E(rp) = W1r1 + W2r2 sp2 = w12s12 + w22s22 + 2W1W2 Cov(r1r2) sp = [w12s12 + w22s22 + 2W1W2 Cov(r1r2)]1/2

TWO-SECURITY PORTFOLIOS WITH DIFFERENT CORRELATIONS 13% 8% 12% 20% St. Dev TWO-SECURITY PORTFOLIOS WITH DIFFERENT CORRELATIONS r = 0

Portfolio Risk/Return Two Securities: Correlation Effects Relationship depends on correlation coefficient -1.0 < r < +1.0 The smaller the correlation, the greater the risk reduction potential If r = +1.0, no risk reduction is possible

Minimum Variance Combination 2 E(r2) = .14 = .20 Sec 2 12 = .2 E(r1) = .10 = .15 Sec 1 s r 1 s 2 - Cov(r1r2) 2 = W1 s 2 s 2 - 2Cov(r1r2) + 2 1 W2 = (1 - W1)

Minimum Variance Combination: r = .2 (.2)2 - (.2)(.15)(.2) = W1 (.15)2 + (.2)2 - 2(.2)(.15)(.2) W1 = .6733 W2 = (1 - .6733) = .3267

Minimum Variance: Return and Risk with r = .2 rp = .6733(.10) + .3267(.14) = .1131 s = [(.6733)2(.15)2 + (.3267)2(.2)2 + p 1/2 2(.6733)(.3267)(.2)(.15)(.2)] 1/2 s = [.0171] = .1308 p

Minimum Variance Combination: r = -.3 (.2)2 - (.2)(.15)(.2) = W1 (.15)2 + (.2)2 - 2(.2)(.15)(-.3) W1 = .6087 W2 = (1 - .6087) = .3913

Minimum Variance: Return and Risk with r = -.3 rp = .6087(.10) + .3913(.14) = .1157 s = [(.6087)2(.15)2 + (.3913)2(.2)2 + p 1/2 2(.6087)(.3913)(.2)(.15)(-.3)] 1/2 s = [.0102] = .1009 p

Extending Concepts to All Securities The optimal combinations result in lowest level of risk for a given return The optimal trade-off is described as the efficient frontier These portfolios are dominant

The minimum-variance frontier of risky assets Efficient frontier Individual assets Global minimum variance portfolio Minimum variance frontier St. Dev.

Extending to Include Riskless Asset The optimal combination becomes linear A single combination of risky and riskless assets will dominate

ALTERNATIVE CALS s CAL (P) CAL (A) E(r) M M P P CAL (Global minimum variance) A A G F s P P&F M A&F

Dominant CAL with a Risk-Free Investment (F) CAL(P) dominates other lines -- it has the best risk/return or the largest slope Slope = (E(R) - Rf) / s [ E(RP) - Rf) / s P ] > [E(RA) - Rf) / sA] Regardless of risk preferences combinations of P & F dominate

Single Factor Model ri = E(Ri) + ßiF + e ßi = index of a securities’ particular return to the factor F= some macro factor; in this case F is unanticipated movement; F is commonly related to security returns Assumption: a broad market index like the S&P500 is the common factor

Single Index Model Risk Prem Market Risk Prem or Index Risk Prem a = the stock’s expected return if the market’s excess return is zero i (rm - rf) = 0 ßi(rm - rf) = the component of return due to movements in the market index ei = firm specific component, not due to market movements

Risk Premium Format Let: Ri = (ri - rf) Risk premium format Rm = (rm - rf) Risk premium format Ri = ai + ßi(Rm) + ei

Estimating the Index Model Excess Returns (i) . . . . . . . . Security Characteristic Line . . . . . . . . . . . . . . . . . . . Excess returns on market index . . . . . . . . . . . . . . . . . . . . . . . Ri = a i + ßiRm + ei

Components of Risk Market or systematic risk: risk related to the macro economic factor or market index Unsystematic or firm specific risk: risk not related to the macro factor or market index Total risk = Systematic + Unsystematic

Measuring Components of Risk si2 = bi2 sm2 + s2(ei) where; si2 = total variance bi2 sm2 = systematic variance s2(ei) = unsystematic variance

Examining Percentage of Variance Total Risk = Systematic Risk + Unsystematic Risk Systematic Risk/Total Risk = r2 ßi2 s m2 / s2 = r2 bi2 sm2 / bi2 sm2 + s2(ei) = r2

Advantages of the Single Index Model Reduces the number of inputs for diversification Easier for security analysts to specialize