Investment Management EffiCient DiversifiCation Efficient-Diversification 1

FINA 4320: Investment Management Efficient Diversification Efficient Diversification Fall 2018 1 / 36 Two-Security Portfolio: Return Expected Return:E [rp ] = w1 E [r1 ] + w2 E [r2 ] Expected Return of portfolio with n securities n E [ rp ] = ∑ w i E [ ri ] i =1 n ∑ wi = 1 i =1 Efficient Diversification Fall 2018 2 / 36 Portfolio Risk Risk factors common to the whole economy lead to market risk, also called systematic risk or nondiversifiable risk Risk that can be eliminated by diversification is called firm-specific risk, also called unique risk, idiosyncratic risk, residual risk, nonsystematic risk or diversifiable risk Efficient Diversification Fall 2018 3 / 36 Portfolio Risk as a Function of the Number of Stocks in the Portfolio Efficient Diversification Fall 2018 4 / 36 Portfolio Variance and Covariance Investing in more securities reduces risk: Portfolio variance: σp2 = n n ∑ ∑ wi wj Cov (ri , rj ) i =1 j =1 How to compute Covariance (Cov ) n Cov (ri , rj ) = (ri (t ) − E [ri ])(rj (t ) − E [rj ]) n−1 t =1 ∑ Efficient Diversification Fall 2018 5 / 36 Naive Diversification Efficient Diversification Fall 2018 6 / 36 Asset Allocation with Two Risky Assets Correlation and Diversification Stock A T-bills Stock B Probability Good 25% 5% 1% 0.5 Bad 10% 5% -5% 0.3 Ugly -25% 5% 35% 0.2 Examples of portfolios: All in A Half A, half T-bills Half A, half B E[r] 10.5% 7.75% 8.25% σ 18.9% 9.45% 4.83% Intuition: Cov (A, B ) = −240.5 ρ(A, B ) = −0.86 Efficient Diversification Fall 2018 7 / 36 Covariance and Correlation 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. Correlation ρ(ri , rj ) = Cov (ri , rj ) σi σj Rewriting portfolio variance in terms of correlation n σp2 = n ∑ ∑ wi wj σi σj ρ(ri , rj ) i =1 j =1 Efficient Diversification Fall 2018 8 / 36 Covariance and Correlation Two stocks case: σp2 = w12 σ12 + w22 σ22 + 2w1 w2 σ1 σ2 ρ(r1 , r2 ) ρ and diversification in a 2 stock portfolio: Diversification improves risk-return trade-off. The lower the correlation b/w assets, the greater the gain from diversification (risk ↓) Portfolios of ρ < +1 always offer better risk-return opportunities than individual securities on their own With perfect positive correlation b/w assets, ρ = +1, no gain from diversification – just weighted avg. of individual security std. deviation σp = w1 σ1 + w2 σ2 Efficient Diversification Fall 2018 9 / 36 Covariance and Correlation Perfect negative correlation b/w assets, ρ = -1, is extreme case of diversification; σp2 = w12 σ12 + w22 σ22 − 2w1 w2 σ1 σ2 = (w1 σ1 − w2 σ2 )2 Create perfectly hedged position with zero portfolio risk using portfolio weights: w1 = w2 = σ2 σ1 + σ2 σ1 σ1 + σ2 ρ(r1 , r2 ) = ρ(r2 , r1 ) (same is true for covariance) Efficient Diversification Fall 2018 10 / 36 Summary Portfolio Risk/Return Two Security Portfolio Amount of risk reduction depends critically on correlation or covariances Adding securities with correlations < 1 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)? Efficient Diversification Fall 2018 11 / 36 Asset Allocation (2 Risky Assets) Portfolio Opportunity Set: Graph of possible combinations of risk and expected return Consider the following example: Bonds: E [rB ] = 5%; σB = 8% Stocks: E [rS ] = 10%; σS = 19% Excel: Efficient Diversification Fall 2018 12 / 36