. wi, wj= percentage of the total portfolio investment in stock i and j respectively. σij= covariance of the returns of stock i and j. If i=j, then σij = σji and σij= σiσjρijFor the two asset portfolio:σp2=w12σ12+2w1w2σ1σ2ρ12+w22σ22σ12 = Variance of Security 1σ22 = Variance of Security 2σij= Covariance of returns for Security 1 and Security 2= σiσjρijNote that for an n-security portfolio the portfolio standard deviation calculation will be comprised of n variance terms only, but n(n-1) covariance terms or the n x n (total # of elements) – n (diagonal terms or variances) = n(n-1) off-diagonal elements (covariance terms). They are symmetric and thus you see 2 times the covariance in the portfolio risk formula. As you add more securities to the portfolio, the covariance terms dominate the risk calculation and the individual security standard deviations become less important.
Chapter 06 - Efficient Diversification 5 σ12=σ11σ12σ13σ14σ15σ21σ22σ23σ24σ25σ31σ32σ33σ34σ35σ41σ42σ43σ44σ45σ51σ52σ53σ54σ55The above is the variance-covariance matrix. σp2=w12σ12+w22σ22++w32σ32+w42σ42+w52σ52+2w1w2σ1σ2ρ12+2w1w3σ1σ3ρ13+2w1w4σ1σ4ρ14+...+2w4w5σ4σ5ρ45When i =/= j, there are 4x5/2 = 10 combinations (i.e. (1,2), (1x3), (1,4), (1,5), (2,3), (2,4), (2,5), (3,4), (3,5), (4,5)). There should be 20 (i.e.2 times the number of combinations terms) for the covariances and 5 terms of variances. Covariances are more important in determining the portfolio risk in a portfolio context given the number of assets is large. The graph depicts return/risk combinations of two securities, B (bonds) and S (stocks) for different hypothetical correlation coefficients. If there is a perfect positive correlation (i.e. ρ= +1) between B and S, combining the two securities yields no diversification benefits and combinations of B and S fall on a straight line because in this case σp=ΣWiσi. However if the assets are perfectly negatively correlated (i.e. ρ= - 1), we can combine the two securities to completely eliminate variance in the combined portfolio. Generally asset correlations will be between 0 and +1 and the combinations can eliminate some risk but not completely remove it. It is critical that you understand that diversification will improve the reward to risk like Sharpe ratio. This is why people diversify.
Chapter 06 - Efficient Diversification 6 Example: We have two assets. Their returns are perfectly negatively correlated. (correlation = -1). Calculate the weight of A in a risk free portfolio. Portfolio risk = 0. Assets Expected Return Standard Deviation of Returns 1 12% 20% 2 4% 7% Portfolio risk σp2=w12σ12+2wiwjσiσjρij+w22σ22=(w1σ1)2+2(wiσ)(wjσj)(−1)+(w2σ2)2=(w1σ1+w2σ2)2σp=w1σ1+(1−w1)σ2=0w1=σ2σ1+σ2=0.070.2+0.07=0.26W2=1−0.26=0.742) Calculate the expected return on this portfolio.
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