What is the portfolios variance and standard deviation Problem is that we also

# What is the portfolios variance and standard

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What is the portfolio’s variance and standard deviation? Problem is that we also need to know the degree to which the 2 stocks either correlate or co-vary The portfolio’s standard deviation is not simply the average of the 2 standard deviations FIN 300 - Risk and Return Pt. 2 14 Variance of a 2-Stock Portfolio If we have 2 stocks in a portfolio and we are given the portfolio weights, the individual stocks’ standard deviations, and either the correlation or the covariance, we can easily solve for the variance and standard deviation of the portfolio FIN 300 - Risk and Return Pt. 2 15 Formula for Portfolio Variance 𝜎𝑝2=𝜔𝐴2𝜎𝐴2+𝜔𝐵2𝜎𝐵2+ 2𝜌𝐴𝐵𝜔𝐴𝜔𝐵𝜎𝐴𝜎𝐵𝜎𝐴& 𝜎𝐵𝑎𝑎𝑎𝑡𝑡𝑎𝑠𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑠.𝑠𝑎𝑑𝑑𝑎𝑡𝑑𝑠𝑑𝑠𝜔𝐴&𝜔𝐵𝑎𝑎𝑎𝑡𝑡𝑎𝑝𝑠𝑎𝑡𝑝𝑠𝑝𝑑𝑠𝑤𝑎𝑑𝑤𝑡𝑡𝑠𝜌𝐴𝐵is the correlation between stocks A & B 𝜎𝑝2𝑑𝑠𝑡𝑡𝑎𝑝𝑠𝑎𝑡𝑝𝑠𝑝𝑑𝑠𝑑𝑎𝑎𝑑𝑎𝑑𝑠𝑎𝑆𝑡𝑠.𝐷𝑎𝑑𝑑𝑎𝑡𝑑𝑠𝑑= 𝜎=𝜎2FIN 300 - Risk and Return Pt. 2 16 Correlation The correlation (ρ) is always in a range from -1 to 1 −𝟏 ≤ 𝝆 ≤ 𝟏When ρ = +1, the two datasets are perfectly positively correlated The 2 datasets peak at the exact same time and they bottom-out at the exact same time – they are perfectly in-sync When ρ= -1, the two datasets are perfectly negatively correlated When one dataset is peaking, the other is at it’s trough, and vice-versa Correlation is a matter of timing – being in-sync – as long as the datasets go up and down at the same time (even if they don’t go up and down by the same amount), they are correlated FIN 300 - Risk and Return Pt. 2 17 Portfolio Variance Example Back to previous numerical example: A: WA = 40% & σΑ = 20% B: WB = 60% & σΒ = 40% Now, what if the correlation of (the returns of) stocks A & B is 0.5 Solve for the portfolio variance and standard deviation: σp2= (0.4)2(0.2)2+ (0.6)2(0.4)2+2(0.4)(0.6)(0.2)(0.4)(0.5) =0.0064 + 0.0576 + 0.0192 = 0.0832 Portfolio Variance = 0.0832 Portfolio Standard Deviation= σp= SQRT(0.0832) = 0.288444 = 28.8444% FIN 300 - Risk and Return Pt. 2 18 Portfolio Variance Example In the previous example, what if instead the correlation is 0.2 σp2= (0.4)2(0.2)2+ (0.6)2(0.4)2+2(0.4)(0.6)(0.2)(0.4)(0.2) =0.0064 + 0.0576 + 0.00768= 0.07168 σp= sqrt(0.07168) = 0.267731 = 26.7731% What if the correlation is -0.5? σp2= (0.4)2(0.2)2+ (0.6)2(0.4)2+2(0.4)(0.6)(0.2)(0.4)(-0.5) = 0.0064 + 0.0576 - 0.0192 = 0.0448 σp= sqrt(0.0448) = 0.21166 = 21.166% Note: The lower the correlation, the lower the variance of the resulting portfolio – resulting in a greater diversification effect! FIN 300 - Risk and Return Pt. 2 19 Covariance Covariance of datasets A & B is calculated as the following: 𝜎𝐴𝐵=1𝑁 −1𝑅𝐴𝐴− 𝑅𝐴𝑅𝐵𝐴− 𝑅𝐵𝑁𝐴=1The above formula (from statistics) is used to calculate the covariance of 2 datasets – assume equal weighting of data Note that the variance of a dataset is really just the  #### You've reached the end of your free preview.

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