CFA CFA Level 1 Portfolio Variance and Portfolio Standard Deviation

# Portfolio Variance and Portfolio Standard Deviation

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• 5

For whatever reason I just cannot get this concept down and it’s killing me due to differences in variance/standard deviation when you’re given weights, asset variance or asset standard deviation and then either covariance or beta. Here’s a great example: Assets A (with variance of 0.25) and B ()variance of 0.40) are perfectly positively correlated. If an investor creates a portfolio using only these two assets with 40% in A and portfolio standard deviation is closest to: a) 0.5795, b) 0.3742 and c) 0.3400…please help!!!!

• 5

@Chevalier exactly. That’s why I solved both ways. Proves their equivalency 🙂

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My approach was to first memorize the correlation formula, then the correlation version of the portfolio sd formula. Also, I have an old habit of converting all variances to sd before attempting to solve anything. Beyond that, it just took a lot of ink and repetition.

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@microeconomist thanks for useful thought. Only problem I have is when you’re given correlation or covariance between A and B, obviously know how to convert correlation to covariance and vice versa given the standard deviations, issue is converting the rest of the formula given radical, etc

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So in this case you need to square the weights (as always), but don’t need to square the variances, you then take the square roots of both variances to get the standard deviations in the final part of the equation and 1 is your correlation number, but you then take the square root of the whole enchilada…i.e. [(0.4)^2(0.25)+(0.6)^2(0.4)+2(0.4)(0.6)1(0.25)^0.5(0.4)^0.5]^0.5….0.5795

@CFA_redemption_12713

Remember that the square root of the variance is the standard deviation. The formulas above all use the standard deviation. That’s why they didn’t square the variances in the first part of the formula.

What I do is always stick to the standard deviations and use them in the formulas.

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If you could memorise the 2 formulas below you should be good.

1. The variance of a portfolio -> w1^2 * SD1^2 + w2^2 * SD2^2 + 2w1 * w2 * r * SD1 * SD2.
2. The formula for correlation (r = Cov / SD1 * SD2) in case they only give you the Covariance.

Also, if you’re very comfortable with the formula and don’t mind using shortcuts.

Notice the last part of the formula goes like this –> 2 * w1 * w2 * r * SD1 * SD2 ?

If the covariance is given you could just do 2 * w1 * w2 * Cov. That way you could save some time since you are not calculating to obtain correlation and then multiply by both the standard deviation (SD).

• 1

So in this case you need to square the weights (as always), but don’t need to square the variances, you then take the square roots of both variances to get the standard deviations in the final part of the equation and 1 is your correlation number, but you then take the square root of the whole enchilada…i.e. [(0.4)^2(0.25)+(0.6)^2(0.4)+2(0.4)(0.6)1(0.25)^0.5(0.4)^0.5]^0.5….0.5795

• 1

@Lagarta I think if it’s perfectly positive correlated you simply take the weighted average of the Std. Devations and don’t have to do the complex calculation. In case one has a Std. Dev. of 0 you only take Std. Dev. x Weight for the other one.

• 0

First Term: (.4)^2 * (.25) = .04
Second Term: (.6)^2 * (.4) = .144
Third Term: 2 * (1) * (.4) * (.6) * (.5) * (.6324555) = .15178932

sqrt(.33578932) = .57947331

OR

perfectly positively correlated means NO CORRELATION EFFECT TERM! 🙂

so

.4(stdev of A) + .6 (stdev of B) = Stdev of portfolio
.4(.5) + .6(.6324555) = .57947331