# Correlation Coefficient and Variance

Which one of the following statements about correlation is NOT correct?

A)

Potential benefits from diversification arise when correlation is less than +1.

B)

If the correlation coefficient were -1, a zero variance portfolio could be constructed.

C)

If the correlation coefficient were 0, a zero variance portfolio could be constructed.

I had to think this one through, is this logic correct? Since the formula for Variance is

and we want 0 variance, that would mean we need the third term 2w1*w2*Corr1,2*stdev1*stdev2 to equal negative some number. That way we can add the other two positive terms to maybe come up with zero variance?

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The correct answer is B, I forgot to mention above.

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Correct answer is C

“Mmmmmm, something…” - H. Simpson

It is.

Simplify the complicated side; don't complify the simplicated side.

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The situation becomes even simpler for c, since the third term becomes 0. The individual asset variances are >=0 and w1+w2=1 with w1,w2>=0. About the only way I can think of to get portfolio variance = 0 is to put 100% weight on an asset with variance = 0.

“Mmmmmm, something…” - H. Simpson

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Thanks for clearing it up! It makes sense now.

I’m too tired to crank the math, but if I remember correctly, with a perfect negative correlation, you get a zero-variance portfolio if w

_{1}/w_{2}= sd_{2}/sd_{1}(ie if you choose weights such that the ratio of the weights is the inverse of the ratios of the standard deviations). In other words if you had standard deviations of 10% for asset 1 and 5% for asset two, weights of 1/3 for asset 1 and 2/3 for asset 2 would give a zero-variance portfolio.You keep using that word. I do not think it means what you think it means.