Correlation Coefficient and Variance

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


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


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


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?

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

Correct answer is C


It is.

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.

Thanks for clearing it up! It makes sense now.

With a perfect negative correlation, you get a zero-variance portfolio if w1/w2 = sd2/sd1 (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 can see this because when the correlation is -1, the formula becomes w12s12 + w22s22 - 2w1w2s1s2 = 0 and this holds when w1/w2 = s2/s1