Standard Deviation of Optimal Portfolio

[jpbcologne]

The curriculum says that the standard deviation of the optimal portfolio is the weighted avarage of standard deviations of the corner portfolios

w1*SD1 + w2*SD2

The standard formula for calculating standard deviation, however, is:

w1²*SD1² + w2²*SD2² (assuming that portfolios are not correlated)

Would somebody kindly explain why the first formula is used in this case?

Thank you in advance!

[cpk123]

sd = sqrt(w1²*SD1² + w2²*SD2² + 2*w1*w2*SD1*SD2*corr12)

corner portfolios have a correlation of 1

so above becomes (w1SD1 + w2SD2)^2

[jpbcologne]

Indee, this will explain the calculation, but why corner portfolios are perfectly correlated?

[cpk123]

the part about the creation of corner portfolios:

Adjacent corner portfolios define a segment of the minimum-variance frontier within which 1) portfolios hold identical assets and 2) the rate of change of asset weights in moving from one portfolio to another is constant.

and of course this is a simplifying assumption. Assuming a correlation of 1 - will make the SD of the weighted portfolio HIGHER, than if the 2 Corner portfolios were less correlated.

When the SD is used in the sharpe ratio computation - you are more conservative, as a result.

[jpbcologne]

Really helpful. Thanks!