Corner Portfolios

Anyone know why when we we calculate the standard deviation for two corner portfolios we do not have to square the weights and square the standard deviation to solve? I understand that in some cases we consider the correlation between the two portfolios to be 1, so standard deviation is slightly overstated…

…but I am more or less asking why in all other cases we need to square the weights and this case we do not.

it is a more conservative thing. You are taking that the two corner portfolios are unrelated - so the Correlation between them is 1.

and it is not SOMETIMES - it is ALL THE TIME.

So the standard portfolio variance formula = (w1sd1)^2 +(w2sd2)^2 + 2 w1w2sd1sd2 rho => becomes (w1sd1)^2 +(w2sd2)^2 + 2 w1w2sd1sd2

and thus std deviation of portfolio becomes w1 sd1 + w2 sd2.


(w1sd1)^2 +(w2sd2)^2 + 2 w1w2sd1sd2 = (w1 sd 1 + w2 sd2 ) ^2

I’m pretty sure that “so” doesn’t remotely belong in that sentence.

The 2 corner portfolios lie on the same line (note a curve anymore) - therefore are perfectly correlated.

Hi Gpessah,

Good question.

If you have done a bit of mathematics, you may know that (a+b)² = a² + b² + 2ab

You may have understood that we consider two adjacent portfolios to be almost perfectly correlated i.e. we assume their coefficient of correlation is 1 (we know in reality that they are less than perfectly positively correlated so our computation will overestimate the standard deviation a bit).

As a result of this assumption, the variance of a portfolio on the efficient frontier between adjacent portfolio 1 (P1) and adjacent portfolio 2 (P2) will be :

w² stddev(P1)² + (1-w)² stddev(P2)² + (1-w) w stddev(P1) stddev(P2)

You will note that, according to the equation given above, this is equal to [w stddev(P1) + (1-w) stddev(P2)] ²

You take the square root of that to go from the variance to the standard deviation and it gives you w stddev(P1) + (1-w) stddev(P2)

I hope this answers to your question.

With all due respect, that’s silly.

_ Any _ two portfolios lie on the same straight line; would you say, therefore, that any two portfolios have correlations of returns of +1.0?

Actually I am reading now the answer of Cpk123. That’s exactly what he was saying (except that a coeff. of correlation of 1 means perfect correlation)

some bits are straight, some bits are curved. but mostly i’d say straight.

Well it’s an approximation (also visually, it looks straight, but it’s not exactly).

So for ANY two corner portfolios, the standard deviation of the portfolio = the weighted average of the standard deviation of the two CP?

For any two _ adjacent _ corner portfolios.