# Corner Portfolios

Can someone help me understand what a corner portfolio is?

Is a corner portfolio a segment along the EF where two portfolios hold the same assets but the weight of said assets vary between the two? Why is this important?

A corner portfolio is either the GMV portfolio (for sake of completeness), or a portfolio at which, as you move up-and-to-the-right along the efficient frontier, the weight on (at least) one asset goes from positive to zero or from zero to positive.

Suppose that you have three securities: A, B, and C.

• The GMV portfolio is 60% A, 40% B; that’s a corner portfolio
• As you move up the efficient frontier, the amount of A increases and the amount of B decreases.
• You hit the spot on the efficient frontier which is 100% A: the weight of B has gone from positive to zero, so that’s a corner portfolio. (As we’ll see, it’s also the point where the weight of C goes from zero to positive, confirming that it’s a corner portfolio.)
• As you move up the efficient frontier, the amount of A decreases and the amount of C increases.
• When you hit the end of the efficient frontier, which is 100% C, the weight of A has gone from positive to zero, so that’s the third (and last) corner portfolio.

I see, thanks S2000. So a corner portfolio can only exist when 1 asset is at 100% and another is at 0%? The objective is to find the ‘optimal’ corner portfolios, and calculate a weight between them to find where the sharpe ratio is maximized?

If that’s true, why do we have to find the corner portfolios in order to solve the highest sharp ratio? Is it because the optimal allocation is not on the EF therefore was never solved originally?

No.

You have to have (at least) one asset at 0%, but the rest can be anything. If you had a portfolio with 30% A, 70% B and 0% C, and when you move up it changes to 29%, 69%, and 2%, then the original portfolio’s a corner portfolio.

The obective is to find all corner portfolios on the efficient frontier, then interpolate for the rest of the efficient frontier.

You don’t. It’s just a nice part of the theory.

If I were looking for the portfolio with the highest Sharpe ratio (and I’ve done this many times), I’d simply put all of the formulae and constraints into Excel and tell Solver to calculate the weights that give the highese Sharpe ratio; I wouldn’t bother with corner portfolios.

My pleasure.

Bump.

The corner portfolio on p. 179-180 of Schweser book 2 is bothering me.

In the blue box example, are they saying the new portfolio is also on the efficient frontier (e.g. a corner portfolio)?

It’s NOT a straight line. Hence, anything in between two points will fall below the efficient frontier making the calculated portfolio suboptimal.

Expected return is a linear function of the returns of the corner portfolios, but expected standard deviation of returns _ is not _ a linear function of the standard deviations of the corner portfolios.

Combinations of those portfolios will, in fact, lie on the effecient frontier, not on a straight line between the corner portfolios.

expected std deviation is NOT a linear function of the standard deviations - that part is very correct as stated by S2000 - but you will see it “closely” approximated to a linear interpolation. (By doing the linear interpolation - the w1 s1 + w2 s2 is used - which means the 2 * corr * w1 * s1 * w2 * s2 is ignored and assumes correlation coeff between the 2 CPs to be 0).

True enough.

The linear interpolation is correct if the correlation of returns is +1.0. Thus, it puts an upper bound on the risk of the (interpolated) portfolio.

I understand what both of you are saying s2k and cpk. However, just look at the graph. Draw a straight line between two points. The EF is a curved line. How can we have calulate a portfolio between the two corner portfolios that could be on the EF?

Naturally there’s a wedge. I’m coming from utility theory (risk neutral vs. risk adverse individuals). There’s a difference in expected utility vs risk.

http://www.fao.org/docrep/w7365e/w7365e0e.htm

take a look at figure 11.3

I’m looking at it from a purely math perspective. How can we say anything on the straight line (e.g. calculated portfolio) is just as good as the curved line (e.g. EF)?

if the CPs are close enough - in terms of % of the assets - the curve would be approximated by a straight line. So the CP is a theoritcal concept - pretty much like differential calculus - the concept of the 1st derivative there - dx -> 0 (tends to 0).

You would need the correlation of returns between the two corner portfolios.

How can we get the corner portfolio? Or this will be given?