Why does a recommended portfolio using the corner portfolio theorem always lie on the efficient frontier?
by definition
- “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. As the minimum-variance frontier passes through a corner portfolio, an asset weight either changes from zero to positive or from positive to zero. The GMV portfolio, however, is included as a corner portfolio irrespective of its asset weights.”
So Corner portfolios are on the minimum variance (efficient frontier) by the above.
By using the Corner portfolio theorem - you are joining two corner portfolios in a linear manner. (by using the weights)
Also the very next para …
“Corner portfolios allow us to create other minimum-variance portfolios. For example, suppose we have a corner portfolio with an expected return of 8 percent and an adjacent corner portfolio with expected return of 10 percent. The asset weights of any minimum-variance portfolio with expected return between 8 and 10 percent is a positive weighted average of the asset weights in the 8%- and 10%-expected- return corner portfolios. In a sign-constrained optimization, the asset weights of any minimum-variance portfolio are a positive linear combination of the corresponding weights in the two adjacent corner portfolios that bracket it in terms of expected return (or standard deviation of return). The foregoing statement is the key observation about the structure of a sign-constrained optimization; we may call it the corner portfolio theorem. Corner portfolios are generally relatively few in number. Knowing the com- position of the corner portfolios allows us to compute the weights of any portfolio on the minimum-variance frontier.”
tries to tell you the answer to what you are asking for
It’s the opposite, there are only several corner portfolios on any efficent frontier, but there needs to be at least one asset class with a zero weight.
Mr Smart - are you trying to contradict what is in the text?
How was it contradictory?
when you start a sentence with “it is the opposite” … do not know what you are talking about, and to whom… esp since your statement came immediately after mine – did not read the rest of the sentence.
my bad!