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andste
Topic Author
Posts: 0
Joined: May 14th, 2006, 3:34 pm

Shortest Proof Mean-Variance Efficiency

Assuming that portfolios must be fully invested (all exposures add to one) and short positions (exposures smaller than zero) and leverage on asset level (individual exposures larger than one) are possible, we know that linear combinations of mean-variance efficient portfolios are again mean-variance efficient...w3 = h*w1 + (1-h)*w2h... mixing parameter, 0<=h<=1w1... exposure vector of a mean-variance efficient portfolio 1w2... exposure vector of a mean-variance efficient portfolio 2w3... exposure vector of a resulting portfolio 3, which will always be mean-variance efficientMy question: What is the shortest, most elegant way to proof this theorem mathematically?Greetings,Andi

Darou
Posts: 2
Joined: August 6th, 2002, 11:03 am

Shortest Proof Mean-Variance Efficiency

You cannot prove: It's not true.You probably mean: mixing efficient Portfolio + risk-free investment. But, the mixture is not on the efficient frontier.
Last edited by Darou on September 28th, 2014, 10:00 pm, edited 1 time in total.

acastaldo
Posts: 14
Joined: October 11th, 2002, 11:24 pm

Shortest Proof Mean-Variance Efficiency

If you assume "unconstrained portfolio weights" for risky assets then it is true; i.e. sum of weights =1 but no constraint on the individual weights. (How realistic that is is another matter, it abstracts from collateral and borrowing/lending issues). I think the proof is in an early Fischer Black paper that I don't have handy at the moment
Last edited by acastaldo on September 28th, 2014, 10:00 pm, edited 1 time in total.

LocalVolatility
Posts: 13
Joined: May 27th, 2009, 10:07 am
Location: Amsterdam
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Shortest Proof Mean-Variance Efficiency

The proof is in Merton (1972) "An Analytical Derivation of the Efficient Portfolio Frontier", Journal of Financial and Quantitative Analysis.Also have a look at Feldman and Reisman (2003) "Simple Construction of the Efficient Frontier", European Financial Management.