Hi.Briefly speaking, the "Two Fund Theorem" states that all the portfolios on the Markowitz Efficient Frontier can be obtained as a one-parameter family of linear combinations of any two of them, i.e., p = s*p1 + (1-s)*p2 for some s. This works when the Markowitz mean-variance optimization is subject only to linear equality constraints. I was hoping to apply this in a case with simple inequality constraints, e.g., bounds of the form 0 <= p <= b (all components of p -- asset weights -- between 0 and some upper bound b). I am only looking at values of s such that 0 <= s <= 1, so that the linear combination still satisfies the bounds. But the theorem doesn't appear to hold in my case (unless, of course, I have a programming error). Can anyone here please tell me whether the Two-Fund Theorem can be extended into the realm of inequality constraints (at least upper and lower bounds)? Thanks!-- TMK --

"2fund th" means that efficient frontier (the best combination of risk-return trade off or simply the best sharpe ratio level) is any linear combination between riskfree asset and tangency portfolio.So you can use any constrain to run optimization and find tangency portfolio (ie no short selling) after that any combination of this optimal portfolio and risk free asset has the same risk/return trade off.Two fund theorem address the best way to invest given a risky market of assets, a riskfree asset and your risk aversion.the way yuo maximize your utility in the risky market is another and different problem: markowitz explain can you solve it if you are a myopic mean variance investor without constrains.but the best risky (not total) portfolio can be found in severl way depending hipotesys and constrains you want use.for example if you are a long run investor and think that same asset class is mean reversion you could ask a positive domand on this asset, also if it has a negative risk premia, if this asset class gives you an intertemporal hedge of market risk.this example is in contrast with myopic markowitz optimization but not in contrast with the two fund theorem.

Hi, fashionmarina.Thank you so much for responding! I guess there's some ambiguity in how different people refer to the "Two Fund Theorem." I was using the terminology as it was defined in the book "Investment Science" by David Luenberger (1998, Oxford University Press, ISBN 0-19-510809-4). Luenberger calls the situation you describe (combination of risk-free asset and tangent portfolio) the "One-Fund Theorem." What he calls the "Two-Fund Theorem" is the use of linear combinations of two portfolios on the efficient frontier to create any other portfolio on the efficient frontier. From what I can tell, the proof of this relies heavily on the use of equality constraints only, so that the quadratic optimization that defines the efficient frontier can be reduced to the solution of a linear system. I read somewhere (I thought it was in Luenberger, but I can't find the passage in there now) that if you have inequality constraints that don't bind, you can ignore them, and if they do bind, you can treat them as equality constraints. That's why I thought there might be some way to resurrect the Two-Fund Theorem in the presence of inequality constraints, although in any case, I don't think it's going to work in the situation I've been toiling on (I suspect that a varying set of inequality constraints bind as you trace the efficient frontier).

i don't the two fund theorem will be true with extra constraints. You 'll get a polygonal boundary (with curved edges), i'd expect two funds for each smooth arc but not everywhere