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I received this interesting problem from a friend today:

Assume that you are a portfolio manager with $10 million to allocate to hedge funds. The due diligence team has identified the following investment opportunities (here Expected Return and Expected StdDev stand for Expected Monthly Return and Expected Standard Deviation of Monthly Return and Price = Price of each investment unit):

Hedge Fund 1: Expected Return = .0101, Expected StdDev = .0212, Price = $2 million

Hedge Fund 2: Expected Return = .0069, Expected StdDev = .0057, Price = $8 million

Hedge Fund 3: Expected Return = .0096, Expected StdDev = .0241, Price = $4 million

Hedge Fund 4: Expected Return = .0080, Expected StdDev = .0316, Price = $1 million

What is the optimal allocation to each hedge fund (use MATLAB)?

Comments: The first thing I noticed is that the covariances across assets are left out, so classical mean variance analysis is out of the question. Next I thought about Lagrange multipliers, but it's not so clear what the objective function should be or how one would incorporate the standard deviation data. So then I turned to general utility theory and thought about stochastic dominance. But for stochastic dominance I would need to assume a specific probability distribution of returns.

Anyone have any hints here? I feel like I'm thinking of several advanced tools but missing a basic insight.

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    You mention that it's not clear what the objective function is. How would you even define "optimal allocation" without an objective function? In other words, I'm not sure your question can be answered unless you decide what your objective function should be.2012-02-10
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    In the absence of any correlation data, it seems reasonable to assume that things are independent, all things being equal. You can then find the efficient frontier contingent on this assumption. Will it be optimal? Probably not. However, it is likely the best you can do in the absence of more data.2012-02-10
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    Might also want to try posting your question here http://quant.stackexchange.com/2012-02-13

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