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.