Portfolio Optimization Problem Without Correlation Info

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. –  Mike Spivey Feb 10 '12 at 18:42
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. –  Aaron Feb 10 '12 at 18:53
Might also want to try posting your question here quant.stackexchange.com –  user2467 Feb 13 '12 at 2:05

This is not really a mathematical question, so I'm not going to give a mathematical answer, but rather a practical one.

Let's say you wanted to apply traditional mean-variance optimization. Then you need to make an assumption about the correlation structure. The most sensible thing would be to assume that all the correlations are zero.

This is because portfolio optimization is error-maximizing in the sense that small errors in estimating the average returns and volatilities will be magnified in the optimizing procedure. As an illustration of the idea, I looked at by how much an error in the estimation of your average returns affects the portfolio allocations output by a simple mean-variance optimizer. It's actually more illustrative to look at the percentage-change Jacobian, which expresses by what factor an error in the inputs is multiplied. If I assume zero correlation between the funds, then the following matrix results:

$$J_p = \left( \begin{matrix} 0.91& -0.82& -0.06& -0.03\\ -0.09& 0.18& -0.06& -0.03\\ -0.09& -0.82& 0.94& -0.03\\ -0.09& -0.82& -0.06& 0.97 \end{matrix} \right)$$

The important fact to notice is that all inputs are below 1 in absolute value. That is, any error in your estimated returns will give you an error in your portfolio weightings which is of a similar (percentage) magnitude.

If instead you assumed a correlation of 50% between each pair of funds, then the following percentage Jacobian results:

$$J_p = \left( \begin{matrix} 43.72& -28.90& -9.07& -5.75\\ -0.08& 0.14& -0.04& -0.01\\ 1.42& 2.32& -4.53& 0.78\\ 0.49& -0.03& 0.42& -0.88 \end{matrix} \right)$$

That is, a relative overestimation of 1% in your estimation of the return of the first fund will result in a 43% over-allocation to that fund. More incredibly, a relative overestimation of 10% (which is entirely within the realms of possibility - most of the time, even getting the sign of the average return right is impressive) would result in you overallocating to that fund by 430%!

This is why you shouldn't use plain vanilla portfolio optimization without being very careful indeed.

This answer is somewhat incomplete at the moment. I may come back and revisit it later, but I wanted to give you a flavor of the kind of thing you need to think about to answer this question (also, I would recommend migrating this to Quant Stack Exchange...)

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I wouldn't suggest that the following is very profound, but it may be as much as can be said given the limited information:

1. If the investor is risk neutral then the lack of information on covariance does not matter and return is maximised by allocating 100% to the fund with the highest expected return, ie Fund 1.

2. If the investor is risk averse then, in the absence of information on covariance, the prudent assumption is the worst case, ie perfect positive correlation between the returns on each pair of funds. Given that assumption, a mean-variance analysis could be undertaken. But the choice of a particular mean-variance combination on the efficient frontier would still depend on the investor's objective function.

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