Sample Question #261 (applied math – optimization with constraints)

You have 30 assets you need to allocate into a portfolio. You’re given the expected returns of the 30 assets as well as the 30×30 covariance matrix. You’re told to use the mean-variance portfolio construction model, and you’re given the appropriate lamda (risk aversion parameter).

Now, you’re also told that 1) you cannot short an asset, and 2) you cannot hold more than 5% of any given asset. How do you solve this portfolio optimization problem? More specifically, (a) can you provide an analytic solution, and (b) how do you solve this numerically?

[Source: a real interview question asked by a UBS derivatives quant]

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Quadratic programming problemLet x_i be the proportion of asset i in the portfolio, i=1,…,30In addition to the normal mean-variance constraints, two more arex_i>=0 (cannot short)x_i<=5%Analytically one can use KKT conditions, numerically Matlab function "quadprog" could do the job.

Another question that can be asked from this is (c) what’s an efficient numerical algorithm for finding the optimal asset mix.