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I’m curious if MILP can be used to optimize portfolio allocations and what inputs, outputs, variables, constraints and objectives are relevant.

For example, is a stock’s average price and variance of central interest? Or is a linear trend and variance there more useful?

I could see maximizing the expectation and minimizing the variance to be desirable. (Not sure how to express this as a linear constraint) but even so if I stock was bought and didn’t grow, was that useful?

Would such a model indicate when to buy and sell and more of just “buy and hold” strategy.

I’m very much a novice in this domain!

jbuddy_13
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    There is a ton of literature on using mixed integer linear (or sometimes quadratic) models for portfolio construction. A web search using those terms will turn up a lot of it. The quadratic part creeps in when you consider variance of returns (and particularly covariances between securities in the portfolio). – prubin Feb 26 '23 at 22:44
  • Add to the above, you may start with basic Markovitz portfolio optimization model.https://en.m.wikipedia.org/wiki/Markowitz_model – Sutanu Majumdar Feb 27 '23 at 00:33
  • https://quant.stackexchange.com/questions/tagged/portfolio-optimization – Rodrigo de Azevedo Feb 27 '23 at 16:06
  • If you replace the L2-norm by an L1-norm you end up with a linear mean-variance problem. – Erwin Kalvelagen Feb 27 '23 at 18:08
  • @ErwinKalvelagen, that’s brilliant! I’ll read up on how MILP handles absolute values. That puzzles me – jbuddy_13 Feb 27 '23 at 19:46
  • I see how it might work for constraints https://or.stackexchange.com/questions/3/working-with-absolute-values-in-constraint-in-a-lp-or-milp but I’m not sure how this would work for objectives. For example ‘min( x-y + y-x)’ is 0 for any combination of x and y – jbuddy_13 Feb 27 '23 at 19:50
  • Actually… https://math.stackexchange.com/questions/1954992/linear-programming-minimizing-absolute-values-and-formulate-in-lp – jbuddy_13 Feb 27 '23 at 19:54
  • See e.g. https://www.jstor.org/stable/2632458 – Erwin Kalvelagen Feb 27 '23 at 20:01
  • https://docs.mosek.com/portfolio-cookbook/index.html has a lot material regarding portfolio optimization including the usage of integer constrained variables to handle cardinality constraints. – ErlingMOSEK Feb 28 '23 at 10:05

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