# Mixing Portfolio Strategies

Given a set of $N$ assets, the amount of strategies proposed in literature to diversify the investors wealth in order to find the 'optimal' portfolio is overwhelming. However, for example DeMiguel et al point out that estimation error is large in general, leading to bad out-of-sample returns for many sophisticated portfolio strategies.

In my mind it should be a straightforward idea to combine several promising allocation candidates such that the estimation risk is 'diversified' as well. In other words, given a set of portfolio weights proposed by $k$ distinct strategies $\{\omega_1,\ldots,\omega_k\}$, it could be beneficial to compute a vector $c\in\mathbb{R}^k, \left( \sum\limits_{i=1}^{k} c_i =1\right)$ based on some criteria and then investing into $\omega^*:=c_1\omega_1+\ldots +c_k\omega_k$.

I am aware of some ideas combining two portfolios via shrinkage approaches (see for example Posterior Inference for Portfolio Weights and the appendix of Optimal Versus Naive Diversification: How Inefficient is the 1/N Portfolio Strategy?)

but I am would like to learn more about those ideas. Any reference or further comment directing in this direction is welcome to answer my question: What are the benefits and gains from combining several allocation strategies?

Not many benefits actually. As you combine different strategies you abstract yourself from what you are trying to reach. E.g. What would you get if you combine a strategy that tries to maximize Sharpe Ratio, with a strategy that maximizes Certain Equivalent? You have no idea on what that would get you. Probably it might even get you a portfolio with high turnover and so high transaction costs.

Even DeMiguel, seems to be moving towards simpler strategies. The 1/N strategy is on example. Another example is to use Parametric Portfolio Policies of Brandt, Santa-Clara and Valkanov. Even DeMiguel on one of his last papers seems to favour this approach: Fifty Ways to Beat the Market? A Portfolio Perspective on Investment Anomalies.

All in all there is no right answer to your question. You should try to mixture several strategies and see out of sample what would you get net of transaction fees. Also, you should clearly define as an investor what is your objective, either by defining that you want the maximum sharpe ratio, or maximum utility given some risk-aversion. Without a clear objective in mind is hard to define what would be the optimal strategy.

• Thank you very much for the interesting abstract and the interesting points you mention in your answer! – muffin1974 Mar 2 '16 at 12:13
• A full version of the DeMiguel "Fifty ways" paper is available here lyxoretf.co.uk/pdfDocuments/… – Alex C Feb 13 '19 at 3:04
• The answer above is one way to look at it. But even if you abstract yourself away from the portfolio objective you're trying to reach, wouldn't the main benefit of mixing strategies be that, where one strategy is known to perform poorly out-of-sample (commonly the case in mean-variance optimization), mixing it with a different strategy would improve out-of-sample performance? Wouldn't an investor be just as much interested in following whichever strategy that has 1% error rate, rather than sticking to an objective known to have 5% error? – develarist Jul 8 at 12:18
• No exactly. Usually more conditioning variables or "mixture" of models, improve in sample performance but decrease out of sample performance. – phdstudent Jul 8 at 12:20
• there are examples where a mixture of portfolios does help though: shrinkage improves portfolios out-of-sample by compromising between the unconstrained and constrained min variance portfolios, while the HRP improves out-of-sample performance by compromising between the min variance and inverse variance portfolios – develarist Jul 8 at 12:23