CO-OPTIMIZATION
Co-Optimization
by Michael de la Maza, Redfire capital Managment Group.
Summary: Improved portfolio performance can be achieved by co-optimizing
trading strategies instead of optimizing them separately and then combining
them into a portfolio using standard asset allocation methods.
The standard method of creating portfolios involves creating trading
strategies first and then combining them using an asset allocation
algorithm. An alternative approach is to co-optimize trading strategies.
When strategies are co-optimized they are created with the understanding
that they will be combined into a portfolio and that it is the portfolio's
performance, not the performance of the individual strategies, that should
be maximized.
To better understand this distinction consider the following two scenarios:
Single strategy/Asset allocation scenario: You ask two experts to provide
you with an optimal trading strategy on a particular market and with
particular parameters. You plan to combine the trading strategies that the
experts provide using your standard asset allocation process. However, the
experts return the same strategy (as they should if they succeed in
optimizing the object function) and, thus, no benefit is derived from
combining the strategies.
Co-optimization strategy: You tell each of two experts to provide you with a
trading strategy on a particular market with particular parameters that has
the property that when it is combined with the strategy provided by the
other expert it will produce an optimal portfolio. You supply the two
experts with a low bandwidth channel with which to communicate. The experts
return two strategies which are very good on a single strategy basis
(although probably not optimal) and also have the property that they have
low correlation/covariance/etc. When you combine these two strategies you
create a portfolio that is far superior to the one created in the single
strategy/asset allocation scenario.
The moral of these two scenarios is that co-optimization improves portfolio
performance by exchanging utility on a single strategy basis for utility at
the portfolio level.
The academic literature as well as the technical analysis literature appears
to consider trading strategy creation and asset allocation as two different
subject with little crossover. What these scenarios suggest is that there
are enormous benefits that can be derived by combining these two processes.
To illustrate these benefits, here is the performance of two portfolios, the
first of which was created using the standard single strategy optimization
followed by asset allocation methodology and the second of which was created
using co-optimization. Both portfolios are for the same set of markets,
using the same optimizer, etc.
Single strategy/Asset allocation Co-optimized
Mean 1.001599 1.001755
Standard Deviation 0.006956 0.006424
Note that the mean-variance performance of the co-optimized portfolio
increased. We have found similar results across many markets.
Co-optimizing portfolios is not straightforward. To the best of my
knowledge there are no publicly available software packages that provide
this feature.
One way to co-optimize is to simply run both strategies through the
optimizer (or inference system or learning algorithm or ...) simultaneously.
Unfortunately, this is computationally expensive. For example, suppose that
the optimizer exhaustively searches n variables each of which can have two
settings. Then the cost of optimizing two trading strategies separately is
on the order of 2*2^n = 2^(n+1) while the cost of co-optimizing the
strategies is 2^(2n). For n = 5 co-optimization is more than an order of
magnitude slower.
A second approach is to iteratively optimize the trading strategies. Begin
by optimizing the first strategy on a single strategy basis. The second
strategy is then optimized using the portfolio utility function. That is to
say, the optimizer searches for the second strategy that, when combined with
the first strategy, optimizes portfolio utility. After the second strategy
is optimized, the first strategy should then be re-optimized using the
portfolio utility function. This procedure should be repeated until some
sort of convergence is reached. The cost of this procedure is 2m*2^n where
m is the number of iterations.
In short, co-optimization appears to provide some benefits at the portfolio
level when it is compared to the standard procedure of single strategy
optimization followed by asset allocation.
Michael de la Maza
Redfire Capital Management Group