Random portfolios as benchmarks for tactical asset allocation strategies
If you are familiar with tactical asset allocation (TAA) strategies, like the Global Equities Momentum (GEM) TAA strategy of Gary Antonacci, you know how hard it can be to find a proper benchmark to evaluate their performances, with their portfolio positions varying wildly from one month to another.
The two standard possibilities have both drawbacks:
 Comparing a TAA strategy with a naive portfolio, let say an equal weighted portfolio of the assets traded by the TAA strategy, is problematic because the naive portfolio cannot compete with the dynamics of the TAA strategy
 Comparing a TAA strategy with other similar TAA strategies is also problematic because “similar” needs to be precisely defined to begin with
So, what to do?
If you consider a TAA strategy as a fund manager  a fund manager who would follow inflexible portfolio construction rules  an accurate benchmark would be the fund manager’s true opportunity set^{1}, that is, all the portfolios that could have been constructed in the same universe of assets and under the same constraints (number of assets, volatility of the portfolio, etc.).
In order to approximate this true opportunity set in practice, it is possible to use a Monte Carlo method to generate thousands of constrained random portfolios^{2}. This simulated peer group will act as a collective benchmark against which the TAA strategy performances can be evaluated.
As Ronald J. Surz puts it^{3}:
performance evaluation should be viewed as a hypothesis test that assesses the validity of the hypothesis “Performance is good”.
To accept or reject this hypothesis, all the possible performance outcomes are constructed to determine where the actual performance result falls.
If the observed performance appears toward the top of all the possibilities, the hypothesis is correct, and performance is good. Otherwise, it is not.
In other words, the hypothesis test compares what actually happened with what could have happened.
Let’s illustrate this process with GEM, which invests monthly in one of the S&P 500 Index, the MSCI ACWI exUSA Index or the Barclays Capital US Aggregate Bond Index, depending on precise rules described in Gary Antonacci’s book^{4}:

First, the universe of assets of the random portfolios needs to be defined. Here, it is pretty simple, it will be the same 3index universe as the GEM strategy.
 Then, the constraints on the random portfolios need to be chosen. For the test below, it will be:
 Long only, no leverage
 Fully invested, with no constraint on the number of indexes in the portfolios (this is a deviation compared to the GEM rules, which might favor the random portfolios)
 Monthly randomly rebalanced, the portfolios being rebalanced every month to a new set of random assets weights satisfying the constraints above
 Finally, a computer needs to be kindly asked to do the heavy lifting. For this part, I will use Node.js^{5} together with the
Portfolio Optimizer API endpoint
/v1/portfolio/simulation/rebalancing/randomweight
to compute the constrained random portfolios. The Gist is available here.
Now, the results!
Because Gary Antonacci worked on his strategy in 2013, let’s first test the performances of GEM during the insample period 19892012 (Google Sheet).
For illustration, here are the equity curves of a couple of random portfolios (in solid) v.s. the equity curve of GEM (in dashed):
For the real thing, here is the probability density of a sample of 10,000 random portfolios Sharpe Ratios, with a red line depicting the GEM Sharpe Ratio:
Pretty amazing! In statistical terms, the pvalue of the Sharpe Ratio of GEM would be close to 0, that is, the Sharpe Ratio of GEM would be among the best obtainable Sharpe Ratios!
Let’s now test the performances of GEM during the outofsample period 2013October 2020 (Google Sheet).
Here is the same probability density as above, for the new period:
Pretty blah this time, with the Sharpe Ratio of GEM being roughly comparable to the Sharpe Ratio of the median random portfolio, which  by definition  displays no particular skills…
Now, while specific conclusions about a particular TAA strategy is out of scope of this blog, the difference in the Sharpe Ratio of GEM relative to its simulated peer group between the insample period and the outofsample period would certainly warrant a detailed investigation.
Anyway, the same kind of random portfolio analysis could be done with any TAA strategy to detect potential regime change, so, don’t hesitate to abuse of the computing power available with today’s computers!
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Ronald J. Surz, Portfolio Opportunity Distributions, The Journal of Investing May 1994, 3 (2) 3641; DOI: 10.3905/joi.3.2.36 ↩

R Stein, Not fooled by randomness: Using random portfolios to analyse investment funds, Investment Analysts Journal, 43:79, 115, DOI: 10.1080/10293523.2014.11082564 ↩

Surz, Ronald. “A Fresh Look at Investment Performance Evaluation: Unifying Best Practices to Improve Timeliness and Reliability.” Journal of Portfolio Management, Vol. 32, No. 4, Summer 2006, pp 5465 ↩

Gary Antonacci, Dual Momentum Investing: An Innovative Strategy for Higher Returns With Lower Risk ↩

Other people are using R, like James Picerno from The Captial Spectator ↩