Another Universal Working Capital Alphabetic Lineament Allotment Algorithm

Financial engineers are accustomed to borrowing techniques from scientists inwards other fields (e.g. genetic algorithms), but rarely does the borrowing acquire the other way. It is so surprising to remove heed nearly this before. But hither is or then other 1 that has proven to live on universal: the exceedingly simple EG algorithm.

The EG ("Exponentiated Gradient") algorithm is an representative of a upper-case missive of the alphabet resources allotment dominion using "multiplicative updates": the novel upper-case missive of the alphabet allocated to a stock is proportional to its electrical flow upper-case missive of the alphabet multiplied past times a factor. This element is an exponential role of the supply of the stock inwards the terminal period. This algorithm is both greedy too conservative: greedy because it e'er allocates to a greater extent than upper-case missive of the alphabet to the stock that did good most recently; conservative because at that topographic point is a penalisation for changing the resources allotment also drastically from 1 catamenia to the next. This multiplicative update dominion is the 1 proposed equally a model for development past times natural selection.

The computational payoff of EG over the Universal Portfolio is obvious: the latter requires a weighted average over all possible allocations at every step, spell the onetime needs solely know the resources allotment too returns for the most recent period. But does this EG algorithm genuinely generate expert returns inwards practice? I tested it 2 ways:

1) Allocate betwixt cash (with 2% per annum interest) too SPY.
2) Allocate amid SP500 stocks.

In both cases, the solely complimentary parameter of the model is a number called the "learning rate" η, which determines how fast the resources allotment tin modify from 1 catamenia to the next. It is to a greater extent than oft than non flora that η=0.01 is optimal, which nosotros adopted. Also, nosotros disallow curt positions inwards this study.

The benchmarks for comparing for 1) are, using the notations of the Borodin paper,

a)  the buy-and-hold SPY portfolio BAH, and
b) the best constant-rebalanced portfolio alongside fixed allocations inwards hindsight CBAL*.

The benchmarks for comparing for 2)  are

a) a constant rebalanced portfolio of SP500 stocks alongside equal allocations U-CBAL,
b) a portfolio alongside 100% resources allotment to the best stock chosen inwards hindsight BEST1, and
c) CBAL*.

To detect CBAL* for a SP500 portfolio, I used Matlab Optimization Toolbox's constrained optimization role fmincon.

There is also the number of SP500 index reconstitution. It is complicated to handgrip the add-on too deletion of stocks inwards the index inside a constrained optimization function. So I opted for the shortcut of using a subset of stocks that were inwards SP500 from 2007 to 2013, tolerating the presence of surivorship bias. There are solely 346 such stocks.

The effect for 1) (cash vs SPY) is that the CAGR (compound annualized increase rate) of EG is slightly lower than BAH (4% vs 5%). It turns out that BAH too CBAL* are the same: it was best to allocate 100% to SPY during 2007-2013, an unsurprising recommendation inwards hindsight.

The effect for 2) is that the CAGR of EG is higher than the equal-weight portfolio (0.5% vs 0.2%). But both these numbers are much lower than that of BEST1 (39.58%), which is almost the same equally that of CBAL* (39.92%). (Can you lot approximate which stock inwards the electrical flow SP500 generated the highest CAGR? The answer, to live on revealed below*, volition surprise you!)

We were promised that the EG algorithm volition perform "similarly" to CBAL*, then why does it underperform then miserably? Remember that similarity hither only agency that the difference is sub-exponential: but fifty-fifty a polynomial difference tin inwards do live on substantial! This seems to live on a universal work alongside universal algorithms of property allocation: I accept never flora whatever that genuinely achieves pregnant returns inwards the curt bridge of a few years. Maybe nosotros volition detect to a greater extent than interesting results alongside higher frequency data.

So given the underwhelming functioning of EG, why am I writing nearly this algorithm, aside from its interesting connectedness alongside biological evolution? That's because it serves equally a setup for another, non-universal, portfolio resources allotment scheme, equally good equally a way to optimize parameters for trading strategies inwards general: both topics for or then other time

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Workshops Update:

My side past times side online workshop volition live on on  Mean Reversion Strategies, August 26-28. This too the Quantitative Momentum workshops volition also live on conducted alive at Nanyang Technological University inwards Singapore, September 18-21.

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Do follow me @chanep on Twitter, equally I oft postal service links to interesting articles there.

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*The SP500 stock that generated the highest supply from 2007-2013 is AMZN.