August 8, 2018

Usually, we plan on having one portfolio playing a specific automated stock trading strategy. We find the best strategy we can and go from there. However, when looking at multiple trading strategies to be applied at the same time, the nature of the problem changes. You now have to take into account how these strategies will behave together. It is where you might have to game the strategies you have within your overall portfolio objectives and limitations.

For the single strategy scenario, the solution is simple, and it has a “simple” mathematical representation:

F(t) = F(0)∙(1 + g_bar)^t = F(0) + Σ(H∙ΔP) - Σ(Exp) = F(0) + n∙x_bar

where g_bar is the portfolio's average growth rate, and Σ(H∙ΔP), the strategy's payoff matrix. It becomes about rates of return, time, and evidently, the initial stake, whatever it is.

In playing multiple strategies, the equation is the same, except for the payoff matrix which is now 3-dimensional: k strategies, by d periods or days, by j stocks for all n trades taken over the portfolio's lifespan. Allocation to each strategy could be as simple as F(0)∙1/k. This, giving each strategy its share of the initial capital. It might help to partially diversify risk, not just over stocks, but over strategies as well. Resulting in reduced volatility, reduced drawdowns, reduced betas, but also, most probably reduced alphas, at a cost of a reduced overall CAGR. As a side effect, you would tend to get a smoother equity curve. And, there is a price for that.

All Strategies Are Not Equal

You have a mix of trading strategies. Certainly, they are not all equal. We can order them by outcome and get something like this:

Σ(H_a∙ΔP) > Σ(H_b∙ΔP) > , … , > Σ(H_spy∙ΔP) > Σ(H_z∙ΔP) > Σ(H_k∙ΔP)

Strategy H_a has more profits than strategy H_b and so on. As such, you know that strategy H_a is the most profitable of the group which is then followed by the others.

The thing to do is to drop all the other strategies and concentrate on your best performer.

H_a performs better than the average indexer too since it produces more than its benchmark H_spy, a surrogate for the market in general. Strategy H_a might not be the most desirable, but it is still the most profitable.

Why play anything else than H_a since this will only reduce the potential portfolio profit generation? That is the thing. However, there are other dimensions to this.

You might do it if there were other benefits to be had, like more diversification, a smoother equity curve, or for spreading risk not only between stocks but to strategies as well. A better risk management kind of thing. Also, a way to scale up and allocate to larger portfolios.

However, there would be an opportunity or convenience cost for this. It is the difference in profit generation: Σ(H_a∙ΔP) – Σ(H_b∙ΔP), and this cost will grow the more you add strategies and the more you add time. The larger the spread between the first two strategies and the others, the higher the cost for this “pseudo-volatility” protection.

It gets worse if Σ(H_a∙ΔP) >> Σ(H_b∙ΔP), meaning that the top trading strategy produces a lot more than the next one in line. It is one thing to mix 10 trading strategies close to the same outcome, it is another to depreciate or downgrade your best performer to what might be a much lower combined CAGR level.

 Gaming Strategy CAGR (click to enlarge)

The top panel in the above chart is a returns table. It gives the future value of \$1.00 invested over some time intervals. The higher the rate of return and the more years provided, the higher the overall performance. Common stuff, used for centuries.

You want to know how much \$10 million invested would generate, choose the cell you want and simply multiply the numbers by \$10 million. A strategy at 10% CAGR will see its \$10M portfolio grow to \$174.5M over 30 years. If you take 10 of those, your initial capital will need to be \$100M which will grow to \$1.745B.

But, and that is the point. You do not have the \$100M. What you have is the \$10M stake which will be divided into the 10 trading strategies. And the outcome will be \$174.5M, the same as if you had done only 1 with the total stake. Take 10 strategies similar to H_b having an average 10% CAGR, divide the initial stake and you get: 10∙[(1/10)∙Σ(H_b∙ΔP)] ≤ Σ(H_b∙ΔP)]. It most likely should be less since the 10 next strategies, even though similar, were ordered by their declining payout. Nonetheless, it is good enough as an approximation.

Strategy Performance

The second panel of the above chart shows how many trading strategies at the same CAGR level would be required to achieve the same performance as the strategy at the 30% CAGR. In the beginning, it is not that many, but as you put in more time, the impact becomes considerable.

The stock market game is still a compounding game. Playing for a couple of years is not enough. That is not where the money is. It is at the other end of the time spectrum, at the 30 years+ level.

We could mix say 29 strategies with an averaged 10% CAGR with a single one at the 30% level, and that single strategy would still be able to pull in 83.8% of all the generated profits all on its own. But even this is distorted. Those 29 strategies, each needed their own \$10M stake. You could put numbers to this for the 29 strategies sharing an initial \$10M stake:

F(t) = \$20M + 29∙[(1/29)∙Σ(H_10∙ΔP)] + 1∙Σ(H_30∙ΔP) – Σ(Exp)

F(t) = \$20M + 16.45∙\$10M + 2,619∙\$10M – Σ(Exp)

which accounts for the initial stake only once.

The 30% CAGR strategy Σ(H_30∙ΔP) would have brought in 99.34% of all the profits of this 30-strategy combination. Should I stress the importance of having at least one such strategy in your portfolio?

But here is the real problem, such strategies are hard to come by, or are they? Evidently, 30% CAGR strategies are. But, not that much. Just over the last week, I took a posted trading strategy on Quantopian and transformed it into such a thing. You can see its conclusion here.

The strategy's code is in the public domain, just follow the above link where it is referenced. It could provide you with a starting point to design your own or transform it to what you want to see. All I did was change some numbers, some strategy assumptions. No trading logic or trading procedures were changed. Note that I have not as yet added protective measures to this strategy which would tend to reduce volatility, drawdowns, and further increase performance. You should do that on your own, if you want, and at your own risks.

Within the Quantopian context, especially its contests, this strategy would not qualify since it is long only, has higher volatility and higher drawdown than desired based on their stated requirements. However, they should realize that having low-volatility, low-beta, low-alpha strategies with a strategy like Σ(H_30∙ΔP) would greatly reduce its volatility and beta while raising the entire portfolio's alpha generation.

But, like in many things: to each their own. Nonetheless, an elementary calculation of volatility for the 30 strategies would give: (29∙0.10 + 1∙0.25)/30 = 0.105 for the case where all 30 strategies got a \$10M allocation. The average portfolio volatility would pass from 0.10 to 0.105. And some would like to count this as too much added risk... Incidentally, the same thing would apply when considering drawdowns: (29∙0.10 + 1∙0.55)/30 = 0.115. It would be expected to have overall portfolio drawdown go from -0.10 to -0.115, again, not a lot of added risk. I think Quantopian might be overlooking the obvious. Unless it was their intent or they were already doing this. However, I have not seen anything to that effect on their site. Even though that is what they should do, game their allocations, strategies, that is.

The above table can be used to answer a number of questions. But that will be left for another time.