Sept. 1, 2020

A lot of emphasis was put on a payoff matrix equation (see my **last article**) to represent a long-term rebalancing stock portfolio. From it, we could estimate the number of trades the rebalancing might generate over the life of the portfolio. However, that was still only half of the solution. What was also needed was an estimate of the profitability of such a trading strategy. That part of the equation is more complicated and has a lot more than just one solution, even though it too, has a simple formulation.

Here is the set of equations related to the portfolio's payoff matrix equation:

F(t) = F_{0} + $X = F_{0} + Σ (**H** ∙ Δ**P**) = F_{0} + n ∙ x_{avg} = F_{0} + y ∙ rb ∙ j ∙ **E**[tr] ∙ u(t) ∙ **E**[PT]

= F_{0} ∙ (1 + r_{m} + α – Σ exp_{t})^{t}

Presently I will concentrate on x_{avg} = u(t) ∙ **E**[PT], the expression for the average net profit per trade. Evidently, x_{avg} could also be expressed as: x_{avg} = Σ (**H** ∙ Δ**P**) / n, which explicitly states that the average net profit per trade is simply the sum of all generated profits and losses from the trading strategy **H**, whatever it is, divided by the total number of executed trades. We already can get a reasonable estimate for n in **E**[n] = y ∙ rb ∙ j ∙ **E**[tr]. We are, therefore, left with finding a long-term estimate for xavg.

Imagine being able to forecast your long-term portfolio performance before you even start. Or, better yet, help you design better trading strategies to achieve prespecified objectives. A solution based on your own trading methods, which could give you a reasonable estimate of where your portfolio might be in some 20^{+} years. This makes the above equation remarkable and also makes it worth investigating in detail. I say it is by understanding the above equations' intricacies and implications that we can design better and more productive trading strategies.

This kind of trading strategy could be operated by hand. It is rebalanced only (rb) times during the year. A simulation is done to illustrate that it was at least possible to do so over an extended time period using historical data. The simulation is there to confirm that the trading procedures used could have worked in the past, and since the same principles would apply going forward, although giving different results, you could feel confident that your trading strategy could survive over the long term and help you prosper.

But even then, it all boils down to your choices, to what you think is appropriate for you. Since a rebalancing portfolio is governed by the above equation, should we not learn how to control it? In a way, you cannot fix or improve on something unless you understand how it works or why and how it is broken. Regardless, the above equations will still prevail.

**The Trading Route**

Trading is different from investing. Trading implies that you can jump in and out of positions at will many times over the life of a portfolio for whatever reason, whereas the buy and holder will do it once over the same period. The buy and holder is expected to get: F(t) = F_{0} ∙ (1 + **E**[r_{m}])^{t}, where **E**[r_{m}] is the expected average market return. Some will do better and some will do worse, it is why the term “expected average” is used. Of note, 70% of long-term market players do not exceed the market average, which is a terrible statistic. Therefore, it is rather imperative that you do not predesign your trading strategy to be part of that group.

For the trader, the expected number of trades **E**[n] takes on an added significance. It can stay relatively small or grow quite large due to the method of play and the continuous rebalancing over extended periods of time.

The rebalancing method, which is an administrative decision, is dictating the number of trades that can be executed over the investment period. For example, rebalancing monthly can generate at most 12 trades per year on a single stock. If you had 400 stocks in your portfolio, you should expect: 1 ∙ 12 ∙ 400 = 4,800 trades per year. However, with a 60% average turnover rate, the expected number of trades would be around: 2,880. Over the same period, the buy and holder would still have 400 opened positions.

Adding one year would generate a total of 5,760 trades for the trader, while the buy and holder would still have 400 positions. Rebalancing for 10 years would put the trader at 28,800 trades while the buy and holder would still have his/her 400 opened positions.

**The Trade Allocation**

The initial equal weight allocation for a 400-stock portfolio is 1 / 400 = 0.0025 or 0.25% per position. This would tend to make the portfolio more than sufficiently diversified and risk-averse, depending on the selection made. Any one stock going bankrupt would only have a minor impact on the whole portfolio, in fact, just -0.25%. Even there, your trading procedures could greatly limit within the stock selection process those potentially going bankrupt stocks. A simple rule like no stock below $10 gets in or stays in the portfolio would tend to do that.

What x_{avg} = u(t) ∙ **E**[PT] is saying is that the trading unit function gains a lot of importance since the expected profit target E[PT] is limited due to the average available monthly price variations. The annual expected appreciation for stocks over the long term (20+ years) has been around 10%, dividends included. On a monthly basis, the expected return should be, on average, about 10%/12 = 0.00833. With the numbers presented, starting with $10M as initial capital, we can make a one-year estimate: 1 ∙ 12 ∙ 400 ∙ 0.60 ∙ F0/400 ∙ 0.00833 = 600,000 or a 6% return. Starting with $1M would make your first-year estimate: 1 ∙ 12 ∙ 400 ∙ 0.60 ∙ F0/400 ∙ 0.00833 = 60,000 or, again, a 6% return which in itself demonstrates the strategy's scalability.

We can see from the above that the expected turnover rate is having an impact to the point of reducing the overall return from an expected 10% to 6%. Should we raise the turnover rate to 100%, the portfolio's expected outcome would rise back to its 10% expected return. And should our trading procedures reduce the turnover rate to 40%, the portfolio's return would correspondingly fall to a 4% expected return. We definitely need to do more in order to gain some long-term return advantage.

**The Rebalancing Procedure**

The rebalancing also says that the expected profit target **E**[PT] might be limited due to the rebalancing procedure itself. There is, on average, just so much the average stock will move in a calendar month. The market does not have a smooth and predictable one-month return that answers to your prescheduled rebalance. However, averaged over the long term, this average monthly return is more subdued. It is like slicing a price series into multiple segments and then tying them back together again. If I buy SPY at the beginning of each month to then sell the position at the end of each month for a year, I should get about the same return as if having bought SPY just once and held it for the year. While you are holding SPY, all you can get is SPY's return, no more and no less, and this, while not accounting for frictional costs.

To generate more than the expected long-term averages, you should expect that you will have to do more than enter into a 1:1 stock return relationship. There are not that many options available. We could select better stocks, or improve on market timing, or leverage the taken positions. We could also complement taken positions with options. The object of the game is to generate some alpha as in F(t) = F_{0} ∙ (1 + r_{m} + α – Σ exp_{t})^{t}, and making sure you get it over the long term.

We already have the information needed to estimate how many trades will be executed over the life of our rebalancing portfolio. We simply fill in the parameters in the equation: **E**[n] = y ∙ rb ∙ j ∙ **E**[tr]. But this will not give us how much money we will be making, only that we will execute the estimated number of trades and that this estimate will be relatively close to the actual executed trades. The first running strategy simulation of our trading program will give us the needed turnover estimate: **E**[tr].

It is a piece of the puzzle, but it still lacks value. This value will have to come from what is being traded at rebalancing time and its preset bet size function. You will have to do with what is available in any of those time slices. You know you will not be able to extract blood from a rock. However, what you can do is select tradable stocks for the purpose you have in mind, and that purpose is to make some money on most of the trades you engage in, or at best, on the overall scheduled rebalancing operations

.Sept. 1, 2020, © Guy R. Fleury. All rights reserved.