## Basic Portfolio Math II¶

The stock market game is relatively simple. You buy some shares, hold them, or resell them later. You intend to invest in worthwhile companies for the duration of whatever holding period you see fit. The main objective remains, in either case, to make a profit. However, this profit should be looked at from a long-term perspective. If you trade, it is not just one trade that you should be concerned with. It is the net result of the sum of all the trades executed over the life of a portfolio. It will be the net balance of the trading account that will matter. The rest might, like which path you took to get there, be just memories of the journey.

$\textbf{Basic Portfolio Math}$ ended with the mathematical expression for a ten-bagger portfolio:$$CAGR = \displaystyle{\left(\frac{10 \cdot F_0}{F_0}\right)^{1/t}-1} = \left(\frac{F_0 + n \cdot \bar x}{F_0 }\right)^{1/t}-1$$and where it was said to put the emphasis on finding a suitable $\,\bar x > 0\,$ that could be repeated $\,n\,$ times to achieve the expected $CAGR$.

This tends to slightly change the game. It is not only finding an average tradable edge: $\,\bar x > 0,\,$ the average net profit per trade, it is also finding the means to execute it $\,n\,$ times over the trading interval $\,t,\,$ and this, within the portfolio's resource management constraints.

The above expression was easily deduced from the simplyfied portfolio payoff matrix value function: $F(t) = F_0 + n \cdot \bar x.$ This payoff could also have been expressed as a matrix integral: $$\displaystyle{ F(t) = F_0 + n \cdot \bar x = F_0 + \int _0^{T}\mathbf{H} \cdot d\mathbf{P} = F_0 + \sum_0^n \mathbf{H} \cdot \Delta\mathbf{P}}$$ These three equalities can help make long-term CAGR estimates since they are all-inclusive. At least, they provide some fundamental basis on which to determine what it might take to get to termination time $T$.

All is accounted for in the above expressions, whatever we do in the trading department or in investing for that matter. Whatever the trading strategy, all the effort should be put on generating a long-term positive edge: $\,\bar x > 0.$ Once we have that, we should seek out how many times it could be executed in real life.

Since $\,n\,$ will tend to be a large number, we should see signs that the law of large numbers will have its importance since from it we will derive that $\,n \cdot \bar x\,$ does tend to a constant: $\,n \cdot \bar x \to constant\,$ for a particular trading strategy.

Start the 10-bagger portfolio example with an initial capital: $F_0 = \$ 1,000,000.\,$Make the objective to reach:$\,10 \cdot F_0 = \$10,000,000\,$ within $\, T = 20\,$ years. This will translate into finding a solution for: $\, n \cdot \bar x = 9 \cdot F_0.\,$ We could take the value of $\,n\,$ and $\,\bar x\,$ from our own trading strategies to make such long-term estimates. Every portfolio simulation software out there give these numbers on every simulation they do. And, any combination of $\,n\,$ times $\,\bar x\,$ equal $\,9 \cdot F_0\,$ will be a solution no matter the trading methods used.

Calculations for the above gives a $CAGR$ estimate of: $\;\hat g = 10^{(1/20)} -1 = 0.1220.\,$ This 10-bagger portfolio would have needed the equivalent of a 12.20$\, \%$ CAGR over those 20 years. Not that high a figure all considered.

Had we started with $F_0 = \$ 10,000,000\,$instead. The outcome at a 12.20$\,\%$CAGR would be:$\, F(t) = 10 \cdot F_0 = \$100,000,000.$ This is the power of compounding at work. Evidently, the size of $\,F_0\,$ matters. It is worth it to make the extra effort to increase the initial capital to be traded. Of course, it goes on the premise that you do have a trading strategy with a positive edge: $\,\bar x > 0.\,$ Otherwise, why trade?

Just as you can scale $\,F_0\,$ up, you can also scale it down, use the formula: $\,F_0 \cdot ( 1 + \hat g)^t$.

### Time¶

Should we be able to reduce the time interval to something like 10 years. Then, things do change.

We still have the same job to do but in half the time.

This would produce: $\, \hat g = 10^{(1/10)} -1 = 0.2589$. Thus, we would have to increase the CAGR to 25.89$\,\%$ to achieve the same outcome as before. But this time, within 10 years.

To execute either scenarios, we have: $\;n \cdot \bar x = 9 \cdot F_0.\,$ That it be done in 10 or 20 years, even if time is implicit in this equality, the expression does say what is needed to make it feasible: new and/or improved trading methods.

Intuitively, we would say that if we trade half the time interval, it should produce about half as much. However, this is a compounding game and it is expressed in terms of CAGR. For instance: $F_0 \cdot (1 + 0.1220)^{20} = 3.16 \cdot F_0 \cdot (1 + 0.1220)^{10}.\,$ If we increase the compounding rate, we could have: $F_0 \cdot (1 + 0.2589)^{20} = 9.99 \cdot F_0 \cdot (1 + 0.2589)^{10}$. This is definitely more that just doubling.

A particular trading strategy might not be able to do both. However, one would choose the higher CAGR scenario with the longer time interval for a specific trading strategy.

Every strategy has its own set of trading characteristics. One of which might be how much it trades, on average, per time unit: $\displaystyle{ \frac{ n \cdot \bar x}{\Delta t}}.\,$ A kind of signature related to its coded procedures and trading methods.

Take for example a market neutral portfolio which intends to hold, on average, some $300$ stocks. Let it rebalance every day based on its particular set of constraints (one of which is market neutrality, half the stocks are long while the other half is short). From 60$\,\%$ to 100$\,\%$ of positions might have adjustments made due to the quasi-randomly changing portfolio weights. Let's make it $200$ stocks in the portfolio have trades, on average, on a daily basis.

This sets the average number of trades to: $\, \bar n = 200.\,$ Over 20 years, some: $\,200 \cdot 252 \cdot 20 = 1,008,000\,$ trades could be executed. This would make $\,\bar x\,= \$ 8.93$per trade to achieve the 10-bagger status over those 20 years. Certainly not much to ask as profit per trade. Note that even if$\, \bar x\,$is relatively small,$ \$9$ M in profits is still money. Furthermore, a machine is doing the work. It could have been $\$ 90$M had we started with$ \$10$ M instead.

Based on the $\$ 1$M setup, the starting bet size would have been:$\;q \cdot p = \frac{1}{300} \cdot F_0 = \$3,333.33$. The initial average percent profit per trade would have been: $\frac{8.93}{3333.33} = 0.268$ $\%$. A quarter of a one percent move. The strategy might not be requiring much but this is making it sensitive to frictional costs, slippage, and trading expenses. A good point, at least in this analysis, is that $\, \bar x\,$ is net of expenses.

As the portfolio grows, so does the bet size. In the end, it will be: $\frac{1}{300} \cdot F(t) = \$ 33,333.33 \,$making the average$ \$8.93$ per trade represent a 0.0268 $\%\,$ profit per trade. The bet size grew gradually over the life of the portfolio. It was controlled by initially setting the total number of stocks to be traded at $300$. The bet size is growing at the same exponential rate as $\,\hat g\,$. We should note that the profit per trade also grew over the trading interval.