May 14, 2009

The Buy & Hold strategy will build equity V(t) at an expected rate of return that will be close to the expected average secular market return:


At least, that is what you should expect. It is, for instance, the whole idea behind index funds. However, the Buy & Hold, as is, is quite inefficient.

The main ingredient of a portfolio is money, and it is how it should be put to use where management comes into play. And this translates to: how should the game be played to optimize the use of available capital as time evolves?

The following equation is my attempt to answer such a question.

Portfolio Formula

(click to enlarge)

The above is permissible under the capital constraints of available equity:


where the equity V(t) builds at a rate λ over the investment period t. As such, λ will be the effective rate of increase generated by the combined efforts of all trading components implemented.


which simply restates:


Even under equity constraints, the formula has two main parts: the accumulation program and the added trading components, which have the main function of supplying additional funds to the accumulation program itself. It's the equity curve that will limit all profit-generating functions. In a scenario where prices, over the long haul, appreciate on average at a 10% rate, the above formula seems to allow a whole lot more by its sheer composition.

Formula Components

Total equity is made available for trading. Funds can bear interest at the risk-free rate while idle. The main objective of the formula is to accumulate shares over time by using the excess equity buildup (making the generated profits also work).

Opened Longs:


The sum of the net profit generated by the accumulated and still opened long positions is governed by the appreciation in price and the rate of increase in the holding inventory. See both my papers, Alpha Power and Jensen Modified Sharpe Ratio, for a more elaborate view of this equation. The profits are generated by increasing the quantity on hand as the price evolves upwards with time. By not increasing inventory, that is making g, it will transform the above equation into a Buy & Hold formulation.

Closed Longs and Shorts:

The sum of net profits generated by the closed long positions provides more capital for the accumulation process.


This equation can be summarized as the number of trades executed times the average profit per trade. The equation also states that as equity increases, the quantity traded should also increase, preferably at an exponential rate, since the equity itself is on a power function. For net profits to be positive requires a trading edge meaning that, on average, a profit is generated. However, once a trading edge has been found, repeat business seems to be the mantra. Doing 10 trades or doing 1,000 can have a major impact on this equation. Therefore, it is not sufficient to find a trading edge; it is more important to find one that is highly repeatable. The same notions apply to the net profits generated by the short side.

Option writing:

The addition of an option writing component simply adds more profits. Since the main purpose of this equity function is to accumulate shares and thereby build an inventory of rising stocks, they might as well produce even more income by using a covered call-writing program. And since there will be unused excess equity, why not use it.

Options are written on the whole inventory on hand. For all called stocks, you repurchase the same amount of stocks and re-issue a new option at a higher strike. And over the investment period, you will have collected what amounts to the option's premium. Here again, as equity rises, inventory rises, and thereby the quantity of contracts rises as well.

I have not included the still-opened long-term short positions in this formula. The reason mostly being that there are only a few available and the expected possible return is less than for the upside. It might have taken less than a year to have Lehman go under, but it took over a hundred years to get there; I'm not that patient. However, such a scenario should be more appropriately handled by short-term short trading procedures.


So, there you have it. This modified Buy & Hold equation has already been provided as part of equation (16) in my first paper: Alpha Power

Equation 16

which looked at a long-term solution to increasing a portfolio at the highest possible rate of return under the constraints of limited capital and an unknown future.

It is by looking at a total solution of what should be done, not only at the beginning of a portfolio but also all along the process of building it that we can get a better view of the possibilities.

The above formula suggests that the Buy & Hold is not dead; it only needs to be adapted for modern times. The Buy & Hold, however, is not a total solution as trading, long and short, can complement the process just as option writing. It is by doing all those things that one can enhance performance to a point it far exceeds the conventional Buy & Hold.

If you take out the volume accelerators from the equation, you will be left with the classic portfolio equation, which states that the average profit per trade times the number of trades is your total profit. The innovation in my formula is in the volume accelerators, which solve many portfolio problems.

Portfolio management has seen many methods trying to optimize performance: Kelly number, optimal-f, fixed ratio, fixed amount, variable ratio, and many others. But most have a deficiency or other. The Kelly number and optimal-f presume that your win rate is constant, which is not the case. The fixed ratio and variable ratio tend to get too risky as all trades are not created equal and should not be treated as such. The fixed amount will underperform as the portfolio grows.

So with one small "innovation" - the volume accelerators - you solve all those questions in a single swoop. You let the market decide who will survive and thrive. It would be your Darwinian approach, where you feed the strong and starve the weak, which will make this a performance reinforcement method that will outperform the market itself.

Created on ... May 14, 2009,   © Guy R. Fleury. All rights reserved.