The Leverage Space Model is a new and superior method for portfolio management that seeks to maximize geometric growth within a given drawdown constraint. This is in stark contrast to more conventional models, which seek to maximize return vs. the variance in those returns.
But before the Leverage Space Model can be explained, we must lay the groundwork. This will be done using a single component case, focusing on the basic concepts. In the second part of this series, we will expand the discussion to include multiple components.
When we speak of a “tradable instrument” it can be stock, a future, an option or a leased car. Tradable instruments have the following attributes: it can be held long or short, it has a yield that can be positive or negative, it has a base currency and it is affected by events.
In addition, there is a timeline, extending forward. This is called the trajectory (the value of the instrument with respect to time) that we expect the instrument to follow. The trajectory is affected by events such as maturity dates, dividends, option expiration, etc., which can be predictable or not, and it is the latter that are of particular concern to the trader. Indeed, the unpredictability of events creates a cone-shaped object, rather than a solitary line, that the trajectory follows (see “Trajectory cone”).
This conic-shaped object is, in reality, more of a bell-shaped manifold across three axes. This implies that if we take discrete snapshots along it, we get something of a probability distribution at each point along the cone (see “Value distributions”). It’s also important to note that these distributions, though seemingly smooth, can be transformed into a series of discrete bins, each bin representing a scenario and outcome with its own probability.
Of course, while the trajectory represents the mode of the probability functions along it and is the single most likely outcome, the probability functions themselves are affected by possible events. In real life, price trajectories are not smooth, but they follow varying points within the cone.
The trader’s exercise traditionally has been to determine a positive mode: the single most likely outcome for this distribution. Going a step further, he may also attempt to determine confidence intervals along the cone for purposes of establishing price targets and stop-loss levels.
However, in either case, the trader is not using the probabilities of that information to direct how much quantity to risk on any specific trade. This is what we will achieve with the Leverage Space Model, which uses these binned distributions, at a given point in time, to determine how much quantity to have on the table.
OPTIMAL F
This brings us to the notion of mathematical expectation, which is the expected value of the entire binned distribution. Mathematical expectation is determined by the following formula:
Where: Pi = The probability associated with the i’th scenario.
Ai = The result of the i’th scenario.
N = The total number of scenarios under consideration.
Essentially, to figure the mathematical expectation of a binned distribution, we take the sum of each bin’s outcome times its probability.
Let’s look at a simplified distribution. Often, we can use a random outcome as a proxy for market situations, at least when it comes to establishing a theoretical understanding of a concept; the mathematics are the same. The difference in market probabilities are chronomorphic, which means they change with time.
Consider a game where a coin is tossed. If it comes up tails, we lose $1. If it comes up heads, we gain $2. There are two bins, two scenarios, and each has a 0.5 probability of occurring.
Our mathematical expectation is 0.5 dollars per play ((-1 * 0.5) + (2 * 0.5)). However, even with this simple game, it is not so straightforward (see “Mistaken impression,” below). The vertical axis is a multiple of the stake ultimately earned, based on a fraction of the stake risked. The horizontal axis is the fraction of the stake being put at risk.
This fraction also is referred to as f. The value known as f is a divisor of the biggest perceived loss (worst-case scenario). So often people harp that the problem with this technique is that it is too sensitive to the biggest loss. But biggest loss is used only to bound the value for f between 0 and 1. We could use any arbitrary value in lieu of biggest loss, but then f would be bounded by 0 and some arbitrary value. This would be of little importance however because when you divide this arbitrary number by the resultant f, you would get the same dollar figure to capitalize the trade with as you would by using the biggest loss. The Optimal f has a value between 0 and 1 that is used to determine how many units to trade. A unit can represent a single vehicle (one contract) or some multiple (100 shares, $1 million). So, in our coin-flip example, with an f of 1.0, you would expect to earn a 1.5 multiple on your stake, given a 0.5 mathematical expectation. This multiple, however, can be deceiving.
The line (the function) depicted is 1+ ME * f. It is considered by most as a straight-line function. It is, but only on a one-shot sense. Here’s why:
1. People trade in quantity relative to the size of their stake.
2. An account grows fastest when traded in quantity relative to equity.
3. Other factors determine the quantity people put on, not just the size of the stake. These may include the extent of the worst-perceived case, dependency to past trades, trader aggressiveness, etc.
Most traders gloss over this decision about quantity. They feel that it is somewhat arbitrary, that the quantity doesn’t ultimately matter. What matters, they assume, is that they are right about the direction of the trade.
However, this emphasis on directional trading is patently false. Being right on the direction of the trade over N trades (that is, garnering a positive mathematical expectation from your analysis), is a necessary prerequisite. But it is only a prerequisite, and it by no means assures success.
The trader’s ultimate goal should be to discern the distribution of the scenario bins. A positive mathematical expectation merely serves to say that the mean of the distribution is greater than zero. Although it must be, that is not nearly enough. In fact, the mean or mode of the distribution can be greater than zero, and you can lose with certainty if you are not able to flesh out the distribution of the outcomes and process it correctly.
If we revisit “Mistaken impression,” and we adjust the chart to accommodate multiple, consecutive plays, where the size of our stake on any given play is a result of the amount won or lost to that point—we no longer have a straight line function. When we make subsequent plays, our world is no longer flat, but curved. The reason is because what we have to bet, to trade with, today, is a function of what happened to our stake yesterday (see “Reality strikes”).
Every trader resides somewhere on this f spectrum, where:
f = (number of contracts * biggest perceived loss per contract) / equity in the account.
This is so because the three input variables (the number of contracts being traded at the moment, the biggest perceived loss per contract and the equity in the account) are all given. Thus, at any point in time, we can take any trader trading any market with any approach or system and assign an f value, based on where he resides on a curve similar to those shown in “Reality strikes.”
UNDERSTANDING F
If a trading approach is profitable, there will always be a curve to this function, and it will have just one peak. This is true for any and every profitable approach to a given market. And, as just stated, because the parameters are given for determining where on an f curve an investor is, we cannot only discern what the f curve is, but precisely where on that curve the investor is trading.
Generally, the more profitable the approach on a given market, the more toward the right the peak will be along the curve. Where the trader’s f is with respect to the peak dictates what kind of gain he is facing and what kind of drawdown, among other things. Different locations on this curve have some pretty dramatic consequences.
For example, trading at the peak (0.25 in the case of our example, which means risking $1 for every $4 stake), will mean that when the worst-case loss is hit, the drawdown will be immediate and at least the same percentage as f. Also, risking anything to the right of the peak makes no more than the equivalent value to the left of the peak, but it brings with it the certainty of a greater drawdown.
Consider the significance of f. At an f of 0.1 or 0.4, which are equidistant from the peak, the expected multiple is the same at 4.66 after 40 plays. This is not even half of what the multiple is at 0.25, yet you are only 15% away from the optimal, and only 40 bets have elapsed. In dollar terms, at f = 0.1, you are trading one unit ($1) for every $10 in your stake, and at f = 0.4 you are trading one unit ($1) for every $2.50. At f = 0.25, you are trading one unit ($1) for every $4.
Clearly, it does not pay to over bet. In a 50/50 game where you win twice the amount that you lose, at an f of 0.5, trading one unit for every $2 in your stake, you are only breaking even. At an f greater than 0.5 you lose money, and it is a matter of time until you are wiped out. If your f in this 50/50, 2:1 payoff game is more than 25% beyond the peak, you will go broke with a probability that approaches certainty.
Everyone is trading at some f value. Note that f describes the percentage of a worst-case scenario. It has nothing to do with margin, per se. If your worst-case scenario is, say, losing $10,000 per unit, and you trade one unit in a $100,000 account, then your f is 0.1. Your f is not determined by how much you have on the table but how you define the “worst-case scenario.”
When you dilute f — trade at lesser levels of aggressiveness than the peak — you decrease your drawdowns arithmetically, but you decrease returns geometrically. This difference grows as time goes by. The peak continues to grow and the price for being short of the peak continues to grow.
It’s critical to remember that the largest loss is the amount you could foresee losing if the worst-case scenario manifests. The purpose is to constrain the x-axis to a rightmost value of 1.0. If you used any arbitrary value for this largest loss, you would still have the same shaped curve, with the same, solitary, optimal peak, only the rightmost value on the x-axis would be a value other than 1. There is no getting around the fact that for every trade, there exists such a curve and you are somewhere on it, reaping the rewards and paying the consequences.
This single trade, single instrument application of f is only the starting point, but that application does not necessarily explain the reality for most traders, who are active both over time and in multiple markets at any one time. However, the elements discussed here provide the building blocks for a far superior portfolio model than what’s common today. In the next installment of this series, we will discuss the Leverage Space Model, expanding this concept to a portfolio and demonstrating its value in the real world.
Ralph Vince is a recognized expert in portfolio analysis. He can be reached at rvince99@hotmail.com For a more technical discussion see “The Handbook of Portfolio Mathematics” (John Wiley & Sons, 2007).
