__Characteristics of Fixed Fractional Trading and Salutary Techniques__

How does a very small account, an account that is going to start out trading 1 contract, use the optimal f approach? One suggestion is that such an account start out by trading 1 contract not for every optimal f amount in dollars (biggest loss/-f), but rather that the drawdown and margin must be considered in the initial phase. The amount of funds allocated towards the first contract should be the greater of the optimal f amount in dollars or the margin plus the maximum historic drawdown (on a 1-unit basis):

A = MAX {(Biggest Loss/-f), (Margin+ABS(Drawdown))}

where,

A = The dollar amount to allocate to the first contract.

f = The optimal f (0 to 1).

Margin = The initial speculative margin for the given contract.

Drawdown = The historic maximum drawdown.

MAX{} = The maximum value of the bracketed values.

ABS() = The absolute value function.

With this procedure an account can experience the maximum draw-down again and still have enough funds to cover the initial margin on another trade. Although we cannot expect the worst-case drawdown in the future not to exceed the worst-case drawdown historically, it is rather unlikely that we will start trading right at the beginning of a new historic drawdown. A trader utilizing this idea will then subtract the amount in Equation from his or her equity each day. With the remainder, he or she will then divide by (Biggest Loss/-f). The answer obtained will be rounded down to the integer, and 1 will be added. The result is how many contracts to trade.

An example may help clarify. Suppose we have a system where the optimal f is .4, the biggest historical loss is -$3,000, the maximum drawdown was -$6,000, and the margin is $2,500. Employing Equation then:

A = MAX{( -$3,000/-.4), ($2,500+ABS( -$6,000))}

= MAX(($7,500), ($2,500+$6,000))

= MAX($7,500, $8,500)

= $8,500

We would thus allocate $8,500 for the first contract. Now suppose we are dealing with $22,500 in account equity. We therefore subtract this first contract allocation from the equity:

$22,500-$8,500 = $14,000

We then divide this amount by the optimal fin dollars:

$14,000/$7,500 = 1.867

Then we take this result down to the integer:

INT( 1.867) = 1

and add 1 to the result (the 1 contract represented by the $8,500 we have subtracted from our equity):

1+1 = 2

We therefore would trade 2 contracts. If we were just trading at the optimal f level of 1 contract for every $7,500 in account equity, we would have traded 3 contracts ($22,500/$7,500). As you can see, this technique can be utilized no matter of how large an account's equity is (yet the larger the equity the closer the two answers will be). Further, the larger the equity, the less likely it is that we will eventually experience a drawdown that will have us eventually trading only 1 contract. For smaller accounts, or for accounts just starting out, this is a good idea to employ.

**THRESHOLD TO GEOMETRIC**

Here is another good idea for accounts just starting out, one that may not be possible if you are employing the technique just mentioned. This technique makes use of another by-product calculation of optimal f called the threshold to geometric. The by-products of the optimal f calculation include calculations, such as the TWR, the geometric mean, and so on, that were derived in obtaining the optimal f, and that tell us something about the system. The threshold to the geometric is another of these by-product calculations. Essentially, the threshold to geometric tells us at what point we should switch over to fixed fractional trading, assuming we are starting out constant-contract trading. Refer back to the example of a coin toss where we win $2 if the toss comes up heads and we lose $1 if the toss comes up tails. We know that our optimal f is .25, or to make 1 bet for every $4 we have in account equity.

If we are starting out trading on a constant-contract basis, we know we will average $.50 per unit per play. However, if we start trading on a fixed fractional basis, we can expect to make the geometric average trade of $.2428 per unit per play. Assume we start out with an initial stake of $4, and therefore we are making 1 bet per play. Eventually, when we get to $8, the optimal f would have us step up to making 2 bets per play. However, 2 bets times the geometric average trade of $.2428 is $.4856. Wouldn't we be better off sticking with 1 bet at the equity level of $8, whereby our expectation per play would still be $.50? The answer is, "Yes." The reason that the optimal f is figured on the basis of contracts that are infinitely divisible, which may not be the case in real life.

We can find that point where we should move up to trading two contracts by the formula for the threshold to the geometric, T:

T = AAT/GAT*Biggest Loss/-f

where

T = The threshold to the geometric.

AAT = The arithmetic average trade.

GAT s The geometric average trade,

f = The optimal f (0 to 1).

In our example of the 2-to-l coin toss:

T = .50/.2428*-1/-.25 = 8.24

Therefore, we are better off switching up to trading 2 contracts when our equity gets to $8.24 rather than $8.00. Shows the threshold to the geometric for a game with a 50% chance of winning $2 and a 50% chance of losing $1.

Notice that the trough of the threshold to the geometric curve occurs at the optimal f. This means that since the threshold to the geometric is the optimal level of equity to go to trading 2 units, you go to 2 units at the lowest level of equity, optimally, when incorporating the threshold to the geometric at the optimal f.

Now the question is, "Can we use a similar approach to know when to go from 2 cars to 3 cars?" Also, 'Why can't the unit size be 100 cars starting out, assuming you are starting out with a large account, rather than simply a small account starting out with 1 car?" To answer the second question first, it is valid to use this technique when starting out with a unit size greater than 1. However, it is valid only if you do not trim back units on the downside before switching into the geometric mode. The reason is that before you switch into the geometric mode you are assumed to be trading in a constant-unit size.

Assume you start out with a stake of 400 units in our 2-to-l coin-toss game. Your optimal fin dollars is to trade 1 contract for every $4 in equity. Therefore, you will start out trading 100 contracts on the first trade. Your threshold to the geometric is at $8.24, and therefore you would start trading 101 contracts at an equity level of $404.24. You can convert your threshold to the geometric, which is computed on the basis of advancing from 1 contract to 2, as:

Converted T = EQ+T-(Biggest Loss/-f)

where

EQ = The starting account equity level.

T = The threshold to the geometric for going from 1 car to 2.

f = The optimal f (0 to 1).

Therefore, since your starting account equity is $400, your T is $8.24, your biggest loss -$1, and your f is .25:

Converted T = 400+8.24-(-1/-.25)

= 400+8.24-4

= 404.24

Thus, you would progress to trading 101 contracts if and when your account equity reached $404.24. We will assume you are trading in a constant-contract mode until your account equity reaches $404.24, at which point you will begin the geometric mode. Therefore, until Your account equity reaches $404.24, you will trade 100 contracts on the next trade regardless of the remaining equity in your account. If, after you cross the geometric threshold (that is, after your account equity hits S404.24), you suffer a loss and your equity drops below $404.24, you will go back to trading on a constant 100-contract basis if and until you cross the geometric threshold again.

This inability to trim back contracts on the downside when you are below the geometric threshold is the drawback to using this procedure when you are at an equity level of trading more than 2 contacts. If you are only trading 1 contract, the geometric threshold is a very valid technique for determining at what equity level to start trading 2 contracts. However, it is not a valid technique for advancing from 2 contracts to 3, because the technique is predicated upon the fact that you are currently trading on a constant-contract basis. That is, if you are trading 2 contracts, unless you are willing not to trim back to 1 contract if you suffer an equity decline, the technique is not valid, and likewise if you start out trading 100 contracts.

You could do just that, in which case the threshold to the geometric, or its converted version in Equation, would be the valid equity point to add the next contract. The problem with doing this in an asymptotic sense. You will not make as much as if you simply traded the full optimal f. Further, your drawdowns will be greater and your risk of ruin higher. Therefore, the threshold to the geometric is only beneficial if you are starting out in the lowest denomination of bet size (1 contract) and advancing to 2, and it is only a benefit if the arithmetic average trade is more than twice the size of the geometric average trade. Furthermore, it is beneficial to use only when you cannot trade fractional units.

**ONE COMBINED BANKROLL VERSUS SEPARATE BANK-ROLLS**

Some very important points regarding fixed fractional trading must be covered before we discuss the parametric techniques. First, when trading more than one market system simultaneously, you will generally do better in an asymptotic sense using only one combined bankroll from which to figure your contract sizes, rather than separate bankrolls for each. It is for this reason that we "recapitalize" the subaccounts on a daily basis as the equity in an account fluctuates. What follows is a run of two similar systems, System A and System B.

Both have a 50% chance of winning, and both have a payoff ratio of 2:1. Therefore, the optimal f dictates that we bet $1 for every S4 units in equity. The first run we see shows these two systems with positive correlation to each other. We start out with $100, splitting it into 2 subaccount units of $50 each. After a trade is registered, it only affects the cumulative column for that system, as each system has its own separate bankroll. The size of each system's separate bankroll is used to determine bet size on the subsequent play:

Both have a 50% chance of winning, and both have a payoff ratio of 2:1. Therefore, the optimal f dictates that we bet $1 for every S4 units in equity. The first run we see shows these two systems with positive correlation to each other. We start out with $100, splitting it into 2 subaccount units of $50 each. After a trade is registered, it only affects the cumulative column for that system, as each system has its own separate bankroll. The size of each system's separate bankroll is used to determine bet size on the subsequent play:

Now we will see the same thing, only this time we will operate from a combined bank starting at 100 units. Rather than betting $1 for every $4 in the combined stake for each system, we will bet $1 for every $8 in the combined bank. Each trade for either system affects the combined bank, and it is the combined bank that is used to determine bet size on the subsequent play:

Notice that using either a combined bank or a separate bank in the preceding example shows a profit on the $100 of $42.38. Yet what was shown is the case where there is positive correlation between the two systems. Now we will look at negative correlation between the same two systems, first with both systems operating from their own separate bankrolls:

As you can see, when operating from separate bankrolls, both systems net out making the same amount regardless of correlation. However, with the combined bank:

With the combined bank, the results are dramatically improved. When using fixed fractional trading you are best off operating from a single combined bank.

**THREAT EACH PLAY AS IF INFINITELY REPEATED**

The next axiom of fixed fractional trading regards maximizing the current event as though it were to be performed an infinite number of times in the future. We have determined that for an independent trials process, you should always bet that f which is optimal and likewise when there is dependency involved, only with dependency f is not constant.

Suppose we have a system where there is dependency in like begetting like, and suppose that this is one of those rare gems where the confidence limit is at an acceptable level for us, that we feel we can safely assume that there really is dependency here. For the sake of simplicity we will use a payoff ratio of 2:1. Our system has shown that, historically, if the last play was a win, then the next play has a 55% chance of being a tin. If the last play was a loss, our system has a 45% chance of the next play being a loss. Thus, if the last play was a win, then from the Kelly formula, Equation, for finding the optimal f :

f = ((2 +1)*.55-1)/2

= (3*.55-1)/2

= .65/2

= .325

After a losing play, our optimal f is:

f = ((2+ l)*.45-l)/2

= (3*.45- l)/2

= .35/2

= .175

Now dividing our biggest losses (-1) by these negative optimal fs dictates that we make 1 bet for every 3.076923077 units in our stake after a win, and make 1 bet for every 5.714285714 units in our stake after a loss. In so doing we will maximize the growth over the long run. Notice that we treat each individual play as though it were to be performed an infinite number of times.

Notice in this example that betting after both the wins and the losses still has a positive mathematical expectation individually. What if, after a loss, the probability of a win was .3? In such a case, the mathematical expectation is negative, hence there is no optimal f and as a result you shouldn't take this play:

ME = (.3*2)+ (.7*-1)

= .6-.7

= -.1

= -.1

In such circumstances, you would bet the optimal amount only after a win, and you would not bet after a loss. If there is dependency present, you must segregate the trades of the market system based upon the dependency and treat the segregated trades as separate market systems. The same principle, namely that asymptotic growth is maximized if each play is considered to be performed an infinite number of times into the future, also applies to simultaneous wagering. Consider two betting systems, A and B. Both have a 2:1 payoff ratio, and both win 50% of the time.

We will assume that the correlation coefficient between the two systems is 0, but that is not relevant to the point being illuminated here. The optimal fs for both systems are .25, or to make 1 bet for every 4 units in equity. The optimal fs for trading both systems simultaneously are .23, or 1 bet for every 4.347826087 units in account equity.1 System B only trades two-thirds of the time, so some trades will be done when the two systems are not trading simultaneously. This first sequence is demonstrated with a starting combined bank of 1,000 units, and each bet for each system is performed with an optimal f of 1 bet per every 4.347826087 units:

Next we see the same exact thing, the only difference being that when A is betting alone (i.e., when B does not have a bet at the same time as A), we make 1 bet for every 4 units in the combined bank for System A, since that is the optimal f on the single, individual play. On the plays where the bets are simultaneous, we are still betting 1 unit for every 4.347826087 units in account equity for both A and B. Notice that in so doing we are taking each bet, whether it is individual or simultaneous, and applying that optimal f which would maximize the play as though it were to be performed an infinite number of times in the future.

As can be seen, there is a slight gain to be obtained by doing this, and the more trades that elapse, the greater the gain. The same principle applies to trading a portfolio where not all components of the portfolio are in the market all the time. You should trade at the optimal levels for the combination of components that results in the optimal growth as though that combination of components were to be traded an infinite number of times in the future.

**EFFICIENCY LOSS IN SIMULTANEOUS WAGERING OR PORTFOLIO TRADING**

Let's again return to our 2:1 coin-toss game. Let's again assume that we are going to play two of these games, which we'll call System A and System B, simultaneously and that there is zero correlation between the outcomes of the two games. We can determine our optimal fs for such a case as betting 1 unit for every 4.347826 in account equity when the games are played simultaneously. When starting with a bank of 100 units, notice that we finish with a bank of 156.86 units:

Now let's consider System C. This would be the same as Systems A and B, only we're going to play this game alone, without another game going simultaneously. We're also going to play it for 8 plays-as opposed to the previous endeavor, where we played 2 games for 4 simultaneous plays. Now our optimal f is to bet 1 unit for every 4 units in equity. What we have is the same 8 outcomes as before, but a different, better end result:

The end result here is better not because the optimal fs differ slightly, but because there is a small efficiency loss involved with simultaneous wagering. This inefficiency is the result of not being able to recapitalize your account after every single wager as you could betting only 1 market system. In the simultaneous 2-bet case, you can only recapitalize 3 times, whereas in the single B-bet case you recapitalize 7 times. Hence, the efficiency loss in simultaneous wagering.

We just witnessed the case where the simultaneous bets were not correlated. Let's look at what happens when we deal with positive (+1.00) correlation:

Notice that after 4 simultaneous plays where the correlation between the market systems employed is+1.00, the result is a gain of 126.56 on a starting stake of 100 units. This equates to a TWR of 1.2656, or a geometric mean, a growth factor per play (even though these are combine plays) of 1.2656^(1/4) = 1.06066.

Now refer back to the single-bet case. Notice here that after 4 plays, the outcome is 126.56, again on a starting stake of 100 units. Thus, the geometric mean of 1.06066. This demonstrates that the rate of growth is the same when trading at the optimal fractions for perfectly correlated markets. As soon as the correlation coefficient comes down below+1.00, the rate of growth increases. Thus, we can state that when combining market systems, your rate of growth will never be any less than with the single-bet case, no matter of how high the correlations are, provided that the market system being added has a positive arithmetic mathematical expectation.

Recall the first example in this section, where there were 2 market systems that had a zero correlation coefficient between them. This market system made 156.86 on 100 units after 4 plays, for a geometric mean of (156.86/100)^(1/4) = 1.119. Let's now look at a case where the correlation coefficients are -1.00. Since there is never a losing play under the following scenario, the optimal amount to bet is an infinitely high amount. But, rather than getting that greedy, we'll just make 1 bet for every 4 units in our stake so that we can make the illustration here:

There are two main points to glean from this section. The first is that there is a small efficiency loss with simultaneous betting or portfolio trading, a loss caused by the inability to recapitalize after every individual play. The second point is that combining market systems, provided they have a positive mathematical expectation, and even if they have perfect positive correlation, never decreases your total growth per time period. However, as you continue to add more and more market systems, the efficiency loss becomes considerably greater. If you have, say, 10 market systems and they all suffer a loss simultaneously, that loss could be terminal to the account, since you have not been able to trim back size for each loss as you would have had the trades occurred sequentially.

There are two main points to glean from this section. The first is that there is a small efficiency loss with simultaneous betting or portfolio trading, a loss caused by the inability to recapitalize after every individual play. The second point is that combining market systems, provided they have a positive mathematical expectation, and even if they have perfect positive correlation, never decreases your total growth per time period. However, as you continue to add more and more market systems, the efficiency loss becomes considerably greater. If you have, say, 10 market systems and they all suffer a loss simultaneously, that loss could be terminal to the account, since you have not been able to trim back size for each loss as you would have had the trades occurred sequentially.

Therefore, we can say that there is a gain from adding each new market system to the portfolio provided that the market system has a correlation coefficient less than 1 and a positive mathematical expectation, or a negative expectation but a low enough correlation to the other components in the portfolio to more than compensate for the negative expectation. There is a marginally decreasing benefit to the geometric mean for each market system added. That is, each new market system benefits the geometric mean to a lesser and lesser degree. Further, as you add each new market system, there is a greater and greater efficiency loss caused as a result of simultaneous rather than sequential outcomes. At some point, to add another market system will do more harm then good.

**TIME REQUIRED TO REACH A SPECIFIED GOAL AND THE TROUBLE WITH FRACTIONAL F**

Suppose we are given the arithmetic average HPR and the geometric average HPR for a given system. We can determine the standard deviation in HPRs from the formula for estimated geometric mean:

EGM = (AHPR^2-SD^2)^(1/2)

where,

AHPR = The arithmetic mean HPR.

SD = The population standard deviation in HPRs.

Therefore, we can estimate the standard deviation, SD, as:

SD^2 = AHPR^2-EGM^2

Returning to our 2:1 coin-toss game, we have a mathematical expectation of $.50, and an optimal f of betting $1 for every $4 in equity, which yields a geometric mean of 1.06066. We can use Equation to determine our arithmetic average HPR:

AHPR = l+(ME/f$)

where,

AHPR = The arithmetic average HPR.

ME = The arithmetic mathematical expectation in units.

f$ = The biggest loss/-f. f = The optimal f (0 to 1).

Thus, we would have an arithmetic average HPR of:

AHPR = 1+(.5/( -1/ -.25))

= 1+(.5/4)

= 1+.125

= 1.125

Now, since we have our AHPR and our ECM, we can employ equation to determine the estimated standard deviation in the HPRs:

SD^2 = AHPR^2-EGM^2

= 1.125^2-1.06066^2

= 1.265625-1.124999636

= .140625364

Thus SD^2, which is the variance in HPRs, is .140625364. Taking the Square root of this yields a standard deviation in these HPRs of .140625364^(1/2) = .3750004853. You should note that this is the estimated standard deviation because it uses the estimated geometric mean as input. It is probably not completely exact, but it is close enough for our purposes.

However, suppose we want to convert these values for the standard deviation (or variance), arithmetic, and geometric mean HPRs to reflect trading at the fractional f. These conversions are now given:

FAHPR = (AHPR-1)*FRAC+1

FSD = SD*FRAC

FGHPR = (FAHPR^2-FSD^2)^(1/2)

where,

FRAC = The fraction of optimal f we are solving for.

AHPR = The arithmetic average HPR at the optimal f.

SD = The standard deviation in HPRs at the optimal f.

FAHPR = The arithmetic average HPR at the fractional f.

FAHPR = The arithmetic average HPR at the fractional f.

FSD = The standard deviation in HPRs at the fractional f

FGHPR = The geometric average HPR at the fractional f.

FGHPR = The geometric average HPR at the fractional f.

For example, suppose we want to see what values we would have for FAHPR, FGHPR, and FSD at half the optimal f (FRAC = .5) in our 2:1 coin-toss game. Here, we know our AHPR is 1.125 and our SD is .3750004853. Thus:

FAHPR = (AHPR-1)*FRAC+1

= (1.125- 1)*.5+1

= .125*.5+1

= .0625+1

= 1.0625

FSD = SD*FRAC

= ,3750004853*.5

= .1875002427

FGHPR = (FAHPR^2-FSD^2)^(1/2)

= (1.0625^2-.1875002427^2)^(1/2)

= (1.12890625-.03515634101)^(1/2)

= 1.093749909^(1/2)

= 1.04582499

Thus, for an optimal f of .25, or making 1 bet for every $4 in equity, we have values of 1.125, 1.06066, and .3750004853 for the arithmetic average, geometric average, and standard deviation of HPRs respectively. Now we have solved for a fractional (.5) f of .125 or making 1 bet for every $8 in our stake, yielding values of 1.0625, 1.04582499, and .1875002427 for the arithmetic average, geometric average, and standard deviation of HPRs respectively. We can now take a look at what happens when we practice a fractional f strategy.

We have already determined that under fractional f we will make geometrically less money than under optimal f. Further, we have determined that the drawdowns and variance in returns will be less with fractional f. What about time required to reach a specific goal? We can quantify the expected number of trades required to reach a specific goal. This is not the same thing as the expected time required to reach a specific goal, but since our measurement is in trades we will use the two notions of time and trades elapsed interchangeably here:

We have already determined that under fractional f we will make geometrically less money than under optimal f. Further, we have determined that the drawdowns and variance in returns will be less with fractional f. What about time required to reach a specific goal? We can quantify the expected number of trades required to reach a specific goal. This is not the same thing as the expected time required to reach a specific goal, but since our measurement is in trades we will use the two notions of time and trades elapsed interchangeably here:

N = ln(Goal)/ln(Geometric Mean)

where,

N = The expected number of trades to reach a specific goal.

Goal = The goal in terms of a multiple on our starting stake, a TWR.

ln() = The natural logarithm function.

Returning to our 2:1 coin-toss example. At optimal f we have a geometric mean of 1.06066, and at half f this is 1.04582499. Now let's calculate the expected number of trades required to double our stake (goal = 2). At full f:

N = ln(2)/ln( 1.06066) = .6931471/.05889134 = 11.76993

Thus, at the full f amount in this 2:1 coin-toss game, we anticipate it will take us 11.76993 plays (trades) to double our stake. Now, at the half f amount:

N = ln(2)/ln(1.04582499) = .6931471/.04480602 = 15.46996

Thus, at the half f amount, we anticipate it will take us 15.46996 trades to double our stake. In other words, trading half f in this case will take us 31.44% longer to reach our goal. Well, that doesn't sound too bad. By being more patient, allowing 31.44% longer to reach our goal, we eliminate our drawdown by half and our variance in the trades by half. Half f is a seemingly attractive way to go. The smaller the fraction of optimal f that you use, the smoother the equity curve, and hence the less time you can expect to be in the worst-case drawdown.

Now, let's look at it in another light. Suppose you open two accounts, one to trade the full f and one to trade the half f. After 12 plays, your full f account will have more than doubled to 2.02728259 (1.06066^12) times your starting stake. After 12 trades your half f account will have grown to 1.712017427 (1.04582499^12) times your starting stake. This half f account will double at 16 trades to a multiple of 2.048067384 (1.04582499^16) times your starting stake. So, by waiting about one-third longer, you have achieved the same goal as with full optimal f, only with half the commotion.

However, by trade 16 the full f account is now at a multiple of 2.565777865 (1.06066^16) times your starting stake. Full f will continue to pull out and away. By trade 100, your half f account should be at a multiple of 88.28796546 times your starting stake, but the full f will be at a multiple of 361.093016! So anyone who claims that the only thing you sacrifice with trading at a fractional versus full f is time required to reach a specific goal is completely correct. Yet time is what it's all about. We can put our money in Treasury Bills and they will reach a specific goal in a certain time with an absolute minimum of drawdown and variance! Time truly is of the essence.

However, by trade 16 the full f account is now at a multiple of 2.565777865 (1.06066^16) times your starting stake. Full f will continue to pull out and away. By trade 100, your half f account should be at a multiple of 88.28796546 times your starting stake, but the full f will be at a multiple of 361.093016! So anyone who claims that the only thing you sacrifice with trading at a fractional versus full f is time required to reach a specific goal is completely correct. Yet time is what it's all about. We can put our money in Treasury Bills and they will reach a specific goal in a certain time with an absolute minimum of drawdown and variance! Time truly is of the essence.