Optimal f. This method of estimating the optimal % of risk has been improved by Raplh Vince. While Kelly’s formula use only average values from past trades, Raplh Vince proposed to take into account all trades, solving the task of optimization of the relative end capital TWR as a function of f.

TWR à Max ₀

where

TWR = P _i=1,..,n (1 – f * Trade_result_i / Max_loss)

We take the negative value of the loss, hence the minus. Actually, this method implies that in the future the trade results will be about the same, but possibly in another order. Solving the TWR maximization, we find the f = f_opt value, where the TWR function reaches its maximum. From f_opt we define position size:

Number_Lots = f_opt * Capital / ( - Max_Loss)

A simple method of calculation is presented in App. 2. According to it the maximal drawdown with optimal f value will be at least f_opt % of account. I.e. if our f_opt is, say, 0.5, then our drawdowns will reach at least 50%. Raplh Vince says that: ”if you are not trading for optimal profits, then you belong in an asylum, not in the market”. Still, he does not consider the fact that a 99% drawdown when trading for an “optimal profit” can land us in asulym – or at least in hospital after trying to explain to the family or investors. It doesn’t help that the capital grows on then.

Besides, the distribution of trade results has a most profound influence on the f_opt value. So f_opt values for two strategies that in the end bring the same profit and have the same maximal loss may be very different.

The weak spot of the optimal f method is that it is fully based on the system’s historical results, on maximal loss to be exact. The risk level set when using f_opt, means we‘ll never have a larger loss.

Unlike gambling, where the outcomes are known and probabilities constant, in trading we have a multitude of random outcomes with undetermined probability of winning. The maximal loss is a nondescreasing step function, with random amplitude leaps occurring at random moments.

So, there is no real evidence to suppose that the maximal loss and maximal drawdown achieved will persist in the future. To calculate f_opt it is possible to use in the TWR formula, instead of the maximal loss a value:

Max_Loss_Evaluation = Avg_Loss – 3.5 Standart_Deviation_of_Loss

But this doesn’t solve the problem yet. The outcome of a future trade is evidently random, so then the optimal f for is must also be random. The f_opt value calculated from previous trades won't be really optimal for future trades, unless we turn to really reckless trading. Let's show an example of this.

We calculate the optimal f for a model system (num = 1) for several trades in a row, as shown in App.1. For the last 10 trades the optimal values would be 0.135, 0.134, 0.131, 0.123, 0.156, 0.142, 0.149, 0.137, 0.155, 0.165. So before the last trade we choose a value of f equal to 0.155 while the optimal value would be 0.165 – we take a less-than-optimal risk. Even worse, the third trade from the right has an optimal f of 0.137, while we consider it to be 0.149, accepting too much risk. So the so-called optimal f is really far from optimal.

The Safe f. Leo Zamansky and David Stendahl tried to overcome large drawdowns by adding a special limit of maximall allowable drawdown:

TWR à Max ₀

If

Max_Drawdown <= Max_Allowed_Drawdown

Another way is to use the maximal drawdown or its estimate instead of the maximal loss in the TWR formula for calculations the safe f.

Optimal f with volatility. Murray Ruggiero proposed to adapt the position size calculated using the optimal f to the current market volatility.. This is founded on the hypothesis that when the market volatility is low, the chance of having a large loss is larger than when the volatility is high. We normalize the volatility from 1 to 0, where 0 is maximal volatility, and 1 – minimal:

Volatility_norm = (Max_Volatility – Current_Volatility / (Max_Volatility – Min_Volatility)

Then

Num_Lots = f_opt * Volatility_norm * Capital /

( -Max_Loss_Estimate)

Here the, f_opt is calculated also using the maximal loss evaluation.

Fixed Ratio. A common problem of all methods using a fixed fraction of the capital is that different methods either maximize the capital growth without relation to risk (i.e. the optimal f) or minimize the risk (i.e. risking not more than x% of capital). Trying to solve this conflict Ryan Jones concludes that the relation of the number of lots traded to capital growth needed to increase the number of lots by one (or the minimal increment) should be a constant⁸:

Previous_Capital+Num_Lots * Delta = Next_Capital

Where Delta defines how aggressive or conservatiove is our application of money management: the more Delta, the larger profit per lot need we receive to increase the number of lots traded. The author proposes using as Delta a part of the maximal drawdown. Our experiments show it’s much better to use volatility.

Table 3 lists the results of testing a basic system with the Jones’ algorithm and Delta proportional to volatility. The method is unprofitable with small Delta values, then leaps to maximal profits, after which both profit and maximal drawdown monotonously decrease as Delta grows.

Table 3. The results of testing the fixed ratio method.

The idea behind this method looks quite doubtful: increasing the number of lots traded from one to two is not equivalent to increasing the number from 10 to 11 (as states Ryan Jones), and an increase from 10 to 20 is a 100% increase. The trader is concerned not about the quantity of contracts, but capital growth and risk in relative values.

Let us perform some manipulations according to Ryan Jones. With a few mathematical transformations (see App.3) we make an expression:

Num_Lots = 0.5 + (2 * Profit/Delta + 0.25)^(0.5).

So the number of lots in fixed ratio trading is proportional to the square root of the capital. All variants of fixed ratio trading define the number of lots traded as depending on the capital linearly. Anyone familiar with the basics of mathematical analysis know that at low X values

y = a * x^(0.5) is larger than the linear y = a * x, and vice versa at high X values (see Ill. 1). Hence with a small capital the fixed relation method prescribes trading a larger number of lots than the fixed share of capital, and a smaller number of lots with a larger capital. In other words, the fixed ratio method recommends higher risks with small capitals than the so- called “risky methods” criticized. Practically, this “new” method also involves risking a fixed part of the capital, in a more aggressive way compared to the original. The book examples showing the advantages have been skillfully selected so that drawdowns occur only after the capital has grown significantly.

Ill. 1. Comparison of function values with different arguments.

As to Ryan Jones’ attempt to break the trading record of Larry Williams in The Robbins 2001 Futures Trading Contest described in the previous article, he failed again... After his account grew by 600% from $15 000 to $107 000, he sent an offer to buy his method, proven by statements capable to bring such profits. Besides, he offered a $299 per month subscription to stay informed of all trades taken in the contest. As a result the drawdown on his capital reached 95%, just what had to be proven.

The method of Larry Williams. During his record-breaking trading Larry Williams used the Kelly’s formula where the starting risk was defined by the size of the margin per futures fontract. The dynamics of the capital were also noteworthy: first a growth from $10 000 to

$210000, then dropped to $700 000 (67% drawdown) and the year was finished at $1100000. By the way, Ralph Wince was working for Williams as programmer. Now Larry Williams recommends the following varian of the fixed fraction method:

Num_Lots = % risk * Capital / (- Max_Drawdown) / 100.

Playing the “market’s money”. As experience shows, for an investor it is much more important not to lose a small part of the starting capital than to lose a substantial part of the profits. The idea is taking smaller risks on starting capital and larger, more aggressive on profits received:

Num_Lots = (%risk_{start_capital}) * (Starting_Capital + MinList(Profit, 0)) + % risk_profit * MaxList(Profit, 0)) / starting_risk_per_unit_of_assets / 100.

Pyramid building. All the methods described above define the starting risk for opening the position. The current or effective risk of an open position is, actually, different. It may be expressed as:

Effective_Risk = MarketPosition * (Entry_Price – Current_Exit_Price) *

Num_Lots * Price_of_a_Point

where MarketPosition equals 1 for long positoins, -1 for short, 0 for no position. Until the trade has no unrealized (paper) profit, the effective risk is positive. A trade protected by a stop-loss order at breakeven level has zero effective risk. As soon as the stop loss is moved past breakeven level, the effective risk becomes negative – which means the position has a guaranteed profit, protected by the stop loss. The capital is no longer subject to risk, so we can risk the guaranteed profit, increasing the position size correspondingly.

Additional_Number_Lots = % risk_guaranteed_profit *

(- MinList(Effective_Risk, 0)) / starting_risk_per_unit_of_assets / 100

Let us now see it all on an example. If we reinvest the guaranteed profits with the same risk, then, as Table 4 shows, profits will increase over 4 times and drawdowns 2.8 times. If we increase the risk for guaranteed profits, net profit skyrockets – but unhappily, drawdowns increase even more.

Table 4. Reinvesting the guaranteed profits.

Regulating the position size on the basis of its risk and volatility. The risk of an open position is usually controlled by exit rules set in the system. For instance, moving stop levels follow the price to increase starting risk or lock down a part of paper profits. But a much more viable idea is to limit the maximal risk and volatility of an open position in relation to the capital. All we need for this is track the values as often as needed.

Excessive_risk = Num_lots * Current_risk_per_unit_of_assets – Max%risk * Capital / 100

and

Excessive_volatility = Num_lots * Current_volatility_of_assets –

Max%volatility * Capital / 100.

As soon as any of those becomes positive, we decrease the position size by a value equal to:

Excessive_num_lots = Excessive_risk / Lot_price

Or correspondingly by:

Excessive_number_lots = Excessive volatility / Lot_price

The practical rationale of this methods is closing a part of the position without waiting for the system signal, when the prices move very fast and far, too far and fast for a trailing stop or a closing point to follow. This solves two tasks at once: first, the risk and volatility are supported at set levels, second, positions frequently close at extreme prices with favourable slippage.

Such methods for one strategy on one asset can be easily applied on the TradeStation platform as shown in App.1. But since real trading involves several strategies applied to portfolios of assets, frequently at different time frames but with a commom portfolio capital. The organization of money management at portfolio level will be discussed in the next article.

Appendix 2. Calculating the optimal f in Excel.

Add the following lines to the code of the system handling the lot:

and launch the system. When we open in Excel the resulting file, we’ll see a table:

Enter =1- C$2*A4/B$2 in B4 and continue till line X.

Enter any value from 0 to 1 in the C2 field, С4 = В4, С5 = С4*В5, D4 =A$2*C4, D2 = COUNTIF(A4:AX,"<>0"), E2 = POWER (CX,1/D$2), E4 = INTEGER (D4/(B$2/-C$2)). Continue the formulas in the С4:Е4 fields till the last non-empty line Х. In the E column we have the number of lots for the f value given in the С2 field.

To calculate the optimal f use the menu: Service à Solution Search à designate target field: $C$Y (where Y – number of the line corresponding to the trade previous to the one we optimize f for); Changing fields: $C$2; Limits: $C$2 >=0; $C$2 <= 1 à Execute.

Excel’s in-built optimizer will find the value of the optimal f, maximizing the TWR function.

Appendix 3. Deriving the formula for the fixed fraction method.

The method states the number of lots traded to capital growth needed to increase the number of lots should be a constant value. This will be written down as:

E_n + n * D = E_n+1

where E_n – current capital, n – current number of lots, D – the Delta parameter. Then, recursively,

E_n = E_{n- 1}+ (n – 1) * D = E_n-2 + (n – 2) * D + (n – 1) * D = …

= E₁ + (1 + 2 + … + (n – 1)) * D,

hence, considering the fact the bracketed expression is a sum of an arithmetical progression members

E_n = E₁ + 0.5 * n * (n – 1) * D.

This equation is a square equation in relation to n:

n^2 – n – 2 * (E_n – E₁ ) / D = 0.

High school analysis course tells us this equation has two roots:

n₁ = 0.5 + (2 * (E_n – E₁ ) / D + 0.25)^(0.5)

and

n₂ = 0.5 – (2 * (E_n – E₁ ) / D + 0.25)^ (0.5),

where n₂ <= 0. If we consider E_n – E₁ to be the profit, then:

Num_lots = 0.5 + (2 * Profit / Delta + 0.25)^(0.5).

This formula starts trading with one lot. If the starting capital allows to trade many lots at once, we must find the E_n – E₁ considering the starting capital. Clearly, E_n = Starting Capital + profit. One Delta can buy k = Price / Delta stocks. The corresponding E_k capital is equal to

k * Price = Price^2 / Delta. Then

And our desired formula will be

Number_lots = 0.5 + (2 * (Starting_capital + profit - 0.5 * Price * (Price + Delta) / Delta) / Delta + 0.25)^ (0.5).