ArticlePDF Available

Money management methods in trading and investing

Authors:

Abstract and Figures

In this paper, we briefly discuss six basic methods of money management in trading and investing and analyze their effectiveness on the Warsaw Stock Exchange. The most efficient methods are the Martingale and Ralph Vince's methods giving profits of 1731% and 1453%, respectively.
No caption available
… 
Content may be subject to copyright.
Money management methods in trading and investing
Michał Wójtowicz
SGB-Bank S.A.
Szarych Szeregów 23a
60-462 Poznań, Poland
michwojt@gmail.com
Abstract. In this paper, we briefly discuss six basic methods of money management in trading and
investing and analyze their effectiveness on the Warsaw Stock Exchange. The most efficient methods
are the Martingale and Ralph Vince’s methods giving profits of 1731% and 1453%, respectively.
Introduction
Most of the theoretical papers about investing are focused on answering the question: in what and
when to invest? But another important issue is money management, which means how much an
investor should invest. In an extreme case, it may happen that the investor, using a strategy with a
positive expected value, will go bankrupt because of poor money management [Wójtowicz 2013,
p.111].
Despite this, the topic of money management is ignored or marginalized in most publications. There
have only been a few papers published on this topic [Thorp 2007].
In this article, we describe the most popular methods of money management in trading and investing:
The Irene Aldridge’s method of money management for a short term speculation is a
compromise between aspiration to full optimization and maintaining relative simple
calculations.
The Ralph Vince’s method relies on maximizing the relative growth rate of the investor’s
capital. This method allows to utilize all the information on a distribution of the investor’s
profit. The solution obtained from this method also has many other advantageous properties
[Ziemba 2005].
The method proposed by Edward Thorp is a simplified version of the Vince method. It permits
the investor to simplify calculations, but it has some adverse properties.
The Van Tharp’s method relies on matching basic methods of money management with the
investor’s preferences.
The Rayan Jones’s method is an improvement of the simplest method of money management,
consisting in buying one contract for a given value of the investor’s wealth.
The Martingale method relies on increasing a bet size after taking a certain series of losses
which is much more frequent than the probability of such an event. Unfortunately, this method
is very risky and can lead to ruin.
In the last part of this article, we present our own example of applying the above-listed money
management methods on the Warsaw Stock Exchange.
1. Money management on a single financial market
Money management on a single financial market includes situations when the investor uses one or
more investments or trading strategies on a single market.
1.1 Money management for short term speculation
The method was proposed by Irene Aldridge [Aldridge 2010] for traders (investors with a very short
time investment horizon) who practice high-frequency trading (i.e. making speculative transactions
implemented in seconds or even less).
In this method, it is assumed that the trader completes transactions on a single financial market using
many strategies simultaneously. They would also like to determine the degree of involvement of their
wealth in a single strategy. The simplest solution is an equal distribution of capital to all the strategies.
For example, for four strategies one should allocate 25% of their wealth in a single strategy.
The advantage of this method is the simplicity of the calculations. Unfortunately, this way does not
utilize any historical information about the results of the strategies. It is obvious that an application of
such data should give a more profitable solution to the investor.
Hence, in a second approach, one studies the historic trends of all the strategies and checks which
distribution of the capital gives the greatest profit. The disadvantages of this method are large
computational complexity and time constraints.
The method proposed by Aldride is a combination of these two solutions. It allows us to find a
relatively optimal solution and, at the same time, it shortens the time needed to find the solution.
At the beginning, we assume we have information about the historic results of the investigated
strategies.
First, we sort all the strategies using the Sharpe ratio, starting from the strategy with the largest value
of the ratio. The Sharpe ratio for a given strategy is defined by the following equation [Aldridge 2010,
p.76]:
  
 
where TR is the estimated number of transactions which are made during a year. This is found through
the product of the mean number of transactions in a day and the number of days in a year.
Second, we include an even number of strategies with the highest Sharp’s ratio in our portfolio,
wherein half of them should be positively correlated with the market and the second half - negatively.
Third, we sort the strategies (those that were included in the portfolio) by their current liquidity (the
number of transactions that were made during a given period of time).
Fourth, we combine the strategies positively correlated with the market with the strategies correlated
negatively into pairs. The combination is made on the basis of the current liquidity, i.e. that a
positively correlated strategy with the highest liquidity is combined with a negatively correlated
strategy with the highest liquidity (among the strategies that are negatively correlated), and so on.
Fifth, for every pair of strategies we separately optimize the variance of the portfolio consisting of
these two strategies alone. The optimization relies on finding the participation of all the strategies in
the portfolio, giving a minimal variance.
As a result, the trader makes transactions for any pair of strategies (of the portfolio), investing the
maximum of their wealth in every strategy, yet less than the limit defined in the fifth step. The total of
the investor’s portfolio involvement should be less than the level defined before, which in turn should
be less than the sum of the limits of all strategies.
The advantage of this method is a relative precise optimization and the simplicity of the calculations.
The disadvantage of this method is that it cannot be use in a quite common situation when the investor
only uses a single strategy. In that case it cannot be combined with any other strategy.
1.2 Maximizing the relative growth of the investor’s capital
The method has been proposed by Ralph Vince [Vince 1990].
It assumes that an investor invests their money on one financial market and only uses one investment
strategy.
Vince introduces the notion of a divisor which is understood as a value which lets the investor to
define how many units (e.g. shares) he should buy with the given capital and the largest possible loss
on a single share [Vince 1990, p. 80]. The divisor should be from the interval (0,1).
For example, assume that the largest possible loss is 100 dollars and the divisor, f, chosen by the
investor is 0.25. Then 
 , so the investor should buy one share for every 400 dollars of their
capital [Vince 1990, p. 88].
At first, the investor should determine the period of time from the historical data, best reflecting the
situation on the given market. Next, the investor should check using the aforementioned data from
the chosen period of time the relative growth of capital using a given strategy and chosen divisors f
between 0.01 and 0.99 at intervals of 0.01. By these calculations, the investor should choose the
divisor generating the biggest relative historical growth of capital. We shall say that the divisor with
this property is an optimal divisor, and we denote it below with the symbol .
With the assumption that the distribution of profits for a given strategy - on a given market - will not
change in the future, the divisor has the following properties: (1) it maximizes the investor’s capital
in the long run [Breiman 1961, p.72], (2) it minimizes the expected time (comparing to portfolios
constructed using other methods) needed to achieve a fixed financial goal [Breiman 1961, p. 68], and
(3) the investor using will not go bankrupt [Hakansson and Miller 1975].
In practice, using the optimal divisor may create a big variance in the investor’s capital [Wójtowicz
2013, p.108-109]. Moreover, if meanwhile the distribution of profits for the given strategy will change
(e.g., the maximal loss may increase drastically) and the investor will still use the old divisor, he may
go bankrupt [Wójtowicz 2013, p. 111].
Ralph Vince has never given a theoretical justification of his method. Due to that, it is neither quoted
nor discussed in scientific papers. In practice, Vince’s method consists of maximizing a function
constructed by means of the distribution of gains and losses on a given market. We discuss it briefly
below.
Let us consider a general case. Let the symbol Y denote a random variable describing an investor’s
financial result: Y = (, , …, , -, …, -), where , , …, are pairwise different non-
negative values (hence, at most one of them equals 0), and , …, are strictly positive, n,m 1.
Further, let the distribution of probability for Y be of the form
P = (,…, , , …, ),
with , 0 for all i,j. In the next part of this paper we assume that E(Y)>0 (that is, the average
financial result of investing is profitable).
If denotes the maximal loss: = max{, …, }, then X denotes the random variable, with the
same distribution of a probability P, of the form Y/bs :
X =
= (
, …,
,
, …,
) = (, , …, , -, …, -),
where =
, =
, i n, j . Then E(X)= 
>0.
The values are non-negative, pairwise different, and belong to the interval (0,1] with = 1.
We also have
E(X) =
 -
 > 0.
The generalized Kelly’s function G is defined by the formula
G(f) =   
   
 .
In [Wójtowicz 2013, p.113-114] it was proved that the function G has the following properties:
(P1) it is concave on the interval [0,1), G(0) = 0 and there exists
(0,1) such that G(
) = 0,
(P2) there exists a global maximum of G with (0,
); it is a solution of the equation
 = 0;
(P3) the number , defined in (P2), is the optimal divisor for the Vince’s method, which
means that = ,
(P4)  = E(X).
An example of G is presented in the second section of this paper.
1.3 A simplified Vince’s method
This method has been proposed by Edward Thorp [Tharp 2008, pp. 214-215], and it is a simplification
of the Vince method.
First, as in Vince’s method, the investor determines a historical period that best reflects the situation
on the given market. Next, the investor calculates a divisor
defined by the formula [Tharp 2008, p.
214]:
 
(1)
where:
 is the quotient of the number of profitable investments by all investments,
T is the quotient of the average profit by the average loss of an investment.
The advantage of this method is a simplification of calculations, which are simpler than in Ralph
Vince’s method.
The disadvantage of Thorp’s method is that, in the long run, the investor applying the parameter
achieves less profit than applying the optimal divisor f* (which maximizes the investor’s
profit, see property (P3) above). Moreover, if
is bigger than f*, the investor should be aware
that larger decreases in their capital would be observed than when using f* [Wójtowicz 2013,
p.108-109].
In an extreme case (i.e., when
is bigger than fc, see property (P1) in Subsection 1.3), the investor
applying the simplified method may go bankrupt [Wójtowicz 2013, p. 111].
1.4 Adapting money management to investment goals
This method was proposed by Van Tharp [Tharp 2008]. The author indicates that the investor may not
be interested in maximizing profit, Sharpe ratio or minimizing their risk. Due to that, every investor
should explicitly specify their investment goals and manage their wealth in a way that allows them to
achieve these goals.
1.4.1 The measure of utility of a strategy in methods of money management
Van Tharp has defined a new ratio (System Quality Number) which, based on historic results of a
strategy, allows one to determine the utility of the strategy in achieving investment goals [Tharp 2008,
p.28]:
System Quality Number

 .
The higher the ratio, the easier it is for an investor to achieve their goals.
1.4.2 Basic methods of money management
Van Tharp discusses five basic methods of money management [Tharp 2008, Section 8]:
1. Buying one contract for a given size of investor’s wealth.
2. Equal division of wealth among all markets.
3. Specifying the percentage of wealth involved in every transaction.
4. Buying such a number of contracts on a given market, for the volatility position sizing of this
market over a fixed period of time (e.g., five days) to be lower than a given percentage of the
share of wealth.
5. Buying such a number of contracts for the share of wealth needed to secure the transaction to
be less than a given percentage value of the wealth.
To illustrate the five, above-listed methods by examples, let us consider five investors Alan, Ben, Carl,
Dirk and Edwin applying each of the methods separately.
Alan practices money management by buying one contract for a given size of his wealth. He would
like to buy one contract for 10 thousand dollars. At the moment, Alan possesses 10 thousand dollars,
and therefore, he can make a transaction to buy only a single contract. When his wealth grows to 20
thousand dollars, Alan can buy 2 contracts. When his wealth decreases to 17 thousand dollars, he may
only complete transactions to buy one contract. This means that, in this method, an investor can buy
only an integer number of contracts.
Ben practices money management by equal division of his wealth onto all markets. He divides his
wealth, which is 2 thousand dollars, onto two markets, shares of A and B companies. Assume that, on
the share market of company A, a signal appeared to buy the stock with one share costing 100 dollars.
According to his plan, Ben should engage half of his wealth and buy 10 shares for 1,000 dollars.
In the next three scenarios we assume that an investor has 1,000 dollars and decides to risk 3 percent
of his wealth in a single transaction; thus they will risk 30 dollars each.
Carl practices money management by risking a fixed percentage of share of wealth in any transaction.
He decides to buy shares of a company which cost 10 dollars per share. When the price will drop to 9
dollars, Carl will withdraw from this transaction. Hence his risk for a single share is 1 dollar, and so
Carl buys 30 shares.
Dirk practices the fourth method of money management. He measures the volatility position sizing on
a given market as the difference between the highest and the lowest price from the previous week.
Dirk decides to buy shares of a company with 5 dollars of volatility; which is also the risk of a single
share. Hence Dirk buys 6 shares.
Edwin practices the fifth method of money management. He decides to buy a contract on shares which
require 10 dollars of margin for a single contract; this is also the risk of a single share. In this case,
Edwin buys 3 contracts.
In practice, the most advised and easiest method of money management is risking a fixed percentage
share of wealth in any transaction.
1.4.3 Optimizing a chosen method of money management
After choosing a method of money management, an investor should define the value of a parameter
optimal for our goals (for example, the percentage of wealth risked in a single transaction).
Next, the investor should determine the historic period of time that best describes the price fluctuations
on a given market. Using this information, the investor should determine the distribution of profit for
their strategy and run numerous simulations (e.g., 10 thousand) of 100 future transactions. Before
running the simulations, the investor should define a satisfying return from 100 transactions and the
biggest acceptable relative decrease of his wealth (let’s call it the investor’s ruin).
In the next step, one should check the results for different values of parameters and find optimal values
for the following six criteria:
- the largest average mean return,
- the largest average median return,
- the greatest probability of reaching an investment goal,
- the biggest value of a parameter for the probability of ruin less than 1,
- the biggest value of a parameter for the probability of ruin equals 0,
- the biggest difference between the probability of ruin and reaching an investment goal.
As a result, we should get 6 different values of parameters for 6 different criteria.
Using this data, the investor should choose the value that best satisfies thier preferences. To make it
easier, for each of the six optimal parameters separately, one should check its probability of ruin,
reaching an investment goal, the average mean return, and the median return.
The advantage of Van Tharp’s method is creating a new measure of utility of a given strategy, which
clearly reflects the specificity of the short-term speculation. Moreover, Van Tharp has defined new
criteria which better allow to describe an investor’s preferences, and has created the tools to meet
them.
1.5 The advanced method of buying one contract for a given size of investor’s wealth
The Rayan Jones’s method [Tharp 2008, p.161-196] is a development of the method of buying one
contract for a given size of investor’s wealth. For example, assume the investor buys one contract for
every 10 thousand dollars of their wealth and that they start with 10 thousand dollars; hence they will
buy another contract but only when their wealth increases by 100%. If the investor had 100 thousand
dollars, they would buy another contract after their wealth increased by 10%. This means that
investors with less money to start have a smaller chance to multiply their capital.
Because of this limit, Jones has proposed a new way of defining the level of capital, which allows a
smaller investor to buy a new contract:
New level = Current number of contracts ∙ Delta + Old level. (2)
Here Delta denotes a monetary value defined by the investor describing risk tolerance (the smaller the
Delta, the bigger the risk).
For example, assume the investor has 25,000 dollars. They invest in one contract and their Delta
equals 2,500 dollars. Therefore, they will buy another contract when their wealth increases to
1∙2,500 + 25,000 = $27,500. For such wealth they should have 2 contracts. When their wealth
increases to 2 ∙ 2,500 + 27 500 = $32,500, they will buy next contract, and so on. However, if their
wealth dropped below 27,500 dollars at an earlier point, they should sell the contract bought before
(and stay with one contract).
In this method, we do not assume anything about the risk associated with investing in a single
contract. But if we make such an assumption and use this method then, along with the increase of
investment engagement, the investor’s risk rapidly grows and next it gradually decreases [Tharp 2008,
p.162].
The advantage of this method is that the investor with a small amount of capital can increase their
commitment to the market at a quicker pace and has a greater possibility to increase their wealth.
Unfortunately, the key elements of this method have never been defined. Jones has neither given us
the method of defining the level of investor’s engagement for a single contract nor pointed on how to
define the parameter Delta.
Moreover, this method requires frequent changing to the investor’s engagement, which can be hard for
big price movements. There is also no assumption about risk connected with a single contract, which
is of crucial importance in practice. For example, assume we buy a contract for $10. If we decide to
close the transaction when the price drops to $9.50, we risk $0.50 per contract. If we assume to close
the transaction when the price drops to $5, we risk $5 per contract and our risk is 10 times higher.
Moreover, a rapid growth of investor’s risk connected with the growth of their involvement on the
market can be a very big threat.
1.6 Martingale strategy
The strategy was defined by Larry Williams [Tharp 2008, pp. 205-207]. This method is additionally
based on the assumption that the investor is aware of the distribution of gains for the strategy. In
addition, in a single transaction, the investor risks a fixed nominal amount or percentage of their
assets.
Let us assume that, on the basis of historical data, the investor estimated the probability of generating
a loss at q. In fact, however, the frequency of occurrence of their losses during the first time interval
was at u, where u is significantly higher than q. In these circumstances, the investor should increase
his involvement on the market. The probability of the investor incurring a loss in the next transaction
is still at q, yet in the long run the frequency of occurrence of the investor’s losses should converge to
q. Such convergence will only occur if the frequency of losses is lower than q in a certain future period
of time. Therefore, through increasing his engagement on the market, the investor will recover the
assets previously lost in the event of higher wins, or even register a certain amount of gain.
The advantage of this method is its simplicity.
Unfortunately, Williams fails to specify when the frequency of loss occurrence can be considered
significantly different from the probability of occurrence of loss according to the distribution. Nor
does she indicate the initial engagement of the investor, or consider the fact that the investor cannot
increase his engagement infinitely, for reason of limited liquidity on the financial market.
An investor using this strategy must take into account major decreases of their wealth and the resulting
mental burden; in the event of occurrence of a longer series of losses, which is always a possibility, the
investor who continuously increases his engagement will ultimately go bankrupt.
2. Examples of money management on the Stock Exchange
The Irene Aldridge method was excluded from the analysis due to the fact that, unlike the other
methods, this one applies only to high frequency data.
The money management methods are illustrated with the example of Pekao SA shares. The examples
do not take into account the money management method for a short-term speculation because it cannot
be applied to daily interval data.
The analysis covers the share prices for the period from 2003-09-05 to 2005-09-16.
Diagram 1. PKO SA share prices during the period from 2003-09-05 to 2005-09-16
Further analysis is based on the investment method, which consists of
(a) buying shares when the momentum indicator with parameter 10 changes from negative to
positive, and
(b) selling shares when the momentum indicator switches from negative to positive.
The value of the momentum indicator with parameter 10 on the given day is the difference between
the closing price of the given day and the closing price ten days prior.
Furthermore, the following assumptions were made:
(i) No transaction costs;
(ii) Access to unlimited leverage, i.e. the ability to invest multiple times higher capital than
actually available;
(iii) No security deposit requirements; and
(iv) Ideal market liquidity.
To determine the optimum parameters, the time series was divided into the following two periods:
from 2003-09-05 to 2004-07-15 and from 2004-07-16 to 2005-09-16. An assumption was made to the
effect that the investor has the initial wealth amounting to a value of PLN 1 million.
The first period was used to determine optimum parameters for the particular methods. During that
time, the investor’s maximum loss was at PLN 5.5 per share. If the investor invests all their wealth in
the company’s shares, without using financial leverage, then according to the adopted strategy, they
would close 16 transactions and generate a 45.94% gain during that period.
The period from 2004-07-16 to 2005-09-16 was used to verify the money management methods based
on the parameters determined according to the first period. If the investor invests all their initial wealth
of PLN 1 million in the company’s shares, without using financial leverage, then according to the
adopted strategy, they would yield a 51.18% gain during that period. In addition, maximum relative
decrease of his wealth would be at 4.06% for the period.
Diagram 2. Relative status of the wealth of an investor that does not use financial leverage
during the period from 2004-07-16 to 2005-09-16 after subsequent transactions.
2.1 Maximization of the relative increase of the investor’s wealth
Based on the investor’s performance during the period from 2003-09-05 from 2004-07-15, the
function G has been estimated as follows:
G(f) =   
   
 =
=
 (       +    
       +
+       +       ).
Through numerically solving the equations G(f) = 0 and  = 0, we get
= 0.9 and = 0.48.
Diagram 3 is a graphic presentation of function G.
Diagram 3. Generalized Kelly’s function G(f), estimated on the basis of the investor’s
performance during the period from 2003-09-05 to 2004-07-15
Using the 0.48 divisor determined in the preceding period, the investor would make a 1453.48% profit
and the maximum relative decrease of his wealth would be at 43.64%.
Diagram 4. Relative status of the wealth of an investor using the relative wealth increase
maximization method during the period from 2004-07-16 to 2005-09-16 after subsequent
transactions.
-0,2
-0,15
-0,1
-0,05
0
0,05
0,1
0 0,2 0,4 0,6 0,8 1 1,2
G(f)
f
2.2 Simplified R. Vince’s method
Probability p of an investor making a profit when using a momentum indicator-based strategy,
estimated on the basis of the initial period, was , with the average profit 7.44, and the average loss
2.56. Therefore, the value of the divisor determined according to the simplified method was
= 0.33
(see (1)).
Using the 0.33 divisor determined according to the simplified method, the investor would make an
878.64% profit and the maximum relative decrease of his wealth would be at 30.48%, occurring
between the first and the second transaction. This is illustrated on diagram 4.
Diagram 5. Relative status of the wealth of an investor using the simplified R. Vince’s method
during the period from 2004-07-16 to 2005-09-16 after subsequent transactions.
2.3 Adapting money management methods to investment goals
Based on the distribution of the initial period’s profits and losses, 10 thousand simulations were
carried out in Excel software. Each simulation consisted of 100 consecutive transactions. On this
basis, optimum divisors were selected on the basis of six different criteria. The overall assumption is
that the investor is striving to accomplish 400% profit after 100 transactions and is willing to risk the
loss of their entire wealth. Optimum divisors for these criteria are presented in the following table.
100,00%
200,00%
300,00%
400,00%
500,00%
600,00%
700,00%
800,00%
900,00%
1000,00%
1100,00%
0 5 10 15 20
Table 1. Optimum divisors in terms of the particular criteria for an investor aiming at gaining
400% profit after 100 transactions and accepting the risk of losing their whole wealth.
Criterion
Divisor
Largest average mean return
0.99
Largest median return
0.99
Maximum divisor at nil probability of
bankruptcy
0.08
Maximum divisor at <1 probability of
bankruptcy
0.99
Maximum probability of achievement of the
goal
0.22
Maximum difference between probability of
goal achievement and bankruptcy
0.21
Consequently:
The maximum average return at 3138% will be achieved by an investor who uses a divisor of
0.99;
The maximum median return at 3762% will be achieved by an investor who uses a divisor of
0.99;
The maximum divisor at nil probability of bankruptcy is at 0.08;
The maximum divisor at <1 probability of bankruptcy is at 0.99 and the probability of
bankruptcy for this value was at 30.65%;
The greatest probability of achieving the goal, at 96.94%, will be accomplished by an investor
who uses a divisor of 0.22;
The largest difference between the probability of achieving the goal and the bankruptcy, at
95.54%, will be accomplished by an investor who uses a divisor of 0.21.
Performance of an investor using each particular divisor is illustrated by diagram 5.
Diagram 6. Relative status of wealth of an investor using divisors optimized for the given
criterion.
In addition:
An investor using the divisor of 0.99 will go bankrupt after the second transaction;
An investor using the divisor of 0.08 will make 216% profit with the largest relative decrease
of wealth at 7.85%;
An investor using the divisor of 0.21 will make 543% profit with the largest relative decrease
of wealth at 19.98%;
An investor using the divisor of 0.22 will make 575% profit with the largest relative decrease
of wealth at 20.88%.
2.4 Advanced method of buying one contract for the given value of the investor’s wealth
During the initial period, the investor’s largest loss was at PLN 5.5. Thus, an investor willing to risk
3% of their wealth in a single transaction should buy   ł
  shares. Therefore, let
us assume that initially the investor having a wealth of PLN 1 million buys one contract in a single
transaction, consisting of 5,400 shares.
Let Delta be at PLN 100,000. Then, the investor will buy a second contract when their wealth reaches
PLN 1,000,000 + 1∙ PLN 100,000 = PLN 1,100,000 (see (2)). The third contract will be bought when
his wealth reaches PLN 1,100,000 + 2∙ PLN 100,000 = PLN 1,300,000.
An investor applying this strategy will make 52.38% profit with the largest relative decrease of wealth
at 4.52%.
-100%
0%
100%
200%
300%
400%
500%
600%
700%
0 5 10 15 20
0,08
0,21
0,22
0,99
Diagram 7. Relative balance of wealth of an investor using the advanced strategy of
buying one contract for the given value of the investor’s wealth
2.5 Martingale strategy
On the basis of data of the first period from 2003-09-05 to 2004-07-15, the probability of loss was
determined at 0.5. A martingale strategy was developed on this basis, where the investor uses the
divisor of 0.48 (see Table 1), determined through maximizing the relative increase of the investor’s
wealth, provided that the investor generated nil or one loss in the last three transactions. However, if
the investor incurred two or three losses in the last three transactions, i.e. the frequency of such losses
was higher than the probability of their occurrence, we expect the investor to increase his commitment
using the 0.6 divisor.
An investor applying this strategy would make 1731.78% profit and the largest relative decrease of his
wealth would be at 43.64%.
Diagram 8. Relative status of the wealth of an investor applying a martingale strategy during the
period from 2004-07-16 to 2005-09-16 after subsequent transactions.
100%
110%
120%
130%
140%
150%
160%
0 5 10 15 20
100%
300%
500%
700%
900%
1100%
1300%
1500%
1700%
1900%
2100%
0 5 10 15 20
Summary
The Irene Aldridge money management method for a short-term speculation is a successful
compromise between aiming at full optimization and maintaining relative simplicity and
comprehensibility of calculations. However, it cannot be applied to the most common case, i.e. an
investor applying a single investment strategy.
Ralph Vince’s method, which consists of maximizing relative growth of the investor’s wealth, has a
number of beneficial characteristics. Yet it exposes the investor to a significant risk of major decreases
of their wealth. Also, this method has its theoretical foundations which have not yet been examined.
The method proposed by Edward Thorpe offers a potential gain which is not higher (and usually is
significantly lower) than that derived from Ralph Vince’s solution.
The Van Tharp method is capable of fulfilling the investor’s expectations. However, it is so general
that it cannot be compared to other methods.
With Rayan Jones’s method, one can bypass one of the key disadvantages of the simplest money
management method, in which an investor with low capital has fewer options of commitment,
compared to an investor with significant wealth. Unfortunately, the key components of this method
have not been determined precisely, which is a restriction of its applicability.
The martingale strategy involves extremely high risks, which renders it highly useless considering the
lack of determination of numerous key issues.
Application of the particular money management methods on the market of Pekao SA shares
generated the following profits:
1731% for the martingale method;
1453% for Vince’s method;
878% for Thorpe’s method;
216% to 575% for van Tharp’s method (disregarding bankruptcy);
52% for Rayan Jones’s method.
The examples of practical implementation of the money management methods have demonstrated that
with the use of a profitable strategy and a high level of awareness of its distribution, money
management can multiply an investor’s profits while simultaneously multiplying his risk. Yet the use
of the wrong method, or inaccurate determination of goals could lead to bankruptcy of an investor
following a strategy with positive expected value.
Bibliography
1. Aldridge, I., 2010, High-Frequency Trading. A Practical Guide to Algorithmic Strategies and Trading
Systems, Willey & Sons, New Jersey.
2. Breiman, L., 1961, Optimal gambling systems for favorable games, Fourth Berkeley Symposium on
Probability and Statistics, no. 1, s.63-78.
3. Cover, T. M., 1991, Universal Portfolios, Mathematical Finance, 1(1), pp. 1-29.
4. Davis, M.H. A., Lleo, S., 2008, A Risk Sensitive Benchmarked Asset Management, Quantitive Finance,
8(4), pp. 415-426.
5. Ethier, S. N., 2004, The Kelly System Maximizes Median Wealth, Journal of Applied Probability, 41(4),
pp. 563-573.
6. Finkelstein M., R. Whitley, 1981, Optimal Strategies for Repeated Games, 1981, Advanced Applied
Probability, 13, pp. 415-428.
7. Hakansson, N. H., 1970, Optimal Investment and Consumption Strategies under Risk for Class of Utility
Functions, Econometrica, 38, pp. 587-607.
8. Hakansson, N. H., 1971, On Optimal Myopic Portfolio Policies, with and without Serial Correlation of
Yields, Journal of Business, 44, pp. 324-334.
9. Hakansson, N. H., Miller, B.L., 1975, Compoundreturn meanvariance efficient portfolios never risk
ruin, Management Science 22, s.391-400.
10. Hens, T., Schenk-Hoppe, K., 2005, Evolutionary Stability of Portfolio Rules in Incomplete Markets,
Journal of Mathematical Economics, 41, pp. 43-66.
11. Kelly, J. L., 1956, A New Interpretation of Information Rate, Bell System Technical Journal, 35, pp. 917-
926.
12. Latane, H. A., 1959, Criteria for Choice among Risky Ventures, Journal of Political Economy, 67, pp.
144-155.
13. MacLean, L. C., W. T. Ziemba, Y. Li, 2005, Time to Wealth Goals in Capital Accumulation, Quantitive
Finance, 5(4), pp. 343-355.
14. Roll, R., 1973, Evidence of the “Growth Optimum” Model, , The Journal of Finance, 28(3), s. 551-566.
15. Sutzer, M., Portfolio Choice with Endogenous Utility: A Large Deviation Approach, 2003, Journal of
Econometrics, 116, pp. 365-386.
16. Tharp, V. K., 2008, Van Tharp’s Definitive Guide to Position Sizing, International Institute of Trading
Mastery, USA
17. Thorp, E. O., 1970, Optimal Gambling Systems for Favorable Games, Review of the International
Statistical Institute, 37(3), pp. 273-293.
18. Thorp, E. O., 1971, Portfolio Choice and the Kelly Criterion, Proceedings of the Business and Economics
Section of the American Statistical Association, pp. 215-224.
19. Thorp, E., O., Zenios, S. A., Ziemba, W. T., 2006, The Kelly Criterion in Blackjack Sports Betting, and
the Stock Market, Handbook of Asset and Liability Management, Volume 1, pp. 385-428.
20. Thorp, E. O., 2008, Understanding the Kelly Criterion, Willmott, May and September.
21. Rudolf, M., Ziemba, T., 2004, Intetemporal Surpulus Management, Journal of Economic Dynamics and
Control, 28, pp. 975-990.
22. Vince, R., 1990, Portfolio management formulas: mathematical trading methods for the futures, options,
and stock markets, John Willey & Sons, New York.
23. Vince, R., 2007, The Handbook of Portfolio Mathematics, John Willey & Sons, New Jersey.
24. Wójtowicz, M., 2013, Konstrukcja portfela inwestycji z zastosowaniem kryterium Kelly’ego, Studia
Oeconomica Posnaniensia, vol. 1, no. 9 (258), pp. 102-119.
25. Ziemba, W. T., 2005, The Symmetric Downside-Risk Sharpe Ratio and the Evaluation of Great Investors
and Speculators, Journal of Portfolio Management, 32(1), pp. 108-122.
... Hal ini sejalan dengan penelitian Wojtowicz (2016), yang melakukan penelitian berdasarkan perbandingan return antara beberapa metode money management dan salah satunya adalah martingale dimana metode ini memberikan return nyata sebesar 1.731,8% pada WARSAW stock exchange. Dan hipotesis ketiga (H3) yang menyatakan bahwa strategi metode money management martingale memberikan return yang lebih optimal dibandingkan fixed lot (tanpa money management) diterima. ...
... Hasil ini mendukung hipotesis keempat yaitu strategy Money management paling optimal memberikan return yang lebih baik dibandingkan return strategi buy and hold, maka dari itu H4 diterima. Hasil eksperimen ini juga selaras dengan penelitian Wojtowicz (2016), yang melakukan penelitian berdasarkan perbandingan return antara beberapa metode money management dan salah satunya adalah martingale dimana metode ini memberikan return nyata sebesar 1.731,8% pada WARSAW stock exchange. ...
Article
In this study an experimental study was using automatic trading by develop an expert advisors to works with backtesting simulation from January 2010 to December 2019 to research the performance returns of the double moving average cross strategy with 6 pairs from SMA (10.30), SMA (10.50) and SMA (10,100) and EMA (10,30), EMA (10.50), and EMA (10,100). EMA performance (10.30) that given treated 3 types of money management methods, namely fixed lot, fixed % lot, and martingale (1.5x) in the GOLD futures market (XAUUSD) at 1 hour timeframe which will be compared with descriptive analysis. This study shows that the EMA (10.30) without using money management (fixed lot) method shows the most optimal results with a total return 63.5% in the futures market which is higher than the passive strategy. The experimental results show that the fixed % lot method decreases performance with lower returns and increases risk when compared without using money management (fixed lot). While the most optimal money management method is martingale (1.5x) with the achievement of a total return 6,610.56% and a risk adjusted ratio (RAR) at 5.02%. Individually, the method that gives the highest yield is shown by EMA (10.30) on the fixed lot method with a total return of 63.5% and RAR at 2.41%, EMA (10.30) on the fixed sum lot method with a total return of 62.77% and RAR 1.52% and EMA (10.100) with a total return 6.610.56% and RAR 5.02% on the martingale method simulated using expert advisors in the futures market.Keywords: Moving Average, Money Management, Gold, Return, risk adjusted return, Martingale, Fixed Lot, Fixed Ratio, Expert Advisor, Automatic Trading
... Este método foi desenvolvido por Ryan Jones e sua base está ancorada na relação da quantidade de contratos a serem negociados com o montante dos lucros necessários para aumentar em um contrato adicional, e tal relação devese permanecer fixa ao longo das negociações (WÓJTOWICZ, 2016). ...
Article
Full-text available
Profitable trading systems with high hit rates can become losers when position sizing (PS) is not done correctly. In this research, a profitable trend following trading system was used in which 8 position size methods were implemented to be applied to the brazilian stock exchange futures market, from 05/01/2005 to 05/01/2016. As the performance of the PS methods is closely related to the choice of its parameters, a methodology based in Monte Carlo simulation (MC) has been implemented. The definition of the most adequate parameter was obtained by limiting the drawdown and maximizing the return. The performance analysis of these PS methods is performed based on the return risk ratio (CAR/MDD) and the results indicated that the PS fixed size presented the best result for the methodology using the simulation of MC.
Article
Full-text available
Au cours de la dernière décade on a constaté que le joueur pouvait avoir l'avantage dans certains jeux de hasard. On verra que le "blackjack", la mise latérale au Baccara - tel qu'il est joué dans le Névada - la roulette et la "roue de la fortune", peuvent tous offrir au joueur une espérance de gain positive. La Bourse a beaucoup de traits communs avec ces jeux de hasard [5]. Elle offre des situations particulières avec des gains attendus allant au-delà d'un taux annuel de 25% [23]. Dès que la théorie particulière d'un jeu a été utilisée pour identifier des situations favorables, se pose le problème de savoir comment répartir au mieux nos ressources. Parallèlement à la découverte de situations favorables dans certains jeux, les grandes lignes d'une théorie mathématique générale pour exploiter ces opportunités s'est développée [2. 3. 10. 13.]. On décrira d'abord les jeux favorables mentionnés ci-dessus: ce sont ceux que l'auteur connaît le mieux. On discutera ensuite la théorie mathématique générale, telle qu'elle s'est développée jusqu'à maintenant, et son application à ces jeux. Une connaissance détaillée d'un jeu particulier n'est pas nécessaire pour suivre l'explication. Chaque discussion portant un jeu favorable dans la partie I est suivie d'un résumé donnant les probabilités correspondantes. Ces résumés sont suffisants pour la discussion de la partie II de sorte qu'un lecteur qui n'a aucun intérêt dans un jeu particulier peut passer directement au résumé. Des références sont données pour ceux qui désirent étudier certains jeux en détail. Pour l'instant, "jeu favorable" veut dire, jeu dans lequel la stratégie est telle que $P({\rm lim}\,S_{n}=\infty)>0$ où Sn est le capital du joueur après n essais.
Article
Full-text available
This paper considers the problem of investment of capital in risky assets in a dynamic capital market in continuous time. The model controls risk, and in particular the risk associated with errors in the estimation of asset returns. The framework for investment risk is a geometric Brownian motion model for asset prices, with random rates of return. The information filtration process and the capital allocation decisions are considered separately. The filtration is based on a Bayesian model for asset prices, and an (empirical) Bayes estimator for current price dynamics is developed from the price history. Given the conditional price dynamics, investors allocate wealth to achieve their financial goals efficiently over time. The price updating and wealth reallocations occur when control limits on the wealth process are attained. A Bayesian fractional Kelly strategy is optimal at each rebalancing, assuming that the risky assets are jointly lognormal distributed. The strategy minimizes the expected time to the upper wealth limit while maintaining a high probability of reaching that goal before falling to a lower wealth limit. The fractional Kelly strategy is a blend of the log-optimal portfolio and cash and is equivalently represented by a negative power utility function, under the multivariate lognormal distribution assumption. By rebalancing when control limits are reached, the wealth goals approach provides greater control over downside risk and upside growth. The wealth goals approach with random rebalancing times is compared to the expected utility approach with fixed rebalancing times in an asset allocation problem involving stocks, bonds, and cash.
Article
The Sharpe ratio, a most useful measure of investment performance, has the disadvantage that it is based on mean-variance theory and thus is valid basically only for quadratic preferences or normal distributions. Hence skewed investment returns can engender misleading conclusions. This is especially true for superior investors with a number of high returns. Many of these superior investors use capital growth wagering ideas to implement their strategies, which means higher growth rates but also higher variability of wealth. A simple modification of the Sharpe ratio to assume that the upside deviation is identical to the downside risk gives more realistic results.
Article
We extend the optimal strategy results of Kelly and Breiman and extend the class of random variables to which they apply from discrete to arbitrary random variables with expectations. Let Fn be the fortune obtained at the nth time period by using any given strategy and let Fn * be the fortune obtained by using the Kelly-Breiman strategy. We show (Theorem 1(i)) that Fn/Fn * is a supermartingale with E(Fn/Fn *)≤ 1 and, consequently, E(lim Fn/Fn *)≤ 1. This establishes one sense in which the Kelly-Breiman strategy is optimal. However, this criterion for 'optimality' is blunted by our result (Theorem 1(ii)) that E(Fn/Fn *)=1 for many strategies differing from the Kelly-Breiman strategy. This ambiguity is resolved, to some extent, by our result (Theorem 2) that Fn */Fn is a submartingale with E(Fn */Fn)≥ 1 and E(lim Fn */Fn)≥ 1; and E(Fn */Fn)=1 if and only if at each time period j,1≤ j≤ n, the strategies leading to Fn and Fn * are 'the same'.
Article
The implications of concentrating on the lowest moment(s) of average compound return over N periods in making investment decisions have recently been examined. In particular, maximization of expected average compound return has been shown to imply the existence of a utility of wealth function in each period with the "right" properties for all finite N \ge 2 as well as in the limit. More importantly, for large N a close (or exact) approximation to the set of mean-variance efficient portfolios (with respect to average compound return) is obtainable via a subset of the isoelastic class of utility of wealth functions. The properties of this class render it both empirically plausible and highly attractive analytically: among them are monotonicity, strict concavity, and decreasing risk aversion; moreover, the optimal mix of risky assets is independent of initial wealth (providing a basis for the formation of mutual funds) and the optimal investment policy is myopic. The purpose of this paper is to extend the class of return distributions for which the preceding results hold and to demonstrate that portfolios which are efficient with respect to average compound return, at least for large N, do not risk ruin either in a short-run or a long-run sense.
Article
This paper develops a sequential model of the individual's economic decision problem under risk. On the basis of this model, optimal consumption, investment, and borrowing-lending strategies are obtained in closed form for a class of utility functions. For a subset of this class the optimal consumption strategy satisfies the permanent income hypothesis precisely. The optimal investment strategies have the property that the optimal mix of risky investments is independent of wealth, noncapital income, age, and impatience to consume. Necessary and sufficient conditions for long-run capital growth are also given.
Article
This chapter focuses on Kelly's capital growth criterion for long-term portfolio growth. The Kelly (–Breiman–Bernoulli–Latané or capital growth) criterion is to maximize the expected value E log X of the logarithm of the random variable X, representing wealth. The chapter presents a treatment of the Kelly criterion and Breiman's results. Breiman's results can be extended to cover many if not most of the more complicated situations which arise in real-world portfolios Specifically, the number and distribution of investments can vary with the time period, the random variables need not be finite or even discrete, and a certain amount of dependence can be introduced between the investment universes for different time periods. The chapter also discusses a few relationships between the max expected log approach and Markowitz's mean-variance approach. It highlights a few misconceptions concerning the Kelly criterion, the most notable being the fact that decisions that maximize the expected log of wealth do not necessarily maximize expected utility of terminal wealth for arbitrarily large time horizons.