Warwick Schneller

COMPUTATIONAL FINANCE – POSTGRADUATE ASSIGNMENT

Money Management

This paper is a review of the recent literature on money management. The paper examines existing gambling position management techniques, namely Optimal f. The potential applications of Particle Swarm Optimization and Bayesian Statistics to position management are also examined.

 

 

Warwick Schneller Semester 093

 

 

1.     INTRODUCTION

 

In an influential 1932 essay, Lionel Robbins defined economics as “the science which studies human behaviour as a relationship between ends and scarce means which have alternative uses.” (Robbins, 1932). Despite economist early recognition of the importance of how to allocate scarce resources finance in the context of trading and money management has largely left this question unanswered. Instead finance faculties have focussed on market efficiency and how financial agents behave.

There is now a well established body of literature on the efficiency of financial markets including research by Fama (1965) , Roche(1995), Malkiel {2003} Brock, Lakonishok, & LeBaron (1992) and Lo, Mamaysky, & Wang (2000). The research undertaken into examining market efficiency adopts a static approach to trading and largely ignores money management issues. Research to date assumes that only fixed (or single) sized positions are held and that profits (accumulated capital) cannot be reinvested (Anderson & Faff, 2004) . This method although effective for determining the pricing efficiency of markets gives no consideration to position sizing.

Finance practitioners on the other hand give considerable thought to how best to allocate funds when trading. Some common approaches include fixed positions sizing, percentage of equity, martingale and maximum risk percentage. Despite the common usage of such methods by traders these methods do not necessarily lead the optimal allocation of capital. The methods commonly used by traders although intuitively appealing have largely no theoretical basis.

To a large extent the position management literature is underdeveloped and in its infancy.

This paper provides a review of the literature on money management and is organised as follows: Section 2 examines and defines money management. Section 3 examines alternate money management techniques. Section 3 are the paper’s conclusions.

 

2.     MONEY MANAGEMENT

 

Money management[1] relates to how much risk a decision maker should take relative to the expected reward. Money management in an economics context is what percentage of wealth should be risked in order to maximize the decision makers utility function (Balsara, 1992).  Despite the importance of position management traders tend to focus most of their efforts on developing and testing trading strategies, often overlooking the fact that even potentially profitable entry and exit rule may end up losing money should money management not be properly implemented (Vanstone & Hahn, 2010).

Vanstone and Hahn (2010) state that position sizing strategies should give consideration to the quality of the buy/sell signal, the state of the market and the amount of capital available. This criterion formalises expected utility theory and places it in a trading context.

Prior to examining alternate position sizing strategies it is necessary to determine whether or not a trading system has a mathematical edge? If a system has a negative expectation  then the optimal money management approach is to expose zero capital (Gehm, 1995). Vince (1992) developed this further and found that the next best strategy in the negative expectation scenario is the “maximum boldness strategy’ in which the trader bets on as few as possible trials. This is analogous to a casino type environment where gambling in more trails with smaller sums of capital will results in the eventually lose of all capital.

If a trader has identified an exploitable market opportunity then it is necessary to determine what approach will maximise the expected returns.  The paper now briefly examines gambling mathematics and its application to money management in trading

 

3.     MONEY MANAGEMENT TECHNIQUES

3.1. GAMING MATHEMATICS AND f

 

The finance literature for position management has drawn a number of developments from gambling environments. In the context of gambling extensive research has been undertaken as to how to maximise the expected payoffs in an environment in which the probabilities are known.

Kelly[2] (1956) found that if the outcome is known with certainty the ‘gambler’ should bet 100% of their available capital on the outcome of each event. Therefore the value of a gambler’s stating capital V_o, would grow, G, over n trials exponentially.

 

If the probability, p, of an error is introduced, then the gambler must rationally adjust the betting strategy or else the probability of error would lead to exhaustion of capital once the first loss is encountered. Therefore for all non-zero values of p the probability of losing all available capital is one.  The implication of this is that all rational agents will adjust their betting strategy to avoid intentional self-destruction and hence will only bet a fraction, l, of available capital per individual bet (Anderson & Faff, 2004).

Since this is a gambling environment the probability of the outcomes is pre-defined and can take on only one of two outcomes, namely a Win, W, or a loss, L. (where the probability of either outcome sums to 1 i.e. . The value after n trials is found to be

The growth, G, of the portfolio can be then stated with a probability of one as:

This insight by Kelly (1956) provided a critical development into the allocation of capital in betting game type environments, namely f. Where f is the percentage of capital invested on each trail to maximise the expected value of the logarithm of the starting capital.

Where  p = Probability of a winning bet

q = Probability of a losing bet (This is the complement of p where q = 1-p)

Equation 5 is only applicable in games where the amount won or loosed are equal. In non-equal gaming environments the formula is modified as follows

Where  p = Probability of a winning bet

b = Ratio of the amount won on a winning bet to the amount lost on a losing bet

The Kelly Formula is generally expressed as follows

The Kelly Criterion is valid when operating in a Bernoulli distribution environment. However, securities markets do not follow such distributions and hence the direct application of the Kelly criteria is limited.

Vince (1992) modified Kelly’s formula to devise what is commonly known at Optimal f. It aims to identify the optimum fixed fraction to bet on any individual outcome (Anderson & Faff, 2004). At the Optimal f, the rate of reinvestment is found that would maximize the geometric rate of return and so dominate all trading strategies.  The Optimal f method does not require a simply win/loss type outcome (i.e. No Bernoulli distribution assumption is applied).

The method proposed by Vince (1992) maximizing the geometric rate of return is achieved by modelling the largest observed loss and trading the portfolio reinvestment rate on this basis and determining which multiple of that largest loss would have produced the largest return on the funds invested.

To derive the Optimal f Vince made a number of developments. The first is to modify the traditional Holding Period Return formula to include f , Thus

 

TWR is the Terminal Wealth Relative. It represents the return on your stake as a multiple. For example a TWR of 10.55 means a return of 10.55 on the original investment or viewed another way 955% profit (R Vince, 1995) .

 

 

 

1. Optimal f * (Geometric Mean HPR -1)

The Optimal f approach resolved two key issues for financial practitioners. First, how much money should be allocated per trade. Secondly, how many trades should be traded at any one time with a given portfolio allocation (R Vince, 1995).

A criticism of this fixed fractional trading approach which is acknowledged by its developer is the severity of the draw downs. For example, if Optimal f is 0.55 then your drawdown would have been at least 55% of equity i.e. trading at the Optimal f will potentially lead to drawdown’s equal to f. This creates a situation where the better the system the higher the f value and higher the drawdown. This creates a paradox whereby if the system is successful then through its success it increases the probability for the ‘risk of ruin’. The risk of ruin is an extreme case where the success of the system and the associated draw downs lead to capital inadequacy and being unable to trade out of the situations.

A subsequent development by Zamansky and Stendahl is Secure f. This method places a constraint of the maximum drawdown. Optimal f identifies what fraction of starting capital to invest in each trade in order to maximise expected returns. This can represent a high percentage of available capital and hence returns can be volatile. The Secure f will always be less than or equal to the Optimal f because the optimiser limit the maximum fraction which can be allocated to each trade to the value which gives a drawdown equal to the constraint (Zamansky & Stendahl). Secure f will not lead to the optimal allocation of resources due to the capital constraint imposed, however, it does mitigate the severity of draw downs which maybe preferential depending on attitudes to risk.

 

3.2. PARTICLE SWARM OPTIMIZATION

 

Particle Swarm Optimization (PSO) is a population based search algorithm based on the simulation of the social behaviour of individuals moving through a multidimensional space (Nenortaite & Simutis, 2004). A non linear technique which is linked to genetic algorithms and ‘evolutionary computing’ it was first introduced by Kennedy and Eberhart (1995) and to date has largely been applied to modelling nature and animal life behaviour (Fourie & Groenwold, 2002). In a finance context Nenortaite and Simutis (2004) applied the technique in combination with artificial neural networks to develop a rule based trading system. To date the PSO technique is still in its infancy and has not been applied to any real extent in financial research.

Fourie & Groenwold (2002) applied the PSO technique to model the behaviour of bird flocks. This research took the individual behaviour of birds including; position, velocity and fitness and then the actual behaviour of the flock to determine migratory patterns and flock fitness levels.  The application of this to position management is based off the criteria of an effective position management system outlined by Vanstone and Hahn (2010);

• Quality of the signal
• State of the market
• The amount of equity in a portfolio

The PSO’s ability to model individual bird position including fitness levels is analogous to calculating the quality of the signal. Secondly, the ability to apply this to the flock is equivalent to applying it on a market wide basis and taking account of market wide behaviour. Lastly, the model factors in the fitness of the flock and the number of birds which make up the flock and therefore shows consideration for what in a finance context is the portfolio and its underlying characteristics including the amount of capital available.

The formal PSO model as developed by Kennedy & Eberhart (1995) and modified by Fourie & Groenwold(2002) is briefly outlined[3].

 

Where a & x are column vectors

f*is the optimal weight allocation

 

A factor which PSO is able to include and has applicability to financial markets is “craziness”. This was modelled by  Fourie & Groenwold(2002) whereby a random operator was included in the functional form to take account of temporary departures of birds from a flock. In a finance context this could potentially be representative of the “random” price/position movements.

 

With

3.3. BAYESIAN PROBABILITY

 

In financial markets news constantly arrives that affects the outlook on securities’ returns. Hence, it is crucial to understand how to incorporate such information in forming an updated opinion on the statistical properties of returns and how best to allocate capital to trade. Bayesian statistical theory allows for the updating of prior beliefs about the distribution of a random variable after observing new information.

The Theorem[4] is briefly outlined;

Let A and B be sets. Then we know that we can obtain the probability of an event within the union of both sets using conditional probabilities:

1. P(AΩB) = P(B|A) P(A)

= P(B| A) = P(A|B) P(B).

Therefore,

1. P(B|A) = P(AΩB)/ P(A)

= P(B|A) P(A) / P(A)

This yields Bayes’ theorem:

 

Bayesian probability interprets probability as a “measure of a state of knowledge” rather than the frequency of an event (Jaynes, Bretthorst, & ebrary Inc., 2003). The literature presents two opposing views on the interpretation of Bayesian statistics. The Objectivist View which interprets the probabilities as an extension of logic and are rational expectations. Alternatively, the Subjectivist View interprets the probabilities as a measure of a “personal belief”.(Jaynes, Bretthorst, & ebrary Inc., 2003). The application of Bayesian probability to machine learning problems is an example of the Objectivist View.  Almgren & Lorenz (2006) Zürich (2008)have been applied Bayesian probability to trading environments, however the focus of the research was into trading system optimisation [5]

In a trading context the decisions made are often based on experience and knowledge. Bayes’ formula is a rational method for making adjustments. The rational basis for decision making potentially allows for its application to position management. For example, commonly used technique such as pyramiding and martingale sequences make adjustments to a position based on a particular sequence of events. However, these methods do not consider the quality of the signal or take account of market conditions. A Bayesian approach allows for a more statistically rigorous method to make position adjustments.

 

4.     CONCLUSION

 

This paper has conducted a brief review of the recent literature on money management. Position management techniques have drawn a number of developments from gambling mathematics. However, the complexity and uncertainty of securities markets means that the direct application of these techniques is not ideal. Vince’s (1992) development of Optimal f is a substantial development in the literature but the high risk of drawdowns reduces its suitability.

Two areas which are in their infancy in their application to trading problems and particularly position managements are; Particle Swarm Optimisation and Bayesian Probability. PSO is a sophisticated approach for modelling nature and animal life. Its ability to model dynamic systems such as bird flocks and the allocation of resources in such systems indicates potential relevance for future research.

Bayesian Probability is a developed statistical technique which has widely been applied to mathematical and engineering type problems but has limited application to trading problems. Future research into its application to position management systems is based on its ability to utilise probabilities to make rational decisions.

5.     BIBLIOGRAPHY

Almgren, R., & Lorenz, J. (2006). Bayesian adaptive trading with a daily cycle. The Journal of Trading, 1(4), 38-46.

Anderson, J. A., & Faff, R. W. (2004). Maximizing futures returns using fixed fraction asset allocation. Applied Financial Economics, 14(15), 1067-1073.

Balsara, N. J. (1992). Money management strategies for futures traders. New York: Wiley.

Brock, W., Lakonishok, J., & LeBaron, B. (1992). Simple Technical Trading Rules and the Stochastic Properties of Stock Returns. Journal of Finance, 47(5), 1731-1764.

CFA Institute. (2009). Quantitative Methods (Vol. 1): Chartered Financial Analyst Society.

Fama, E. F. (1965). THE BEHAVIOR OF STOCK-MARKET PRICES. Journal of Business, 38(1), 34-105.

Fourie, P. C., & Groenwold, A. A. (2002). The particle swarm optimization algorithm in size and shape optimization. Structural and Multidisciplinary Optimization, 23(4), 259-267.

Gehm, F. (1995). Quantitative trading & money management : a guide to risk analysis and trading survival (Rev. ed.). Chicago ; London: Irwin.

Jaynes, E. T., Bretthorst, G. L., & ebrary Inc. (2003). Probability theory the logic of science. Cambridge, UK ; New York, NY: Cambridge University Press.

Kelly Jr, J. (1956). A new interpretation of information rate. Information Theory, IRE Transactions on, 2(3), 185-189.

Kennedy, J., & Eberhart, R. C. (1995). Particle swarm optimization.

Kochieva, E., Mitra, G., & Lucas, C. A. Algorithmic Trading: Market Impact Models and Trade Scheduling.

Lo, A. W., Mamaysky, H., & Wang, J. (2000). Foundations of Technical Analysis: Computational Algorithms, Statistical Inference, and. Journal of Finance, 55(4), 1705-1765.

Malkiel, B. G. (2003). A random walk down Wall Street : the time-tested strategy for successful investing (Completely rev. and updated . ed.). New York: W.W. Norton.

Nenortaite, J., & Simutis, R. (2004). Stocks’ Trading System Based on the Particle Swarm Optimization Algorithm. Lecture Notes in Computer Science, 843-850.

Nofsinger, J. R., & Sias, R. W. (1999). Herding and Feedback Trading by Institutional and Individual Investors. Journal of Finance, 54(6), 2263-2295.

Robbins, L. (1932). An essay on the nature & significance of economic science: Macmillan & co., limited.

Roche, J. (1995). Forecasting commodity markets : using technical, fundamental and econometric analysis. London: Probus.

Vanstone, B., & Hahn, T. (2010). Designing stockmarket trading systems: with and without soft  computing (1 ed.): Harriman house.

Vince, R. (1992). The mathematics of money management: risk analysis techniques for traders: John Wiley & Sons Inc.

Vince, R. (1995). The new money management : a framework for asset allocation. New York: Wiley.

Zamansky, L. J., & Stendahl, D. C. Evaluating System Efficiency. Stocks & Commodities, V15:10 461-464.

Zürich, E. T. H. (2008). Optimal Trading Algorithms: Portfolio Transactions, Multiperiod Portfolio Selection, and Competitive Online Search.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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