Showing papers on "Kelly criterion published in 2016"
01 Dec 2016
TL;DR: It is shown that in many cases, the theoretical Kelly-based results may lead to bets which are “too conservative” rather than too aggressive, and a result is provided which is called the Restricted Betting Theorem.
Abstract: Kelly betting is a prescription for optimal resource allocation among a set of gambles which are typically repeated in an independent and identically distributed manner. In this setting, there is a large body of literature which includes arguments that the theory often leads to bets which are “too aggressive” with respect to various risk metrics. To remedy this problem, many papers include prescriptions for scaling down the bet size. Such schemes are referred to as Fractional Kelly Betting. In this paper, we take the opposite tack. That is, we show that in many cases, the theoretical Kelly-based results may lead to bets which are “too conservative” rather than too aggressive. To make this argument, we consider a random vector X with its assumed probability distribution and draw m samples to obtain an empirically-derived counterpart X. Subsequently, we derive and compare the resulting Kelly bets for both X and X with consideration of sample size m as part of the analysis. This leads to identification of many cases which have the following salient feature: The resulting bet size using the true theoretical distribution for X is much smaller than that for X. If instead the bet is based on empirical data, “golden” opportunities are identified which are essentially rejected when the purely theoretical model is used. To formalize these ideas, we provide a result which we call the Restricted Betting Theorem. An extreme case of the theorem is obtained when X has unbounded support. In this situation, using X, the Kelly theory can lead to no betting at all.
29 citations
TL;DR: In this article, an analogy between information engines and gambling is presented, where bits representing measurements are converted into energy and bits representing currency exchange rates or in the stock market using options.
Abstract: In information theory, one area of interest is gambling, where mutual information characterizes the maximal gain in wealth growth rate due to knowledge of side information; the betting strategy that achieves this maximum is named the Kelly strategy. In the field of physics, it was recently shown that mutual information can characterize the maximal amount of work that can be extracted from a single heat bath using measurement-based control protocols, i.e. using 'information engines'. However, to the best of our knowledge, no relation between gambling and information engines has been presented before. In this paper, we briefly review the two concepts and then demonstrate an analogy between gambling, where bits are converted into wealth, and information engines, where bits representing measurements are converted into energy. From this analogy follows an extension of gambling to the continuous-valued case, which is shown to be useful for investments in currency exchange rates or in the stock market using options. Moreover, the analogy enables us to use well-known methods and results from one field to solve problems in the other. We present three such cases: maximum work extraction when the probability distributions governing the system and measurements are unknown, work extraction when some energy is lost in each cycle, e.g. due to friction, and an analysis of systems with memory. In all three cases, the analogy enables us to use known results in order to obtain new ones.
12 citations
TL;DR: In this paper, the authors proposed a method to obtain the maximum growth while staying above an exante discrete time wealth path with high probability, where shortfalls below the path are penalized with a convex function of the shortfall.
Abstract: The optimal capital growth strategy or Kelly strategy has many desirable properties such as maximizing the asymptotic long-run growth of capital. However, it has considerable short-run risk since the utility is logarithmic, with essentially zero Arrow–Pratt risk aversion. It is common to control risk with a Value-at-Risk (VaR) constraint defined on the end of horizon wealth. A more effective approach is to impose a VaR constraint at each time on the wealth path. In this paper, we provide a method to obtain the maximum growth while staying above an ex-ante discrete time wealth path with high probability, where shortfalls below the path are penalized with a convex function of the shortfall. The effect of the path VaR condition and shortfall penalties is a lower growth rate than the Kelly strategy, but the downside risk is under control. The asset price dynamics are defined by a model with Markov transitions between several market regimes and geometric Brownian motion for prices within a regime. The stochast...
11 citations
TL;DR: This work solves the stochastic equation for two classes of carrying capacity functions (power laws and logarithmic), and in both cases the optimal trajectories of the control parameter are computed.
Abstract: We consider the problem of finding optimal strategies that maximize the average growth rate of multiplicative stochastic processes. For a geometric Brownian motion, the problem is solved through the so-called Kelly criterion, according to which the optimal growth rate is achieved by investing a constant given fraction of resources at any step of the dynamics. We generalize these finding to the case of dynamical equations with finite carrying capacity, which can find applications in biology, mathematical ecology, and finance. We formulate the problem in terms of a stochastic process with multiplicative noise and a nonlinear drift term that is determined by the specific functional form of carrying capacity. We solve the stochastic equation for two classes of carrying capacity functions (power laws and logarithmic), and in both cases we compute the optimal trajectories of the control parameter. We further test the validity of our analytical results using numerical simulations.
8 citations
Dissertation•
01 Jan 2016
TL;DR: This paper aims to provide a chronology of the events leading up to and including the publication of the Declaration of the Republic of Ireland and its contents and aims to clarify the aims and aims of this work.
Abstract: 8 Declaration 9 Copyright 11 Acknowledgements 13
7 citations
TL;DR: In this paper, an extension to the growth optimal derivative that can accommodate risk preferences differing from those of logarithmic utility is presented. And the authors show that power investors under the real-world probability can be viewed as logrithmic investors under myopic probability of Guasoni and Robertson.
Abstract: This article presents an extension to the growth optimal derivative that can accommodate risk preferences differing from those of logarithmic utility. Analysis of the optimal derivative provides interesting insights into the behaviour of power investors. We show that power investors under the real-world probability can be viewed as logarithmic investors under the myopic probability of Guasoni and Robertson [(2012). “Portfolios and Risk Premia for the Long Run.” Annals of Applied Probability, 22 (1), 239–284]. Furthermore, this intuition provides criteria for establishing whether fractional Kelly betting is optimal for power investors. Finally, the Black–Scholes model is used to demonstrate how the optimal derivative can be implemented and we show that our approach is consistent with classical techniques.
4 citations
TL;DR: In this article, the authors compared the performance of three portfolio construction methods: an optimized portfolio, an equally-weighted portfolio, and a nonlinear nonparametric model for constructing hedge funds.
4 citations
Posted Content•
TL;DR: This note generalises the Kelly strategy (optimal for simple investment games with log utility) to a large class of practical utility functions and including the effect of extraneous wealth, demonstrating the practical calculation of the optimal action at every stage is made possible through use of the binomial expansion.
Abstract: Prompted by a recent experiment by Victor Haghani and Richard Dewey, this note generalises the Kelly strategy (optimal for simple investment games with log utility) to a large class of practical utility functions and including the effect of extraneous wealth.
A counterintuitive result is proved : for any continuous, concave, differentiable utility function, the optimal choice at every point depends only on the probability of reaching that point.
The practical calculation of the optimal action at every stage is made possible through use of the binomial expansion, reducing the problem size from exponential to quadratic.
Applications include (better) automatic investing and risk taking under uncertainty.
3 citations
TL;DR: In this paper, the authors restrict the Kelly optimization by security measures Value at Risk (VaR) or Expected Shortfall as special cases of the Spectral risk measure, which allows to increase the growth-criterion for a given security level.
Abstract: Investing in a portfolio strategy, which maximizes the the expected logarithm of capital implies outperforming any other significantly different strategy, given knowledge of the true underlying process (Breiman, 1961). But, the sole use of the Kelly Criterion implies larger bets than a risk-averse investor would accept (Clark and Ziemba, 1987; Hausch and Ziemba, 1985). This paper restricts the Kelly optimization by security measures Value at Risk (VaR) or Expected Shortfall as special cases of the Spectral risk measure.As financial market returns are, with large probability, non-Gaussian, this paper utilizes stable laws and its scaling behavior to model financial market returns for a given horizon in an i.i.d. world. Instead of simulating from the class of elliptically stable distributions, a nonparametric scaling approximation, based on the dataset itself, is proposed. Additionally, assets with non-linear payoff structure, long put-options, are incorporated into the nonlinear optimization to allow for asymmetric payoffs. For chosen special cases of the Spectral measure, this allows to increase the growth-criterion for a given security level.
2 citations
TL;DR: Haghani et al. as discussed by the authors derived optimal strategy for a game with known payouts and probability, but uncertain limits on total payout and number of betting opportunities, and further developed some general gambling principles applicable to practical risk taking situations in which all parameters are uncertain and robust approximate simple solutions are required.
Abstract: This paper was inspired by "Rational Decision-Making Under Uncertainty: Observed Betting Patterns on a Biased Coin" (Haghani, Victor and Dewey, Richard, https://ssrn.com/abstract=2856963). It derives optimal strategy for a game with known payouts and probability, but uncertain limits on total payout and number of betting opportunities. It further develops some general gambling principles applicable to practical risk taking situations in which all parameters are uncertain and robust approximate simple solutions are required.
2 citations
Posted Content•
TL;DR: In this paper, the influence of Commodity Trading Advisors (CTA) on the price process is explored with the help of a simple model, where CTA managers are taken to be Kelly optimisers, which invest a fixed proportion of their assets in the risky asset and the remainder in a riskless asset.
Abstract: The influence of Commodity Trading Advisors (CTA) on the price process is explored with the help of a simple model. CTA managers are taken to be Kelly optimisers, which invest a fixed proportion of their assets in the risky asset and the remainder in a riskless asset. This requires regular adjustment of the portfolio weights as prices evolve. The CTA trading activity impacts the price change in the form of a power law. These two rules governing investment ratios and price impact are combined and lead through updating at fixed time intervals to a deterministic price dynamic. For different choices of the model parameters one gets qualitatively different dynamics. The result can be expressed as a phase diagram. Meta-CTA strategies can be devised to exploit the predictability inherent in the model dynamics by avoiding critical areas of the phase diagram or by taking a contrarian position at an opportune time.
28 Feb 2016
TL;DR: In this article, an asymptotisch optimale Kelly portfolio is implemented and tested in a simulation study and on empirical basis, and closed-form solutions for the growth-optimum strategy are presented under different parametric distribution assumptions.
Abstract: In dieser Masterarbeit wird das asymptotisch optimale Kelly Portfolio, im Gegensatz zum Mittelwert/Varianz Ansatz, implementiert und in einer Simulationsstudie, wie auch auf empirischer Basis getestet. Das hauptsachliche Ziel von Kelly (1956) ist die Maximierung des erwarteten Logarithmus des Vermogens, welche, wie Breiman (1961) bewiesen hat, zu einer asymptotisch, optimalen Strategie in einer unabhangigen, stationaren Welt fuhrt, welche aber auf verschieden verteilte, zeitabhangige Renditen erweitert werden kann.%%%%In this thesis the Kelly growth-optimum criterion, as one strand of portfolio theory, besides the widely used mean-variance approach, is implemented and tested in a simulation study and on empirical basis. The main objective of Kelly (1956) is the maximization of the expected logarithm of growth, leading to, as Breiman (1961) proves, the asymptotically optimal strategy in an i.i.d. world, which can be extended to arbitrary and time-dependent returns. Under different parametric distribution assumptions for the outcomes, closed-form solutions for the growth-optimum strategy will be presented. Within a simulation study it will be shown that, sampling from the assumed data generating process naturally supports the asymptotic outperformance. As the assumption of the known process is loosened and the Kelly strategy needs to be implemented upon past, limited data, draw-down risks are increased and the portfolio maximizing the expected logarithm of end wealth is shifted to fractional Kelly bets. This holds for the empirical out-of-sample test. As the main statistical focus remains the improvement of the moment estimates in terms of errors, conditional moments are estimated by econometric time-series models. Setting the conditional mean forecast to zero if the conditional volatility forecasts surpasses the unconditional volatility, leads to a cancellation of positions in times of high uncertainty, which, one the one hand, decreases errors in the mean estimate and on the other hand, decreases portfolio draw-downs substantially.
Posted Content•
TL;DR: In this paper, the authors explore how uncertainty beyond risk determines consistent leverage in a simple model of the world with fat tails due to significant, not fully quantifiable and not too rare events.
Abstract: Risk and uncertainty will always be a matter of experience, luck, skills, and modelling. Leverage is another concept, which is critical for the investor decisions and results. Adaptive skills and quantitative probabilistic methods need to be used in successful management of risk, uncertainty and leverage. The author explores how uncertainty beyond risk determines consistent leverage in a simple model of the world with fat tails due to significant, not fully quantifiable and not too rare events. Among particular technical results, for the single asset fractional Kelly criterion is derived in the presence of the fat tails associated with subjective uncertainty. For the multi-asset portfolio, Kelly criterion provides an insightful perspective on Risk Parity strategies, which can be extended for the assets with fat tails.
Posted Content•
TL;DR: In this paper, the problem of finding optimal strategies that maximize the average growth rate of multiplicative stochastic processes with finite carrying capacity is considered. But the authors focus on the case of dynamical equations with finite carry capacity, which can find applications in biology, mathematical ecology and finance.
Abstract: We consider the problem of finding optimal strategies that maximize the average growth-rate of multiplicative stochastic processes. For a geometric Brownian motion the problem is solved through the so-called Kelly criterion, according to which the optimal growth rate is achieved by investing a constant given fraction of resources at any step of the dynamics. We generalize these finding to the case of dynamical equations with finite carrying capacity, which can find applications in biology, mathematical ecology, and finance. We formulate the problem in terms of a stochastic process with multiplicative noise and a non-linear drift term that is determined by the specific functional form of carrying capacity. We solve the stochastic equation for two classes of carrying capacity functions (power laws and logarithmic), and in both cases compute optimal trajectories of the control parameter. We further test the validity of our analytical results using numerical simulations.
01 Jan 2016
TL;DR: In this paper, the Kelly criterion is used to allocate capital to a project with the intent of maximizing the return on the capital employed and limiting exposure to a critical shortfall in the total capital available for other projects.
Abstract: The Kelly Criterion, developed in 1956 by John Kelly at the Bell Laboratories, provides a method to allocate capital to a project with the intent of maximizing the return on the capital employed and limiting exposure to a critical shortfall in the total capital available for other projects. This shortfall can occur when projects that are funded early in the funding cycle are subjected to a run of bad luck and both the corporate success rate and value added from exploration falls significantly lower than expected. This disappointment could cause a tactical revision to the budget and diminish the pool of capital available for the remaining projects. Even when this criterion has already been applied to balance the portfolio with the corporate risk attitude and the capital available, the budget may be subjected to a sudden reduction in the remaining funds available due to reasons beyond their control. This constraint may possibly be due to temporary cash flow shortages, another corporate division with a sudden need for capital or as we have seen in the last six months the need to pay down debt. Because the constrained budget is not a change in corporate attitude regarding money to be placed at risk, but rather a temporary economic remedy to a shortage of cash currently available, the company may prefer to reduce the budget year allocation but maintain the corporate risk attitude. To do this the company must determine which projects in the portfolio best meet the corporate objectives for maximizing long term return at an appropriate level of risk and either reduce equity or postpone some projects to meet the cash flow constraints. This paper will suggest one method to make the required adjustments based on a linear programming model. A linear program solution is similar to a marble dropped into a tilted box. The marble will come to rest at the intersection of the two sides that form the lowest location in the box. It will not find a solution if, for instance, one side is perfectly aligned with the low point such that all the points on that edge are equally low or if there are baffles that prevent the marble from continuing to roll to the lowest point. Other more robust models such as non-linear or integer programming might find a solution in these more complex situations.
TL;DR: This paper applied the Kelly criterion to games that have several outcomes, where the organizer charges a rake, and used a formula-based approach to address the challenge of identifying games where the player's current bankroll has the highest expected growth rate.
Abstract: This study focuses on Sit and go poker tournaments, where a player with an advantage over her opponents needs to manage her bankroll properly The study applies the Kelly criterion to games that have several outcomes, where the organizer charges a rake The premise is that an advantage itself is not enough for a poker player to play at any stakes, because risking too large a fraction of the bankroll will result in a negative expected growth rate, even though the game itself is characterized by a positive expected value Accordingly, this study uses a formula-based approach to address the challenge of identifying games where the player’s current bankroll has the highest expected growth rate, while also considering differences in the rake
Posted Content•
TL;DR: In this article, the authors presented a new approach in the framework of a biased roulette wheel and used the approach of a quantitative trading strategy, commonly used in quantitative finance, in order to assess the profitability of the strategy in the short term.
Abstract: The purpose of this research paper it is to present a new approach in the framework of a biased roulette wheel. It is used the approach of a quantitative trading strategy, commonly used in quantitative finance, in order to assess the profitability of the strategy in the short term. The tools of backtesting and walk-forward optimization were used to achieve such task. The data has been generated from a real European roulette wheel from an on-line casino based in Riga, Latvia. It has been recorded 10,980 spins and sent to the computer through a voice-to-text software for further numerical analysis in R. It has been observed that the probabilities of occurrence of the numbers at the roulette wheel follows an Ornstein-Uhlenbeck process. Moreover, it is shown that a flat betting system against Kelly Criterion was more profitable in the short term.
10 May 2016