Kelly Strategies Research Articles

The reinforcement learning Kelly strategy

The full Kelly portfolio strategy's deficiency in the face of estimation errors in practice can be mitigated by fractional or shrinkage Kelly strategies. This paper provides an alternative, the RL Kelly strategy, based on a reinforcement learning (RL) framework. RL algorithms are developed for the practical implementation of the RL Kelly strategy. Extensive simulation studies are conducted, and the results confirm the superior performance of the RL Kelly strategies.

High Risk Premia with Modest Risk Aversion

Standard finance theory has long identified equity investment with aggregate consumption and equity investors with average consumers, while treating most growth risks as iid. The combination makes it impossible to reconcile high equity risk premia with low risk aversion. Reinterpreting equity markets as gambling casinos with unstable risks cuts a Gordian knot. Enrichment-seeking gamblers pursuing fractional Kelly strategies need a risk premium to maintain net long positions in equities. In simulations that use GDP disaster risks to proxy dividend trends, rational learning induces enough volatile excess volatility to account for risk premia of hundreds of basis points.

Awareness of Crash Risk Improves Kelly Strategies in Simulated Financial Time Series

We simulate a simplified version of the price process including bubbles and crashes proposed in Kreuser and Sornette (2018). The price process is defined as a geometric random walk combined with jumps modelled by separate, discrete distributions associated with positive (and negative) bubbles. The key ingredient of the model is to assume that the sizes of the jumps are proportional to the bubble size. Thus, the jumps tend to efficiently bring back excess bubble prices close to a “normal” or fundamental value (“efficient crashes”). This is different from existing processes studied that assume jumps that are independent of the mispricing. The present model is simplified compared to Kreuser and Sornette (2018) in that we ignore the possibility of a change of the probability of a crash as the price accelerates above the normal price. We study the behaviour of investment strategies that maximize the expected log of wealth (Kelly criterion) for the risky asset and a risk-free asset. We show that the method behaves similarly to Kelly on Geometric Brownian Motion in that it outperforms other methods in the long-term and it beats classical Kelly. As a primary source of outperformance, we determine knowledge about the presence of crashes, but interestingly find that knowledge of only the size, and not the time of occurrence, already provides a significant and robust edge. We then perform an error analysis to show that the method is robust with respect to variations in the parameters. The method is most sensitive to errors in the expected return.

Multivariate Volatility Regulated Kelly Strategy: A Superior Choice in Low Correlated Portfolios

We propose a Multivariate Volatility Regulated Kelly strategy, which has extra penalization on variance compared to the Kelly criterion. The objective function is constructed and solved. We show the superiority of our method in relative low correlated portfolios, relatively to fractional Kelly and full Kelly strategies. Our strategy reduce the short-term risk without sacrificing growth rate to invest proportion in risk free assets. Simulation results and China commodity futures empirical results strongly support our method.

Multivariate Volatility Regulated Kelly Strategy: A Superior Choice in Low Correlated Portfolios

We propose a Multivariate Volatility Regulated Kelly strategy, which has extra penalization on variance compared to the Kelly criterion. The objective function is constructed and solved. We show the superiority of our method in relatively low correlated portfolios, relative to the fractional Kelly and full Kelly strategies. Our strategy reduces the short-term risk without sacrificing the growth rate to invest more in risk-free assets. Simulation results and Chinese commodity future empirical results strongly support our method.

Fractional Kelly strategies with low-risk stocks

Fractional Kelly strategies with low-risk stocks

A Response to Professor Paul A. Samuelson’s Objections to Kelly Capital Growth Investing

The Kelly capital growth investment strategy maximizes the expected utility of final wealth with a Bernoulli logarithmic utility function. In 1956, Kelly showed that static expected log maximization yields the maximum asymptotic long-run growth. Good properties include minimizing the time to large asymptotic goals, maximizing the median, and being ahead (on average) after the first period. Bad properties include extremely large bets for short-term favorable investment situations because the Arrow-Pratt risk aversion index is essentially zero. Paul Samuelson was a critic of this approach, and his criticism is partially responsible for the fact that most finance academics and investment professionals do not recommend Kelly strategies. Samuelson’s points are theoretically correct and sharpen the theory, cautioning Kelly investors to understand the strategy’s true characteristics, including ways to lower investment exposure. His objections help us better understand the theory without detracting from its numerous valuable applications.

Kelly Criterion for Multivariate Portfolios: A Model-Free Approach

The Kelly criterion is a money management principle that beats any other approach in many respects. In particular, it maximizes the expected growth rate and the median of the terminal wealth. However, until recently application of the Kelly criterion to multivariate portfolios has seen little analysis. We briefly introduce the Kelly criterion and then present its multivariate version based only on the first and the second moments of the asset excess returns. Additionally, we provide a simple numerical algorithm to manage virtually arbitrarily large portfolios according to so-called fractional Kelly strategies.

Response to Paul a Samuelson Letters and Papers on the Kelly Capital Growth Investment Strategy

The Kelly Capital Growth Investment Strategy (KCGIS) is to maximize the expected utility of nal wealth with a logarithmic utility function. This approach dates to Bernoulli's 1738 suggestion of log as the utility function arguing that marginal utility was proportional to the reciprocal of current wealth. In 1956 Kelly showed that static expected log maximization yields the maximum asymptotic long run growth. Later, others added more good properties such as minimizing the time to large asymptotic goals, maximizing the median, and being ahead on average for the fi rst period. But there are bad properties as well such as extremely large bets for short term favorable investment situations because the Arrow-Pratt risk aversion index is near zero. Paul Samuelson was a critic of this approach and here we discuss his various points sent in letters to Ziemba and papers reprinted in the recent book, MacLean, Thorp and Ziemba (2010). Samuelson's opposition has prevented many fi nance academics and professionals from using Kelly strategies. For example, Ziemba was asked to explain this to Fidelity Investments, a major Boston investment fi rm close to and influenced by Samuelson at MIT. I agree that these points of Samuelson are correct and argue that they all make sense and caution users of this approach to be careful and understand the true characteristics of these investments including ways to lower the investment exposure.

How Does the Fortune’s Formula Kelly CapitalGrowth Model Perform?

William Poundstone’s book, <i>Fortune’s Formula</i>, brought the Kelly capital growth criterion to the attention of investors. But how do full and fractional Kelly strategies perform in practice? The authors study three simple investment situations and simulate the behavior of these strategies over medium-term horizons using a large number of scenarios. The results show 1) the great superiority of full Kelly and close-to-full Kelly strategies over longer horizons in that they earn very large gains a large fraction of the time, 2) the very risky short-term performance of Kelly and high-fractional Kelly strategies, 3) a consistent trade-off of growth versus security as a function of the bet size determined by the various strategies, and 4) no matter how favorable the investment opportunities are, or how long the finite horizon is, a sequence of bad scenarios can lead to very poor final wealth outcomes with a loss of most of the investor’s initial capital. Hence, in practice, financial engineering is important for dealing with the short-term volatility and long-run situations in a sequence of bad scenarios. Properly used, however, the strategy has much to commend it, especially in trading with many repeated investments. <b>TOPICS:</b>Financial crises and financial market history, volatility measures, statistical methods

Risk management: Survival of the fittest

This paper responds to three questions not well answered for many students of finance: (1) How do investors choose where to aim on an efficient frontier of trade-offs between portfolio expected return and risk? (2) What do investors do when return variance is inadequate as a summary of risk? (3) How can investors take into account the need to avoid interim shortfalls on the way to a more distant goal? To have great impact, the answers must be simple enough to be carried in the heads of practical investors. Since vivid examples help, the last part of this paper examines five cases: the option income fund, extreme CPPI (Constant Proportion Portfolio Insurance), the Long-Term Capital Management case, the Enron case and split capital trust funds. In each case, the discretionary wealth hypothesis is used, a reformulation of what are sometimes known as fractional Kelly strategies, to pinpoint the easily remedied blind spots that led to unexpected disappointments.

Growth versus security tradeoffs indynamic investment analysis

This paper presents an approach to the problem of optimal dynamic choice in discrete orcontinuous time where there is a direct tradeoff of growth versus security. In each period,the investor must allocate the available resources among various risky assets. The maximizationof the expected logarithm of the period‐by‐period wealth, called the capital growthor the Kelly criterion, has many desirable properties such as maximizing the asymptoticrate of asset growth. However, this strategy has low risk aversion and typically has verylarge wagers which yield high variance of wealth. With uncertain parameters, this can leadto overbetting and loss of wealth. Using fractional Kelly strategies leads to a less volatileand safer sequence of wealth levels with less growth. The investor can choose a desirabletradeoff of growth and security appropriate for the problem under consideration. Thisapproach yields simple two‐dimensional graphs analogous to static mean variance analysisthat capture the essence of the dynamic problem in a form useful for sound investmentanalysis. Use of the approach in practice is illustrated on favorable investments in blackjack,horse racing, lotto games, index and commodity futures and options trading.

Growth Versus Security in Dynamic Investment Analysis

This paper concerns the problem of optimal dynamic choice in discrete time for an investor. In each period the investor is faced with one or more risky investments. The maximization of the expected logarithm of the period by period wealth, referred to as the Kelly criterion, is a very desirable investment procedure. It has many attractive properties, such as maximizing the asymptotic rate of growth of the investor's fortune. On the other hand, instead of focusing on maximal growth, one can develop strategies based on maximum security. For example, one can minimize the ruin probability subject to making a positive return or compute a confidence level of increasing the investor's initial fortune to a given final wealth goal. This paper is concerned with methods to combine these two approaches. We derive computational formulas for a variety of growth and security measures. Utilizing fractional Kelly strategies, we can develop a complete tradeoff of growth versus security. The theory is applicable to favorable investment situations such as blackjack, horseracing, lotto games, index and commodity futures and options trading. The results provide insight into how one should properly invest in these situations.