This is explained by the fact that a couple of price paths, “the happy few”, are realizing most of the gains. After 10 years we notice a difference between the average level (red dot) and the median level (red cross). Figure 1 shows the evolution of several GBM price paths with a mean of 8% and a volatility of 25% over a period of 10 years. The following example highlights the impact of time on long-term results. Multiplicative dynamics infer that the future change in the level of wealth is dependent on the current level of wealth. The choice of this strategy equally leads to the portfolio with the highest geometric mean.Īs investment and risk always involve a time dimension, the concept of time plays a crucial role in risk-taking. This is equivalent to the choice of strategy that has a greater probability of leading to as much or more wealth than any other different strategy. (In the case of a large number of independent trials, he advised using the simple arithmetic mean.) This approach starts from a probabilistic Markovian framework to determine the growth optimal portfolio. Latane advised using the Kelly approach for investment situations that concern a significant portion of wealth which have a cumulative effect. Their findings are consistent with those of in 1738 in an influential paper called “Exposition of a New Theory on the Measurement of Risk” in which he advises maximizing the expected logarithmic utility of wealth. In between these two propositions lies the proportional solution presented by, who recommend optimizing the geometric mean of outcomes. One can also wager a small proportion of starting capital, which already seems more reasonable than the first suggestion. ![]() However, it seems a bit like Russian roulette if you played the game many times. When presented with favorable opportunities over time, the most extreme is to risk your entire wealth. (For a discussion of the Kelly criterion see for instance ).) Thorp shows that under certain circumstances, the optimal Kelly portfolio is approximately one of the Markowitz efficient portfolios but that this approximation can break down badly in practice. The growth optimal portfolio (GOP) considers investor behavior under multi-period dynamics with typically an infinite investment horizon. While the mean–variance theory initially was a one-period static theory, Markowitz recognized the relevance of the geometric mean return from the Kelly criterion. In the landscape of risk and return there are two main paradigms within portfolio theory: Markowitz’s parametric mean–variance framework and the Kelly criterion (also labeled the capital growth criterion or the growth optimal portfolio (GOP) Theory), which found its origin in information entropy. The second objective is to use this framework and show the relationship between leverage, optimizing for long-term growth and drawdown risk. The first objective is to develop a framework that can generate insights into possible drawdown characteristics for a risk-taker’s portfolio. In this article, drawdown risk is studied from a portfolio perspective. We show that in the case of heavier tailed outcomes, extra care is needed, and optimal might not be so optimal in the end. Based on Monte Carlo simulation, we analyze the medium-term behavior of different cumulative return paths and study the impact of different return outcome distributions. We demonstrate, through a series of experiments, the importance of path dependent risks in the case of outcomes subject to various return distributions. Given a certain set of profitable trading characteristics, a risk-taker who maximizes expected growth can still be faced with significant drawdowns to the point where a strategy becomes unsustainable. ![]() In this paper, we provide a flexible framework for assessing path dependent risk for a trading or investment operation. Path-dependent risk measures, such as drawdown risk, provide a means to assess the risk of significant portfolio retracements. While growth is definitely an important consideration, the focus on growth alone can lead to significant drawdowns, leading to psychological discomfort for a risk-taker. ![]() The Kelly criterion determines optimal bet sizes that maximize long-term growth.
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