# Tag Info

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The short answer is that: Maximizing the expected logarithm leads to more wealth almost surely in the long run. In contrast, maximizing expected return can easily lead to going broke almost surely in the long run! Maximizing expected return results in betting everything on your highest expected return investment. Repeatedly doing that over time typically ...

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Maximizing $E[\log(G)]$ which corresponds to a concave utility function is a subtle way of incorporating risk aversion in the utility. Maximizing $E[G]$ is basically saying that you have linear utility which corresponds to infinite risk appetite because as soon as you have positive expectation you are willing to bet as much capital as possible no matter the ...

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The original paper was concerned with optimizing the long run geometric return. In fact, he does not explicitly optimize either $\mathbb{E}(G)$ or $\mathbb{E}(\log(G))$. He also assumes the probabilities are known. The expectation is implicit in his assumption that $$G=\lim_{N\to\infty}\frac{1}{N}\log\left(\frac{V_N}{V_0}\right).$$ He notes that $$V_N=(... 8 The Kelly criterion is just one approach to portfolio construction (or bet sizing) that considers the risk-return tradeoff. There are many possible strategies (static or dynamic) that incorporate other criteria such as the maximum drawdown, probability of ruin, etc. As pointed out by @John, Kelly is maximizing the log of wealth, which is equivalent to ... 8 This problem can be expressed as the original Merton's portfolio problem. Consider wealth process defined by SDE$$ d X _ { t } = \frac { X _ { t } \alpha _ { t } } { S _ { t } } d S _ { t } + \frac { X _ { t } \left( 1 - \alpha _ { t } \right) } { S _ { t } ^ { 0 } } d S _ { t } ^ { 0 } $$where \alpha_t is proportion of the investment in the risky ... 5 This is indeed a very good question! There were (and still are) very hefty debates, where even academic champions like Paul Samuelson were involved! A very good starting point to get some main arguments is the following chapter 4 from the book "Fortunes Formula" by William Poundstone: https://books.google.de/books?id=xz4y3u-qM04C&lpg=PA179&dq=... 5 The Kelly Criterion aims to maximise the expected value of the logarithm of terminal wealth. The derivation starts off by assuming that there is a risky asset that is following a Geometric Brownian Motion:$$ \frac{\,dS}{S} = \mu \,dt + \sigma \,dZ_t $$This is combined with a riskless asset that is continuously compounding:$$ \frac{dB}{B} = r \,dt $$... 4 I hope this help you. We have to start from the very first step, namely how the Kelly formula is calculated. We have the chance to make a bet on a event A that as an odd (decimal odds) O_A. We want bet only a fraction f of our capital V_0. How much of our capital we have to bet? Well, if we win will face with a capital V_1$$ V_1=(1+(O_A-1)f)V_0 $$... 3 Kelly is mostly based upon assets with zero correlation made independent of each other. The way I approximate Kelly for multiple bets with correlation is: Assume after your first bet the capital is gone. Place a second bet based upon the Kelly of the remaining capital. Factor in correlation.. Part 3 is the challenging part. I assume that with multiple bets ... 3 Maybe you will find the following papers pretty interesting: Laureti, P., Medo, M., and Zhang, Y.-C. (2010). Analysis of Kelly-optimal portfolios. Quantitative Finance, 10(7): 689–697. and Nekrasov, Vasily, Kelly Criterion for Multivariate Portfolios: A Model-Free Approach (September 30, 2014). The last one is available at SSRN. Particularly, ... 3 @chrisaycock raises a valid point about the process not being particular rigorous. There are reasons why it is not a rigorous or optimized sizing mechanism. 1) most market makers are quoting many strikes & expirations, usually multiple underlying assets these days. Depending on venue there may be ways to limit the number of executions, eg, exchange ... 2 In the following blog post the analytical results of the following paper and reproduced by simulation with real market data: Michael Stutzer: The Paradox of Diversification Abstract The current market malaise may keep some investors on the sidelines. The benefits of diversification may not seem as appealing in situations where the constituent investments ... 2 For a single period, I would consider scenario optimisation: simulate your assets' returns (which you can do since you know their statistical properties, including correlations), and in this way create a large number of scenarios. Collect these scenarios in a matrix, R, say, with scenarios in rows, assets in columns. The portfolio returns for a given weight ... 2 They are the same. The maximum growth rate is achieved when the Sharpe ratio is maximized. For the proof, see here. 2 There are two communities that parametrize the random variables of the returns and levels differently. And unfortunately, they both use the same notation. One community uses \mu to denote the mean of the logarithm of the level. The other uses \tilde{\mu} to denote the logarithm of the mean of the level. Yes, the order in which you apply transformations ... 2 When it comes to Kelly, I've always liked the explanation at http://www.financialwisdomforum.org/gummy-stuff/kelly-ratio.htm. At this link there are three versions of Kelly explained, if you don't mind the "chatty" style. The third version brings in the standard deviation of wins and losses, which I think is very useful. 2 According to Skiena (link page 21) the Kelly fraction in the case of wins all equal to W and losses all equal to L is:$$f=\frac{pW-qL}{WL}=p/L-q/W$$where q=1-p and p is the probability of a win. When the wins and losses are random, with average W and L respectively, I am not sure this formula is completely justified. But it might be a good ... 2 ok I found it 🙂and this works for any distribution, not just the normal distribution f^*=\frac μ {σ^2 + μ^2} \approx \frac μ {σ^2} \space if μ\llσ here the steps: https://www.dropbox.com/s/4nqd5yfk2xcuag5/kelly.pdf 2 Unfortunately, the solution isn't simple in that you can pick up a piece of paper or pencil, but with software it isn't actually as bad as it is about to sound. To begin with, note that the Kelly Criterion is precisely equivalent to assuming logarithmic utility and maximizing the utility of wealth. This is valuable in two ways. First, it allows you to ... 2 Luenberger's book has a discussion on growth-optimal (i.e. Kelly) portfolios, also for the multivariate case with correlated assets. @BOOK{Luenberger1998, title = {Investment Science}, publisher = {Oxford University Press}, year = 1998, author = {David G. Luenberger} } 2 Thorp defines g_{\infty} as the mean long run logarithmic portfolio return. He argues that this is maximized when the portfolio is set$$F^{*}=C^{-1}(M-R)$$Here is M a vector of drift rates m_i (these are usually denoted by \mu in a Geometric Brownian motion). This is nowadays a fairly standard formula, see e.g. here https://faculty.chicagobooth.edu/... 1 Full Kelly bet criteria maximizes the expected logarithmic rate of return. In your example, you propose to reach a specific rate of return. If ever the target is to achieve a specific rate of return which is less than maximal, then the optimal bet size is said to be fractional Kelly. In other words, the fractional Kelly bet which achieves the target rate of ... 1 The Kelly Criterion would tend to create very few trades as it is a maximal solution as t\to\infty. If you have included the cost of liquidity in your calculations, the drag would minimize trades. That is a good thing though. The Kelly Criterion is equivalent to the logarithmic utility of wealth. Any code would be specific to you as you know your ... 1 Suppose we decompose each \phi_i into the mean \mu plus some extra bit \kappa_i. Then what would the squared term look like?$$\phi_i^2 = (\mu + \kappa_i)^2 = \mu^2 + \kappa_i^2 + 2\mu\kappa_i$$Now we've been told that the mean is small, which is generally code for "the square can be neglected", leaving us with \kappa^2 and 2\mu\kappa. We are ... 1 No real answer here, but rather some thoughts: Kelly can deal with any distribution, not necessarily binary outcomes. See examples here. The more severe issue IMO is the fact that investment opportunity arrival times are unknown. Knowing this sequence in advance would substantially improve the gains. There is a method in computer science called "... 1 I think your calculation is right and the Kelly criterion is very aggressive. Note however that it is meant to apply to the situation where you win exactly your last bet times 299 84% of the time and you lose exactly your bet times 1181.4 the other percentage of the time. This is not the case here so this is at best an estimation and it's somewhat self ... 1 I hope my computations are correct. Let u(t,x)=\max_{(f_s)_{s\geq t}}\mathbb{E}[(b+X^{f_.}_T)^\gamma]. Using HJB (you have to prove that it is ok to use it).$$0=\max_{f}\partial_t u(t,x)+(\mu f x - C)\partial_x u(t,x)+\frac{\sigma^2}{2}f^2x^2\partial_{xx}u(t,x) Since $\partial_{xx}u(t,x)<0$ (prove it), maximum is hit at $f=\frac{\mu x \partial_xu(... 1 Kelly calculates optimal leverage for maximising geometric growth. At the same time, any change in leverage does not lead to a change in a risk-adjusted return (i.e. Sharpe). Therefore Kelly cannot be used to improve risk-adjusted return. Talking about the excess vola, in practive one rarely applies Kelly. The bet is usually Kelly/2, Kelly/4 or even less. 1 What are you saying is not completely correct. What kelly criterion maximizes is the average growth of the capital invested. In fact, if I want to invest a fraction$f$of my 1000 units the amount that I will have after$M$trades will be$1000\Pi_{i=1}^{M} (1+f\phi_i)$What we need to maximize is expected long-term growth rate. Growth rate is given by$\...

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As the paper suggests, the results that are shown in table 2 are taken from (if you read the caption) Ziemba, William T., and Donald B. Hausch, Betting at the Racetrack (New York: Norris M. Strauss, 1986) The citation is not included for some reason, hence your confusion. Your code works fine by the way. Thanks

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