<p>Yes.  I just published a paper deriving the distributions of returns for all asset and liability classes.  It isn't one distribution or one family of distributions.  Stocks are a mixture of multiple distributions.  Some of these distributions have covariance structures, and some do not.  You may have to do a bunch of your own math though.  I have been contemplating writing a Kelly bet paper, but have not had the time.  I am about to present a distribution-free replacement for Ito calculus that does not assume the existence of first or higher moments at the Southwestern Finance Association Conference.  As such, I have been busy.  It is nearly independent of the economic model in use.  I say that because it would cover a model built around panda bears if that were the true model in nature but nobody knew it, but it may not hold if certain implicit biological assumptions became untrue in the future about humans.</p>

<p>What you will want to do is account for bankruptcy risk, cash merger risk, stock-for-stock merger risk and liquidity risk in addition to return risk.  There are distributions for each in the paper.  Because a Kelly bet is equivalent to having logarithmic utility, my recommendation is to solve it as a log problem.</p>

<p>The difficulty you will find in the paper is that for stocks, there is no equivalent to a covariance matrix.  As you add dimensions, you do not add terms for the scale parameter.  If you ignore liquidity, the limitation on liability, bankruptcy and mergers then the returns for one asset are: $$\frac{1}{\pi}\frac{\sigma}{\sigma^2+(r-\mu)^2},$$ but for two assets it becomes $$\frac{1}{2\pi}\frac{\gamma}{(\gamma^2+\sum_{i=1}^2(r_i-\mu_i)^2)^{3/2}}$$ and for three assets it becomes $$\frac{1}{\pi^2}\frac{\delta^2}{\sqrt{\delta^3}\left(\delta+\sum_{i=1}^3(r_i-\mu_i)^2\right)^2}.$$</p>

<p>Notice that the scale parameter changes as assets are added which I denote by simply changing the variable name $(\sigma,\gamma,\delta)$.  This is really challenging to work with because of the poor form.  Also, I didn't work out the correct constant of integration since I didn't want to sit around working out the integrals under the limitation of liability.  It is late, and I teach in the morning.  These assume up to infinitely negative returns.</p>

<p>There is a practical alternative as well here, and you should assume no correlation as well.  Graham and Dodd investing provide the clue.  If you assume an all-or-none utility function, that you receive a utility of one if your return exceeds a threshold and a zero otherwise, then you get a simple answer.</p>

<p>First calculate the Kelly bet on a binomial, which is trivial, for your threshold rate of return.  That becomes your allocation level.  Second, as price falls, the probability of meeting the level increases and so your variance falls.  So pay as little as possible for a bet.  As you are reducing your price, you are also exposing less money to each bet, so you are reducing your risk in three ways, both through the probability of loss is reduced, the variance is reduced and the exposure size being reduced per unit of risk.</p>

<p>You can find the paper at </p>

<blockquote>
  <p><a href="https://file.scirp.org/pdf/JMF_2017083015172459.pdf" rel="nofollow noreferrer">Harris, D.E. (2017) The Distribution of Returns. Journal
  of Mathematical Finance, 7, 769-804</a>.</p>
</blockquote>

<p>The paper is built around the observation that questions such as $\text{var}\left(\frac{S_{t+1}}{S_t}\right)$ are built around two observable stock prices.  This implies that stock prices and stock volumes are data, but returns are a function of data and therefore a statistic and not data itself.</p>

<p>Since the return is future value divided by present value minus one, it follows that returns are a ratio distribution of future prices divided by current prices.  Under mild assumptions about how stocks are traded and the number of investors involved, stock prices should be normally distributed at any static moment.  More precisely, the limit order book should be.  These distributions are just the ratios of two normal distributions for each stock.  It also nicely fits the data if you make adjustments for things like bankruptcy and so forth.</p>

<p>By converting your problem to a binomial you tame it and put yourself in Warren Buffett's shoes with a mathematical explanation as to why that is the case.  The downside is you now have all the terrible reading to do of annual reports as you have to restate them from their official values to their economic values for this to work.  So a ten year deferred tax liability for $100 million needs reduced to present value since this is an interest-free loan from the Treasury and the difference added to equity.  Still, once you do the analysis, you can make a good return.</p>

<p>Also note that these distributions lack both a mean and a sufficient statistic. In the real world they are truncated at -100% and so no non-Bayesian solution exists for any of these that is unbiased and admissible.  Don't plug in your R function. It won't work.  You have to build this from scratch if you are going to make predictions.  Nicely, Bayesian predictions are of the form:</p>

<p>$$\Pr(\tilde{x}|\mathbf{X})=\int_{\theta\in\Theta}\Pr(\tilde{x}|\theta)\Pr(\theta|\mathbf{X})\mathrm{d}\theta,$$ so you can form predictions that do not depend upon the value of the parameters.  Use the CDF of $\tilde{x}$ to form your binomial density.  </p>

<p>Technically, this is not quite correct because the multi-asset form is not independent.  No two returns can ever be independent, even if there is no correlation.  Still, it is a reasonable heuristic.</p>

CC BY-SA 3.0
2018-02-01T11:30:43.103Z
Thanks so much. I'm gonna read the paper and implement it. :)
dev85
CC BY-SA 3.0
2018-01-26T06:31:27.783Z
2018-01-26T07:17:55.843Z
2018-01-26T07:17:55.843Z
Dave Harris
<p>Thanks for your patience with answering my question.</p>

<p>I am interested in building an optimal wealth allocation strategy across multiple betting opportunities, correlated or uncorrelated and with different types of constraints.</p>

<p>I've been looking for relevant papers and literature.</p>

<p>What I've found so far is all Markowitz theory, Kelly criterion related. My question is if there are other alternatives to such strategies that may have very different idea behind them.</p>

<p>Also if there are some papers that you yourself found very relevant to Markowitz and Kelly. Papers about informed betting strategies, investments across multiple opportunities, wealth allocation etc.</p>

<reference-request><investment>
Kelly criterion wealth allocation alternatives
dev85
# Kelly criterion wealth allocation alternatives

(Asked by: username_0 on 2018-01-25)

Thanks for your patience with answering my question.

I am interested in building an optimal wealth allocation strategy across multiple betting opportunities, correlated or uncorrelated and with different types of constraints.

I've been looking for relevant papers and literature.

What I've found so far is all Markowitz theory, Kelly criterion related. My question is if there are other alternatives to such strategies that may have very different idea behind them.

Also if there are some papers that you yourself found very relevant to Markowitz and Kelly. Papers about informed betting strategies, investments across multiple opportunities, wealth allocation etc.

**Tags:** reference-request, investment

---

## 1 answers

### Answer by username_1 on 2018-01-26. Score: 0 | Accepted answer

Yes. I just published a paper deriving the distributions of returns for all asset and liability classes. It isn't one distribution or one family of distributions. Stocks are a mixture of multiple distributions. Some of these distributions have covariance structures, and some do not. You may have to do a bunch of your own math though. I have been contemplating writing a Kelly bet paper, but have not had the time. I am about to present a distribution-free replacement for Ito calculus that does not assume the existence of first or higher moments at the Southwestern Finance Association Conference. As such, I have been busy. It is nearly independent of the economic model in use. I say that because it would cover a model built around panda bears if that were the true model in nature but nobody knew it, but it may not hold if certain implicit biological assumptions became untrue in the future about humans.

What you will want to do is account for bankruptcy risk, cash merger risk, stock-for-stock merger risk and liquidity risk in addition to return risk. There are distributions for each in the paper. Because a Kelly bet is equivalent to having logarithmic utility, my recommendation is to solve it as a log problem.

The difficulty you will find in the paper is that for stocks, there is no equivalent to a covariance matrix. As you add dimensions, you do not add terms for the scale parameter. If you ignore liquidity, the limitation on liability, bankruptcy and mergers then the returns for one asset are: $$\frac{1}{\pi}\frac{\sigma}{\sigma^2+(r-\mu)^2},$$ but for two assets it becomes $$\frac{1}{2\pi}\frac{\gamma}{(\gamma^2+\sum\_{i=1}^2(r\_i-\mu\_i)^2)^{3/2}}$$ and for three assets it becomes $$\frac{1}{\pi^2}\frac{\delta^2}{\sqrt{\delta^3}\left(\delta+\sum\_{i=1}^3(r\_i-\mu\_i)^2\right)^2}.$$

Notice that the scale parameter changes as assets are added which I denote by simply changing the variable name $(\sigma,\gamma,\delta)$. This is really challenging to work with because of the poor form. Also, I didn't work out the correct constant of integration since I didn't want to sit around working out the integrals under the limitation of liability. It is late, and I teach in the morning. These assume up to infinitely negative returns.

There is a practical alternative as well here, and you should assume no correlation as well. Graham and Dodd investing provide the clue. If you assume an all-or-none utility function, that you receive a utility of one if your return exceeds a threshold and a zero otherwise, then you get a simple answer.

First calculate the Kelly bet on a binomial, which is trivial, for your threshold rate of return. That becomes your allocation level. Second, as price falls, the probability of meeting the level increases and so your variance falls. So pay as little as possible for a bet. As you are reducing your price, you are also exposing less money to each bet, so you are reducing your risk in three ways, both through the probability of loss is reduced, the variance is reduced and the exposure size being reduced per unit of risk.

You can find the paper at

> [Harris, D.E. (2017) The Distribution of Returns. Journal
> of Mathematical Finance, 7, 769-804](https://file.scirp.org/pdf/JMF_2017083015172459.pdf).

The paper is built around the observation that questions such as $\text{var}\left(\frac{S\_{t+1}}{S\_t}\right)$ are built around two observable stock prices. This implies that stock prices and stock volumes are data, but returns are a function of data and therefore a statistic and not data itself.

Since the return is future value divided by present value minus one, it follows that returns are a ratio distribution of future prices divided by current prices. Under mild assumptions about how stocks are traded and the number of investors involved, stock prices should be normally distributed at any static moment. More precisely, the limit order book should be. These distributions are just the ratios of two normal distributions for each stock. It also nicely fits the data if you make adjustments for things like bankruptcy and so forth.

By converting your problem to a binomial you tame it and put yourself in Warren Buffett's shoes with a mathematical explanation as to why that is the case. The downside is you now have all the terrible reading to do of annual reports as you have to restate them from their official values to their economic values for this to work. So a ten year deferred tax liability for $100 million needs reduced to present value since this is an interest-free loan from the Treasury and the difference added to equity. Still, once you do the analysis, you can make a good return.

Also note that these distributions lack both a mean and a sufficient statistic. In the real world they are truncated at -100% and so no non-Bayesian solution exists for any of these that is unbiased and admissible. Don't plug in your R function. It won't work. You have to build this from scratch if you are going to make predictions. Nicely, Bayesian predictions are of the form:

$$\Pr(\tilde{x}|\mathbf{X})=\int\_{\theta\in\Theta}\Pr(\tilde{x}|\theta)\Pr(\theta|\mathbf{X})\mathrm{d}\theta,$$ so you can form predictions that do not depend upon the value of the parameters. Use the CDF of $\tilde{x}$ to form your binomial density.

Technically, this is not quite correct because the multi-asset form is not independent. No two returns can ever be independent, even if there is no correlation. Still, it is a reasonable heuristic.

> **Comments:**
> **username_0**: Thanks so much. I'm gonna read the paper and implement it. :)  

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