Historical Kelly Optimization and the Limits of Empirical Returns

# Can I apply the Kelly criterion directly, without fitting any distributions?


# Can I apply the Kelly criterion directly, without fitting any distributions?












#### Problem

I want to apply the Kelly criterion to asset returns, so that I know how much to hold of each, ideally (and how much I should keep as a cash reserve).

As far as I understand the Kelly criterion, it's about maximizing the expected logarithmic returns - which computes as $$\frac{1}{n}\sum_{t=1}^{n} log(\frac{wealth_t}{wealth_{t-1}})$$

This correctly weighs losses, since summing in log-space is equivalent to multiplying in regular-space: even just one complete loss bankrupts my portfolio with no chance of recovery (it has a log of $-\infty$). Thanks to Matthew Gunn for explaining.

#### Research

In many "tricks" I've found online, people fit distributions to the returns, or find a neat formula.

What I dislike is there's some assumption of normality, or appeal to the law of large numbers - but I am skeptical they hold. For example, this answer and this page mention the variance - but variance can't capture fat tails.

Taleb shows that a fat-tailed distribution finds it much harder to converge to a mean, so the "large numbers" need to be much larger.

Another problem is the multivariate scenario: n-dimensional distributions are a headache to get right. Many mention covariance, but I have the same objection as for variance: it does not apply for non-normal distributions.

#### My question

To get to the point: Is it a good idea to side-step fitting any distribution and optimize directly on the data?

For example, say I compare stocks and bonds. Stocks have a high return, but also high risk, while bonds have a low return and low risk.

I'd get the two series, compute their log-returns, and find a linear combination with a maximum average log-return.

The coefficients of the linear combination will be the amount I invest in each.

Perhaps I divide the result by 2, to get half-Kelly, because past performance is no guarantee of future results.

But is skipping any distribution modelling a good idea?

I am inclined to believe it is, at least in the case when the data includes several instances of crashes. But I have a hunch I might be missing something, maybe related to ergodicity. Can you invalidate my belief?

##### Results

In case anyone is curious, I did it, and it seems very strange:

- when using monthly returns, the algorithm likes crazy leverage, like 3x stocks and 6.67x bonds.

- when using yearly rolling returns, it only goes 1.7x stocks and 2.34x bonds.

- 5-year returns make it go 2x stocks and .16x bonds.

- 10-year returns, both go up again to 2.45x and bonds go down to .263x.

I guess I don't even know what to ask. What is this? Serial correlation?

In any case, I suspect tail risk is not captured properly, so I can't recommend my method as it is. It doesn't seem to even converge.

## Answer by FP0 (score 2, accepted)

https://quant.stackexchange.com/a/71666

I think that your worries regarding normality, in general the right choice of a distribution, and the headache regarding high-dimensional distributions are valid.

However, how could you maximise the expected value of your total log-returns without modelling these individual log-returns?

By using a couple of series of log-returns and creating combinations of them, you are already modelling your log-returns. Instead of having a "usual" distribution like a normal distribution derived from some fitting procedure, you are using some kind of empirical distribution derived from past observations of the log-returns. Depending on how much data you have on hand and whether you think that observations of past returns will be representative of future returns, you could give it a try. But I do not think that there is a definitive answer to your question since there are too many questions being raised. I think that the main issue will be the stability of the distribution of the returns, and of their correlations, in the end.

## Answer by danuker (score 0)

https://quant.stackexchange.com/a/71698

My allocations varied wildly from 3x stocks/6x bonds to 2x stocks/.16x bonds, depending on how exactly I made the fit.

This is a red flag. Extreme data in the tails were very affected by what I believed to be minimal processing. But the "tails" were wagging the dog. So I dug deeper.

I read more from Taleb, and he refers to this as the "Lucretius fallacy, which as we saw can be paraphrased as: the fool believes that the tallest river and tallest mountain there is equals the tallest ones he has personally seen".

I am the fool, and I thank Taleb for letting me know!

Here is his monograph, "Statistical Consequences of Fat Tails", which proved to me an illuminating piece of work. Chapter 9 begins with the previous quote.

Needless to say, I have more studying to do before I juggle with estimating my exposures.