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Markowitz, Kelly and the Reckless Explorers

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When you think about optimizing your portfolio often two seemingly different methodologies come to mind: Mean-Variance Optimization and the Kelly formula.

Mean-Variance optimization was invented by Harry Markowitz. In a famous paper published in the Journal of Finance in 1952 he was the first one to analyze the allocation of funds in a portfolio. The problem interpretation by Markowitz was really novel at the time long before the Efficient Market Hypothesis. His research in portfolio optimization has been recognized with a Nobel Prize in Economics in 1990.

So what is Mean-Variance Optimization about?

At first glance an investor who wants to make the maximum amount of profit should invest in the security with the highest expected value. However, this is not what real investors are doing: they are rather trying to spread their capital in small separate investments in such a way that the total risk of the whole is as small as possible. Investing with a single instrument is just too risky and one can do better by selecting a diversified blend of many securities. This is what is known as an Expected Utility Theory when instead of the expected value of return, investors maximize the expectation of some function, known as a utility function.

According to Markowitz there can be many portfolios that yield the same expected return and the solution to the optimization problem would be to select the one portfolio which results in the minimal possible risk. Risk in this case is expressed as an expected volatility of portfolio. This is a really powerful idea.

Another famous line of thought is an Optimal Capital Growth theory and the Kelly formula. In 1956 the scientist from the Bell Labs, John Larry Kelly has published a paper where he analyzed the rate at which capital can grow given a risky opportunity. His main result is a Kelly criterion for the fraction of one’s capital that one should bet each time to achieve the highest growth of wealth. The formula is obtained for the sequence of bets on an outcome of a biased coin but can be generalized to other risky bets, for example stocks or indices.

To illustrate the Kelly formula let’s assume that we propose to bet multiple times in a game of chance when the odds are in our favor. Each time we can bet any amount of our existing capital and the goal is to grow as quickly as possible. Clearly, betting zero in each bet is not optimal as well as betting all of the existing capital. In the latter case the moment we lose the bet we are bankrupt. Therefore, there is some optimal intermediate fraction that produces the best result. This number is known as an  optimal f.

Kelly criterion can also be interpreted from the point of view of an Expected Utility Theory, in this case logarithmic utility.

Both the Kelly theory and Markowitz optimization would size positions optimally but from a totally different viewpoint. Markowitz would focus on the one-time best optimization while Kelly would optimize the positions over the lifetime of the investment. Markowitz would optimize over the set of potential outcomes while Kelly would optimize over the particular realization of ones wealth.

Surprisingly, these two different theories ARE the same and their unification is given by what is called by mathematicians an Ergodic Theorem. Allow me a little digression and I will explain the ergodic theorem in simple terms. Imagine a perfect world where everyone lives forever. The planet is inhabited by reckless explorers who travel the world visiting new places. But since the world is finite and explorers live forever eventually they will see it all. When we ask ourselves “what is the chance of finding a particular explorer at a certain place?” it turns out to be the same as to ask, “how often this explorer likes to visit the place?”. This is the essence of the ergodic theorem – the chance of finding the explorer at some place is proportional to the average frequency of visits to the place.

A particular example of an ergodic theorem is the Law of Large Numbers. This is a core result that makes all statistical estimation possible. Assume that we want to calculate the expected value of some random variable. Statistics teaches us that in order to do this one needs to make many observations of the same random variable and take an average. But a priori, it is not clear why the result of such computation will coincide with expected value. The law of large numbers asserts it is true exactly when the size of the sample goes to infinity – observed frequencies in the sample will approach the true probabilities of the values. Here we can see the ergodic theorem above – the probability to draw a certain value is equal to the fraction of occurrences of this value in the sample. This explanation might look trivial and only technical but it is in fact quite philosophical. Imagine, that drawing one observation takes a unit of time and the sample mean is an average taken over the trajectory in the time-domain while the expectation is an average taken in the state domain.

That is exactly what happens in Kelly vs Markowitz. There is the same random variable – the utility of one’s wealth. Markowitz will optimize the expectation of this utility over the state domain of all possible tomorrow outcomes and their probabilities. Kelly will optimize this utility over the trajectory of one’s wealth as it evolves over time. From the ergodic theorem, when the time passed is long enough and all possible outcomes are observed, both optimal values will coincide. That is how the Markowitz theory from the world of economics is tied to the Kelly theory from the world of signal processing by the mathematical theorem about brave people exploring our wonderful planet.

To infinity and beyond…

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