Portfolio Optimization, When the Underlying Asset is Subject to a Multiplicative Continuous Brownian Motion With Gaussian Price Fluctuations

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Imagine that you are an investor with some starting capital, which you can invest in just one risky asset. You decided to use the following simple strategy: you always maintain a given fraction 0 < r < 1 of your total current capital invested in this asset, while the rest (given by the fraction 1 − r) you wisely keep in cash. You select a unit of time (say a week, a month, a quarter, or a year, depending on how closely you follow your investment, and what transaction costs are involved) at which you check the asset’s current price, and sell or buy some shares of this asset. By this transaction you adjust the current money equivalent of your investment to the above pre-selected fraction of your total capital.

The question that is interesting is: which investment fraction provides the optimal typical long-term growth rate of investor’s capital? By typical, it is meant that this growth rate occurs at large-time horizon in majority of realizations of the multiplicative process. By extending time-horizons, one can make this rate to occur with probability arbitrary close to one. Contrary to the traditional economics approach, where the expectation value of an artificial “utility function” of an investor is optimized, the optimization of a typical growth rate does not contain any ambiguity.

Let us assume that during the timescale, at which the investor checks and readjusts his asset’s capital to the selected investment fraction, the asset’s price changes by a random factor, drawn from some probability distribution, and uncorrelated from price dynamics at earlier intervals. In other words, the price of an asset experiences a multiplicative random walk with some known probability distribution of steps. This assumption is known to hold in real financial markets beyond a certain time scale. Contrary to continuum theories popular among economists our approach is not limited to Gaussian distributed returns: indeed, we were able to formulate our strategy for a general probability distribution of returns per capital (elementary steps of the multiplicative random walk).

Thus risk-free interest rate, asset’s dividends, and transaction costs are ignored (when volatility is large they are indeed negligible). However, the task of including these effects in our formalism is rather straightforward. The quest of finding a strategy, which optimizes the long-term growth rate of the capital is by no means new: indeed it was first discussed by Daniel Bernoulli in about 1730 in connection with the St. Petersburg game. In the early days of information sciences, C. F. Shannon has considered the application of the concept of information entropy in designing optimal strategies in such games as gambling. Working from the foundations of Shannon, J. L. Kelly Jr. has specifically designed an optimal gambling strategy in placing bets, when a gambler has some incomplete information about the winning outcome (a “noisy information channel”). In modern day finance, especially the investment in very risky assets is no different from gambling. The point Shannon and Kelly wanted to make is that, given that the odds are slightly in your favor albeit with large uncertainty, the gambler should not bet his whole capital at every time step. On the other hand, he would achieve the biggest long-term capital growth by betting some specially optimized fraction of his whole capital in every game. This cautious approach to investment is recommended in situations when the volatility is very large. For instance, in many emergent markets the volatility is huge, but they are still swarming with investors, since the long-term return rate in some cautious investment strategy is favorable.

Later on Kelly’s approach was expanded and generalized in the works of Breiman. Our results for multi-asset optimal investment are in agreement with his exact but non-constructive equations. In some special cases, Merton has considered the problem of portfolio optimization, when the underlying asset is subject to a multiplicative continuous Brownian motion with Gaussian price fluctuations.

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