Overview
Maximizing the expected growth rate of a portfolio has very nice long term properties. In particular, it can be shown to beat other strategies with a high probability. (see Cover for formal defintions and discussion)
Simple Justification
This argument follows the discussion in luenberger. Designate the value of an asset at time {% i %} as {% P_i %}, then define the arithmetic return of the asset in period {% t+1 %} as
{% R_{i+1} = \frac{P_{i+1}}{P_{i}} %}
The value of asset at time {% t+1 %} is
{% P_{i+1} = P_i \times R_{i+1} %}
Then by running this equation backward you get
{% P_{i} = P_0 \times R_{1} \times R_{2} \times R_{1} ... \times R_{i} %}
Taking the logarithm of both sides.
{% ln P_{i} = ln P_0 + \sum_0^i ln R_j %}
{% ln (\frac{P_n}{P_0}) ^{1/n} = \frac{1}{n} \sum_0^i ln R_j %}
By the
law of large numbers
{% \frac{1}{n} \sum_0^i ln R_j \rightarrow \mathbb{E}(ln R_i) %}
{% ln (\frac{P_n}{P_0}) ^{1/n} \rightarrow \mathbb{E}(ln R_i) %}
This would seem to indicate that any strategy that is not the log-optimal strategy will
lose to the strategy that maximizes the expected log returns.
Continuous Time
The following equation purports to show that the growth rate determines the long term behaviour of an asset. (using defintions given in stochastic portfolio theory)
{% lim_{T \to \infty} \frac{1}{T} (log S(T) - \int_0^T \alpha(t)dt ) = 0 \;\;\; a.s. %}
(see fernholz section 1.3)
Demonstration
This demonstration shows the performance of a rebalanced growth optimal portfolio versus a stock. In this example, there are two assets, a stock that doubles with probability 1/2, and halves with probability 1/2. The second asset is a risk free asset with zero return. That is, you can keep your money under the mattress.
The growth optimal portfolio will put half its money in the stock, and half its money under the mattress, and then rebalance the portfolio back to half and half after each period. (That is, after the stock moves up or down, the money is no longer invested half and half and needs to be reablanced) (see Luenberger).
This demonstration shows the performance of the portfolio versus the stock alone. It can be re-run by clicking the button. The vast majority of trials will show that the portfolio with half its money on the table outperforms the stock.