Long Term Growth
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 Argument
(see
luenberger)
ASsuming that we designate the value of an asset at time i as {% P_i %}, then if we define the arithmetic return of the asset in period
t+1 as
{% R_{i+1} = \frac{P_{i+1}}{P_{i}} %}
Then the value of asset at 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} %}
Then by 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) %}
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.
