Another alternative to mean-variance is to select the portfolio that has the highest expected geometric mean return. This, in effect, maximizes the expected value of terminal wealth.
The geometric mean is defined as:
where Rij is the ith possible return on the jth portfolio and each outcome is equally likely.
If the likelihood of each outcome is different and Pij is the probability of the ith outcome for the jth portfolio, then
and can be written as,
The resulting portfolio is usually very well diversified and extreme values have a tendency to be eliminated. If a strategy has a probability of bankruptcy then the whole product will become zero.
The geometric mean is a measure of central tendency, just like a median. It is different from the traditional mean (which we sometimes call the arithmetic mean) because it uses multiplication rather than addition to summarize data values. Geometric means are often useful summaries for highly skewed data.
The geometric mean for any time period is less than or equal to the arithmetic mean. The two means are equal only for a return series that is constant (i.e., the same return in every period). For a non-constant series, the difference between the two is positively related to the variability or standard deviation of the returns.
The main problem with this method is that it does not differentiate between investors and thereby does not explicitly refer to risk. If our expected return forecasts were the same, then every investor, irrespective of their circumstances, would hold the same portfolio.
Arguably, this method could be used by a mutual fund that has a broadly diversified group of investors. It is very quick and easy to use.
Maximizing the geometric mean is equivalent to maximizing the expected value of a log utility
Wednesday, January 20, 2010
Here is an excellent overview of risk aversion.