The arithmetic mean (or just the mean) is what is relevant for thinking about the trade-off between risk and return:
where 
indicates the
sum.
The geometric mean gives the equivalent constant return to a buy-and-hold strategy with reinvestment of dividends:
![geometric mean = [prod_t=1...T (1+R_t)]^(1/T)-1](latex/eq118.png.0)
where 
indicates the
product. Over many short periods, the geometric mean is
approximately the arithmetic mean minus half the variance.
Phil Dybvig
|
On average, how much is this asset's value expected to increase this period? The answer is the arithmetic mean return. Or, for past data the sample mean is the answer to the question, ``How much did the investment grow per period, on average?'' When we talk about a trade-off between risk and return, we use the arithmetic mean return for the ``return'' part of ``risk and return''. This is the sensible measure, and this is what is proportional to the risk exposure in pricing models, such as the CAPM or the APT. |