## Annualized Geometric Return

The annualized return $y$ is the rate at which a portfolio compounds each year. If the rate of return $r_i$ observed varies between period $i$ the compounding rate is obtained by taking the geometric average $y$ of the single period returns.

$$y = \exp\left(\frac{1}{n} \sum_i \ln (1+r_i)\right)-1$$

A mathematical property of geometric average is that it is always smaller than the simple (arithmetic) average return $r$ given by:

$$r = \frac{1}{n} \sum_i r_i $$

## Cases where the Geometric Return should be used

Annualized return is what is required to know how a given investment is expected to grow over the years. The annualized return is linked to the average log return which makes it suitable for multi-period compounding computation. It drives long term return. When returns are volatile, it is much lower than the simple average return, hence the proclivity of asset managers to advertise their performance as simple average rather than compound rate.

## Cases where the Simple Return should be used

The simple return of a portfolio is the average of the simple returns of the portfolio components. For this reason, the simple return computed over the portfolio rebalancing interval needs to be used as input in Markowitz MVO. Our service automatically converts the annualized return into a simple return. Another case where the simple return needs to be computed is the sharpe ratio and kelly ratio.

## Relation Between Annualized Return and Simple Return

If the simple return has average $r$ and standard deviation $\sigma$, the expected annualized return $y$ under a normally distributed assumption is

$$\ln(1+y) = r - \frac{1}{2} \sigma^2$$