The expected return is the predicted return. It can, for example, be deduced from a model such as a factor model ($\approx$ regression).

Portfolio

The expected return of a portfolio is the weighted average of the expected returns for each asset in the portfolio.

\[\mathbb{E}[R_p] = \Sigma_{i=1}^d \omega_i \mathbb{E}[R_i]\]

Indeed: \(R_p = \sum_{i=1}^N \omega_i R_i\)

Warning if continuous returns:

\[r_p = ln(1+R_p) \color{red}{\neq} \sum_{i=1}^N \omega_i r_i\] \[r_p = ln \big( \sum_{i=1}^N \omega_i e^{r_i} \big)\]

Strategy

The expected return is also used to assess the profitability of a strategy. Example:

A hourly strategy consists in trading an asset that has a hourly volatility of 1.22\%.

The strategy has a success rate of 55\%.

Say the strategy runs without interruption for one month. It means that, in one month, \(30*24*0.55=396\) trades are correct and \(30*24*0.45=324\) are wrong.

$\mathbb{E}[R] = (1+1.22\%)^{396}(1-1.22\%)^{324}-1=128\%$ !

This looks very impressive, however assuming 0.2% fees paid at each trade:

$\mathbb{E}[R] = (1+1.22\%-0.2\%)^{396}(1-(1.22\%+0.2\%))^{324}-1=-46\%$

In order to achieve a minimum return, one could solve the following equation to find the required success rate:

$\mathbb{E}[R] = (1+1.22\%-0.2\%)^{720x}(1-(1.22\%+0.2\%))^{720-720x}-1>2\%$ => $x = 63\%$.