Now this is a farily basic question, but since I see professionals having trouble with this all the time, let us go through it
Simple returns aggregate nicely (linearly) across trades but not time, whereas log-returns aggregate nicely across time but not trades. So when you take arithmetic average of simple returns across time, you make a minor travesty...
1. The "return" of the returns, simple vs log
If $S$ denotesthe asset value, then the
simple return between day $d-1$ and day $d$ is $R_d := S_d/S_{d-1} - 1$, and log-return $r_d := \ln(S_d/S_{d-1})$, such that the relation $r_d=\ln(1+R_d) \iff R_d=\exp(r_d)-1$ holds.
1.1. Why log-returns are Normal, and simple returns are "abnormal"...
If the asset price $S$ is Log-normally distributed (e.g. stocks), then log-returns are Normally distributed and additive. That is, if $S_d=S_{d-1}\exp(\nu \Delta t + \sigma \sqrt{\Delta t}Z_d)$, then
\begin{align}
R_d&=\exp(\nu \Delta t + \sigma \sqrt{\Delta t}Z_d)-1, \\
r_d&=\nu \Delta t + \sigma \sqrt{\Delta t}Z_d.
\end{align}
That is, log-returns are nicely Normally distributed, whereas simple returns are shifted Log-normally distributed, and less nice...
1.2. Averaging log-returns makes sense across time, simple returns do not
The arithmetic mean of the daily log-returns is an unbiased estimator of the drift term
$\hat{\nu}\Delta t:=\frac{1}{n}\sum_{d=1}^n r_d$, whereas for the simple returns, we would have to do a geometric mean, which will be a biased estimate of the drift term. Arithmetic mean of the simple returns does not have the same practical meaning in this case.
2. Relationship therapy between simple and log returns
To simplify notation let $X_{day}^{(trade)}:=X_{day,trade}$, and let $X_{day}$ be aggregated over trades, and $X^{(trade)}$ be a trade aggregated over time. Aggregation over trades and time is $X$.
2.1. Fixed time, across trades
\begin{align}
R_d &= \sum_{i=1}^m w_i R_d^{(i)}, \\
r_d :&= \ln(1 + R_d) = \ln(1 + \sum_{i=1}^m w_i R_d^{(i)}) = \ln \left(\sum_{i=1}^m w_i \exp(r_d^{(i)})\right),
\end{align}
where we can see that simple returns linearly aggregate across trades, whereas log-returns do not.
2.2. Fixed trade, across time
\begin{align}
R^{(i)} &= \prod_{d=1}^n (1+R_d^{(i)})-1, \\
r^{(i)} &:= \ln(1 + R^{(i)}) = \sum_{d=1}^n \ln(1+R_d^{(i)}) = \sum_{d=1}^n r_d^{(i)},
\end{align}
where we can see that simple returns do not linearly aggregates across trades, whereas log-returns do.
2.3. Across trades and across time
Now in order to get the total return for the portfolio, you have two options
- Aggregate across trades, then time
- Aggregate across trades in simple returns $R_d=\sum_{i=1}^m w_i R_d^{(i)}$ for all $d$ays.
- Convert to log-returns $r_d := \ln(1+R_d)$.
- Aggregate (sum) across time $r=\sum_{d=1}^n r_d$.
- Optionally convert back to simple return via $R=\exp(r)-1$.
- Aggregate across time, then trades
- Aggregate across time in log-returns $r^{(i)}=\sum_{d=1}^n r_d^{(i)}$.
- Convert to simple returns $R^{(i)} := \exp(r^{(i)})-1$.
- Aggregate across trades in simple returns $R=\sum_{i=1}^m w_iR^{(i)}$.
- Optionally convert back to log-returns via $r=\ln(1+R)$.
Now it is unfortunately common to see people that purely work with log-returns by linearly aggregating across trades and time. Implicitly they are assuming that $r_d^{(i)} \approx R_d^{(i)}$, which is true for small returns. However, when this is not the case, or when summing many small returns e.g. return over large horizons, this might cause large errors.
