Aggregating Portfolio Returns Across Trades and Time

# How to calculate the log return of portfolio?


# How to calculate the log return of portfolio?












Suppose that we have five trades each day with these returns ($R_{day,trade}$) and we have 300 days in total:

$R_{1,1}$, $R_{1,2}$, $R_{1,3}$, $R_{1,4}$, $R_{1,5}$

$R_{2,1}$, $R_{2,2}$, $R_{2,3}$, $R_{2,4}$, $R_{2,5}$

For each day, we calculate the return of that day by weighting these returns (these weights are coming from outside of the portfolio and summation of these weights are 1 for each day), so for days 1 and 2, the returns are $R_{1}$ and $R_{2}$ and later for calculating the average return of this portfolio for 300 days we calculate the average of these returns ($R_{1}$, $R_{2}$, ..).

- Is this procedure correct?

- Now, suppose that we want to have log returns. There are two ways to reach log returns. First, calculate the log return of each trade $(ln(Pt/Pt−1)$ and continue the mentioned steps. The other one is when we reach the daily returns, we use $Rn=ln(1+R)$ for calculating daily log returns, and the average is the log return of the portfolio (daily).

## Answer by Pontus Hultkrantz (score 7)

https://quant.stackexchange.com/a/71078

Now this is a fairly 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 aggregate across time, 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.

## Answer by user70865 (score -3)

https://quant.stackexchange.com/a/78119

Heyya Hawaii, Get into returns world. Do some proper calculations than just sending a formula. Use a real time example.