How to calculate Sharpe Ratio from $ returns?

Question

I have a pairs strategy that I am trying to calculate the sharpe ratio for. Currently I am using python for my analysis and calculation. I have a dataframe that contains the cumulative returns in $'s for each day. I am confused on how to convert this information into something that I can calculate the sharpe ratio from.

Could anyone point me in the right direction on how to use cumulative returns (in $'s) to find the sharpe ratio? Any help is appreciated! Thanks

Answer 1

Let's say your cumulative return series is $\{R_i \mid i=0,1,...,N-1\}$ of length $N$ days.

There's 3 conventional ways to do this at this stage. You may convert the cumulative dollar return curve into arithmetic returns:

$\displaystyle{r_i}= \dfrac{R_i-R_{i-1}}{R_{i-1}}$

Or dollar returns:

$\displaystyle{r_i=R_i-R_{i-1}}$

Then take the ratio:

$\displaystyle{SR_{1d} = \dfrac{E\{r_i\}-r_f}{std\{r_i\}} }$

where the risk-free rate $r_f$ is often taken to be $0$. Finally, you annualize it:

$\displaystyle{SR_{1y}=SR \cdot \sqrt{252}}$

Here's an example of how you can do it in Python:

import numpy as np
import pandas as pd

# Simulate cumulative returns of 100 days
N = 100
R = pd.DataFrame(np.random.normal(size=100)).cumsum()

# Approach 1
r = (R - R.shift(1))/R.shift(1)

# Approach 2
r = R.diff()

sr = r.mean()/r.std() * np.sqrt(252)

Answer 2

With the other answer, the Sharpe ratio might be positive although the returns are negative. This comes from the fact that geometric mean and arithmetic mean are different.

The following approach could be used:

returns = pd.DataFrame(np.random.normal(size=100))
mean_log_returns = (np.mean(np.log(returns + 1)))
mean_returns = np.exp(mean_log_returns) - 1
std = returns.std()
sharpe_ratio = mean_returns / std * np.sqrt(252)