def ssd(returns):
"""
Calculate semi standard deviation of returns
"""
ret = np.copy(returns)
mean_ret = nanmean(ret)
ret[ret > nanmean(ret)] = mean_ret
std = 0
for curr_ret in ret:
std += (curr_ret - mean_ret) * (curr_ret - mean_ret)
std /= (ret.shape[0] - 1)
return np.sqrt(std)
def information_ratio(returns, factor_returns):
"""
Determines the Information ratio of a strategy.
Parameters
----------
returns : pd.Series or np.ndarray
Daily returns of the strategy, noncumulative.
- See full explanation in :func:`~empyrical.stats.cum_returns`.
factor_returns: float / series
Benchmark return to compare returns against.
Returns
-------
float
The information ratio.
Note
-----
See https://en.wikipedia.org/wiki/information_ratio for more details.
"""
if len(returns) < 2:
return np.nan
active_return = _adjust_returns(returns, factor_returns)
tracking_error = nanstd(active_return, ddof=1)
if np.isnan(tracking_error):
return 0.0
if tracking_error == 0:
return np.nan
return nanmean(active_return) / tracking_error
def alpha_aligned(returns,
factor_returns,
risk_free=0.0,
period=DAILY,
annualization=None,
_beta=None):
"""Calculates annualized alpha.
If they are pd.Series, expects returns and factor_returns have already
been aligned on their labels. If np.ndarray, these arguments should have
the same shape.
Parameters
----------
returns : pd.Series or np.ndarray
Daily returns of the strategy, noncumulative.
- See full explanation in :func:`~empyrical.stats.cum_returns`.
factor_returns : pd.Series or np.ndarray
Daily noncumulative returns of the factor to which beta is
computed. Usually a benchmark such as the market.
- This is in the same style as returns.
risk_free : int, float, optional
Constant risk-free return throughout the period. For example, the
interest rate on a three month us treasury bill.
period : str, optional
Defines the periodicity of the 'returns' data for purposes of
annualizing. Value ignored if `annualization` parameter is specified.
Defaults are:
'monthly':12
'weekly': 52
'daily': 252
annualization : int, optional
Used to suppress default values available in `period` to convert
returns into annual returns. Value should be the annual frequency of
`returns`.
- See full explanation in :func:`~empyrical.stats.annual_return`.
_beta : float, optional
The beta for the given inputs, if already known. Will be calculated
internally if not provided.
Returns
-------
float
Alpha.
"""
if len(returns) < 2:
return np.nan
ann_factor = annualization_factor(period, annualization)
if _beta is None:
_beta = beta_aligned(returns, factor_returns, risk_free)
adj_returns = _adjust_returns(returns, risk_free)
adj_factor_returns = _adjust_returns(factor_returns, risk_free)
alpha_series = adj_returns - (_beta * adj_factor_returns)
return nanmean(alpha_series) * ann_factor
def sortino_ratio(returns,
required_return=0,
period=DAILY,
annualization=None,
_downside_risk=None):
"""
Determines the Sortino ratio of a strategy.
Parameters
----------
returns : pd.Series or pd.DataFrame
Daily returns of the strategy, noncumulative.
- See full explanation in :func:`~empyrical.stats.cum_returns`.
required_return: float / series
minimum acceptable return
period : str, optional
Defines the periodicity of the 'returns' data for purposes of
annualizing. Value ignored if `annualization` parameter is specified.
Defaults are:
'monthly':12
'weekly': 52
'daily': 252
annualization : int, optional
Used to suppress default values available in `period` to convert
returns into annual returns. Value should be the annual frequency of
`returns`.
_downside_risk : float, optional
The downside risk of the given inputs, if known. Will be calculated if
not provided.
Returns
-------
depends on input type
series ==> float
DataFrame ==> pd.Series
Annualized Sortino ratio.
"""
if len(returns) < 2:
return np.nan
ann_factor = annualization_factor(period, annualization)
if len(returns) < 2:
return np.nan
adj_returns = _adjust_returns(returns, required_return)
mu = nanmean(adj_returns, axis=0)
dsr = (_downside_risk if _downside_risk is not None else downside_risk(
returns, required_return))
sortino = mu / dsr
if len(returns.shape) == 2:
sortino = pd.Series(sortino, index=returns.columns)
return sortino * ann_factor
def alpha_aligned(returns, factor_returns, risk_free=0.0, period=DAILY,
annualization=None, _beta=None):
"""Calculates annualized alpha.
If they are pd.Series, expects returns and factor_returns have already
been aligned on their labels. If np.ndarray, these arguments should have
the same shape.
Parameters
----------
returns : pd.Series or np.ndarray
Daily returns of the strategy, noncumulative.
- See full explanation in :func:`~empyrical.stats.cum_returns`.
factor_returns : pd.Series or np.ndarray
Daily noncumulative returns of the factor to which beta is
computed. Usually a benchmark such as the market.
- This is in the same style as returns.
risk_free : int, float, optional
Constant risk-free return throughout the period. For example, the
interest rate on a three month us treasury bill.
period : str, optional
Defines the periodicity of the 'returns' data for purposes of
annualizing. Value ignored if `annualization` parameter is specified.
Defaults are:
'monthly':12
'weekly': 52
'daily': 252
annualization : int, optional
Used to suppress default values available in `period` to convert
returns into annual returns. Value should be the annual frequency of
`returns`.
- See full explanation in :func:`~empyrical.stats.annual_return`.
_beta : float, optional
The beta for the given inputs, if already known. Will be calculated
internally if not provided.
Returns
-------
float
Alpha.
"""
if len(returns) < 2:
return np.nan
ann_factor = annualization_factor(period, annualization)
if _beta is None:
_beta = beta_aligned(returns, factor_returns, risk_free)
adj_returns = _adjust_returns(returns, risk_free)
adj_factor_returns = _adjust_returns(factor_returns, risk_free)
alpha_series = adj_returns - (_beta * adj_factor_returns)
return nanmean(alpha_series) * ann_factor
def downside_risk(returns,
required_return=0,
period=DAILY,
annualization=None):
"""
Determines the downside deviation below a threshold
Parameters
----------
returns : pd.Series or np.ndarray or pd.DataFrame
Daily returns of the strategy, noncumulative.
- See full explanation in :func:`~empyrical.stats.cum_returns`.
required_return: float / series
minimum acceptable return
period : str, optional
Defines the periodicity of the 'returns' data for purposes of
annualizing. Value ignored if `annualization` parameter is specified.
Defaults are:
'monthly':12
'weekly': 52
'daily': 252
annualization : int, optional
Used to suppress default values available in `period` to convert
returns into annual returns. Value should be the annual frequency of
`returns`.
Returns
-------
float, pd.Series
depends on input type
series ==> float
DataFrame ==> pd.Series
Annualized downside deviation
"""
if len(returns) < 1:
return np.nan
ann_factor = annualization_factor(period, annualization)
downside_diff = _adjust_returns(returns, required_return).copy()
mask = downside_diff > 0
downside_diff[mask] = 0.0
squares = np.square(downside_diff)
mean_squares = nanmean(squares, axis=0)
dside_risk = np.sqrt(mean_squares) * np.sqrt(ann_factor)
if len(returns.shape) == 2 and isinstance(returns, pd.DataFrame):
dside_risk = pd.Series(dside_risk, index=returns.columns)
return dside_risk
def downside_risk(returns, required_return=0, period=DAILY,
annualization=None):
"""
Determines the downside deviation below a threshold
Parameters
----------
returns : pd.Series or np.ndarray or pd.DataFrame
Daily returns of the strategy, noncumulative.
- See full explanation in :func:`~empyrical.stats.cum_returns`.
required_return: float / series
minimum acceptable return
period : str, optional
Defines the periodicity of the 'returns' data for purposes of
annualizing. Value ignored if `annualization` parameter is specified.
Defaults are:
'monthly':12
'weekly': 52
'daily': 252
annualization : int, optional
Used to suppress default values available in `period` to convert
returns into annual returns. Value should be the annual frequency of
`returns`.
Returns
-------
float, pd.Series
depends on input type
series ==> float
DataFrame ==> pd.Series
Annualized downside deviation
"""
if len(returns) < 1:
return np.nan
ann_factor = annualization_factor(period, annualization)
downside_diff = _adjust_returns(returns, required_return).copy()
mask = downside_diff > 0
downside_diff[mask] = 0.0
squares = np.square(downside_diff)
mean_squares = nanmean(squares, axis=0)
dside_risk = np.sqrt(mean_squares) * np.sqrt(ann_factor)
if len(returns.shape) == 2 and isinstance(returns, pd.DataFrame):
dside_risk = pd.Series(dside_risk, index=returns.columns)
return dside_risk
def sortino_ratio(returns, required_return=0, period=DAILY,
annualization=None, _downside_risk=None):
"""
Determines the Sortino ratio of a strategy.
Parameters
----------
returns : pd.Series or np.ndarray or pd.DataFrame
Daily returns of the strategy, noncumulative.
- See full explanation in :func:`~empyrical.stats.cum_returns`.
required_return: float / series
minimum acceptable return
period : str, optional
Defines the periodicity of the 'returns' data for purposes of
annualizing. Value ignored if `annualization` parameter is specified.
Defaults are:
'monthly':12
'weekly': 52
'daily': 252
annualization : int, optional
Used to suppress default values available in `period` to convert
returns into annual returns. Value should be the annual frequency of
`returns`.
_downside_risk : float, optional
The downside risk of the given inputs, if known. Will be calculated if
not provided.
Returns
-------
float, pd.Series
depends on input type
series ==> float
DataFrame ==> pd.Series
Annualized Sortino ratio.
"""
if len(returns) < 2:
return np.nan
ann_factor = annualization_factor(period, annualization)
adj_returns = _adjust_returns(returns, required_return)
mu = nanmean(adj_returns, axis=0)
dsr = (_downside_risk if _downside_risk is not None
else downside_risk(returns, required_return))
sortino = mu / dsr
return sortino * ann_factor