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ENH: add expected max drawdown function (#90)
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@pepicello
pepicello committed Aug 9, 2020
1 parent b026b4e commit 8db7656
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1 change: 1 addition & 0 deletions empyrical/__init__.py
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Expand Up @@ -43,6 +43,7 @@
downside_risk,
excess_sharpe,
max_drawdown,
expected_max_drawdown,
omega_ratio,
roll_alpha,
roll_alpha_aligned,
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115 changes: 114 additions & 1 deletion empyrical/stats.py
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Expand Up @@ -19,7 +19,7 @@
import pandas as pd
import numpy as np
from math import pow
from scipy import stats, optimize
from scipy import stats, optimize, interpolate
from six import iteritems
from sys import float_info

Expand Down Expand Up @@ -405,6 +405,119 @@ def max_drawdown(returns, out=None):
roll_max_drawdown = _create_unary_vectorized_roll_function(max_drawdown)


def expected_max_drawdown(mu, sigma, t, gbm=False):
"""
Determines the expected maximum drawdown of a brownian motion,
given drift and diffusion
If a geometric Brownian motion with stochastic differential equation
dS(t) = Mu0 * S(t) * dt + Sigma0 * S(t) * dW(t) ,
it converts to the form here by Ito's Lemma with X(t) = log(S(t)) such that
Mu = Mu0 - 0.5 * Sigma0^2
Sigma = Sigma0 .
Parameters
----------
mu : float
The drift term of a Brownian motion with drift.
sigma : float
The diffusion term of a Brownian motion with drift.
t : float
A time period of interest
gbp : bool
If true, compute for geometric brownian motion
Returns
-------
expected_max_drawdown : float
Note
-----
See http://www.cs.rpi.edu/~magdon/ps/journal/drawdown_journal.pdf
for more details.
"""

if gbm:
new_mu = mu - 0.5 * sigma**2
else:
new_mu = mu

def emdd_qp(x):
""" Q function based on lookup table, for positive drifts """
A = [0.0005, 0.001, 0.0015, 0.002, 0.0025, 0.005,
0.0075, 0.01, 0.0125, 0.015, 0.0175, 0.02,
0.0225, 0.025, 0.0275, 0.03, 0.0325, 0.035,
0.0375, 0.04, 0.0425, 0.045, 0.0475, 0.05,
0.055, 0.06, 0.065, 0.07, 0.075, 0.08,
0.085, 0.09, 0.095, 0.1, 0.15, 0.2,
0.25, 0.3, 0.35, 0.4, 0.45, 0.5,
1.0, 1.5, 2.0, 2.5, 3.0, 3.5,
4.0, 4.5, 5.0, 10.0, 15.0, 20.0,
25.0, 30.0, 35.0, 40.0, 45.0, 50.0,
100.0, 150.0, 200.0, 250.0, 300.0, 350.0,
400.0, 450.0, 500.0, 1000.0, 1500.0, 2000.0,
2500.0, 3000.0, 3500.0, 4000.0, 4500.0, 5000.0]
B = [0.01969, 0.027694, 0.033789, 0.038896, 0.043372, 0.060721,
0.073808, 0.084693, 0.094171, 0.102651, 0.110375, 0.117503,
0.124142, 0.130374, 0.136259, 0.141842, 0.147162, 0.152249,
0.157127, 0.161817, 0.166337, 0.170702, 0.174924, 0.179015,
0.186842, 0.194248, 0.201287, 0.207999, 0.214421, 0.220581,
0.226505, 0.232212, 0.237722, 0.24305, 0.288719, 0.325071,
0.355581, 0.382016, 0.405415, 0.426452, 0.445588, 0.463159,
0.588642, 0.668992, 0.72854, 0.775976, 0.815456, 0.849298,
0.878933, 0.905305, 0.92907, 1.088998, 1.184918, 1.253794,
1.307607, 1.351794, 1.389289, 1.42186, 1.450654, 1.476457,
1.647113, 1.747485, 1.818873, 1.874323, 1.919671, 1.958037,
1.991288, 2.02063, 2.046885, 2.219765, 2.320983, 2.392826,
2.448562, 2.494109, 2.532622, 2.565985, 2.595416, 2.621743]
if x > 5000:
return 0.25 * np.log(x) + 0.49088
elif x < 0.0005:
return 0.5 * np.sqrt(np.pi * x)
else:
return interpolate.interp1d(A, B)([x])[0]

def emdd_qn(x):
""" Q function based on lookup table, for negative drifts """
A = [0.0005, 0.001, 0.0015, 0.002, 0.0025, 0.005,
0.0075, 0.01, 0.0125, 0.015, 0.0175, 0.02,
0.0225, 0.025, 0.0275, 0.03, 0.0325, 0.035,
0.0375, 0.04, 0.0425, 0.045, 0.0475, 0.05,
0.055, 0.06, 0.065, 0.07, 0.075, 0.08,
0.085, 0.09, 0.095, 0.1, 0.15, 0.2,
0.25, 0.3, 0.35, 0.4, 0.45, 0.5,
0.75, 1.0, 1.25, 1.5, 1.75, 2.0,
2.25, 2.5, 2.75, 3.0, 3.25, 3.5,
3.75, 4.0, 4.25, 4.5, 4.75, 5.0]
B = [0.019965, 0.028394, 0.034874, 0.040369, 0.045256, 0.064633,
0.079746, 0.092708, 0.104259, 0.114814, 0.124608, 0.133772,
0.142429, 0.150739, 0.158565, 0.166229, 0.173756, 0.180793,
0.187739, 0.194489, 0.201094, 0.207572, 0.213877, 0.220056,
0.231797, 0.243374, 0.254585, 0.265472, 0.27607, 0.286406,
0.296507, 0.306393, 0.316066, 0.325586, 0.413136, 0.491599,
0.564333, 0.633007, 0.698849, 0.762455, 0.824484, 0.884593,
1.17202, 1.44552, 1.70936, 1.97074, 2.22742, 2.48396,
2.73676, 2.99094, 3.24354, 3.49252, 3.74294, 3.99519,
4.24274, 4.49238, 4.73859, 4.99043, 5.24083, 5.49882]
if x > 5:
return x + 0.5
elif x < 0.0005:
return 0.5 * np.sqrt(np.pi * x)
else:
return interpolate.interp1d(A, B)([x])[0]

if ((not np.isfinite(new_mu)) | (not np.isfinite(sigma)) | (sigma <= 0)):
return np.nan
else:
alpha = new_mu / (2 * sigma**2)
if new_mu == 0:
return - np.sqrt(np.pi / 2) * sigma * np.sqrt(t)
elif new_mu > 0:
return - emdd_qp(alpha * new_mu * t) / alpha
else:
return emdd_qn(alpha * new_mu * t) / alpha


def annual_return(returns, period=DAILY, annualization=None):
"""
Determines the mean annual growth rate of returns. This is equivilent
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13 changes: 13 additions & 0 deletions empyrical/tests/test_stats.py
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Original file line number Diff line number Diff line change
Expand Up @@ -282,6 +282,19 @@ def test_max_drawdown_translation(self, returns, constant):
assert max_dd <= raised_dd
assert depressed_dd <= max_dd

@parameterized.expand([
(0, 1, 1, False, -1.2533141373155001),
(1, 1, 1, False, -0.926318),
(-1, 1, 1, False, -1.769186),
(0, 1, 1, True, -1.477444),
])
def test_expected_max_drawdown(self, mu, sigma, t, gbm, expected):
assert_almost_equal(
self.empyrical.expected_max_drawdown(mu, sigma, t, gbm),
expected,
DECIMAL_PLACES,
)

@parameterized.expand([
(mixed_returns, empyrical.DAILY, 1.9135925373194231),
(weekly_returns, empyrical.WEEKLY, 0.24690830513998208),
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