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Measured of Risk-adjusted Return
import math
import numpy
import numpy.random as nrand
"""
Note - for some of the metrics the absolute value is returns. This is because if the risk (loss) is higher we want to
discount the expected excess return from the portfolio by a higher amount. Therefore risk should be positive.
"""
def vol(returns):
# Return the standard deviation of returns
return numpy.std(returns)
def beta(returns, market):
# Create a matrix of [returns, market]
m = numpy.matrix([returns, market])
# Return the covariance of m divided by the standard deviation of the market returns
return numpy.cov(m)[0][1] / numpy.std(market)
def lpm(returns, threshold, order):
# This method returns a lower partial moment of the returns
# Create an array he same length as returns containing the minimum return threshold
threshold_array = numpy.empty(len(returns))
threshold_array.fill(threshold)
# Calculate the difference between the threshold and the returns
diff = threshold_array - returns
# Set the minimum of each to 0
diff = diff.clip(min=0)
# Return the sum of the different to the power of order
return numpy.sum(diff ** order) / len(returns)
def hpm(returns, threshold, order):
# This method returns a higher partial moment of the returns
# Create an array he same length as returns containing the minimum return threshold
threshold_array = numpy.empty(len(returns))
threshold_array.fill(threshold)
# Calculate the difference between the returns and the threshold
diff = returns - threshold_array
# Set the minimum of each to 0
diff = diff.clip(min=0)
# Return the sum of the different to the power of order
return numpy.sum(diff ** order) / len(returns)
def var(returns, alpha):
# This method calculates the historical simulation var of the returns
sorted_returns = numpy.sort(returns)
# Calculate the index associated with alpha
index = int(alpha * len(sorted_returns))
# VaR should be positive
return abs(sorted_returns[index])
def cvar(returns, alpha):
# This method calculates the condition VaR of the returns
sorted_returns = numpy.sort(returns)
# Calculate the index associated with alpha
index = int(alpha * len(sorted_returns))
# Calculate the total VaR beyond alpha
sum_var = sorted_returns[0]
for i in range(1, index):
sum_var += sorted_returns[i]
# Return the average VaR
# CVaR should be positive
return abs(sum_var / index)
def prices(returns, base):
# Converts returns into prices
s = [base]
for i in range(len(returns)):
s.append(base * (1 + returns[i]))
return numpy.array(s)
def dd(returns, tau):
# Returns the draw-down given time period tau
values = prices(returns, 100)
pos = len(values) - 1
pre = pos - tau
drawdown = float('+inf')
# Find the maximum drawdown given tau
while pre >= 0:
dd_i = (values[pos] / values[pre]) - 1
if dd_i < drawdown:
drawdown = dd_i
pos, pre = pos - 1, pre - 1
# Drawdown should be positive
return abs(drawdown)
def max_dd(returns):
# Returns the maximum draw-down for any tau in (0, T) where T is the length of the return series
max_drawdown = float('-inf')
for i in range(0, len(returns)):
drawdown_i = dd(returns, i)
if drawdown_i > max_drawdown:
max_drawdown = drawdown_i
# Max draw-down should be positive
return abs(max_drawdown)
def average_dd(returns, periods):
# Returns the average maximum drawdown over n periods
drawdowns = []
for i in range(0, len(returns)):
drawdown_i = dd(returns, i)
drawdowns.append(drawdown_i)
drawdowns = sorted(drawdowns)
total_dd = abs(drawdowns[0])
for i in range(1, periods):
total_dd += abs(drawdowns[i])
return total_dd / periods
def average_dd_squared(returns, periods):
# Returns the average maximum drawdown squared over n periods
drawdowns = []
for i in range(0, len(returns)):
drawdown_i = math.pow(dd(returns, i), 2.0)
drawdowns.append(drawdown_i)
drawdowns = sorted(drawdowns)
total_dd = abs(drawdowns[0])
for i in range(1, periods):
total_dd += abs(drawdowns[i])
return total_dd / periods
def treynor_ratio(er, returns, market, rf):
return (er - rf) / beta(returns, market)
def sharpe_ratio(er, returns, rf):
return (er - rf) / vol(returns)
def information_ratio(returns, benchmark):
diff = returns - benchmark
return numpy.mean(diff) / vol(diff)
def modigliani_ratio(er, returns, benchmark, rf):
np_rf = numpy.empty(len(returns))
np_rf.fill(rf)
rdiff = returns - np_rf
bdiff = benchmark - np_rf
return (er - rf) * (vol(rdiff) / vol(bdiff)) + rf
def excess_var(er, returns, rf, alpha):
return (er - rf) / var(returns, alpha)
def conditional_sharpe_ratio(er, returns, rf, alpha):
return (er - rf) / cvar(returns, alpha)
def omega_ratio(er, returns, rf, target=0):
return (er - rf) / lpm(returns, target, 1)
def sortino_ratio(er, returns, rf, target=0):
return (er - rf) / math.sqrt(lpm(returns, target, 2))
def kappa_three_ratio(er, returns, rf, target=0):
return (er - rf) / math.pow(lpm(returns, target, 3), float(1/3))
def gain_loss_ratio(returns, target=0):
return hpm(returns, target, 1) / lpm(returns, target, 1)
def upside_potential_ratio(returns, target=0):
return hpm(returns, target, 1) / math.sqrt(lpm(returns, target, 2))
def calmar_ratio(er, returns, rf):
return (er - rf) / max_dd(returns)
def sterling_ration(er, returns, rf, periods):
return (er - rf) / average_dd(returns, periods)
def burke_ratio(er, returns, rf, periods):
return (er - rf) / math.sqrt(average_dd_squared(returns, periods))
def test_risk_metrics():
# This is just a testing method
r = nrand.uniform(-1, 1, 50)
m = nrand.uniform(-1, 1, 50)
print("vol =", vol(r))
print("beta =", beta(r, m))
print("hpm(0.0)_1 =", hpm(r, 0.0, 1))
print("lpm(0.0)_1 =", lpm(r, 0.0, 1))
print("VaR(0.05) =", var(r, 0.05))
print("CVaR(0.05) =", cvar(r, 0.05))
print("Drawdown(5) =", dd(r, 5))
print("Max Drawdown =", max_dd(r))
def test_risk_adjusted_metrics():
# Returns from the portfolio (r) and market (m)
r = nrand.uniform(-1, 1, 50)
m = nrand.uniform(-1, 1, 50)
# Expected return
e = numpy.mean(r)
# Risk free rate
f = 0.06
# Risk-adjusted return based on Volatility
print("Treynor Ratio =", treynor_ratio(e, r, m, f))
print("Sharpe Ratio =", sharpe_ratio(e, r, f))
print("Information Ratio =", information_ratio(r, m))
# Risk-adjusted return based on Value at Risk
print("Excess VaR =", excess_var(e, r, f, 0.05))
print("Conditional Sharpe Ratio =", conditional_sharpe_ratio(e, r, f, 0.05))
# Risk-adjusted return based on Lower Partial Moments
print("Omega Ratio =", omega_ratio(e, r, f))
print("Sortino Ratio =", sortino_ratio(e, r, f))
print("Kappa 3 Ratio =", kappa_three_ratio(e, r, f))
print("Gain Loss Ratio =", gain_loss_ratio(r))
print("Upside Potential Ratio =", upside_potential_ratio(r))
# Risk-adjusted return based on Drawdown risk
print("Calmar Ratio =", calmar_ratio(e, r, f))
print("Sterling Ratio =", sterling_ration(e, r, f, 5))
print("Burke Ratio =", burke_ratio(e, r, f, 5))
if __name__ == "__main__":
test_risk_metrics()
test_risk_adjusted_metrics()
@Gabrielvon

This comment has been minimized.

Gabrielvon commented Feb 23, 2018

Some functions using for-loop with append are slow. Using list comprehension will massively enhancing the speed.

Take the prices(returns, base) as an example:

def prices(returns, base):
    # Converts returns into prices
    s = [base]
    for i in range(len(returns)):
        s.append(base * (1 + returns[i]))
    return numpy.array(s)

You could modify it as the following, which is 100 times faster.

def prices(returns, base):
    # Converts returns into prices
    return np.array([100] + [base * (1 + ri) for ri in returns])
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