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edhec_risk_kit_204.py
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edhec_risk_kit_204.py
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import pandas as pd
import numpy as np
def get_ffme_returns():
"""
Load the Fama-French Dataset for the returns of the Top and Bottom Deciles by MarketCap
"""
me_m = pd.read_csv("data/Portfolios_Formed_on_ME_monthly_EW.csv",
header=0, index_col=0, na_values=-99.99)
rets = me_m[['Lo 10', 'Hi 10']]
rets.columns = ['SmallCap', 'LargeCap']
rets = rets/100
rets.index = pd.to_datetime(rets.index, format="%Y%m").to_period('M')
return rets
def get_fff_returns():
"""
Load the Fama-French Research Factor Monthly Dataset
"""
rets = pd.read_csv("data/F-F_Research_Data_Factors_m.csv",
header=0, index_col=0, na_values=-99.99)/100
rets.index = pd.to_datetime(rets.index, format="%Y%m").to_period('M')
return rets
def get_hfi_returns():
"""
Load and format the EDHEC Hedge Fund Index Returns
"""
hfi = pd.read_csv("data/edhec-hedgefundindices.csv",
header=0, index_col=0, parse_dates=True)
hfi = hfi/100
hfi.index = hfi.index.to_period('M')
return hfi
def get_ind_file(filetype, weighting="vw", n_inds=30):
"""
Load and format the Ken French Industry Portfolios files
Variant is a tuple of (weighting, size) where:
weighting is one of "ew", "vw"
number of inds is 30 or 49
"""
if filetype is "returns":
name = f"{weighting}_rets"
divisor = 100
elif filetype is "nfirms":
name = "nfirms"
divisor = 1
elif filetype is "size":
name = "size"
divisor = 1
else:
raise ValueError(f"filetype must be one of: returns, nfirms, size")
ind = pd.read_csv(f"data/ind{n_inds}_m_{name}.csv", header=0, index_col=0, na_values=-99.99)/divisor
ind.index = pd.to_datetime(ind.index, format="%Y%m").to_period('M')
ind.columns = ind.columns.str.strip()
return ind
def get_ind_returns(weighting="vw", n_inds=30):
"""
Load and format the Ken French Industry Portfolios Monthly Returns
"""
return get_ind_file("returns", weighting=weighting, n_inds=n_inds)
def get_ind_nfirms(n_inds=30):
"""
Load and format the Ken French 30 Industry Portfolios Average number of Firms
"""
return get_ind_file("nfirms", n_inds=n_inds)
def get_ind_size(n_inds=30):
"""
Load and format the Ken French 30 Industry Portfolios Average size (market cap)
"""
return get_ind_file("size", n_inds=n_inds)
def get_ind_market_caps(n_inds=30, weights=False):
"""
Load the industry portfolio data and derive the market caps
"""
ind_nfirms = get_ind_nfirms(n_inds=n_inds)
ind_size = get_ind_size(n_inds=n_inds)
ind_mktcap = ind_nfirms * ind_size
if weights:
total_mktcap = ind_mktcap.sum(axis=1)
ind_capweight = ind_mktcap.divide(total_mktcap, axis="rows")
return ind_capweight
#else
return ind_mktcap
def get_total_market_index_returns(n_inds=30):
"""
Load the 30 industry portfolio data and derive the returns of a capweighted total market index
"""
ind_capweight = get_ind_market_caps(n_inds=n_inds)
ind_return = get_ind_returns(weighting="vw", n_inds=n_inds)
total_market_return = (ind_capweight * ind_return).sum(axis="columns")
return total_market_return
def skewness(r):
"""
Alternative to scipy.stats.skew()
Computes the skewness of the supplied Series or DataFrame
Returns a float or a Series
"""
demeaned_r = r - r.mean()
# use the population standard deviation, so set dof=0
sigma_r = r.std(ddof=0)
exp = (demeaned_r**3).mean()
return exp/sigma_r**3
def kurtosis(r):
"""
Alternative to scipy.stats.kurtosis()
Computes the kurtosis of the supplied Series or DataFrame
Returns a float or a Series
"""
demeaned_r = r - r.mean()
# use the population standard deviation, so set dof=0
sigma_r = r.std(ddof=0)
exp = (demeaned_r**4).mean()
return exp/sigma_r**4
def compound(r):
"""
returns the result of compounding the set of returns in r
"""
return np.expm1(np.log1p(r).sum())
def annualize_rets(r, periods_per_year):
"""
Annualizes a set of returns
We should infer the periods per year
but that is currently left as an exercise
to the reader :-)
"""
compounded_growth = (1+r).prod()
n_periods = r.shape[0]
return compounded_growth**(periods_per_year/n_periods)-1
def annualize_vol(r, periods_per_year):
"""
Annualizes the vol of a set of returns
We should infer the periods per year
but that is currently left as an exercise
to the reader :-)
"""
return r.std()*(periods_per_year**0.5)
def sharpe_ratio(r, riskfree_rate, periods_per_year):
"""
Computes the annualized sharpe ratio of a set of returns
"""
# convert the annual riskfree rate to per period
rf_per_period = (1+riskfree_rate)**(1/periods_per_year)-1
excess_ret = r - rf_per_period
ann_ex_ret = annualize_rets(excess_ret, periods_per_year)
ann_vol = annualize_vol(r, periods_per_year)
return ann_ex_ret/ann_vol
import scipy.stats
def is_normal(r, level=0.01):
"""
Applies the Jarque-Bera test to determine if a Series is normal or not
Test is applied at the 1% level by default
Returns True if the hypothesis of normality is accepted, False otherwise
"""
if isinstance(r, pd.DataFrame):
return r.aggregate(is_normal)
else:
statistic, p_value = scipy.stats.jarque_bera(r)
return p_value > level
def drawdown(return_series: pd.Series):
"""Takes a time series of asset returns.
returns a DataFrame with columns for
the wealth index,
the previous peaks, and
the percentage drawdown
"""
wealth_index = 1000*(1+return_series).cumprod()
previous_peaks = wealth_index.cummax()
drawdowns = (wealth_index - previous_peaks)/previous_peaks
return pd.DataFrame({"Wealth": wealth_index,
"Previous Peak": previous_peaks,
"Drawdown": drawdowns})
def semideviation(r):
"""
Returns the semideviation aka negative semideviation of r
r must be a Series or a DataFrame, else raises a TypeError
"""
if isinstance(r, pd.Series):
is_negative = r < 0
return r[is_negative].std(ddof=0)
elif isinstance(r, pd.DataFrame):
return r.aggregate(semideviation)
else:
raise TypeError("Expected r to be a Series or DataFrame")
def var_historic(r, level=5):
"""
Returns the historic Value at Risk at a specified level
i.e. returns the number such that "level" percent of the returns
fall below that number, and the (100-level) percent are above
"""
if isinstance(r, pd.DataFrame):
return r.aggregate(var_historic, level=level)
elif isinstance(r, pd.Series):
return -np.percentile(r, level)
else:
raise TypeError("Expected r to be a Series or DataFrame")
def cvar_historic(r, level=5):
"""
Computes the Conditional VaR of Series or DataFrame
"""
if isinstance(r, pd.Series):
is_beyond = r <= -var_historic(r, level=level)
return -r[is_beyond].mean()
elif isinstance(r, pd.DataFrame):
return r.aggregate(cvar_historic, level=level)
else:
raise TypeError("Expected r to be a Series or DataFrame")
from scipy.stats import norm
def var_gaussian(r, level=5, modified=False):
"""
Returns the Parametric Gauusian VaR of a Series or DataFrame
If "modified" is True, then the modified VaR is returned,
using the Cornish-Fisher modification
"""
# compute the Z score assuming it was Gaussian
z = norm.ppf(level/100)
if modified:
# modify the Z score based on observed skewness and kurtosis
s = skewness(r)
k = kurtosis(r)
z = (z +
(z**2 - 1)*s/6 +
(z**3 -3*z)*(k-3)/24 -
(2*z**3 - 5*z)*(s**2)/36
)
return -(r.mean() + z*r.std(ddof=0))
def portfolio_return(weights, returns):
"""
Computes the return on a portfolio from constituent returns and weights
weights are a numpy array or Nx1 matrix and returns are a numpy array or Nx1 matrix
"""
return weights.T @ returns
def portfolio_vol(weights, covmat):
"""
Computes the vol of a portfolio from a covariance matrix and constituent weights
weights are a numpy array or N x 1 maxtrix and covmat is an N x N matrix
"""
return (weights.T @ covmat @ weights)**0.5
def plot_ef2(n_points, er, cov):
"""
Plots the 2-asset efficient frontier
"""
if er.shape[0] != 2 or er.shape[0] != 2:
raise ValueError("plot_ef2 can only plot 2-asset frontiers")
weights = [np.array([w, 1-w]) for w in np.linspace(0, 1, n_points)]
rets = [portfolio_return(w, er) for w in weights]
vols = [portfolio_vol(w, cov) for w in weights]
ef = pd.DataFrame({
"Returns": rets,
"Volatility": vols
})
return ef.plot.line(x="Volatility", y="Returns", style=".-")
from scipy.optimize import minimize
def minimize_vol(target_return, er, cov):
"""
Returns the optimal weights that achieve the target return
given a set of expected returns and a covariance matrix
"""
n = er.shape[0]
init_guess = np.repeat(1/n, n)
bounds = ((0.0, 1.0),) * n # an N-tuple of 2-tuples!
# construct the constraints
weights_sum_to_1 = {'type': 'eq',
'fun': lambda weights: np.sum(weights) - 1
}
return_is_target = {'type': 'eq',
'args': (er,),
'fun': lambda weights, er: target_return - portfolio_return(weights,er)
}
weights = minimize(portfolio_vol, init_guess,
args=(cov,), method='SLSQP',
options={'disp': False},
constraints=(weights_sum_to_1,return_is_target),
bounds=bounds)
return weights.x
def tracking_error(r_a, r_b):
"""
Returns the Tracking Error between the two return series
"""
return np.sqrt(((r_a - r_b)**2).sum())
def msr(riskfree_rate, er, cov):
"""
Returns the weights of the portfolio that gives you the maximum sharpe ratio
given the riskfree rate and expected returns and a covariance matrix
"""
n = er.shape[0]
init_guess = np.repeat(1/n, n)
bounds = ((0.0, 1.0),) * n # an N-tuple of 2-tuples!
# construct the constraints
weights_sum_to_1 = {'type': 'eq',
'fun': lambda weights: np.sum(weights) - 1
}
def neg_sharpe(weights, riskfree_rate, er, cov):
"""
Returns the negative of the sharpe ratio
of the given portfolio
"""
r = portfolio_return(weights, er)
vol = portfolio_vol(weights, cov)
return -(r - riskfree_rate)/vol
weights = minimize(neg_sharpe, init_guess,
args=(riskfree_rate, er, cov), method='SLSQP',
options={'disp': False},
constraints=(weights_sum_to_1,),
bounds=bounds)
return weights.x
def gmv(cov):
"""
Returns the weights of the Global Minimum Volatility portfolio
given a covariance matrix
"""
n = cov.shape[0]
return msr(0, np.repeat(1, n), cov)
def optimal_weights(n_points, er, cov):
"""
Returns a list of weights that represent a grid of n_points on the efficient frontier
"""
target_rs = np.linspace(er.min(), er.max(), n_points)
weights = [minimize_vol(target_return, er, cov) for target_return in target_rs]
return weights
def plot_ef(n_points, er, cov, style='.-', legend=False, show_cml=False, riskfree_rate=0, show_ew=False, show_gmv=False):
"""
Plots the multi-asset efficient frontier
"""
weights = optimal_weights(n_points, er, cov)
rets = [portfolio_return(w, er) for w in weights]
vols = [portfolio_vol(w, cov) for w in weights]
ef = pd.DataFrame({
"Returns": rets,
"Volatility": vols
})
ax = ef.plot.line(x="Volatility", y="Returns", style=style, legend=legend)
if show_cml:
ax.set_xlim(left = 0)
# get MSR
w_msr = msr(riskfree_rate, er, cov)
r_msr = portfolio_return(w_msr, er)
vol_msr = portfolio_vol(w_msr, cov)
# add CML
cml_x = [0, vol_msr]
cml_y = [riskfree_rate, r_msr]
ax.plot(cml_x, cml_y, color='green', marker='o', linestyle='dashed', linewidth=2, markersize=10)
if show_ew:
n = er.shape[0]
w_ew = np.repeat(1/n, n)
r_ew = portfolio_return(w_ew, er)
vol_ew = portfolio_vol(w_ew, cov)
# add EW
ax.plot([vol_ew], [r_ew], color='goldenrod', marker='o', markersize=10)
if show_gmv:
w_gmv = gmv(cov)
r_gmv = portfolio_return(w_gmv, er)
vol_gmv = portfolio_vol(w_gmv, cov)
# add EW
ax.plot([vol_gmv], [r_gmv], color='midnightblue', marker='o', markersize=10)
return ax
def run_cppi(risky_r, safe_r=None, m=3, start=1000, floor=0.8, riskfree_rate=0.03, drawdown=None):
"""
Run a backtest of the CPPI strategy, given a set of returns for the risky asset
Returns a dictionary containing: Asset Value History, Risk Budget History, Risky Weight History
"""
# set up the CPPI parameters
dates = risky_r.index
n_steps = len(dates)
account_value = start
floor_value = start*floor
peak = account_value
if isinstance(risky_r, pd.Series):
risky_r = pd.DataFrame(risky_r, columns=["R"])
if safe_r is None:
safe_r = pd.DataFrame().reindex_like(risky_r)
safe_r.values[:] = riskfree_rate/12 # fast way to set all values to a number
# set up some DataFrames for saving intermediate values
account_history = pd.DataFrame().reindex_like(risky_r)
risky_w_history = pd.DataFrame().reindex_like(risky_r)
cushion_history = pd.DataFrame().reindex_like(risky_r)
floorval_history = pd.DataFrame().reindex_like(risky_r)
peak_history = pd.DataFrame().reindex_like(risky_r)
for step in range(n_steps):
if drawdown is not None:
peak = np.maximum(peak, account_value)
floor_value = peak*(1-drawdown)
cushion = (account_value - floor_value)/account_value
risky_w = m*cushion
risky_w = np.minimum(risky_w, 1)
risky_w = np.maximum(risky_w, 0)
safe_w = 1-risky_w
risky_alloc = account_value*risky_w
safe_alloc = account_value*safe_w
# recompute the new account value at the end of this step
account_value = risky_alloc*(1+risky_r.iloc[step]) + safe_alloc*(1+safe_r.iloc[step])
# save the histories for analysis and plotting
cushion_history.iloc[step] = cushion
risky_w_history.iloc[step] = risky_w
account_history.iloc[step] = account_value
floorval_history.iloc[step] = floor_value
peak_history.iloc[step] = peak
risky_wealth = start*(1+risky_r).cumprod()
backtest_result = {
"Wealth": account_history,
"Risky Wealth": risky_wealth,
"Risk Budget": cushion_history,
"Risky Allocation": risky_w_history,
"m": m,
"start": start,
"floor": floor,
"risky_r":risky_r,
"safe_r": safe_r,
"drawdown": drawdown,
"peak": peak_history,
"floor": floorval_history
}
return backtest_result
def summary_stats(r, riskfree_rate=0.03):
"""
Return a DataFrame that contains aggregated summary stats for the returns in the columns of r
"""
ann_r = r.aggregate(annualize_rets, periods_per_year=12)
ann_vol = r.aggregate(annualize_vol, periods_per_year=12)
ann_sr = r.aggregate(sharpe_ratio, riskfree_rate=riskfree_rate, periods_per_year=12)
dd = r.aggregate(lambda r: drawdown(r).Drawdown.min())
skew = r.aggregate(skewness)
kurt = r.aggregate(kurtosis)
cf_var5 = r.aggregate(var_gaussian, modified=True)
hist_cvar5 = r.aggregate(cvar_historic)
return pd.DataFrame({
"Annualized Return": ann_r,
"Annualized Vol": ann_vol,
"Skewness": skew,
"Kurtosis": kurt,
"Cornish-Fisher VaR (5%)": cf_var5,
"Historic CVaR (5%)": hist_cvar5,
"Sharpe Ratio": ann_sr,
"Max Drawdown": dd
})
def gbm(n_years = 10, n_scenarios=1000, mu=0.07, sigma=0.15, steps_per_year=12, s_0=100.0, prices=True):
"""
Evolution of Geometric Brownian Motion trajectories, such as for Stock Prices through Monte Carlo
:param n_years: The number of years to generate data for
:param n_paths: The number of scenarios/trajectories
:param mu: Annualized Drift, e.g. Market Return
:param sigma: Annualized Volatility
:param steps_per_year: granularity of the simulation
:param s_0: initial value
:return: a numpy array of n_paths columns and n_years*steps_per_year rows
"""
# Derive per-step Model Parameters from User Specifications
dt = 1/steps_per_year
n_steps = int(n_years*steps_per_year) + 1
# the standard way ...
# rets_plus_1 = np.random.normal(loc=mu*dt+1, scale=sigma*np.sqrt(dt), size=(n_steps, n_scenarios))
# without discretization error ...
rets_plus_1 = np.random.normal(loc=(1+mu)**dt, scale=(sigma*np.sqrt(dt)), size=(n_steps, n_scenarios))
rets_plus_1[0] = 1
ret_val = s_0*pd.DataFrame(rets_plus_1).cumprod() if prices else rets_plus_1-1
return ret_val
import statsmodels.api as sm
def regress(dependent_variable, explanatory_variables, alpha=True):
"""
Runs a linear regression to decompose the dependent variable into the explanatory variables
returns an object of type statsmodel's RegressionResults on which you can call
.summary() to print a full summary
.params for the coefficients
.tvalues and .pvalues for the significance levels
.rsquared_adj and .rsquared for quality of fit
"""
if alpha:
explanatory_variables = explanatory_variables.copy()
explanatory_variables["Alpha"] = 1
lm = sm.OLS(dependent_variable, explanatory_variables).fit()
return lm
def portfolio_tracking_error(weights, ref_r, bb_r):
"""
returns the tracking error between the reference returns
and a portfolio of building block returns held with given weights
"""
return tracking_error(ref_r, (weights*bb_r).sum(axis=1))
def style_analysis(dependent_variable, explanatory_variables):
"""
Returns the optimal weights that minimizes the Tracking error between
a portfolio of the explanatory variables and the dependent variable
"""
n = explanatory_variables.shape[1]
init_guess = np.repeat(1/n, n)
bounds = ((0.0, 1.0),) * n # an N-tuple of 2-tuples!
# construct the constraints
weights_sum_to_1 = {'type': 'eq',
'fun': lambda weights: np.sum(weights) - 1
}
solution = minimize(portfolio_tracking_error, init_guess,
args=(dependent_variable, explanatory_variables,), method='SLSQP',
options={'disp': False},
constraints=(weights_sum_to_1,),
bounds=bounds)
weights = pd.Series(solution.x, index=explanatory_variables.columns)
return weights
def ff_analysis(r, factors):
"""
Returns the loadings of r on the Fama French Factors
which can be read in using get_fff_returns()
the index of r must be a (not necessarily proper) subset of the index of factors
r is either a Series or a DataFrame
"""
if isinstance(r, pd.Series):
dependent_variable = r
explanatory_variables = factors.loc[r.index]
tilts = regress(dependent_variable, explanatory_variables).params
elif isinstance(r, pd.DataFrame):
tilts = pd.DataFrame({col: ff_analysis(r[col], factors) for col in r.columns})
else:
raise TypeError("r must be a Series or a DataFrame")
return tilts
def weight_ew(r, cap_weights=None, max_cw_mult=None, microcap_threshold=None, **kwargs):
"""
Returns the weights of the EW portfolio based on the asset returns "r" as a DataFrame
If supplied a set of capweights and a capweight tether, it is applied and reweighted
"""
n = len(r.columns)
ew = pd.Series(1/n, index=r.columns)
if cap_weights is not None:
cw = cap_weights.loc[r.index[0]] # starting cap weight
## exclude microcaps
if microcap_threshold is not None and microcap_threshold > 0:
microcap = cw < microcap_threshold
ew[microcap] = 0
ew = ew/ew.sum()
#limit weight to a multiple of capweight
if max_cw_mult is not None and max_cw_mult > 0:
ew = np.minimum(ew, cw*max_cw_mult)
ew = ew/ew.sum() #reweight
return ew
def weight_cw(r, cap_weights, **kwargs):
"""
Returns the weights of the CW portfolio based on the time series of capweights
"""
w = cap_weights.loc[r.index[0]]
return cap_weights.loc[r.index[0]]
def backtest_ws(r, estimation_window=60, weighting=weight_ew, verbose=False, **kwargs):
"""
Backtests a given weighting scheme, given some parameters:
r : asset returns to use to build the portfolio
estimation_window: the window to use to estimate parameters
weighting: the weighting scheme to use, must be a function that takes "r", and a variable number of keyword-value arguments
"""
n_periods = r.shape[0]
# return windows
windows = [(start, start+estimation_window) for start in range(n_periods-estimation_window)]
weights = [weighting(r.iloc[win[0]:win[1]], **kwargs) for win in windows]
# convert List of weights to DataFrame
weights = pd.DataFrame(weights, index=r.iloc[estimation_window:].index, columns=r.columns)
returns = (weights * r).sum(axis="columns", min_count=1) #mincount is to generate NAs if all inputs are NAs
return returns