ml-finance-python

edhec_risk_kit_127.py

(22747B)

 import pandas as pd
 import numpy as np
 import math
 
 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_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):
     """
     Load and format the Ken French 30 Industry Portfolios files
     """
     known_types = ["returns", "nfirms", "size"]
     if filetype not in known_types:
         sep = ','
         raise ValueError(f'filetype must be one of:{sep.join(known_types)}')
     if filetype is "returns":
         name = "vw_rets"
         divisor = 100
     elif filetype is "nfirms":
         name = "nfirms"
         divisor = 1
     elif filetype is "size":
         name = "size"
         divisor = 1
     ind = pd.read_csv(f"data/ind30_m_{name}.csv", header=0, index_col=0)/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():
     """
     Load and format the Ken French 30 Industry Portfolios Value Weighted Monthly Returns
     """
     return get_ind_file("returns")
 
 def get_ind_nfirms():
     """
     Load and format the Ken French 30 Industry Portfolios Average number of Firms
     """
     return get_ind_file("nfirms")
 
 def get_ind_size():
     """
     Load and format the Ken French 30 Industry Portfolios Average size (market cap)
     """
     return get_ind_file("size")
 
                          
 def get_total_market_index_returns():
     """
     Load the 30 industry portfolio data and derive the returns of a capweighted total market index
     """
     ind_nfirms = get_ind_nfirms()
     ind_size = get_ind_size()
     ind_return = get_ind_returns()
     ind_mktcap = ind_nfirms * ind_size
     total_mktcap = ind_mktcap.sum(axis=1)
     ind_capweight = ind_mktcap.divide(total_mktcap, axis="rows")
     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 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
 
 
 def discount(t, r):
     """
     Compute the price of a pure discount bond that pays a dollar at time period t
     and r is the per-period interest rate
     returns a |t| x |r| Series or DataFrame
     r can be a float, Series or DataFrame
     returns a DataFrame indexed by t
     """
     discounts = pd.DataFrame([(r+1)**-i for i in t])
     discounts.index = t
     return discounts
 
 def pv(flows, r):
     """
     Compute the present value of a sequence of cash flows given by the time (as an index) and amounts
     r can be a scalar, or a Series or DataFrame with the number of rows matching the num of rows in flows
     """
     dates = flows.index
     discounts = discount(dates, r)
     return discounts.multiply(flows, axis='rows').sum()
 
 def funding_ratio(assets, liabilities, r):
     """
     Computes the funding ratio of a series of liabilities, based on an interest rate and current value of assets
     """
     return pv(assets, r)/pv(liabilities, r)
 
 def inst_to_ann(r):
     """
     Convert an instantaneous interest rate to an annual interest rate
     """
     return np.expm1(r)
 
 def ann_to_inst(r):
     """
     Convert an instantaneous interest rate to an annual interest rate
     """
     return np.log1p(r)
 
 def cir(n_years = 10, n_scenarios=1, a=0.05, b=0.03, sigma=0.05, steps_per_year=12, r_0=None):
     """
     Generate random interest rate evolution over time using the CIR model
     b and r_0 are assumed to be the annualized rates, not the short rate
     and the returned values are the annualized rates as well
     """
     if r_0 is None: r_0 = b 
     r_0 = ann_to_inst(r_0)
     dt = 1/steps_per_year
     num_steps = int(n_years*steps_per_year) + 1 # because n_years might be a float
     
     shock = np.random.normal(0, scale=np.sqrt(dt), size=(num_steps, n_scenarios))
     rates = np.empty_like(shock)
     rates[0] = r_0
 
     ## For Price Generation
     h = math.sqrt(a**2 + 2*sigma**2)
     prices = np.empty_like(shock)
     ####
 
     def price(ttm, r):
         _A = ((2*h*math.exp((h+a)*ttm/2))/(2*h+(h+a)*(math.exp(h*ttm)-1)))**(2*a*b/sigma**2)
         _B = (2*(math.exp(h*ttm)-1))/(2*h + (h+a)*(math.exp(h*ttm)-1))
         _P = _A*np.exp(-_B*r)
         return _P
     prices[0] = price(n_years, r_0)
     ####
     
     for step in range(1, num_steps):
         r_t = rates[step-1]
         d_r_t = a*(b-r_t)*dt + sigma*np.sqrt(r_t)*shock[step]
         rates[step] = abs(r_t + d_r_t)
         # generate prices at time t as well ...
         prices[step] = price(n_years-step*dt, rates[step])
 
     rates = pd.DataFrame(data=inst_to_ann(rates), index=range(num_steps))
     ### for prices
     prices = pd.DataFrame(data=prices, index=range(num_steps))
     ###
     return rates, prices
 
 def bond_cash_flows(maturity, principal=100, coupon_rate=0.03, coupons_per_year=12):
     """
     Returns the series of cash flows generated by a bond,
     indexed by the payment/coupon number
     """
     n_coupons = round(maturity*coupons_per_year)
     coupon_amt = principal*coupon_rate/coupons_per_year
     coupons = np.repeat(coupon_amt, n_coupons)
     coupon_times = np.arange(1, n_coupons+1)
     cash_flows = pd.Series(data=coupon_amt, index=coupon_times)
     cash_flows.iloc[-1] += principal
     return cash_flows
     
 def bond_price(maturity, principal=100, coupon_rate=0.03, coupons_per_year=12, discount_rate=0.03):
     """
     Computes the price of a bond that pays regular coupons until maturity
     at which time the principal and the final coupon is returned
     This is not designed to be efficient, rather,
     it is to illustrate the underlying principle behind bond pricing!
     If discount_rate is a DataFrame, then this is assumed to be the rate on each coupon date
     and the bond value is computed over time.
     i.e. The index of the discount_rate DataFrame is assumed to be the coupon number
     """
     if isinstance(discount_rate, pd.DataFrame):
         pricing_dates = discount_rate.index
         prices = pd.DataFrame(index=pricing_dates, columns=discount_rate.columns)
         for t in pricing_dates:
             prices.loc[t] = bond_price(maturity-t/coupons_per_year, principal, coupon_rate, coupons_per_year,
                                       discount_rate.loc[t])
         return prices
     else: # base case ... single time period
         if maturity <= 0: return principal+principal*coupon_rate/coupons_per_year
         cash_flows = bond_cash_flows(maturity, principal, coupon_rate, coupons_per_year)
         return pv(cash_flows, discount_rate/coupons_per_year)
 
 def macaulay_duration(flows, discount_rate):
     """
     Computes the Macaulay Duration of a sequence of cash flows, given a per-period discount rate
     """
     discounted_flows = discount(flows.index, discount_rate)*pd.DataFrame(flows)
     weights = discounted_flows/discounted_flows.sum()
     return np.average(flows.index, weights=weights.iloc[:,0])
 
 def match_durations(cf_t, cf_s, cf_l, discount_rate):
     """
     Returns the weight W in cf_s that, along with (1-W) in cf_l will have an effective
     duration that matches cf_t
     """
     d_t = macaulay_duration(cf_t, discount_rate)
     d_s = macaulay_duration(cf_s, discount_rate)
     d_l = macaulay_duration(cf_l, discount_rate)
     return (d_l - d_t)/(d_l - d_s)
 
 def bond_total_return(monthly_prices, principal, coupon_rate, coupons_per_year):
     """
     Computes the total return of a Bond based on monthly bond prices and coupon payments
     Assumes that dividends (coupons) are paid out at the end of the period (e.g. end of 3 months for quarterly div)
     and that dividends are reinvested in the bond
     """
     coupons = pd.DataFrame(data = 0, index=monthly_prices.index, columns=monthly_prices.columns)
     t_max = monthly_prices.index.max()
     pay_date = np.linspace(12/coupons_per_year, t_max, int(coupons_per_year*t_max/12), dtype=int)
     coupons.iloc[pay_date] = principal*coupon_rate/coupons_per_year
     total_returns = (monthly_prices + coupons)/monthly_prices.shift()-1
     return total_returns.dropna()