Exercise week 5

import pandas as pd import numpy as np import matplotlib.pyplot as plt import seaborn as sns import scipy.stats from sklearn.linear_model import LinearRegression from scipy.interpolate import UnivariateSpline from sympy import symbols, solve

# Load data df = pd.read_csv('StockData.txt', delimiter = ",") # Convert to one date column df['Date'] = pd.to_datetime(df[['Year', 'Month', 'Day']]) df = df.drop(columns = ['Year', 'Month', 'Day']) # Extract sub df with only stock values in USD and EUR df_stocks_USD = df.iloc[:,:5] df_stocks_EUR = df_stocks_USD*df['USDEUR'][:,np.newaxis] # Define portfolio #alpha = 2000*np.ones(5)

alpha = 2000*np.ones(5)

# VaR on the final day def simple_method(data, alpha, CI = 0.99): #Percentage change between the current and a prior element. return_data = data.pct_change() # Standard deviation of portfolio portfolio_std = (alpha@return_data.cov(ddof = 0)@alpha.T)**(1/2) # Quantile of CI Q = scipy.stats.norm.ppf(CI) # Calculate VaR = portfolio_std * Q return VaR print(f'VaR = {simple_method(df_stocks_EUR, alpha):.2f}€')

def fx_method(data_fx, alpha, CI = 0.99): # Add sum og portfolio alpha_comb = np.append(alpha,sum(alpha)) #Percentage change between the current and a prior element. return_data = data_fx.pct_change() portfolio_std = (alpha_comb@return_data.cov(ddof = 0)@alpha_comb.T)**(1/2) Q = scipy.stats.norm.ppf(CI) # Calculate VaR VaR = portfolio_std * Q return VaR print(f"VaR = {fx_method(df[['Stock 1','Stock 2','Stock 3','Stock 4','Stock 5','USDEUR']], alpha):.2f}€")

def CAPM(data, fx, index, alpha, CI = 0.99): # Merge the stock dataframe with indicies comb_df = pd.concat((data,index), axis=1) # Get covariance matrix cov_mat = comb_df.pct_change().cov(ddof = 0) # Get variance of market (index) and exchange rate r_m = cov_mat.loc['Index','Index'] r_fx = fx.pct_change().var() # Calculate beta from the formula beta = np.array([cov_mat.loc[stock,'Index']/r_m for stock in data.columns]) # Exposure is the dotproduct of alpha and beta and the investment in foreign currency (all) omega = np.append(sum(alpha*beta),sum(alpha)) # Get variance as sum of both risk facots ret_std = np.sqrt(omega@np.diag(np.append(r_m,r_fx))@omega.T) # Calculate VaR as ususal VaR = ret_std * scipy.stats.norm.ppf(CI) return VaR print(f"VaR = {CAPM(df[['Stock 1','Stock 2','Stock 3','Stock 4','Stock 5']], df['USDEUR'], df['Index'], alpha, CI = 0.99):.2f}€")

def Reg_method(df_stock, fx, index, alpha, CI = 0.99): # Change in fortfolio PF_change = (df_stocks_EUR.pct_change()@alpha)[1:] # Get variance of market (index) and exchange rate r_m = index.pct_change().var() r_fx = fx.pct_change().var() # Get regressors X = np.vstack((index.pct_change(), fx.pct_change())).T[1:] # Get risk factors from a linear regression reg = LinearRegression().fit(X, PF_change) omega = reg.coef_ # Get variance as sum of both risk factors ret_std = np.sqrt(omega@np.diag(np.append(r_m,r_fx))@omega.T) # Calculate VaR as ususal VaR = ret_std * scipy.stats.norm.ppf(CI) return VaR print(f"VaR = {Reg_method(df[['Stock 1','Stock 2','Stock 3','Stock 4','Stock 5']], df['USDEUR'], df['Index'], alpha, CI = 0.99):.2f}€")

df_bond = pd.read_csv('BondPrices.txt', delimiter = ",") df_yield = pd.read_csv('Yields.txt', delimiter = ",")

Delta_bond = df_bond.pct_change() #return of bond cov_mat = Delta_bond.cov() #covariance Vol = Delta_bond.std() corr_mat = Delta_bond.corr() #correlation plt.figure(figsize = (12,8)) sns.heatmap(corr_mat, xticklabels=corr_mat.columns, yticklabels=corr_mat.columns) plt.tight_layout() plt.show()

plt.figure(figsize = (14,8)) Vol.plot() plt.show()

def bond_mapping(df_yields, Vols, w, corr, cov, n, N): # Helper function that can linear interpolate def interpolate(x,y,z): return y[0]+((z-x[0]) / (x[1]-x[0]))*(y[1]-y[0]) # Find interpolate exchange rate Rate1 = interpolate([float(x) for x in df_yields.columns[:2]],df_yields.iloc[-1,:2].values,w[0]) Rate2 = interpolate([float(x) for x in df_yields.columns[2:4]],df_yields.iloc[-1,2:4].values,w[1]) # Value of cash flows PV1 = n/(1+Rate1)**w[0] PV2 = (n+N)/(1+Rate2)**w[1] #Interpolate volatility to yields Vol1 = interpolate([float(x) for x in df_yields.columns[:2]],Vols[:2].values,w[0]) Vol2 = interpolate([float(x) for x in df_yields.columns[2:4]],Vols[2:4].values,w[1]) #Find alpha to allocate value to different bonds a_0 = symbols('a_0') eoq_1 = a_0**2*Vols[0]**2 + (1-a_0)**2*Vols[1]**2+2*corr.loc[str(0.25),str(0.5)]*Vols[0]*Vols[1]*a_0*(1-a_0) - Vol1**2 sol1 = solve(eoq_1) sol1 = [i for i in sol1 if i<1][0] a_1 = symbols('a_1') eoq_2 = a_1**2*Vols[2]**2 + (1-a_1)**2*Vols[3]**2+2*corr.loc[str(0.75),str(1)]*Vols[2]*Vols[3]*a_1*(1-a_1) - Vol2**2 sol2 = solve(eoq_2) sol2 = [i for i in sol2 if i<1][0] #Calculate amounts placed in bonds Amount = [sol1*PV1, (1-sol1)*PV1, sol2*PV2, (1-sol2)*PV2] #Caluclate std PF_std = (Amount@cov.iloc[:4,:4]@Amount)**0.5 #Caluclate VaR VaR = scipy.stats.norm.ppf(0.99)*PF_std return VaR print(f"VaR = {bond_mapping(df_yield, Vol, [0.3, 0.8], corr_mat, cov_mat, 50000, 1000000):.2f}€")

# 10-day var is calculated in the usual fashion using the fact it scales with the square root bond_VaR = bond_mapping(df_yield, Vol, [0.3, 0.8], corr_mat, cov_mat, 50000, 1000000) print(f"VaR = {np.sqrt(10)*bond_VaR:.2f}€")