By Christopher van Hoecke, Maxwell Margenot, and Delaney Mackenzie
https://www.quantopian.com/lectures/variance
This lecture corresponds to the Variance lecture, which is part of the Quantopian lecture series. This homework expects you to rely heavily on the code presented in the corresponding lecture. Please copy and paste regularly from that lecture when starting to work on the problems, as trying to do them from scratch will likely be too difficult.
Part of the Quantopian Lecture Series:
# Useful Libraries
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
X = np.random.randint(100, size = 100)
Using the skills aquired in the lecture series, find the following parameters of the list X above:
# Range of X
range_X = np.ptp(X)
## Your code goes here
print 'Range of X: %s' %(range_X)
# Mean Absolute Deviation
# First calculate the value of mu (the mean)
mu = np.mean(X)
## Your code goes here
abs_dispersion = [np.abs(mu - x) for x in X]
MAD = np.sum(abs_dispersion)/len(abs_dispersion)
print 'Mean absolute deviation of X:', MAD
# Variance and standard deviation
## Your code goes here
print 'Variance of X:', np.var(X)
print 'Standard deviation of X:', np.std(X)
# Semivariance and semideviation
## Your code goes here
lows = [e for e in X if e <= mu]
semivar = np.sum( (lows - mu) ** 2 ) / len(lows)
print 'Semivariance of X:', semivar
print 'Semideviation of X:', np.sqrt(semivar)
# Target variance
## Your code goes here
B = 60
lows_B = [e for e in X if e <= B]
semivar_B = sum(map(lambda x: (x - B)**2,lows_B))/len(lows_B)
print 'Target semivariance of X:', semivar_B
print 'Target semideviation of X:', np.sqrt(semivar_B)
Using the skills aquired in the lecture series, find the following parameters of prices for AT&T stock over a year:
att = get_pricing('T', fields='open_price', start_date='2016-01-01', end_date='2017-01-01')
# Rolling mean
variance = att.rolling(window = 30).var()
# Rolling standard deviation
std = att.rolling(window = 15).std()
The portfolio variance is calculated as
$$\text{VAR}_p = \text{VAR}_{s1} (w_1^2) + \text{VAR}_{s2}(w_2^2) + \text{COV}_{S_1, S_2} (2 w_1 w_2)$$Where $w_1$ and $w_2$ are the weights of $S_1$ and $S_2$.
Find values of $w_1$ and $w_2$ to have a portfolio variance of 50.
asset1 = get_pricing('AAPL', fields='open_price', start_date='2016-01-01', end_date='2017-01-01')
asset2 = get_pricing('XLF', fields='open_price', start_date='2016-01-01', end_date='2017-01-01')
cov = np.cov(asset1, asset2)[0,1]
w1 = 0.87
w2 = 1 - w1
v1 = np.var(asset1)
v2 = np.var(asset2)
pvariance = (w1**2)*v1+(w2**2)*v2+(2*w1*w2)*cov
print 'Portfolio variance: ', pvariance
Congratulations on completing the Variance exercises!
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