Exercises: Variance - Answer Key

By Christopher van Hoecke, Maxwell Margenot, and Delaney Mackenzie

https://www.quantopian.com/lectures/variance

IMPORTANT NOTE:

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:


In [1]:
# Useful Libraries
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt

Data:

In [2]:
X = np.random.randint(100, size = 100)

Exercise 1:

Using the skills aquired in the lecture series, find the following parameters of the list X above:

  • Range
  • Mean Absolute Deviation
  • Variance and Standard Deviation
  • Semivariance and Semideviation
  • Target variance (with B = 60)
In [3]:
# Range of X

print 'Range of X: %s' %(np.ptp(X))
Range of X: 99
In [4]:
# Mean Absolute Deviation
# First calculate the value of mu (the mean)

mu = np.mean(X)

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
Mean absolute deviation of X: 24.6858
In [5]:
# Variance and standard deviation

print 'Variance of X:', np.var(X)
print 'Standard deviation of X:', np.std(X)
Variance of X: 817.7859
Standard deviation of X: 28.5969561317
In [6]:
# Semivariance and semideviation

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)
Semivariance of X: 804.51664902
Semideviation of X: 28.3640026974
In [7]:
# Target variance

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)
Target semivariance of X: 1182
Target semideviation of X: 34.3802268753

Exercise 2:

Using the skills aquired in the lecture series, find the following parameters of prices for AT&T stock over a year:

  • 30 days rolling variance
  • 15 days rolling Standard Deviation
In [8]:
att = get_pricing('T', fields='open_price', start_date='2016-01-01', end_date='2017-01-01')
In [9]:
# Rolling mean
variance = att.rolling(window = 30).var()
In [10]:
# Rolling standard deviation
std = att.rolling(window = 15).std()

Exercise 3 :

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.

In [11]:
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
Portfolio variance:  50.1862438743

Congratulations on completing the Variance answer key!

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