Exercises: Means¶
By Christopher van Hoecke and Max Margenot
Lecture Link :¶
https://www.quantopian.com/lectures/means
IMPORTANT NOTE:¶
This lecture corresponds to the Means 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:
Key Concepts¶
# Useful Functions
def mode(l):
# Count the number of times each element appears in the list
counts = {}
for e in l:
if e in counts:
counts[e] += 1
else:
counts[e] = 1
# Return the elements that appear the most times
maxcount = 0
modes = {}
for (key, value) in counts.iteritems():
if value > maxcount:
maxcount = value
modes = {key}
elif value == maxcount:
modes.add(key)
if maxcount > 1 or len(l) == 1:
return list(modes)
return 'No mode'
# Useful Libraries
import scipy.stats as stats
import numpy as np
Data:¶
l=[]
for x in range(1,100):
x=np.random.randint(1,100)
l.append(x)
np.mean(l)
b. Mean of returns¶
Find the mean of the returns of Iteris Inc. (ITI).
price = get_pricing('ITI', fields='price', start_date='2005-01-01', end_date='2010-01-01')
returns = price.pct_change()[1:]
np.mean(returns)
np.median(l)
b. Median of the returns.¶
Find the median associated with the returns of Bank of America Corp. (BAC).
price = get_pricing('BAC', fields='open_price', start_date='2005-01-01', end_date='2010-01-01')
returns = price.pct_change()[1:]
np.median(returns)
mode(l)
b. Mode of the returns.¶
Find the mode associated with the returns of Goldman Sachs Corp. (GS). Recall with returns, there may not be any values that appear more than once.
start = '2014-01-01'
end = '2015-01-01'
pricing = get_pricing('GS', fields='price', start_date=start, end_date=end)
returns = pricing.pct_change()[1:]
print mode(returns)
# Since all of the returns are distinct, we use a frequency distribution to get an alternative mode.
# np.histogram returns the frequency distribution over the bins as well as the endpoints of the bins
hist, bins = np.histogram(returns, 20)
maxfreq = max(hist)
print 'Mode of bins:', [(bins[i], bins[i+1]) for i, j in enumerate(hist) if j == maxfreq]
stats.gmean(l)
b. Geometric Mean of returns.¶
Find the Geometric Mean of the price of Citi bank (C) for the last 5 years.
price = get_pricing('C', fields='open_price', start_date='2005-01-01', end_date='2010-01-01')
stats.gmean(price)
stats.hmean(l)
b. Harmonic Mean of stock returns.¶
Find the Harmonic Mean of the financial ETF (XLF) over the last 2 years.
price = get_pricing('XLF', fields='open_price', start_date='2016-01-01', end_date='2018-01-01')
stats.hmean(price)
Exercise 6 : Skewness and why it matters.¶
Skewness in a probability distribution is the measure of asymmetry. Negative skew has fewer low values and a longer left tail, whereas positive skew has fewer high values and a longer right tail. In asset pricing, skewness is an important information, naimly in risk assessment. Knowledge that the market has a 60% chance of going down and a 40% chance of going up apears helpfull but only if we know the market is obeying a normal distrubtuion. If we are told that the market will go up 2% but down 18%, we can see how skewness would give us better information.
Determine if the returns of SPY from 2010 to 2017 is positivly or negativly skewed. Recall a data set is positivly skewed if the mode is smaller than the median, which is smaller than the mean. A data set is negativly skewed in the event of the reverse (i.e: the mean is greater than the median, which is greater than the mode)
import matplotlib.pyplot as plt
# Collect Data.
price = get_pricing('SPY', fields='volume', start_date='2016-01-01', end_date='2017-01-01')
returns = price.pct_change()[1:]
# Calculate Mean, Median and Mode.
mean = np.mean(returns)
median = np.median(returns)
mode = stats.mode(returns)[0][0]
print 'mean:', mean
print 'median:', median
print 'mode:', mode
print len(returns)
# Setting parameters and print skewness outcome.
if mode < median < mean:
print 'The returns are positivly skewed.'
if mean<median<mode:
print 'The returns are negativly skewed.'
if mean == median == mode:
print 'There is no Skewness: the returns are symetricaly distributed'
We can clearly see positive skewing from the histogram of the returns. We see fewer higher values and a longer right tail. Plot the histograms of the returns now.
plt.hist(returns, bins = 50);
plt.xlabel('Returns');
plt.ylabel('Frequency');
plt.title('Histogram of Returns');
Congratulations on completing the Means exercises!
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