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The negative skewness of SPY ETF returns
We analize the growth of 1000$ invested in the SPY ETF between 1993 (inception) and January 2020.¶
The annualized compound total return from the inception of the ETF to January 2020 is about 9,5%.¶
In our analysis we used the log returns to exploit the additive property, but the conclusion would be the same.¶
What would have been the growth if we had excluded the best days and the worst days from our trading strategy?¶
We took the data from the yahoo finance database.
We have replaced the missed worst days and the missed best days with a return equal to 0.
import matplotlib.pyplot as plt
import numpy as np, pandas as pd
import warnings
warnings.filterwarnings('ignore')
import seaborn as sns
plt.style.use('bmh')
from scipy.stats import skew
df1 = pd.read_csv('SPY.csv')
df1.set_index('Date', inplace = True)
df1.index = pd.to_datetime(df1.index)
df = df1.copy()
log_rets = (np.log(df['Adj Close']) - np.log(df['Adj Close'].shift(1)))[1:]
length_period = len(df.resample('Y')) - 1
cagr = ((df['Adj Close'][-1] - df['Adj Close'][0]) / df['Adj Close'][0])**(1/length_period) - 1
three_moments = pd.DataFrame([np.mean(log_rets), np.std(log_rets), skew(log_rets)], index = ['Mean', 'Std', 'Skewness']\
, columns = ['1993-2020 SPY daily returns stats'])
pd.DataFrame(dict(CAGR = cagr), index = ['1993-2020'])
three_moments
The effects of the negative skewness can be further investigated by wondering:
1) What would our final capital have been if we had excluded the top $n$ worst days?
2) What would our final capital have been if we had excluded the top $n$ best days?
def growth_without_worstdays(df1):
wealths = []
total_log_ret = np.sum(df1)
final_wealth = np.exp(total_log_ret)*1000
print('Growth of 1000$ from 1993 to 2020: {}$'.format((np.exp(total_log_ret)*1000).round(2)))
print()
df = df1.copy()
wealths.append(final_wealth)
missed_worst_days = [1,5,15,25]
xticks_ = ['Top ' + str(i) for i in [1,5,15,25]]
for n in missed_worst_days:
values_to_convert_in_zero = df.sort_values().values[:n]
log_rets_minus_n_worstdays = df.where(df > values_to_convert_in_zero.max(),0.)
total_log_ret = np.sum(log_rets_minus_n_worstdays)
final_wealth = np.exp(total_log_ret)*1000
wealths.append(final_wealth)
print('Growth of 1000$ without {} top worst days: {}$'.format(n, final_wealth.round(2)))
print()
fig, ax = plt.subplots(figsize=(10,8))
xpos = [i+1 for i,_ in enumerate(missed_worst_days)]
sns.barplot([0]+xpos, wealths, ax = ax)
for i,t in zip(ax.patches,wealths):
# get_x pulls left or right; get_height pushes up or down
ax.text(i.get_x()+.2, i.get_height()+10, \
str(int(t))+' $', fontsize=15,
color='yellow', backgroundcolor = 'black' )
plt.xticks([0]+xpos, ['Total period'] + xticks_, fontsize = 'large')
plt.xlabel('Missed worst days', fontsize = 'xx-large')
plt.ylabel('Growth of 1000$', fontsize = 'xx-large')
plt.title('Performance of SPY 1993-2020 ',fontsize = 'x-large',color = 'yellow',backgroundcolor = 'black')
return None
Let's start excluding the worst days of the SPY¶
growth_without_worstdays(log_rets)
def growth_without_bestdays(df1):
wealths = []
total_log_ret = np.sum(df1)
final_wealth = np.exp(total_log_ret)*1000
print('Growth of 1000$ from 1993 to 2020 {}$'.format((np.exp(total_log_ret)*1000).round(2)))
print()
df = df1.copy()
wealths.append(final_wealth)
missed_best_days = [1,5,15,25]
xticks_ = ['Top ' + str(i) for i in [1,5,15,25]]
for n in missed_best_days:
values_to_convert_in_zero = df.sort_values(ascending = False).values[:n]
log_rets_plus_n_bestdays = df.where(df < values_to_convert_in_zero.min(),0.)
total_log_ret = np.sum(log_rets_plus_n_bestdays)
final_wealth = np.exp(total_log_ret)*1000
wealths.append(final_wealth)
print('Growth of 1000$ without {} best days: {}$'.format(n, final_wealth.round(2)))
print()
fig, ax = plt.subplots(figsize=(10,8))
xpos = [i+1 for i,_ in enumerate(missed_best_days)]
sns.barplot([0]+xpos, wealths, ax = ax)
for i,t in zip(ax.patches,wealths):
# get_x pulls left or right; get_height pushes up or down
ax.text(i.get_x()+.2, i.get_height()+10, \
str(int(t))+' $', fontsize=15,
color='yellow', backgroundcolor = 'black' )
plt.xticks([0]+xpos, ['Total period'] + xticks_, fontsize = 'large')
plt.xlabel('Missed best days', fontsize = 'xx-large')
plt.ylabel('Growth of 1000$', fontsize = 'xx-large')
plt.title('Performance of SPY, 1993-2020 ',fontsize = 'x-large', color = 'yellow',backgroundcolor = 'black')
return None
Now we exclude the best days of the S&P 500 Index¶
growth_without_bestdays(log_rets)
Let's suppose that we have a risk-neutral prototype investor under a single bet scenario.
In other words, under a single gamble scenario, such investor is indifferent to risk to lose \$100 in order to win \$100 if the probability is the same (0.5).
And let's suppose that we have the following gamble: with probability 0.5 our investor will obtain the final capital without the top worst $n$ days or with the same probability he will obtain the final capital without the top best $n$ days.
The graphs above show that our prototype investor would refuse the gamble for $n$ equal to 1.
Indeed $12548$ (the final capital without the gamble) $> 0.5 \times 10957 + 0.5 \times 13918$.
But for $n$ >1 our investor will take the gamble because the gain he would obtain if he missed the $n$ worst days would exceed the money he would lose if he missed the $n$ best days (and this fact is increasingly evident as $n$ increases).
This is a natural consequence of the negative skewness of historical daily returns of the S&P 500 index.