Value at Risk, VaR¶
In this code, using Geometric Brownian Motion, simulate returns and create distribution. And then find the VaR.
VaR¶
Value at risk (VaR): Loss that will be incurred in the event of an extreme adverse price change with some given, usually low, probability. i.e., the worst-case outcome. For example, 1% VaR means that 99% of returns will exceed the VaR and 1% of returns will be worse.
Assuming that portfolio returns are normally distributed, the VaR is fully determined by the mean and standard deviation of the distribution.
For example, VaR(1%, normal) = Mean - 2.33SD
To obtain a sample estimate of 1% VaR, we sort the observations from high to low. The VaR is the return at the 1st percentile of the sample distribution.
For example, with 95% confidence, we expect that our worst daily loss will not exceed 4%. If we invest 100 dollars, we are 95% confident that our worst daily loss will not exceed 4 dollars (100 dollars x -4%).
| Confidence Level | Two sided CV | One sided CV |
|---|---|---|
| 90% | 1.64 | 1.28 |
| 95% | 1.96 | 1.64 |
| 99% | 2.58 | 2.33 |
In [49]:
!pip install pandas-datareader
!pip install --upgrade pandas-datareader
!pip install yfinance
#Diable the warnings
#import warnings
#warnings.filterwarnings('ignore')
Looking in indexes: https://pypi.org/simple, https://us-python.pkg.dev/colab-wheels/public/simple/ Requirement already satisfied: pandas-datareader in /usr/local/lib/python3.7/dist-packages (0.10.0) Requirement already satisfied: lxml in /usr/local/lib/python3.7/dist-packages (from pandas-datareader) (4.9.0) Requirement already satisfied: pandas>=0.23 in /usr/local/lib/python3.7/dist-packages (from pandas-datareader) (1.3.5) Requirement already satisfied: requests>=2.19.0 in /usr/local/lib/python3.7/dist-packages (from pandas-datareader) (2.27.1) Requirement already satisfied: python-dateutil>=2.7.3 in /usr/local/lib/python3.7/dist-packages (from pandas>=0.23->pandas-datareader) (2.8.2) Requirement already satisfied: pytz>=2017.3 in /usr/local/lib/python3.7/dist-packages (from pandas>=0.23->pandas-datareader) (2022.1) Requirement already satisfied: numpy>=1.17.3 in /usr/local/lib/python3.7/dist-packages (from pandas>=0.23->pandas-datareader) (1.21.6) Requirement already satisfied: six>=1.5 in /usr/local/lib/python3.7/dist-packages (from python-dateutil>=2.7.3->pandas>=0.23->pandas-datareader) (1.15.0) Requirement already satisfied: idna<4,>=2.5 in /usr/local/lib/python3.7/dist-packages (from requests>=2.19.0->pandas-datareader) (2.10) Requirement already satisfied: charset-normalizer~=2.0.0 in /usr/local/lib/python3.7/dist-packages (from requests>=2.19.0->pandas-datareader) (2.0.12) Requirement already satisfied: urllib3<1.27,>=1.21.1 in /usr/local/lib/python3.7/dist-packages (from requests>=2.19.0->pandas-datareader) (1.24.3) Requirement already satisfied: certifi>=2017.4.17 in /usr/local/lib/python3.7/dist-packages (from requests>=2.19.0->pandas-datareader) (2022.5.18.1) Looking in indexes: https://pypi.org/simple, https://us-python.pkg.dev/colab-wheels/public/simple/ Requirement already satisfied: pandas-datareader in /usr/local/lib/python3.7/dist-packages (0.10.0) Requirement already satisfied: pandas>=0.23 in /usr/local/lib/python3.7/dist-packages (from pandas-datareader) (1.3.5) Requirement already satisfied: lxml in /usr/local/lib/python3.7/dist-packages (from pandas-datareader) (4.9.0) Requirement already satisfied: requests>=2.19.0 in /usr/local/lib/python3.7/dist-packages (from pandas-datareader) (2.27.1) Requirement already satisfied: pytz>=2017.3 in /usr/local/lib/python3.7/dist-packages (from pandas>=0.23->pandas-datareader) (2022.1) Requirement already satisfied: numpy>=1.17.3 in /usr/local/lib/python3.7/dist-packages (from pandas>=0.23->pandas-datareader) (1.21.6) Requirement already satisfied: python-dateutil>=2.7.3 in /usr/local/lib/python3.7/dist-packages (from pandas>=0.23->pandas-datareader) (2.8.2) Requirement already satisfied: six>=1.5 in /usr/local/lib/python3.7/dist-packages (from python-dateutil>=2.7.3->pandas>=0.23->pandas-datareader) (1.15.0) Requirement already satisfied: idna<4,>=2.5 in /usr/local/lib/python3.7/dist-packages (from requests>=2.19.0->pandas-datareader) (2.10) Requirement already satisfied: charset-normalizer~=2.0.0 in /usr/local/lib/python3.7/dist-packages (from requests>=2.19.0->pandas-datareader) (2.0.12) Requirement already satisfied: urllib3<1.27,>=1.21.1 in /usr/local/lib/python3.7/dist-packages (from requests>=2.19.0->pandas-datareader) (1.24.3) Requirement already satisfied: certifi>=2017.4.17 in /usr/local/lib/python3.7/dist-packages (from requests>=2.19.0->pandas-datareader) (2022.5.18.1) Looking in indexes: https://pypi.org/simple, https://us-python.pkg.dev/colab-wheels/public/simple/ Requirement already satisfied: yfinance in /usr/local/lib/python3.7/dist-packages (0.1.70) Requirement already satisfied: requests>=2.26 in /usr/local/lib/python3.7/dist-packages (from yfinance) (2.27.1) Requirement already satisfied: numpy>=1.15 in /usr/local/lib/python3.7/dist-packages (from yfinance) (1.21.6) Requirement already satisfied: pandas>=0.24.0 in /usr/local/lib/python3.7/dist-packages (from yfinance) (1.3.5) Requirement already satisfied: lxml>=4.5.1 in /usr/local/lib/python3.7/dist-packages (from yfinance) (4.9.0) Requirement already satisfied: multitasking>=0.0.7 in /usr/local/lib/python3.7/dist-packages (from yfinance) (0.0.10) Requirement already satisfied: python-dateutil>=2.7.3 in /usr/local/lib/python3.7/dist-packages (from pandas>=0.24.0->yfinance) (2.8.2) Requirement already satisfied: pytz>=2017.3 in /usr/local/lib/python3.7/dist-packages (from pandas>=0.24.0->yfinance) (2022.1) Requirement already satisfied: six>=1.5 in /usr/local/lib/python3.7/dist-packages (from python-dateutil>=2.7.3->pandas>=0.24.0->yfinance) (1.15.0) Requirement already satisfied: urllib3<1.27,>=1.21.1 in /usr/local/lib/python3.7/dist-packages (from requests>=2.26->yfinance) (1.24.3) Requirement already satisfied: idna<4,>=2.5 in /usr/local/lib/python3.7/dist-packages (from requests>=2.26->yfinance) (2.10) Requirement already satisfied: charset-normalizer~=2.0.0 in /usr/local/lib/python3.7/dist-packages (from requests>=2.26->yfinance) (2.0.12) Requirement already satisfied: certifi>=2017.4.17 in /usr/local/lib/python3.7/dist-packages (from requests>=2.26->yfinance) (2022.5.18.1)
In [4]:
import math
import numpy as np
import numpy.random as npr
import pandas as pd
import pandas_datareader as pdr
import yfinance as yf
import scipy as sp
from scipy import stats
from pylab import plt, mpl
plt.style.use('fivethirtyeight')
#plt.style.use('seaborn')
mpl.rcParams['font.family'] = 'DejaVu Sans'
pd.set_option('precision', 3)
pd.set_option('display.max_colwidth', 100)
%matplotlib inline
In [5]:
print("At 95% confidence level {:.2f}".format(sp.stats.norm.ppf(1-.10/2)))
print("At 97.5% confidence level {:.2f}".format(sp.stats.norm.ppf(1-.05/2)))
print("At 90% confidence level {:.2f}".format(sp.stats.norm.ppf(1-.10)))
print("At 99% confidence level {:.2f}".format(sp.stats.norm.ppf(1-.01)))
At 95% confidence level 1.64 At 97.5% confidence level 1.96 At 90% confidence level 1.28 At 99% confidence level 2.33
In [6]:
def get_prices(tickers, freq_p, st_day, end_day):
mystock = pd.DataFrame()
for t in tickers:
mystock[t] = yf.download(t, start=st_day, end=end_day, interval=freq_p)['Adj Close']
return mystock
In [16]:
# for short time horizons er will be small, and therefore VaR estimations
# will not be much influenced by it
# assuming that portfolio is normally distributed,
def var_calc(CL, days, p_val, vol, t):
VaR = p_val * vol * np.sqrt(t/days) * sp.stats.norm.ppf(CL)
percent_loss = -VaR/p_val *100
print("Assuming that we invest {:.2f}, for the next {:.1f} trading days".format(p_val, days))
print("At {:.3f} confidence level, loss will not exceed {:,.2f}".format(CL, VaR))
print("This represents a move of {:.2f} standard deviations below the expected return,\
or a loss of {:.2f}%.".format(sp.stats.norm.ppf(CL), percent_loss))
return
In [14]:
tic=['SPY', 'TLT', 'TSLA', 'AAPL', 'VNQ', 'BAC', 'WMT', 'AMD', 'JNJ', 'GM', 'CMG', 'SHV']
prices= get_prices(tic, freq_p='1wk', st_day="2011-01-01", end_day="2022-05-31") # id , 1wk, 1mo
prices.info()
prices.tail(3)
[*********************100%***********************] 1 of 1 completed [*********************100%***********************] 1 of 1 completed [*********************100%***********************] 1 of 1 completed [*********************100%***********************] 1 of 1 completed [*********************100%***********************] 1 of 1 completed [*********************100%***********************] 1 of 1 completed [*********************100%***********************] 1 of 1 completed [*********************100%***********************] 1 of 1 completed [*********************100%***********************] 1 of 1 completed [*********************100%***********************] 1 of 1 completed [*********************100%***********************] 1 of 1 completed [*********************100%***********************] 1 of 1 completed <class 'pandas.core.frame.DataFrame'> DatetimeIndex: 640 entries, 2011-01-01 to 2022-05-21 Data columns (total 12 columns): # Column Non-Null Count Dtype --- ------ -------------- ----- 0 SPY 595 non-null float64 1 TLT 595 non-null float64 2 TSLA 595 non-null float64 3 AAPL 595 non-null float64 4 VNQ 595 non-null float64 5 BAC 595 non-null float64 6 WMT 595 non-null float64 7 AMD 595 non-null float64 8 JNJ 595 non-null float64 9 GM 595 non-null float64 10 CMG 595 non-null float64 11 SHV 595 non-null float64 dtypes: float64(12) memory usage: 65.0 KB
Out[14]:
| SPY | TLT | TSLA | AAPL | VNQ | BAC | WMT | AMD | JNJ | GM | CMG | SHV | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Date | ||||||||||||
| 2022-05-07 | 401.72 | 115.98 | 769.59 | 147.11 | 96.78 | 35.17 | 148.05 | 95.12 | 175.721 | 38.21 | 1318.28 | 110.22 |
| 2022-05-14 | 389.63 | 118.51 | 663.90 | 137.59 | 94.83 | 33.86 | 119.20 | 93.50 | 175.850 | 35.40 | 1294.11 | 110.23 |
| 2022-05-21 | 415.26 | 119.08 | 759.63 | 149.64 | 100.30 | 37.02 | 128.48 | 102.26 | 179.934 | 38.57 | 1402.42 | 110.27 |
In [17]:
for t in tic:
confidence_l = .95
annual_volatility = np.std(prices[t].pct_change()*np.sqrt(252))
most_recent_p = prices[t][-1]
holding_period = 252
frequency = 21
print("Historical annual volatility of {} = {:.4f}".format(t, annual_volatility))
var_calc(confidence_l, holding_period, most_recent_p, annual_volatility, frequency)
print(50 * "-")
Historical annual volatility of SPY = 0.3427 Assuming that we invest 415.26, for the next 252.0 trading days At 0.950 confidence level, loss will not exceed 67.56 This represents a move of 1.64 standard deviations below the expected return, or a loss of -16.27%. -------------------------------------------------- Historical annual volatility of TLT = 0.2965 Assuming that we invest 119.08, for the next 252.0 trading days At 0.950 confidence level, loss will not exceed 16.77 This represents a move of 1.64 standard deviations below the expected return, or a loss of -14.08%. -------------------------------------------------- Historical annual volatility of TSLA = 1.1661 Assuming that we invest 759.63, for the next 252.0 trading days At 0.950 confidence level, loss will not exceed 420.60 This represents a move of 1.64 standard deviations below the expected return, or a loss of -55.37%. -------------------------------------------------- Historical annual volatility of AAPL = 0.5918 Assuming that we invest 149.64, for the next 252.0 trading days At 0.950 confidence level, loss will not exceed 42.05 This represents a move of 1.64 standard deviations below the expected return, or a loss of -28.10%. -------------------------------------------------- Historical annual volatility of VNQ = 0.4469 Assuming that we invest 100.30, for the next 252.0 trading days At 0.950 confidence level, loss will not exceed 21.28 This represents a move of 1.64 standard deviations below the expected return, or a loss of -21.22%. -------------------------------------------------- Historical annual volatility of BAC = 0.6979 Assuming that we invest 37.02, for the next 252.0 trading days At 0.950 confidence level, loss will not exceed 12.27 This represents a move of 1.64 standard deviations below the expected return, or a loss of -33.14%. -------------------------------------------------- Historical annual volatility of WMT = 0.4125 Assuming that we invest 128.48, for the next 252.0 trading days At 0.950 confidence level, loss will not exceed 25.17 This represents a move of 1.64 standard deviations below the expected return, or a loss of -19.59%. -------------------------------------------------- Historical annual volatility of AMD = 1.2085 Assuming that we invest 102.26, for the next 252.0 trading days At 0.950 confidence level, loss will not exceed 58.68 This represents a move of 1.64 standard deviations below the expected return, or a loss of -57.38%. -------------------------------------------------- Historical annual volatility of JNJ = 0.3385 Assuming that we invest 179.93, for the next 252.0 trading days At 0.950 confidence level, loss will not exceed 28.92 This represents a move of 1.64 standard deviations below the expected return, or a loss of -16.08%. -------------------------------------------------- Historical annual volatility of GM = 0.7413 Assuming that we invest 38.57, for the next 252.0 trading days At 0.950 confidence level, loss will not exceed 13.58 This represents a move of 1.64 standard deviations below the expected return, or a loss of -35.20%. -------------------------------------------------- Historical annual volatility of CMG = 0.7466 Assuming that we invest 1402.42, for the next 252.0 trading days At 0.950 confidence level, loss will not exceed 497.18 This represents a move of 1.64 standard deviations below the expected return, or a loss of -35.45%. -------------------------------------------------- Historical annual volatility of SHV = 0.0086 Assuming that we invest 110.27, for the next 252.0 trading days At 0.950 confidence level, loss will not exceed 0.45 This represents a move of 1.64 standard deviations below the expected return, or a loss of -0.41%. --------------------------------------------------
Monte Carolo approach using stochastic geometric Borwnian motion,
In [18]:
S0 = prices['AAPL'][-1]
print(S0)
r = 0.05
sigma = np.std(prices['AAPL'].pct_change()*np.sqrt(252))
T = 21.0
I = 10000
ST1 = S0 * np.exp((r - 0.5 * sigma ** 2) * T +
sigma * math.sqrt(T) * np.random.standard_normal(I))
ST2 = S0 * np.random.lognormal((r - 0.5 * sigma ** 2) * T,
sigma * math.sqrt(T), size=I)
149.63999938964844
In [19]:
def print_statistics(a1, a2):
'''
Parameters
==========
a1, a2: ndarray objects
results objects from simulation
'''
sta1 = sp.stats.describe(a1)
sta2 = sp.stats.describe(a2)
print('%14s %14s %14s' % ('statistic', 'data set 1', 'data set 2'))
print(45 * "-")
print('%14s %14.3f %14.3f' % ('size', sta1[0], sta2[0]))
print('%14s %14.3f %14.3f' % ('min', sta1[1][0], sta2[1][0]))
print('%14s %14.3f %14.3f' % ('max', sta1[1][1], sta2[1][1]))
print('%14s %14.3f %14.3f' % ('mean', sta1[2], sta2[2]))
print('%14s %14.3f %14.3f' % ('std', np.sqrt(sta1[3]), np.sqrt(sta2[3])))
print('%14s %14.3f %14.3f' % ('skew', sta1[4], sta2[4]))
print('%14s %14.3f %14.3f' % ('kurtosis', sta1[5], sta2[5]))
In [20]:
print_statistics(ST1, ST2)
statistic data set 1 data set 2
---------------------------------------------
size 10000.000 10000.000
min 0.000 0.001
max 300993.143 114507.025
mean 445.331 313.031
std 5473.487 2648.420
skew 36.528 28.691
kurtosis 1653.676 1042.670
In [ ]:
Let's assume that we buy 5000 shares of AAPL. And using historical returns, let's estimate mean and standard deviation of returns on AAPL. In order to save time, let's assume they are 19% and 30.7%, respectively.¶
In [21]:
aapl = 1
aapl_price = pdr.get_quote_yahoo('AAPL')['price'] # the most recent price
aapl_value = aapl * aapl_price
aapl_value = aapl_value.at['AAPL']
t = 21/252
mu = .05
volatility = .307
iterations = 10000
In [22]:
type(aapl_price)
Out[22]:
pandas.core.series.Series
In [23]:
aapl_price.describe()
Out[23]:
count 1.00 mean 148.84 std NaN min 148.84 25% 148.84 50% 148.84 75% 148.84 max 148.84 Name: price, dtype: float64
In [24]:
# checking to see what get_quote_yahoo is retreveing
pdr.get_quote_yahoo('AAPL')
Out[24]:
| language | region | quoteType | typeDisp | quoteSourceName | triggerable | customPriceAlertConfidence | currency | marketState | gmtOffSetMilliseconds | ... | earningsTimestamp | earningsTimestampStart | earningsTimestampEnd | trailingAnnualDividendRate | trailingPE | trailingAnnualDividendYield | epsTrailingTwelveMonths | epsForward | displayName | price | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| AAPL | en-US | US | EQUITY | Equity | Nasdaq Real Time Price | True | HIGH | USD | PREPRE | -14400000 | ... | 1651163400 | 1658779200 | 1659124800 | 0.88 | 24.253 | 0.006 | 6.137 | 6.56 | Apple | 148.84 |
1 rows × 78 columns
In [25]:
type(aapl_price)
Out[25]:
pandas.core.series.Series
In [26]:
type(aapl_value)
Out[26]:
numpy.float64
In [27]:
aapl_price
Out[27]:
AAPL 148.84 Name: price, dtype: float64
In [28]:
# aapl_value = # of shares times price
aapl_value
Out[28]:
148.84
Below, let's simulate future value from Geometric Brownian Motion series.
In [29]:
def VaR(pv, mu, vol, T, iterations):
end = pv * np.exp((mu - .5 * vol ** 2) * T +
vol * np.sqrt(T) * np.random.standard_normal(iterations))
ending_values = end - pv
return ending_values
In [30]:
at_risk = VaR(aapl_value, mu, volatility, t, iterations)
at_risk
Out[30]:
array([ 12.27010694, -3.17392878, -1.72863707, ..., -13.28831742,
-6.68840767, 17.63879795])
In [31]:
type(at_risk)
Out[31]:
numpy.ndarray
In [32]:
np.shape(at_risk)
Out[32]:
(10000,)
In [33]:
np.ndim(at_risk)
Out[33]:
1
In [34]:
at_risk.mean()
Out[34]:
0.6267317841989819
In [35]:
at_risk.std()
Out[35]:
13.313012258045937
In [36]:
plt.hist(at_risk,bins=100)
Out[36]:
(array([ 1., 2., 2., 3., 1., 8., 5., 6., 7., 9., 13.,
12., 19., 19., 30., 39., 29., 39., 58., 68., 74., 72.,
104., 117., 101., 126., 137., 144., 163., 178., 189., 181., 219.,
233., 245., 272., 240., 270., 253., 283., 264., 290., 262., 269.,
275., 250., 271., 288., 240., 278., 225., 271., 228., 205., 194.,
200., 167., 181., 149., 154., 153., 115., 127., 112., 109., 95.,
69., 60., 58., 65., 45., 47., 37., 38., 26., 27., 24.,
18., 15., 19., 15., 18., 5., 7., 13., 9., 7., 2.,
9., 6., 2., 0., 4., 3., 1., 1., 4., 2., 0.,
1.]), array([-40.65270376, -39.72052762, -38.78835148, -37.85617535,
-36.92399921, -35.99182307, -35.05964693, -34.12747079,
-33.19529466, -32.26311852, -31.33094238, -30.39876624,
-29.4665901 , -28.53441397, -27.60223783, -26.67006169,
-25.73788555, -24.80570941, -23.87353328, -22.94135714,
-22.009181 , -21.07700486, -20.14482872, -19.21265259,
-18.28047645, -17.34830031, -16.41612417, -15.48394803,
-14.5517719 , -13.61959576, -12.68741962, -11.75524348,
-10.82306734, -9.89089121, -8.95871507, -8.02653893,
-7.09436279, -6.16218665, -5.23001052, -4.29783438,
-3.36565824, -2.4334821 , -1.50130596, -0.56912983,
0.36304631, 1.29522245, 2.22739859, 3.15957473,
4.09175086, 5.023927 , 5.95610314, 6.88827928,
7.82045542, 8.75263155, 9.68480769, 10.61698383,
11.54915997, 12.48133611, 13.41351224, 14.34568838,
15.27786452, 16.21004066, 17.1422168 , 18.07439293,
19.00656907, 19.93874521, 20.87092135, 21.80309749,
22.73527362, 23.66744976, 24.5996259 , 25.53180204,
26.46397818, 27.39615431, 28.32833045, 29.26050659,
30.19268273, 31.12485887, 32.057035 , 32.98921114,
33.92138728, 34.85356342, 35.78573956, 36.71791569,
37.65009183, 38.58226797, 39.51444411, 40.44662025,
41.37879638, 42.31097252, 43.24314866, 44.1753248 ,
45.10750094, 46.03967707, 46.97185321, 47.90402935,
48.83620549, 49.76838163, 50.70055776, 51.6327339 ,
52.56491004]), <a list of 100 Patch objects>)
In [37]:
percentiles = [1,5,10]
np.percentile(at_risk, percentiles)
Out[37]:
array([-27.7934676 , -20.33712168, -16.12524849])
In [ ]: