Empirical Analysis of S&P500 Returns

This document provides an empirical analysis of S&P500 returns using RiskMetrics and Historical Simulation methods. It includes plots of daily returns, histogram, autocorrelations, and VaRs. The document also tests the validity of 1% and 5% VaRs using a binomial test.

3. Empirical
Use S&P500 returns for September 1, 2017 through December 31, 2018. (1 point)
Plot the daily returns, and the histogram of daily returns and PDF of Normal distribution.
(1 point)
Plot of daily returns
01-09-17
17-09-17
03-10-17
19-10-17
04-11-17
20-11-17
06-12-17
22-12-17
07-01-18
23-01-18
08-02-18
24-02-18
12-03-18
28-03-18
13-04-18
29-04-18
15-05-18
31-05-18
16-06-18
02-07-18
18-07-18
03-08-18
19-08-18
04-09-18
20-09-18
06-10-18
22-10-18
07-11-18
23-11-18
09-12-18
25-12-18
-6.00%
-4.00%
-2.00%
0.00%
2.00%
4.00%
6.00%
Daily Returns
Days
Returns

Histogram of daily returns
-0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05
0
20
40
60
80
100
120
140
160
0
20
40
60
80
100
120
140
160
Histogram of Daily S&P 500 Returns and the
Normal Distribution
September 1, 2017 - December 31, 2018
Returns
Probability distribution
Plot autocorrelations, 1 to 100 lags. (1 point)
1
6
11
16
21
26
31
36
41
46
51
56
61
66
71
76
81
86
91
96
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
Lag order
Autocorrelation of Daily Returns

Plot squared return autocorrelation. (1 point)
1
5
9
13
17
21
25
29
33
37
41
45
49
53
57
61
65
69
73
77
81
85
89
93
97
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
Lag order
Autocorrelation of Squared Daily Returns
3.1. For each day, from September 1, 2018 through December 31, 2018, calculate the 10-
day 1% VaRs for S&P500 returns using the following methods: (a) RiskMetrics, that is,
normal distribution with an exponential smoother on variance using the weight, λ=0.94,
and (b) Historical Simulation. Use a 250-day moving sample. Compute the 10-day VaR
from 1-day VaR just by multiplying by square root of 10. Plot the VaRs. (10 points)
Calculations are shown in attached excel file.

05-09-17
22-09-17
09-10-17
26-10-17
12-11-17
29-11-17
16-12-17
02-01-18
19-01-18
05-02-18
22-02-18
11-03-18
28-03-18
14-04-18
01-05-18
18-05-18
04-06-18
21-06-18
08-07-18
25-07-18
11-08-18
28-08-18
14-09-18
01-10-18
18-10-18
04-11-18
21-11-18
08-12-18
25-12-18
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
Dates
VaR
3.2. Assume that each day a trader has a 10-day, 1% dollar VaR limit of $100,000. Assume
that the trader each day invests the maximum amount possible in the S&P 500 between
September 1, 2018 and December 31, 2018. Construct daily P&Ls of this position for 10-
day 1% VAR using (a) RiskMetrics and (b) Historical Simulation. Plot them. (10 points)
What do you observe? What do you conclude?
a) Value A= $100,000
Return per year R= 1-0.94 = 0.06 = 6%
Confidence level c = 99%
Volatility v = 1.20%
Trading days T = 252
We start by evaluating the present volatility given by;

Volatility (t+1) V= √ ( 0.94∗v2 ) +0.06∗( R
T )2
(Chen, 2012)
Substituting we have;
volatility ( t+1 ) = √ ( 0.94∗0.0122 ) +0.06∗( 0.06
252 )
2
=0.01163 ≈ 1.163 %
Inverse probability p=NORM . S . INV ( 1−c ) =−2.3264
We therefore obtain VaR by the formula;
VaR=− p∗V
¿−(−2.3264∗0.001163)
¿ 2.7066 %
10 day VaR = 2.7066 %∗√10
= 8.559%
Inverse in VaR = e−0.08559=0.9180
VaR $ = 100,000∗( 1−0.9180 ) =$ 82,000
b) Historical simulation (Paul Embrechts, 2013)
A = $100,000
10 day VaR = 8.559%
Mean = A∗( R∗10
T )+ A
¿ 100,000 ( 0.09∗10
250 )+100,000=$ 100,360

0 2 4 6 8 10 12
0.00%
200.00%
400.00%
600.00%
800.00%
1000.00%
1200.00%
Daily Returns
Days
Returns
Historical simulation results to a higher value as compared to RiskMetrics. Thus, RiskMetrics is
the best method to adopt.
3.3. Test the validity of 1% VaR and 5% VaR for (a) RiskMetrics and (b) Historical
Simulation for the same time period using binomial test (and 5% p-value significance
level). What do you conclude?
a) Solved in excel file.
Its observed that 1% VaR yields a larger value as compared to 5% VaR. This indicates that 5%
VaR yields acceptable values as compared to 1% VaR. Thus, adopt 5% VaR.

References
Chen, C. W., Gerlach, R., Hwang, B. B., & McAleer, M. (2012). Forecasting Value-at-Risk using nonlinear
regression quantiles and the intra-day range. International Journal of Forecasting, 28(3), 557-
574.
Paul Embrechts, C. K. (2013). Modelling Extremal Events: for Insurance and Finance. Springer Science &
Business Media.