Much of the modeling illustrated in previous courses and carried out in practice is based
on a model of risk factor increments distributed identically and independently (iid)
normal.
For example it is common when considering individual equity returns to assume that the
price appreciation component of continuously compounded returns, xt,t+h = ln(St+h/St)
over horizon (h) will be identically and independently distributed N(u, 2).
This assumption implies that for each successive period of length (h) the xj,j+h is an
independent random draw from the normal distribution.
The (iid) assumption has implications for aggregating risk factor increments to longer
horizons H > h. For instance if we have a sample of trading day closing equity prices and
want to know properties of a 5-trading day return.
xt,t+5 = ln(St+5/St) = ln[S1/S0 * S2/S1 * S3/S2 * S4/S3 * S5/S4] =
ln(S1/S0 ) + ln(S2/S1) + ln(S3/S2) + ln(S4/S3) + ln(S5/S4)
The (iid) property implies that
 2 ( xt  5, t )  5   2 ( xt 1, t ) or
 ( xt  5, t )   ( xt 1, t ) 5
(iid) distributed variables exhibit the absence of autocorrelation.
positive autocorrelation:  > 0; xt+1 =  * xt + ~t 1
negative autocorrelation  < 0; xt+1 =  * xt + ~t 1
In the presence of positive autocorrelation the true standard deviation of xt+h,t will be
greater than that implied by the (iid) assumption, i.e.
 ( xt  h, t )   ( xt 1 )  h
In the presence of negative autocorrelation the true standard deviation of xt+h,t will be
less than that implied by the (iid) assumption, i.e.
 ( xt  h, t )   ( xt 1 )  h
When computing statistics from sample data to inform risk statements it is important to
consider the precision with which the statistic estimates the true population characteristic.
1
For instance the Central Limit Theorem applied to the sample mean, m, indicates that for
samples drawn from a normally distributed random variable or as the sample size
increases, is itself normal and the variance of the distribution of the sample mean
decreases with sample size, T.

ˆ
m ~ N (u, 2 )  se(m) 
T
T
The sampling distribution of the sample variance:
ˆ 2 ~ N ( 2 ,  4
2
1
)  se(ˆ )  ˆ
T 1
2T
Notice that for a given sample size, T, the estimate of standard deviation is more precise
than the estimate of the mean. Additionally, as the sample size is increased the precision
of the estimate of standard deviation improves at a faster rate than the precision of the
estimate of the mean.
2
Common risk factors historical time series: euro – 1/3/99-8/6/04, wti – 1/2/86-8/9/04,
spx – 1/2/80/8/16/04, t10y – 1/4/80-8/6/04
Descriptive statistics for increments of (1,5,10) days:
euro, wti, spx: increments defined as logarithmic changes (continuously compounded
returns).
t10y: increments defined as simple changes
euro
1-day
5-day
10-day
count
1407
281
140
max
min
mean
stdev
0.027089 -0.02114 4.80E-05 0.005879
0.034379 -0.0419 0.000189 0.014333
0.051321 -0.05817 0.000342 0.020965
wti
1-day
5-day
10-day
count
4698
939
469
max
0.14119
0.21016
0.27682
spx
1-day
5-day
10-day
count
6213
1242
621
max
min
mean
stdev
0.087089 -0.07633 0.000364 0.00902
0.098566 -0.18293 0.001836 0.023567
0.13025 -0.23547 0.003671 0.033192
t10y
1-day
5-day
10-day
count
6143
1228
614
max
0.0065
0.0109
0.0114
skewness kurtosis
-0.02984 4.2749
-0.23071
2.732
-0.17898 2.8418
min
mean
stdev skewness kurtosis
-0.4064 0.000187 0.022219 -1.7203
32.062
-0.33144 0.000586 0.053865 -0.7576
7.2538
-0.29578 0.001231 0.072609 -0.31464 4.6256
skewness kurtosis
0.015626 9.5031
-0.57046 7.3057
-0.81819 9.0398
min
mean
stdev skewness kurtosis
-0.0075 -1.05E-05 0.000844 -0.21168 9.5927
-0.0136 -5.16E-05 0.00197 -0.30407 7.7887
-0.0235 -0.0001 0.002899 -0.8591
11.519
3
Precision estimate of mean and standard deviation
mean 95% confidence interval u
t: H(0)
low
high
0.3065
-0.0003
0.0004
0.2215
-0.0015
0.0019
0.1931
-0.0031
0.0038
std
95% confidence interval for 
se(s) z: H(0)
low
high
0.0001 53.0471
0.0057
0.0057
0.0006 23.7065
0.0131
0.0135
0.0013 16.7332
0.0185
0.0193
euro
1-day
5-day
10-day
se(m)
0.0002
0.0009
0.0018
wti
1-day
5-day
10-day
se(m) H(0)
0.0003 0.5774
0.0018 0.3335
0.0034 0.3672
-0.0004
-0.0029
-0.0053
0.0008
0.0040
0.0078
se(s)
H(0)
0.0002 96.9330
0.0012 43.3359
0.0024 30.6268
0.0218
0.0514
0.0680
0.0219
0.0523
0.0695
spx
1-day
5-day
10-day
se(m) H(0)
0.0001 3.1781
0.0007 2.7448
0.0013 2.7560
0.0001
0.0005
0.0011
0.0006
0.0031
0.0063
se(s)
H(0)
0.0001 111.4720
0.0005 49.8397
0.0009 35.2420
0.0089
0.0226
0.0313
0.0089
0.0230
0.0320
t10y
1-day
5-day
10-day
se(m) H(0)
0.0000 -0.9702
0.0001 -0.9185
0.0001 -0.8826
0.0000
0.0001
0.0001
se(s)
H(0)
0.0000 110.8422
0.0000 49.5580
0.0001 35.0428
0.0008
0.0019
0.0027
0.0008
0.0019
0.0028
0.0000
-0.0002
-0.0003
Autocorrelation of underlying processes
euro
1-day
5-day
10-day
est.
stdev stdev(1-day)*sqrt(h)
0.0059
0.0143
0.0131
0.0210
0.0186
wti
1-day
5-day
10-day
stdev
0.0222
0.0539
0.0726
0.0497
0.0703
spx
1-day
5-day
10-day
stdev
0.0090
0.0236
0.0332
0.0202
0.0285
t10y
1-day
5-day
10-day
stdev
0.0008
0.0020
0.0029
0.0019
0.0027
4
When interpreting daily logarithmic returns remember that
simple return = St/St-1 -1 = eX – 1. For instance the minimum daily logarithmic return of
–.4064 for wti(1-day) is equivalent to a simple return of –33.40%.
For all 1-day return series examined, sample kurtosis is greater than 3.0. Distributions of
daily logarithmic returns are highly kurtotic relative to the normal distribution. Notice
that as the horizon is extended to 5 and 10 days the estimated kurtosis declines.
The distribution of daily logarithmic returns is fat-tailed relative to the normal. The
frequency of extreme price changes is far greater than expected by a normally distributed
random variable.
A simple test of the normality hypothesis.
The studentized range is a test statistic with known quantiles under the null hypothesis of
normality. The definition of the studentized range is simply the sample range divided by
the sample standard deviation.
SR = (Max – Min)/SD
H0 = normality
HA = not H0
5
The studentized range test is a two-sided test, the null hypothesis is rejected if the test
statistic is greater in absolute value than the critical value for the chosen confidence level.
Fractiles of the distribution of the Studentized Range from samples of size, T
Source: H.A. David, H.O. Hartley and E.S. Pearson, "The distribution of the ratio, in a single normal sample, of range to standard deviation,"
Biometrika 61 (1954) pg. 491
0.9
0.95
0.975
0.99
0.995
T
3
1.997
1.999
2.000
2.000
2.000
3
4
2.409
2.429
2.439
2.445
2.447
4
5
2.712
2.753
2.782
2.803
2.813
5
6
2.949
3.012
3.056
3.095
3.115
6
7
3.143
3.222
3.282
3.338
3.369
7
8
3.308
3.399
3.471
3.543
3.585
8
9
3.449
3.552
3.634
3.720
3.772
9
T
0.005
0.01
0.025
0.05
0.1
10
2.470
2.510
2.590
2.670
2.770
3.570
3.685
3.777
3.875
3.935
10
11
2.530
2.580
2.660
2.740
2.840
3.680
3.800
3.903
4.012
4.079
11
12
2.590
2.650
2.730
2.800
2.910
3.780
3.910
4.010
4.134
4.208
12
13
2.650
2.700
2.780
2.860
2.970
3.870
4.000
4.110
4.244
4.325
13
14
2.700
2.750
2.830
2.910
3.020
3.950
4.090
4.210
4.340
4.431
14
15
2.750
2.800
2.880
2.960
3.070
4.020
4.170
4.290
4.430
4.530
15
16
2.800
2.850
2.930
3.010
3.130
4.090
4.240
4.370
4.510
4.620
16
17
2.840
2.900
2.980
3.060
3.170
4.150
4.310
4.440
4.590
4.690
17
18
2.880
2.940
3.020
3.100
3.210
4.210
4.380
4.510
4.660
4.770
18
19
2.920
2.980
3.060
3.140
3.250
4.270
4.430
4.570
4.730
4.840
19
20
2.950
3.010
3.100
3.180
3.290
4.320
4.490
4.630
4.790
4.910
20
30
3.220
3.270
3.370
3.460
3.580
4.700
4.890
5.060
5.250
5.390
30
40
3.410
3.460
3.570
3.660
3.790
4.960
5.150
5.340
5.540
5.690
40
50
3.570
3.610
3.720
3.820
3.940
5.150
5.350
5.540
5.770
5.910
50
60
3.690
3.740
3.850
3.950
4.070
5.290
5.500
5.700
5.930
6.090
60
80
3.880
3.930
4.050
4.150
4.270
5.510
5.730
5.930
6.180
6.350
80
100
4.020
4.000
4.200
4.310
4.440
5.680
5.900
6.110
6.360
6.540
100
150
4.300
4.360
4.470
4.590
4.720
5.960
6.180
6.390
6.640
6.840
150
200
4.500
4.560
4.670
4.780
4.900
6.150
6.380
6.590
6.850
7.030
200
500
5.060
5.130
5.250
5.370
5.490
6.720
6.940
7.150
7.420
7.600
500
1000
5.500
5.570
5.680
5.790
5.920
7.110
7.330
7.540
7.800
7.990
1000
6