International Journal of Computational Intelligence Systems 
Vol. XX(Z); Month (Year), pp. xx–yy

DOI: 10.1080/XXXXXXXXXXXXXX; ISSN XXXX–XXXX online 
https://www.atlantis-press.com/journals/ijndc

Journal of Risk Analysis and Crisis Response 
Vol. 10(1); April (2020), pp. 6–11

DOI: https://doi.org/10.2991/jracr.k.200421.003; ISSN 2210-8491; eISSN 2210-8505 
https://www.atlantis-press.com/journals/jracr

Research Article

Kernel Density Estimation of White Noise for Non-diversifiable 
Risk in Decision Making

Emma Anyika Shileche1,*, Patrick Weke2, Thomas Achia3

1School of Computing and Mathematics, The Co-operative University of Kenya, Nairobi, Kenya
2School of Mathematics, University of Nairobi, Nairobi, Kenya
3School of Public Health, University of Witwatersrand, Johannesburg, South Africa 

A R T I C L E  I N F O
Article History

Received 20 October 2019 
Accepted 14 December 2019 

Keywords

Curve estimation
non-diversifiable risk
random variable
probability estimates

A B S T R A C T
Many businesses make profit yearly and tend to invest some of the profit so that they can cushion their organizations against any 
future unknown events that can affect their current profit making. Since future happenings in businesses cannot be predicted 
accurately, estimates are made using experience or past data which are not exact. The probability element (which is normally 
determined by experience or past data) is important in investment decision making process since it helps address the problem 
of uncertainty. Many of the investment decision making methods have incorporated the expectation and risk of an event in 
making investment decisions. Most of those that use risk account for diversifiable risk (non-systematic risk) only thus limiting 
the predictability element of these investment methods since total risk are not properly accounted for. A few of these methods 
include the certainty (probability) element. These include value at risk method which uses covariance matrices as total risk 
and the binning system which always assumes normal distribution and thus does not take care of discrete cases. Moreover 
comparison among various entities lacks since the probabilities derived are for individual entities and are just quantile values. 
Finite investment decision making using real market risk (non-diversifiable risk) was undertaken in this study. Non-diversifiable 
risk (systematic risk) estimates of a portfolio of stocks determined by a real risk weighted pricing model are used as initial data. 
The variance of non-diversifiable risk is estimated as a random variable referred to as random error (white noise). The estimator 
is used to calculate estimates of white noise (wn). A curve estimation of the wn is made using Kernel Density Estimation (KDE). 
KDE is a non-parametric way to estimate the probability density function of a random variable. KDE is a fundamental data 
smoothing problem where inferences about the population are made, based on a finite data sample. This is used to derive 
probability estimates of the non-diversifiable risks of the various stocks. This enables determination of total risk with given 
probabilities of its occurrence thus facilitating decision making under risky and uncertain situations as well as accentuating 
comparison among the portfolio of stocks.

© 2020 The Authors. Published by Atlantis Press SARL.  
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

*Corresponding author. Email: mmnk55378@gmail.com

1. INTRODUCTION

In the past few years there has been evidence of collapse of well- 
established business entities. This has been attributed to lack of 
accurate methods of preventing or measuring risk and uncertainty 
as opposed to lack of the same methods. Many companies on Wall 
Street in 2008 went under despite having extensive measures of 
mitigating risk such as futures and forward. An investigation into 
some of these methods reveals the lack of a well-estimated market 
risk measure in the models. It is an obvious fact now that it was the 
external reactions that brought down the companies on Wall Street. 
Once the markets got a hint of the internal financial and invest-
ment affairs of the companies this spread so rapidly and in a matter 
of hours these companies had collapsed. A good example is the 
Lehman Brothers Holdings limited, Merrill Lynch and companies, 
and American Investment group as explained by Lucchetti et al. [1]. 
These indicate that market environments are so critical in the exis-
tence of business entities such that variables affecting the business 
entities from the market environments should be estimated with a 

lot of precision. This paper determines total risk which has both the 
systematic and non-systematic components. It should be noted that 
the non-systematic risk is internal in nature, and in most cases well 
known and relatively less difficult to estimate while systematic is 
external and in most cases is embodied in market risk.

This paper will determine the risk factor of systematic risk a phe-
nomenon lacking in many risk models. Since we have seen from 
most examples that market risk is the precursor of most companies 
down falls, it is hoped that investors and companies will be able to 
easily estimate riskiness of the risk measures thus enabling them 
make informed decisions.

Jorion [2] determines the Value at Risk (VAR) measure as the 
forecasted volatility, St multiplied by standard normal deviate, a 
for the selected confidence level (e.g. a = 2.33 for a one-tailed 
confidence level of 99%). The portfolio variance then becomes 
S w wt t t t

2 = ¢S  where Σt is the forecasted covariance matrix for the 
market risk factors as of the close day t. Hence we have, VARt = 
a St. Although this research takes care of all the other shortcom-
ings of previous researches, the portfolio variance is determined 
as a covariance which goes against the definition of market risk as 

https://doi.org/10.2991/jracr.k.200421.003
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mailto:mmnk55378%40gmail.com?subject=


 E.A. Shileche et al. / Journal of Risk Analysis and Crisis Response 10(1) 6–11  7

that which cannot be diversified. This has been clearly addressed 
in this paper by using non-diversifiable risk which is determined 
without covariance.

Dennis et al. [3] in estimating density dependence process noise 
and observation error offers a statistical approach for jointly esti-
mating density dependence, process error and observation error. 
Although this model is relatively easy for ecologists to use and is 
applicable in many population systems, this process noise has a 
normal distribution with mean m and variance s 2 (Et ~ N(0, s 

2)). 
This paper looks at a case of no assumption of normality for the 
noise process. White noise is determined as a random variable on 
the precincts of Sklar [4] where he says no common probability 
space can be found for a given set of random variables, but such 
common probability spaces exists for arbitrary proper subsets of 
the given set. In this study the subsets were the portfolios of dif-
ferent companies used giving a common probability space that is 
estimated. The results of Wu [5] show that for finite parameters 
the consistency of the least squares estimator is equivalent to the 
existence of a consistent estimator thus the estimator of white noise 
derived in this paper is an unbiased estimator.

2. RISK OF NON-DIVERSIFIABLE RISK

2.1.  Determination of White Noise of 
Non-diversifiable Risk

White noise refers to a purely random process whose random vari-
ables are a sequence of mutually independent, identically distrib-
uted random variables. Thus it describes an event and is a function 
with a domain that makes some real number correspond to each 
outcome of the experiment. In this paper white noise is taken as the 
random error of non-diversifiable risk NGwi of an investment i. To 
add credence to this study it is imperative to show that white noise 
is a random variable. Proposition 2.1 below seeks to do so.

Proposition 2.1: Let Vi be the white noise of the non-diversifiable 
risk NGwi, then Vi(.) is a random variable.

Proof: Given NGwi and 

 i. The domain Ω V Vi n j
j i

1£ £ -¥£ £¥
¹

{ }
ì
í
î

ü
ý
þ

,  

 ii. The counter domain r is such that 0 ≤ r ≤ 1. 

 iii. The range of returns i is −∞ ≤ i ≤ ∞.

then Vi(.) is an event. That is Vi(.) is such that the subset wr = {s: 
Vi(s) ≤ r}, where s is a subset of the domain. This is true since 0 ≤ 
Vi(s) ≤ 1. If wr belongs to W for every real number r, where W is the 
set of all outcomes of event Vi(.).

Then the probability of Vi(.), P[Vi(.)] is a set function having 
domain Vi(.) and counter domain the interval [0,1]. Therefore Vi(.) 
has a probability space (Ω, W, P[Vi(.)]). Also W consists of four  

subsets; f, ,V Vi n j
j i

1£ £ -¥£ £¥
¹

{ }
ì
í
ï

îï

ü
ý
ï

þï
and Ω.

Such that if: 

 i. r < 0, then s:Vi(s) ≤ r = ϕ. 

 ii. 0 ≤ r < 1, then s:Vi(s) ≤ r, where V j
i j

-¥£ £¥
¹

. 

 iii. r ≥ 1, then {s:Vi(s) ≤ r} = Ω = V Vi n j
i j

1£ £ -¥£ £¥
¹

, .

Since Vi(.) has a probability space, and W consists of the four sub-
sets above. Then for each r the set {s:Vi(s) ≤ r} belongs to W, thus 
Vi(.) is a random variable. Since Vi(.) is a random variable and it is 
independent with unique parameters. Therefore the parameters of 
white noise for example its mean and variance as well as its unique 
probability distribution can be determined. The probability esti-
mates of non-diversifiable risk for investment decisions are then 
estimated as shown in the following subsections.

2.1.1. Determination of random error

The non-diversifiable variance estimator

 N w s s
Gwi i i

i

n

e
i

n

i
2

2 2

1

2

1

= +
= =
å å   (1)

derived from the non-diversifiable risk estimated in Anyika et al. 
[6] indicates the presence of random error in the risk estimator. 
This error is taken to be white noise (wn) thus it can be said to be a 
random variable V1,V2,V3, ...,V∞ which is mutually independent and 
identically distributed. This is estimated from a sample of data by 
first varying the variance of individual return values of ri resulting in

 wn T s si
r gw

i

n

i

 =
=
å 2 2

1

  (2)

where T
z
z

=
-

-( )
2
1

2
, z being the total number of returns and (2) is 

the predicted random error.

From (2) the actual value of wni  is given by

 wn w s C Li i ri
i

n

= +
=
å 2 2

1

  (3)

where C and L are values representing the location (mean) and 
scale (variance) parameters. These parameters are determined such 
that the bias and variance of the actual and predicted values of wn 
are minimized as follows;

Let the variance between actual and sample white noise be

 var ( , )wn wn
z

wn
z

wn wn
z

wni i i i i i  = -
-

-
æ
è
ç

ö
ø
÷ + -

2
1

2
1

2
1

2
2

2   (4)

The values of C and L which will minimize variance are given by 
the partial derivatives of C and L, fC and fL respectively. After several 
iterations;

 
f wn

z
wn LC i i= - -

=2
2

1


 

 
f

z
wn wn LL i i= -

- =
2

1


 



8 E.A. Shileche et al. / Journal of Risk Analysis and Crisis Response 10(1) 6–11

Thus the value of

 wn
z

wni i= -
4

3 1( )


  (5)

Proposition 2.2: wni  is an unbiased estimator of wni.

Proof: From Equation (2)

 

wn T s s

E wn
z
z

E V V

z

i r gw
i

z

i i r
i

z

i
�

�

2 2

1

2
2 2

1

2
1

2

( )
( )

(zz z
E V E V

z
z

wn
z

V

i

z

i r

i i

1
1
1

2
1

1

2
1

2 2

2
2

)

( )
( )� 22

11

2
22

1

VV

z
z

zu zwn

i j

z

i

z

i

z

i
( )

� 11 1

2
1

2 2

2
2 2

z
zu zwn z z

z
z

zu zwn u wn z

i

i i

( ) ( )

( )

�

� � (( )

( )
( )

( )

z

z
z

zu zwn u wn z z

z
z

i i

1

2
1

1

2
1

2

2
2 2 2

2

� �

zzu zwn u wn z

z z
z

wn

z
z

wn

i i

i

2 2 2 2

2
2 1
1

2
1

� �

�

�

( )
( )

( )
ii

  (6)

where wni and m are the actual variance and mean of non-diversifiable 
risk respectively.

Dividing Equation (6) by z results in

 

=
-

æ
è
ç

ö
ø
÷

-

=
-

æ
è
ç

ö
ø
÷

-

z
z z

wn

z
z z

z
wn

z

i

i

2

1

1
2

1
1





 

lim as z → ∞

 

=
-
¥

æ
è
ç

ö
ø
÷

-
¥

æ
è
ç

ö
ø
÷

=
-
-

=

1
2

1
1

1 0
1 0

2

wn

wn

wn

i

i

i







( )
( )

 

Thus wni  is an unbiased estimator of wni.

From the results of Wu [7], Equation (6) and proposition 2.2, wni  
is a consistent estimator of wni.

3.  DERIVATION OF A PROBABILITY DENSITY 
FUNCTION FOR RANDOM ERROR OF 
NON-DIVERSIFIABLE RISK

3.1.  Kernel Density Estimation of White 
Noise of Non-diversifiable Risk

Let V1,V2, ...,Vn denote a sample of size n from the random variable 
Vi(.) with density f. The kernel density estimates of f at the point v 
is given by

 f v
nh

k
v V

hn
i

i

n
 ( )

( )
=

-æ

è
ç

ö

ø
÷

=
å

1
1

  (7)

where the kernel k satisfies k v dv( ) =
-¥

¥

ò 1  and the smoothing 

parameter h is known as the bandwidth. Vi  is the mean of v.

Proposition 3.1: vi
i

n

=
å

1
 is a minimum sufficient statistics of Vi

_

.

Proof:

Given the function f
hn

v V
hcn

i

i

nL

= -
-

é

ë

ê
ê

ù

û

ú
ú

-

=
å

1
1

1

( ) , the likelihood (L) of v is

 

L v V
nh

v V

hc

L v V
v

In
nh

v

i
i

i

n

i
i

i

i( : )

( : )

= -
-( )é

ë

ê
ê

ù

û

ú
ú

=
¶
¶

-
-

å
å
=

1
1

1
1

1

VV

hc

i
i

n

i

n
=

=

å
å

é

ë

ê
ê
ê
ê

ù

û

ú
ú
ú
ú

1

1

  (8)

since, v Vi iå = , then V Vi i- = 0
Let v v v vn. ( , ,..., )

. . .= 1 2  be a point in v v v v vn: ( , ,..., )= 1 2

Then 
L v V

L v V
vi

In
hn

v i
In

hn

i

i

i

n

i

n

( : )
( . : . )

( )

.
( )

=

¶
¶
¶
¶

=

=

å

å

1
1

1
1

1

1

and therefore, 
L v V

L v V
i

i

( : )
( . : . )

= 1

Meaning that it is independent of Vi  and thus, vi
i

n

=
å

1
 is a minimum 

sufficient statistics of Vi
_

. Lehmann and Scheffé [8] remarks that if 
the sample space is discrete or a finite dimensional Euclidean space 
then a minimal sufficient statistic will always exists. Since a mini-
mum sufficient statistic exists then the sample space is discrete and 
the probability density function exists.



 E.A. Shileche et al. / Journal of Risk Analysis and Crisis Response 10(1) 6–11  9

It is generally known that the value of the bandwidth is of critical 
importance while the shape of the kernel function has little prac-
tical impact. Thus we estimate bandwidth and use a given kernel 
function to get the density estimation of the white noise of non- 
diversifiable risk.

3.2.  Bandwidth Selection for Kernel  
Density Estimation of the wn of  
Non-diversifiable Risk

Assuming that the underlying density is sufficiently smooth and 
that the kernel has fourth moment using the Taylor series

 Bias f v
h

k f v hh
� ( ) ( ) ( ) ( ){ } = +

2

2
2

2
m ²   (9)

 Var f v
nh

R k f v
nh

( ) ( ) ( ){ } = +1 0 1   (10)

where R k k v dv( ) ( )= ò 2  [9]. Adding the leading variance and 
squared bias terms produces the Asymptotic Mean Squared bias 
squared Error (AMSE)

 AMSE ={ } = + éë ùûf v nh R k f v
h

k f vn
� ( ) ( ) ( ) ( ) ( )

1
4

4

2
2 2m ²   (11)

The overall measure of the discrepancy between f
L

and f is the 
Mean Integrated Squared Error (MISE) which is given by

 
MISE

Bias Var

f E f y f y dy

f v dv f v

n h

h h

 ( ) = -( ){ }
= +

ò

ò ò

( ) ( )

( ( )) ( ( )
L L

2 ))dv
  (12)

Under an integrability assumption on f, integrating the expression 
for AMSE gives the expression for the Asymptotic MISE (AMISE), 
i.e.,

 AMISE f nh
R k

h
k R fh

Læ

è
ç
ç

ö

ø
÷
÷
= +

1
4

4

2
2( ) ( ) ( )m ²   (13)

where

 R f f v dv( ) [ ( ) ] .² ²= ò 2  
The value of the bandwidth that minimizes the AMISE is given by

 h
R k
k R f

nAMISE =
é

ë
ê
ê

ù

û
ú
ú

-( )
( ) ( )m2

2

1
5 1

5
²

  (14)

Using the rule of thumb method a global bandwidth h is based on 
replacing R(f ″) the unknown part of hAMISE, by its value for a para-
metric family expressed as a multiple of a scale parameter, which is 
then estimated from the data. The method dates back to Deheuvels 
[10] and Scott [11]. It has been popularized for kernel estimates by 
Silverman [12].

The plug-in method is used to estimate hAMISE in this study. Here 
the unknown quantity R(f ″) in the expression for hAMISE is replaced 

by an estimate. The “solve - the - equation” plug-in approach devel-
oped by Sheather and Jones [13] is based on deriving, the pilot 
bandwidth for the estimate R(f ″), as a function of h, namely

 g h C k
R f
R f

h( ) ( )
( )
( )

=
é

ë
ê

ù

û
ú

²
¢²

1
7 5

7   (15)

The unknown functions of f are estimated using kernel density esti-
mates with bandwidth based on normal rules of thumb resulting in

 h
R k

k R f g
ns j

h

=
( )
é

ë

ê
ê
ê

ù

û

ú
ú
ú

-( )

( ) ( )( )m2
2

1
5 1

5
�²

  (16)

where hSJ is known as the Sheather–Jones plug-in bandwidth. 
Under smoothness assumption on the underlying density, n5/14 
(hSJ/hAMISE−1) has an asymptotic N(0, sSJ

2) distribution. Thus, the 
Sheather–Jones plug-in bandwidth has a relative convergence rate 
of order n−5/14, which is much higher than that of biased cross- 
validation.

The triangle kernel is used for smoothing

This is given by

 K t
t c t c

t ctri
( )

/ , /
, /

=
- £

>
ì
í
î

1 1
0 1

  (17)

where c is the constant used to scale the resulting kernel so that the 
upper and lower quartiles occur at ±0.25. Substituting the kernel in 
Equation (16) and the unknowns, h and n into the density function 
(7) gives the function

 f hn
v V

hcn
i

i

nL

= -
-

é

ë

ê
ê

ù

û

ú
ú

-

=
å

1
1

1

( )
  (18)

This function is used to generate probabilities of non-diversifiable 
risks of given portfolio thus ensuring that actual systematic risk is 
determined.

4. RESULTS

4.1.  Calculating Actual Non-diversifiable 
Risk

The sample white noise is estimated by varying the variance of 
individual return values ri as given by Equation (2). The non- 
diversifiable risk estimates in Table 1 are substituted into Equation (3)  
to give Equation (5).

Sample wn estimates are then substituted into Equation (5) to 
determine the actual white noise.

4.2. Density Estimates of Actual White Noise

R statistical software is used to calculate Sheather–Jones (sj) bandwidth 
and hence the density estimates of actual wn as plotted in Figure 1.



10 E.A. Shileche et al. / Journal of Risk Analysis and Crisis Response 10(1) 6–11

A summary of statistics resulting from sj density estimation in 
Table 2 enables us apportion densities of the different quartile 
ranges.

F-values are calculated as follows:

 
F

v vi
v

=
-
s  

where vi = white noise of portfolio i, v  = mean of white noise of 
all the portfolios, and sv = variance of all the portfolios. The prob-
ability density estimate of a portfolio i is determined by comparing 
the F-values with the apportioned densities of the different quartile 

Figure 1 | A plot of the density estimates of actual white noise. Call: Density  
(x = x, bw = 0.7559, x lim = c(−2, 2)). Data: x (20 obs.); Bandwidth “bw” = 
0.7559.

Table 1 | The non-diversifiable risks of 20 stocks 
from NYSE

Company rn
YH 29.33
TIF 63.5
TM 538.1
HM 28.23
PONARD 27.97
VIC 0.561
DAWT 25.58
BP 105.2
SUNTB 108.3
PNC 7.876
AIG 7164
FORD 1898
AMR 25.17
BPH 1.752
CTL 5.547
PFE 46.58
RTI 9.054
GSK 35.81

Table 2 | Final results of white noise and kernel density estimation of 
portfolios of stocks

Company wn F Probabilities Actual sn
YH 0.000729 0.827722 0.63566 18.64391
TIF 0.00027 −0.27869 0.491 31.1785
TM 0.00011 −0.66196 0.4414 237.5173
HM 0.000128 −0.62098 0.4467 12.61034
PONARD 0.001551 2.809139 0.979 27.38263
VIC 0.00046 0.179302 0.5511 0.309222
DAWT 0.001456 2.580143 0.9388 24.0145
BP 0.000113 −0.65714 0.442 46.4984
SUNTB 0.00011 −0.66437 0.441 47.7603
PNC 0.000142 −0.58723 0.4511 3.552864
AIG 0.000657 0.654167 0.613 4390.919
FORD 0.000308 −0.18709 0.5033 954.7601
AMR 0.000491 0.254027 0.521 13.11357
BPH 0.000227 −0.38234 0.4778 0.837106
CTL 0.0000884 −0.71655 0.4342 2.408507
PFE 0.0000988 −0.69146 0.4375 20.37875
RTI 0.000238 −0.35582 0.4813 4.35769
GSK 0.0000872 −0.71933 0.4339 15.53796
BCE 0.00022 −0.39921 0.4756 99.44796
STGI 0.000227 −0.38234 0.4778 7.601798

ranges and the maximum value. An F-score of positive 0.827722 
has its density calculated as follows:

0.527674 + (1 – 0.827722)0.130402 = 0.63556

where: 0.527674 is the maximum value, 0.130402 is the value 
apportioned to the first quartile and 1 − 0.827722 represents the 
fraction occupied in the first quartile.

Final results of the survey are tabulated in Table 3.

4.3. Wilcoxon Signed Rank Test

Wilcoxon signed rank test of hypothesis is used to compare the VaR 
method of determining risk and Kernel white noise method.

Here we test the hypothesis that risks obtained by Kernel white 
noise are a reflection of actual risks than those obtained by VaR.

Ho: The population difference are centered at 0.

Ha: The population differences are centered at a value <0.

Based on a significance level of a = 0.01, the proper test is to reject 

Ho if Z < −Za. Determining Z and Za
1 0
0 1
æ

è
çç

ö

ø
÷÷

Table 3 | A summary of the results of a kernel density 
estimation of a portfolio white noise

X Y

Min.:−2.2676128 Min.: 0.005892
1st Qu.: 1.133396 1st Qu.: 0.041992
Median: 0.00081 Median: 0.170879
Mean: 0.0008191 Mean: 0.219614
3rd Qu.: 1.135035 3rd Qu.: 0.397272
Max.: 2.269251 Max.: 0.527674



 E.A. Shileche et al. / Journal of Risk Analysis and Crisis Response 10(1) 6–11  11

 Z
t t

t
= =

-åtest statistic m
s  

where mt
n n

=
+( )1

4
, st

n n n
=

+ +( )( )1 2 1
24

, and 

t
n

a n d= + -( )å 2 2 1( ) .

Using normal tables −Za = −1.645, using the difference in risks and 
their ranks in Table 4 (Z = −3.88).

Since, Z < Za we reject the null hypothesis, so there is sufficient evi-
dence to conclude that the kernel white noise risks are a reflection 
of actual risks as compared with those obtained by VaR.

5. CONCLUSION

An estimate of random error is made with the least bias and vari-
ance. Probability estimates of the asset parameters are made thus 
boosting the level of surety. These are made in comparison i.e. 
looking at given portfolios one is able to make a decision among 
a variety of them. Methods like VaR use generated variances  
to give probability estimates using extreme values. This lacks the  
comparability factor and assumes the central limit theory lead-
ing to application of normality conditions. They also use covari-
ance parameters as market risks thus going against its definition. 
A case study of New York Stock Exchange (NYSE) Dow index in 
2008 indicates that the portfolios with the highest actual non- 
diversifiable risks were AIG with 4390.919%, FORD; 954.7661%, 
and TM; 237.5173%. These are corporates which experienced 
financial difficulties during the credit crunch in the USA in 2008. 
AIG and TM had to be given some financial rescue packages to stay 

afloat until the financial crump was reversed. From the analysis of 
the results from the NYSE case study of the Dow index in 2008 it 
is clear that there is a relationship between the determined actual 
non-diversifiable and the actual market risk on the ground over 
the past 2 years. These research findings can aid investors make 
solid investment decisions as well as the different corporate cut ion 
themselves against any financial stress currently and in future.

CONFLICTS OF INTEREST

The authors declare they have no conflicts of interest.

AUTHORS’ CONTRIBUTION

The original idea and subsequent analysis of the research was 
undertaken by EAS with the guidance of PW and TA.

ACKNOWLEDGMENTS

I wish to acknowledge the University of Nairobi staff in the school 
of Mathematics as well as my family members for their encourage-
ment and support during the period of undertaking this research.

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Table 4 | A table of Wilcoxon signed rank test for large samples paired

Company Actual rn(i)
VaR at ` = 

0.01 (ii)
(i) – (ii) Rank

Yahoo 18.64 0.4699 18.17 10
Tiffany 31.18 0.2693 30.91 14
Toyota 237.52 0.1790 237.34 18
HM 12.61 0.1688 12.44 7
Ponardph 27.38 0.8349 26.55 13
Vical Inc 0.31 0.4629 −0.15 1
Data Watch 24.01 0.8398 23.17 12
Bp 46.50 0.1391 46.36 15
Suntrust 47.76 0.2145 47.55 16
Pnc 3.55 0.1914 3.36 4
AIG 4390.92 0.5951 4390.32 20
Ford 954.76 0.3662 954.39 19
Amr 13.11 0.4806 12.63 8
Bph 0.84 0.2036 0.63 2
Ctl 2.41 0.1978 2.211 3
Pfe 20.38 0.1607 20.22 11
Rti 4.358 0.3184 4.039 5
Gsk 15.54 0.1297 15.41 9
Bce 99.45 0.1775 99.27 17
Sbg 7.602 0.3714 7.230 6

https://doi.org/10.1890/0012-9615(2006)76[323:EDDPNA]2.0.CO;2
https://doi.org/10.1890/0012-9615(2006)76[323:EDDPNA]2.0.CO;2
https://doi.org/10.1890/0012-9615(2006)76[323:EDDPNA]2.0.CO;2
https://doi.org/10.1890/0012-9615%282006%2976%5B323:EDDPNA%5D2.0.CO%3B2
https://doi.org/10.1890/0012-9615%282006%2976%5B323:EDDPNA%5D2.0.CO%3B2
https://www.jstor.org/stable/25048038
https://www.jstor.org/stable/25048038