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A project assignment on Bayesian Regression Analysis submitted to the Department of Mathematics, faculty of science, University of Lagos.
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BAYESIAN LINEAR REGRESSION MODEL:
ANALYSIS OF THE FINANCIAL ACTIVITIES
OF THE BANKING SECTOR OF THE NIGERIA
STOCK EXCHANGE.
AN ASSIGNMENT SUBMITTED
BY
ABDULFATAI SHAKIRUDEEN 060806002
(Mathematics Department, University of Lagos)
SUBMITTED TO
DR M. ADAMU-IRIA
STATISTICAL PACKAGES
MAT 829
MAY 2013
ABSTRACT
Bayesian statistics is an approach to statistics which formally seeks use of prior information with
the data. Bayes Theorem provides the formal basis for making use of both sources of
information in a formal manner. The Bayesian analysis is the study of
different features of posterior density. In this study, Bayesian Regression analysis using R
software (with MCMC pack) is used to explore data extracted from the Nigeria Stock Exchange
in the Capital market (case study of two banks; Access Bank PLC and United Bank of Africa
PLC). Data collected for this project is a secondary data from Nigeria stock Exchange; All-share
Index, daily price of stock, interest rate, exchange rate and daily oil price for the period of five
years 2005-2009. For this purpose, we define the response variable as the Nigeria Stock
Exchange All-share Index (NSEAI). The covariates are the Daily price of stock, Interest rate
(Lending rate), Exchange rate and Oil price. Simulation approach of Bayesian analysis was
found to be the most useful one in this study.
BAYESIAN REGRESSION METHODS
1. Introduction
Prior Probabilities and Bayes Theorem
The task of Bayesian analysis is to build a model for the relationship between parameters () and
observables (y), and then calculate the probability distribution of parameters conditional on the
data, p(|y). In addition, the Bayesian analysis may calculate the predicted distribution of
unobserved data.
Bayesian statistics begins with a model for the joint probability distribution of and y, p(,y).
may be a single parameter or a vector of many parameters, and y may be a vector of
observations of a single variable or a matrix with multiple observations of many variables. The
function p is a probability distribution. An example of a model is the familiar one for estimating
the mean and variance of a normally distributed population, in which p(,y) is a normal
distribution with mean and variance given by the parameter vector , and y is a sample of
independent measurements. Using the definition of conditional probability (Mangel and Clark
1988, Howson and Urbach 1989), p(,y) can be decomposed into two components:
p(,y) = p() p(y| ).. .1
By convention, p() is called the prior distribution of (i.e. the distribution prior to observing the
data y) and p(y| ) is called the likelihood function (i.e. the likelihood of observing the data given
a particular parameter value ). Bayes theorem provides the posterior probability distribution
p(|y) (i.e. the distribution of obtained after observing y and combining the information in the
data with the information in the prior distribution):
p(|y) = p() p(y| ) / p(y)..2
Equation (2) provides a probability distribution of given observations of the data y.
In this equation, p(y) is the sum (or integral) of p() p(y| ) over all possible values of -Mangel
and Clark (1988) or Howson and Urbach (1989).
2. Subjectivity
Bayesian probabilities are sometimes called subjective probabilities. It is important to
understand exactly what is meant by subjective in this context. Decision analyses are often
unique. The situation in which one is making the decision may occur only once. It cannot be
replicated, so there is no possibility for measuring probabilities by repeated sampling.
Nevertheless, Bayesian analysis may be used to compute the probabilities needed to make
decisions. Because these probabilities cannot be measured by repeated sampling, they are called
subjective and they represent a degree of belief in a particular outcome-Lindley (1985),
Howson and Urbach (1989) and Pratt et al. (1995).
Also, if there is no basis in observed data for estimating the prior probability distribution, then
the analyst may simply assume a particular prior distribution. The consequences of this
assumption can be tested by sensitivity analyses that compare the response of the posterior
distribution to different assumptions about the prior distribution. Most commonly, a non-
informative prior distribution is assumed. A non-informative prior distribution assigns the same
probability to each possible value of the parameters. If the number of observations is at least
moderately large, a non-informative prior distribution will have negligible impact on the
posterior distribution. If the data y is limited, however, the choice of prior distribution may have
a substantial impact on the posterior distribution. In this case, sensitivity analysis is needed to
evaluate the consequences of different assumptions about the prior distribution.
3. Linear Regression with Non-informative Prior
In linear regression, the observations consist of a response variable in a vector y and one or more
predictor variables in a matrix X. The vector y has n elements, corresponding to n observations.
The matrix X has n rows, corresponding to the observations, and k columns corresponding to the
number of predictors. If the regression includes an intercept, one of the columns of X is a column
of ones. The parameters are the regression coefficients and the error variance of the fitted
model, 2. The model that relates observations and parameters is written:
(y | , 2, X) ~ Normal(X , 2 I)..3
In words, this model states that the distribution of y given parameters and 2 and predictors X
is a normal distribution with mean X and variance 2. The identity matrix is I. A normal
distribution is completely specified by its mean and variance.
Once the model is specified, the Bayesian analysis seeks the posterior distribution for the
parameters and a predictive distribution for the models predictions. The analysis begins with a
prior distribution. A non-informative prior distribution that is commonly used for linear
regression is p(, 2) 1/2..4
In words, this expression means that the joint probability distribution of and 2 given X is a
flat surface with a constant level proportional to 1/2.
The posterior distribution of given 2 is | 2, y ~ Normal (E, V 2) (A.5)
Expression (5) states that the probability distribution of given 2 and y is normal with mean E
and variance V 2. The parameters of this normal distribution are computed from
E = (X X)-1 X y.6
V = (X X)-1.7
The apostrophe () denotes matrix transposition. The marginal posterior distribution of 2 (i.e.
the integral over all possible values of of the joint distribution of and 2) is
2 | y ~ Inverse 2 (n - k, s2).8
Expression (8) says that the probability distribution of 2 given y follows an inverse 2
distribution. The inverse 2 distribution, presented by Gelman et al. (1995), is fully defined by
two parameters, the degrees of freedom and the scale factor. In this case there are n k degrees
of freedom (where n is the number of observations of y and k is the number of parameters to be
estimated, i.e. the number of columns of X). The scale factor s2 is computed by
s2 = (y - X E) (y - X E) / (n k)..10
Note that y - X E is the vector of residuals, or deviations of observations from predictions.
The marginal posterior distribution of given y is written
| y ~ Multivariate Student t (n k, E, s2)11
The multivariate Student t distribution (presented by Gelman et al. 1995) has three parameters,
the degrees of freedom n k, the mean E, and the scale factor s2. This distribution is derived
by integrating the posterior distribution of given 2 (5) over all possible values of 2 (8).
Regressions are often fitted in order to make predictions. The predictive distribution, yp, given a
new set of predictors Xp has mean
E(yp | y) = Xp E..12
The marginal posterior distribution of the variance of this prediction is
var(yp | 2, y) = (I + Xp V Xp) 2.13
where I is the identity matrix. This variance formula has two components, I 2 for sampling
variance of the new observations and Xp V Xp 2 for uncertainty about . The marginal
posterior distribution of yp given y is
p(yp | y ) ~ Multivariate Student-t [n k, Xp E, (I + Xp V Xp) s2]..14
Bayesian analysis using the non-informative prior of equation (4). The classical estimates of
and 2 are E and s2, respectively. The classical standard error estimate for is V s2. The
classical prediction for new data is yp = Xp E with variance (I + Xp V Xp) s2.
4. Linear Regression with Informative Prior
Bayesian analysis can be used to combine two different sources of information in a single model
to estimate parameters or make predictions. The results can then be combined with a third
source of information to improve the parameter estimates or predictions. This process can be
repeated over and over again to combine information from any number of sources. Combining
multiple sources of information is one of the most important uses of Bayesian statistics (Hilborn
and Mangel 1997). In linear regression with an informative prior distribution, there are two
statistically-independent data sets that provide information about the model to be analyzed. We
assign one data set to be the prior, and use the other data set for the likelihood. Usually it is
convenient to assume that the prior distribution of the k regression parameters is multivariate
Student-t. This distribution has three parameters, a vector of mean regression parameters, a
matrix with variances along the main diagonal and covariance elsewhere, and degrees of
freedom. In this case, the mean vector contains the k prior estimates of the mean regression
parameters (B0) and the k x k parameter covariance matrix S0, model variance s02 and degrees
of freedom n0. For the second data set, we have a n1 x 1 response vector y1 and a n1 x k matrix
of predictors X1.
The posterior can be computed by treating the prior as additional data points, and then weighting
their contribution to the posterior (Gelman et al. 1995).
5. DATA ANALYSIS
ACCESS BANK PLC.
5.1 Simple Linear Regression analysis using R-package.
R version 2.15.2 (2012-10-26) -- "Trick or Treat"
Copyright (C) 2012 The R Foundation for Statistical Computing
ISBN 3-900051-07-0
Platform: x86_64-w64-mingw32/x64 (64-bit)
> data1 data1
> attach(data1)
> lm(NSEASI~PRICE+INTEREST+EXCHANGE+OIL)
Call:
lm(formula = NSEASI ~ PRICE + INTEREST + EXCHANGE + OIL)
Coefficients:
(Intercept) PRICE INTEREST EXCHANGE OIL
3.3104 0.4697 2.3299 -5.3469 0.3923
Table 1. The estimated coefficients for Access Bank.
> summary(lm(NSEASI~PRICE+INTEREST+EXCHANGE+OIL))
Call:
lm(formula = NSEASI ~ PRICE + INTEREST + EXCHANGE + OIL)
Residuals:
Min 1Q Median 3Q Max
-0.41806 -0.16074 -0.00902 0.15483 0.46518
Table 2.
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 3.31044 0.52732 6.278 5.71e-08 ***
PRICE 0.46975 0.09486 4.952 7.35e-06 ***
INTEREST 2.32985 0.52988 4.397 5.07e-05 ***
EXCHANGE -5.34694 0.77386 -6.909 5.31e-09 ***
OIL 0.39228 0.09805 4.001 0.00019 ***
---Table 3. The estimated coefficients with standard error for Access Bank.
Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1
Residual standard error: 0.2258 on 55 degrees of freedom
Multiple R-squared: 0.827, Adjusted R-squared: 0.8144
F-statistic: 65.71 on 4 and 55 DF, p-value: < 2.2e-16
> anova(lm(NSEASI~PRICE+INTEREST+EXCHANGE+OIL))
Analysis of Variance Table
Response: NSEASI
Df Sum Sq Mean Sq F value Pr(>F)
PRICE 1 7.2169 7.2169 141.5014 < 2.2e-16 ***
INTEREST 1 0.3667 0.3667 7.1897 0.0096600 **
EXCHANGE 1 5.0055 5.0055 98.1435 7.798e-14 ***
OIL 1 0.8164 0.8164 16.0077 0.0001902 ***
Residuals 55 2.8051 0.0510
Table 4. Anova table for Access bank.
---
Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1
> plot(NSEASI~PRICE+INTEREST+EXCHANGE+OIL)
1.0 1.5 2.0 2.5
1.0
1.5
2.0
2.5
OIL
NS
EA
SI
5.3 Bayesian Linear Regression Analysis using R-package.
> library(MCMCpack)
Loading required package: coda
Loading required package: lattice
Loading required package: MASS
##
## Markov Chain Monte Carlo Package (MCMCpack)
## Copyright (C) 2003-2013 Andrew D. Martin, Kevin M. Quinn, and Jong Hee Park
##
## Support provided by the U.S. National Science Foundation
## (Grants SES-0350646 and SES-0350613)
##
Warning messages:
1: package MCMCpack was built under R version 2.15.3 2: package coda was built under R version 2.15.3 > M6 summary(M6)
Iterations = 1001:11000
Thinning interval = 1
Number of chains = 1
Sample size per chain = 10000
1. Empirical mean and standard deviation for each variable, plus standard error of the mean:
Mean SD Naive SE Time-series SE
(Intercept) 3.31131 0.53659 0.0053659 0.0053659
PRICE 0.47069 0.09694 0.0009694 0.0009694
INTEREST 2.33433 0.53859 0.0053859 0.0056140
EXCHANGE -5.35102 0.78375 0.0078375 0.0078375
OIL 0.39044 0.09988 0.0009988 0.0009988
sigma2 0.05303 0.01061 0.0001061 0.0001168
Table 5. The estimated coefficients for Access Bank using Bayesian Regression Model.
2. Quantiles for each variable:
2.5% 25% 50% 75% 97.5%
(Intercept) 2.25156 2.95241 3.31539 3.66211 4.36502
PRICE 0.28003 0.40412 0.47118 0.53532 0.66045
INTEREST 1.28593 1.97042 2.34178 2.70126 3.38008
EXCHANGE -6.88844 -5.87923 -5.35285 -4.83245 -3.82332
OIL 0.19330 0.32454 0.39170 0.45712 0.58544
sigma2 0.03612 0.04546 0.05168 0.05898 0.07756
>plot(M6,trace=FALSE)
6.0 UBA PLC
6.1 Simple Linear Regression using R-package.
> data2 data2
> attach(data2)
> lm(NSEASI~PRICE+INTEREST+EXCHANGE+OIL)
Call:
lm(formula = NSEASI ~ PRICE + INTEREST + EXCHANGE + OIL)
Coefficients
(Intercept) PRICE INTEREST EXCHANGE OIL
1.6753 0.4002 1.3577 -2.7630 0.2790
Table 8. The estimated coefficients for UBA.
> summary(lm(NSEASI~PRICE+INTEREST+EXCHANGE+OIL))
Call:
lm(formula = NSEASI ~ PRICE + INTEREST + EXCHANGE + OIL)
Residuals:
Min 1Q Median 3Q Max
-0.21473 -0.07511 -0.01378 0.05101 0.36808
Table 9.
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.67530 0.30128 5.561 8.15e-07 ***
PRICE 0.40024 0.02590 15.455 < 2e-16 ***
INTEREST 1.35771 0.27000 5.028 5.60e-06 ***
EXCHANGE -2.76303 0.43410 -6.365 4.12e-08 ***
OIL 0.27902 0.05058 5.516 9.59e-07 ***
---Table 10. The estimated coefficients with standard error for UBA.
Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1
Residual standard error: 0.1164 on 55 degrees of freedom
Multiple R-squared: 0.9532, Adjusted R-squared: 0.9497
F-statistic: 279.8 on 4 and 55 DF, p-value: < 2.2e-16
> anova(lm(NSEASI~PRICE+INTEREST+EXCHANGE+OIL))
Analysis of Variance Table
Response: NSEASI
Df Sum Sq Mean Sq F value Pr(>F)
PRICE 1 13.8116 13.8116 1018.7482 < 2.2e-16 ***
INTEREST 1 0.0006 0.0006 0.0418 0.8387
EXCHANGE 1 0.9472 0.9472 69.8640 2.280e-11 ***
OIL 1 0.4126 0.4126 30.4308 9.587e-07 ***
Residuals 55 0.7457 0.0136
Table 11. Anova table for UBA.
---
Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1
> plot(NSEASI~PRICE+INTEREST+EXCHANGE+OIL)
1.0 1.5 2.0 2.5
1.0
1.5
2.0
OIL
NS
EA
SI
6.2 Bayesian Linear Regression Analysis using R-package.
> library(MCMCpack)
Loading required package: coda
Loading required package: lattice
Loading required package: MASS
##
## Markov Chain Monte Carlo Package (MCMCpack)
## Copyright (C) 2003-2013 Andrew D. Martin, Kevin M. Quinn, and Jong Hee Park
##
## Support provided by the U.S. National Science Foundation
## (Grants SES-0350646 and SES-0350613)
##
Warning messages:
1: package MCMCpack was built under R version 2.15.3 2: package coda was built under R version 2.15.3 > M6 summary(M6)
Iterations = 1001:11000
Thinning interval = 1
Number of chains = 1
Sample size per chain = 10000
1. Empirical mean and standard deviation for each variable, plus standard error of the mean:
Mean SD Naive SE Time-series SE
(Intercept) 1.67580 0.306723 3.06e-03 3.067e-03
PRICE 0.40046 0.026465 2.647e-04 2.647e-04
INTEREST 1.35941 0.275462 2.755e-03 2.755e-03
EXCHANGE -2.76445 0.440716 4.407e-03 4.407e-03
OIL 0.27802 0.051560 5.156e-04 5.156e-04
Sigma2 0.01411 0.002822 2.822e-05 3.109e-05
Table 12. The estimated coefficients with standard error for UBA using Bayesian Regression Model.
2. Quantiles for each variable:
2.5% 25% 50% 75% 97.5%
(Intercept) 1.070025 1.4706 1.67813 1.87632 2.27812
PRICE 0.348276 0.3827 0.40066 0.41820 0.45158
INTEREST 0.824582 1.1724 1.36324 1.54945 1.89541
EXCHANGE -3.623964 -3.0635 -2.76876 -2.46834 -1.90566
OIL 0.175582 0.2432 0.27886 0.31263 0.37938
sigma2 0.009611 0.0121 0.01375 0.01569 0.02064
>plot(M6,trace=FALSE)
0.5 1.0 1.5 2.0 2.5 3.0
0.0
0.6
1.2
Density of (Intercept)
N = 10000 Bandw idth = 0.05086
0.30 0.35 0.40 0.45 0.50
05
10
15
Density of PRICE
N = 10000 Bandw idth = 0.004446
0.0 0.5 1.0 1.5 2.0 2.5
0.0
0.6
1.2
Density of INTEREST
N = 10000 Bandw idth = 0.04628
-4 -3 -2 -1
0.0
0.4
0.8
Density of EXCHANGE
N = 10000 Bandw idth = 0.07404
0.0 0.1 0.2 0.3 0.4 0.5
02
46
Density of OIL
N = 10000 Bandw idth = 0.008662
0.005 0.010 0.015 0.020 0.025 0.030
050
150
Density of sigma2
N = 10000 Bandw idth = 0.000451
7.0 Summary of finds, Conclusion and Recommendation.
7.1 Summary of finds and Conclusion
In this study, analysis of data extracted from Nigeria Stock Exchange was subjected to
investigate the relationship between the respond variable All-share Index and daily price of
stock, interest rate, exchange rate and oil price of the banking sector of the Nigeria Stock
Exchange; case study of two banks Access Bank Plc, and United Bank For Africa (UBA) Plc.
The result of the simple linear regression is very similar to the Bayesian Regression as shown in
the table below. Although more computationally intensive, the Bayesian Regression is similarly
easy to implement and automatically provides interval estimates for all parameters, including the
standard error. By the results of the analysis for the two banks the estimated coefficients are
statistically significant and the price of stock, Lending rate and oil price have positive effect on
the response variable All-share Index that is, as all the performance metrics increase All-share
Index increases except the Exchange rate; as the exchange rate goes down All-share Index
increases and vice versa.
SLR Std. Error BLR Std. Error
Intercept 3.31044 0.52732 3.31131 0.53659
Price of stock 0.46975 0.09486 0.47069 0.09694
Interest Rate 2.32985 0.52988 2.33433 0.53859
Exchange Rate -5.34694 0.77386 -5.35102 0.78375
Oil price 0.39228 0.09805 0.39044 0.09988
Table 14. The estimated coefficients with standard error (Std. Error) for Access Bank Plc. Using Simple Linear
Regression (SLR) and Bayesian Linear Regression (BLR).
SLR Std. Error BLR Std. Error
Intercept 1.67530 0.30128 1.67580 0.306723
Price of stock 0.40024 0.02590 0.40046 0.026465
Interest Rate 1.35771 0.27000 1.35941 0.275462
Exchange Rate -2.76303 0.43410 -2.76445 0.440716
Oil price 0.27902 0.05058 0.27802 0.051560
Table 15. The estimated coefficients with standard error (Std. Error) for UBA Plc. Using Simple Linear Regression
(SLR) and Bayesian Linear Regression (BLR).
Note: that All-share Index is an arbitrary number used in Stock Exchange for evaluation
purpose.
7.2 Recommendation.
Based on the information gathered and findings from the analysis carried out in this study, the
following recommendations are suggested.
1) There is need for the federal government to commence buying of shares of the banks on
the Nigeria Stock Exchange (NSE). When these shares are purchased, they will serve
twin purpose- being investment for the government which it can hold, earn returns and
later resell and increase the demand segment of the capital market following the upward
movement of both the market capitalization and the All-share Index.
2) There is need for advertisement for the banks involve being their public offer (IPO) so
that more people may participate in the purchasing of their shares which will increase the
percentage of their All-share Index.
3) The banks should not be over expose to the capital market because this significantly
increases the quantum of banks non- performing loans which invariably led to loss of
depositors fund with the banks.
4) The government fiscal policy on exchange rate, bank lending rate (interest rate) and oil
price should be look into because of the significant of its values in the profit of the banks.
REFERENCE
[1] Box, G. E. P., Hunter W. G., and Hunter J. S. (1978): Statistics for Experimenters. John
Wiley.
[2] Gelman, A., Carlin, J. B., Stern H. S. and Rubin, D. B. (1995): Bayesian Data Analysis.
Chapman and Hall.
[3] Kass, R. E. and Steffy, D. (1989): Approximate Bayesian inference in conditionally
independent hierarchical models (parametric empirical Bayes models). J. Amer. Statist. Assoc.,
84:717-726.
4] Lindley, D. V. and Smith, A. F. M. (1972): Bayes estimates for the linear model (with
discussion). J. R. Statist. Soc . Ser B 34: 1-41.
[5] R Development Core Team (2007). R: A language and environment for statistical
computing. R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0,
URL http://www.R-project.org.
[6] Snedecor, G. W. and Cochran, W. G. (1989). Statistical Methods, 8th edition. IOWA State
University Press, Ames. IOWA.
[7] Tanner, M. A. (1996): Tools for Statistical Inference . Springer-Verlag
[8] Venables, W. N. and Replay, D. B. (2002). Modern Applied Statistics with S-PLUS .
Springer, New York.
APENDIX
ACCESS BANK PLC
S/NO NSEASI PRICE INTEREST RATE EXCHANGE RATE OIL PRICE
1 1.0000 1.0000 1.0000 1.0000 1.0000
2 0.8738 0.9503 0.9739 0.9999 1.1156
3 0.7844 0.8776 0.9077 0.9999 1.1531
4 0.7686 0.832 0.9899 0.9999 1.0375
5 0.9518 1.0103 0.9723 0.9997 1.0656
6 1.0761 1.2459 0.9947 1.0001 1.0953
7 0.9427 1.0279 1.0096 1.0001 1.1143
8 0.9008 0.8267 0.9184 0.9940 0.9589
9 0.8212 0.8136 0.9899 0.9747 1.0848
10 0.8520 0.8375 1.0096 0.9749 1.0381
11 0.8035 0.7890 1.0091 0.9736 1.0075
12 0.7818 0.7899 0.9189 0.9708 1.1192
13 0.8993 0.1564 0.9723 0.9708 1.0836
14 0.8989 0.1492 0.9675 0.9657 1.1076
15 0.8853 0.1395 0.9541 0.9593 1.2333
16 0.8752 0.1324 0.9616 0.9572 1.2461
17 0.9135 0.1337 0.9792 0.9571 1.2364
18 0.9464 0.1396 0.9685 0.9571 1.3185
19 1.0183 0.1404 0.9925 0.9566 1.3169
20 1.1920 0.1563 0.9808 0.9562 1.1357
21 1.2383 0.1653 0.9877 0.9559 1.0521
22 1.2427 0.3212 0.9979 0.9558 1.0607
23 1.2235 0.4102 0.9984 0.9558 1.1095
24 1.2300 0.3908 0.9952 0.9559 0.9721
25 1.3183 0.3987 0.9973 0.9558 1.0442
26 1.5093 0.5361 0.9941 0.9557 1.1213
27 1.5560 0.6228 1.0091 0.955 1.2163
28 1.7363 0.8112 0.9909 0.9537 1.2341
29 1.7938 0.9895 0.9157 0.9505 1.2802
30 1.9220 1.0771 0.9995 0.9496 1.3759
31 1.9379 1.0650 0.9792 0.9476 1.3150
32 1.9594 1.0548 0.9744 0.9256 1.4197
33 1.9373 1.0548 0.9744 0.9377 1.5181
34 1.9170 1.0938 0.9712 0.9269 1.5181
35 1.9876 1.0954 0.9755 0.8967 1.7003
36 2.0661 1.1952 0.9712 0.8798 1.6660
37 2.1883 1.3234 0.9883 0.8787 1.6909
38 2.3710 1.3491 0.9776 0.8786 1.7347
39 2.4384 1.3349 0.9419 0.8783 1.8953
40 2.3281 1.2094 0.9984 0.8780 2.0126
41 2.3017 1.1146 0.9552 0.8777 2.2850
42 2.1249 1.0065 0.9109 0.8775 2.4561
43 2.0158 0.9542 0.9557 0.8772 2.5114
44 1.8036 0.7896 0.9157 0.8770 2.1514
45 1.8048 0.7161 1.0251 0.8769 1.8536
46 1.5682 0.5817 1.0256 0.8768 1.3236
47 1.3137 0.4611 1.0117 0.8770 0.9523
48 1.1261 0.3596 1.1296 0.9714 0.7388
49 1.0000 0.3277 1.0789 1.0748 0.7950
50 0.8738 0.2448 1.2592 1.0968 0.7925
51 0.7844 0.2743 1.2752 1.1015 0.8762
52 0.7686 0.2795 1.2357 1.0975 0.9608
53 0.9518 0.4188 1.2192 1.1021 1.0905
54 1.0761 0.5341 1.2075 1.1086 1.3083
55 0.9427 0.3909 1.2160 1.1117 1.2362
56 0.9008 0.3475 1.2293 1.1357 1.3656
57 0.8212 0.3272 1.2251 1.1417 1.2856
58 0.8520 0.3797 1.2267 1.1173 1.3908
59 0.8035 0.3666 1.2320 1.1302 1.4601
60 0.7818 0.4034 1.2512 1.1202 1.4165
UBA PLC
S/NO NSEASI PRICE INTEREST RATE EXCHANGE RATE OIL PRICE
1 1.0000 1.0000 1.0000 1.0000 1.0000
2 0.8631 0.9331 0.9739 0.9999 1.1156
3 0.7837 0.8727 0.9077 0.9999 1.1531
4 0.7587 0.8247 0.9899 0.9999 1.0375
5 0.9272 0.9766 0.9723 0.9997 1.0656
6 1.0714 1.2379 0.9947 1.0001 1.0953
7 0.9368 1.0323 1.0096 1.0001 1.1143
8 0.8968 0.8232 0.9184 0.9940 0.9589
9 0.8157 0.8049 0.9899 0.9747 1.0848
10 0.8449 0.8283 1.0096 0.9749 1.0381
11 0.7979 0.7837 1.0091 0.9736 1.0075
12 0.7754 0.7820 0.9189 0.9708 1.1192
13 0.8913 0.7062 0.9723 0.9708 1.0836
14 0.8909 0.6640 0.9675 0.9657 1.1076
15 0.8774 0.6498 0.9541 0.9593 1.2333
16 0.8674 0.6974 0.9616 0.9572 1.2461
17 0.9054 0.7420 0.9792 0.9571 1.2364
18 0.9380 0.7697 0.9685 0.9571 1.3185
19 1.0092 0.8027 0.9925 0.9566 1.3169
20 1.1814 1.0475 0.9808 0.9562 1.1357
21 1.2273 1.2664 0.9877 0.9559 1.0521
22 1.2316 1.4314 0.9979 0.9558 1.0607
23 1.2126 1.3792 0.9984 0.9558 1.1095
24 1.2191 1.3788 0.9952 0.9559 0.9721
25 1.3066 1.6050 0.9973 0.9558 1.0442
26 1.4959 2.0831 0.9941 0.9557 1.1213
27 1.5422 2.0972 1.0091 0.9550 1.2163
28 1.7208 2.0972 0.9909 0.9537 1.2341
29 1.7778 2.1281 0.9157 0.9505 1.2801
30 1.9049 2.5679 0.9995 0.9496 1.3759
31 1.9207 2.9766 0.9792 0.9476 1.3150
32 1.942 2.9416 0.9744 0.9256 1.4197
33 1.9201 2.9345 0.9744 0.9377 1.5181
34 1.8999 2.9213 0.9712 0.9269 1.5181
35 1.9699 2.9358 0.9755 0.8967 1.7003
36 2.0477 2.7062 0.9712 0.8798 1.6660
37 2.1689 2.7634 0.9883 0.8787 1.6909
38 2.3499 2.761 0.9776 0.8786 1.7347
39 2.4167 2.7232 0.9419 0.8783 1.8953
40 2.3074 2.8921 0.9984 0.8780 2.0126
41 2.2812 3.2763 0.9552 0.8777 2.2850
42 2.106 1.8548 0.9109 0.8775 2.4561
43 1.9978 1.7688 0.9557 0.8772 2.5114
44 1.7876 1.5453 0.9157 0.8770 2.1514
45 1.7887 1.5439 1.0251 0.8769 1.8536
46 1.5543 1.2372 1.0256 0.8768 1.3236
47 1.3021 0.9342 1.0117 0.8770 0.9523
48 1.1161 0.7788 1.1296 0.9714 0.7388
49 0.9911 0.5766 1.0789 1.0748 0.7950
50 0.866 0.4917 1.2592 1.0968 0.7925
51 0.7774 0.4541 1.2752 1.1015 0.8762
52 0.7618 0.4883 1.2357 1.0975 0.9608
53 0.9433 0.7802 1.2192 1.1021 1.0905
54 1.0666 0.8142 1.2075 1.1086 1.3083
55 0.9343 0.6857 1.2160 1.1117 1.2362
56 0.8928 0.6524 1.2293 1.1357 1.3656
57 0.8139 0.6755 1.2251 1.1417 1.2856
58 0.8444s 0.7248 1.2267 1.1173 1.3908
59 0.7963 0.6494 1.2320 1.1302 1.4601
60 0.7748 0.6168 1.2512 1.1202 1.4165