Medium-Term Forecasting Of Investment Portfolio Profitability

International Journal of Management and Economics Fundamental

108

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VOLUME

Vol.05 Issue 06 2025

PAGE NO.

108-116

DOI

10.37547/ijmef/Volume05Issue06-22

Medium-Term Forecasting Of Investment Portfolio
Profitability

Tursunkhodjayeva Shirin

Ph.D., Doctoral Student Of The “Finance And Financial Technologies” Department, Tashkent State University Of Economics,
Uzbekistan

Received:

23 April 2025;

Accepted:

19 May 2025;

Published:

30 June 2025

Abstract:

This study explores medium-term forecasting of investment portfolio profitability by analyzing the stock

prices of six Uzbek joint-stock companies using time series models. The research compares classical statistical
models such as ARIMA with nonlinear models like GARCH and LSTM to determine their accuracy in volatile market
conditions. Over 848 ARIMA model combinations were tested, and the most optimal models were selected based
on statistical indicators such as AIC, BIC, and significance of parameters. Findings revealed that combining ARIMA
with GARCH models improves forecast precision due to the volatility observed in stock returns. The study also
highlights that while residuals exhibit autocorrelation and non-normality, the models remain statistically robust
for forecasting daily prices from August 2024 to December 2027. The research supports the need for hybrid
approaches to better capture the dynamics of financial markets.

Keywords:

ARIMA, GARCH, Stock Price Forecasting, Time Series Models, Investment Portfolio, Financial Market

Volatility, Uzbekistan Stock Market, Forecast Accuracy, Nonlinear Models, Econometric Analysis, ARCH Effect,
Neural Networks (LSTM, ANN).

Introduction:

Stock price forecasting has a huge impact

on the country's economy. After all, the financial
market plays an important role in the country's
economy. Being able to forecast market movements
increases interest in it, thereby contributing to the
development of the financial market. Data on the
financial market mainly consists of time series data.
Therefore, financial market forecasting is carried out
based on historical data. Based on the principle that

“history repeats itself” in the financial market,

investors and financial analysts forecast stock returns
based on the current market situation. Choosing the
optimal model is important when forecasting the
return on an investment portfolio, stocks, and the
financial market in general. Because, accordingly, the
investor determines the entry and exit points of the
market, which, based on sound information, helps to
make the right decision to invest capital in the financial
market and get high profits. Different economists have

used different models to implement this forecast.
However, prioritizing any one model still remains a
complex process. Because the financial market is a non-
linear, highly volatile market, the uncertainty of the
data in it and the shortcomings of forecasting models
complicate the forecasting process. In addition, the
presence of various factors such as the irrational or
rational behavior of investors, their emotional and
psychological state make the movement in the financial
market more dynamic. The fact that stock prices also
have sharp and unstable fluctuations under the
influence of internal and external factors such as
various news and published reports can lead to errors
in forecasting. According to Shah, the growth of social
and Internet-based media has had a significant impact
on the interaction between public opinion and stock
market dynamics.

The following figure shows the main models used in
financial market forecasting: (See Figure 4.9)

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International Journal of Management and Economics Fundamental (ISSN: 2771-2257)

Figure 4.9. Models used in time series forecasting 430

Autoregressive (linear statistical) models study the
coefficients that model the relationship between
several time steps for a target percentage
characteristic. One of the popular autoregressive
methods is the autoregressive integrated moving
average (ARIMA) model. It was proposed by Kumar and
Jain in 2010. ARIMA predicts the future as a linear
combination of historical values and errors, eliminating
the trending nature of the variables by implementing
differentiation. It is especially effective for short-term
forecasting. The advantages of these models are their
short-term forecasting efficiency, ease of description,
and ability to detect seasonality. However, the inability
to model nonlinear relationships between variables in
multivariate forecasts is considered their main
disadvantage.

Most researchers use ARIMA and LSTM models to
forecast financial markets. However, these models are
also not without their drawbacks. For example, Islam
and Nguyen point out that the most popular ARIMA
model has some limitations in dealing with nonlinear,
non-stationary and seasonal data in time series . In
addition, it is difficult to perform long-term forecasting
using this model. According to Banerjee and Nayak, the
LSTM model does not have parameters predetermined
like ARIMA, and hyperparameters must be properly
tuned to use the model. A group of scientists led by
Agrawal proved that the LSTM model is superior to MA,
LR and ARIMA models , while a group of scientists led

by Srivastava found the LSTM model to be the most
suitable model for working with time series data among
other neutral network models. A number of other
scientists have compared neutral network models with
classical statistical models and noted that neutral
network models are more powerful in many respects.
For example, scientists such as Namini and Rhanoui
have shown in their studies that LSTM is superior to
ARIMA, and Gurushin has proven that even models
that combine statistical and neutral network models
(GARCH-ANN, EGRACH-ANN) are less effective than a
simple ANN model. The fact that stock prices are
associated with volatility makes it possible to forecast
them using the GARCH model. Since GARCH is the most
effective method for forecasting volatility. According to
a study conducted by a group of scientists led by
Zareemba, volatility is considered very important for
the functioning of financial markets, as it is an indicator
of stress associated with financial investments,
uncertainty, and financial risk.

According to Cont, the most valuable characteristic of
financial risk is the presence of variability in it. Because
this variability has a structure such as volatility and
clustering tendency. To better assess this effect, the
corresponding family of autoregressive conditional
heteroskedasticity models is used. The following table
presents the characteristics of heteroskedastic models.
(See Table 4.8)

Table 4.8

Heteroscedasticity models

Model

Year

Scientist

Formula

Limitation

ARCH

1

1982

Angle

𝜎

𝑡

2

= 𝛼

0

+ 𝛼

1

𝑢

𝑡−1

2

𝛼

0

> 0,

𝛼

1

≥ 0

1

Engle, RF Autoregressive conditional heteroskedasticity with estimates of the variance of the United Kingdom inflation // Econometrica –

1982 – Vol. 50, Issue 4. – P. 987-1007. - New York, Cambridge University Press, 1982.

Vaqtli qatorlarni prognozlashda foydalaniladigan modellar

Chiziqli modellar

Chiziqsiz modellar

Statistik

Stoxastik

Statistik

Neytral tarmoqlar

ARIMA, VAR,

VEC

Geometrik brown

harakati

ARCH, GARCH

ANN, CNN, RNN,

LSTM

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International Journal of Management and Economics Fundamental (ISSN: 2771-2257)

ARCH

2

1986 Bollerslev

𝜎

𝑡

2

= 𝛼

0

+ 𝛼

1

𝑢

𝑡−1

2

+ 𝛽

1

𝜎

𝑡−1

2

𝛼

0

> 0,

𝛼

1

≥ 0,

𝛽

1

≥ 0

Integrated GARCH

(IGARCH)

3

1986

Engle and

Bollerslev

𝜎

𝑡

2

= 𝛼

0

+ (1 + 𝛽

1

)𝑢

𝑡−1

2

+ 𝛽

1

𝜎

𝑡−1

2

𝛼

0

> 0,

𝛽

1

≥ 0

Exponential GARCH

(EGARCH)

4

1991

Nelson

𝑙𝑜𝑔𝜎

𝑡

2

= 𝛼

0

+ 𝛾(|𝑧

𝑡−1

|

− 𝐸[|𝑧

𝑡−1

|])

+ 𝜓𝑧

𝑡−1

+ 𝛽

1

𝑙𝑜𝑔𝜎

𝑡−1

2

-

Glosten-Jagannathan-

Runkle GARCH

(GJR-GARCH)

5

1993

Glosten

and

others

𝜎

𝑡

2

= 𝛼

0

+ 𝛼

1

𝑢

𝑡−1

2

+ 𝛾

1

𝑃

𝑡−1

−

𝑢

𝑡−1

2

+ 𝛽

1

𝜎

𝑡−1

2

𝛼

0

> 0,

𝛼

1

≥ 0,

𝛽

1

≥ 0,

𝛼

1

+ 𝛾

1

≥ 0

Threshold GARCH

(TGARCH)

6

1994

Zakoian

𝜎

𝑡

= 𝛼

0

+ 𝛼

1

|𝑢

𝑡−1

|

+ 𝛾

1

𝑃

𝑡−1

−

|𝑢

𝑡−1

|

+ 𝛽

1

𝜎

𝑡−1

2

𝛼

0

> 0,

𝛼

1

≥ 0,

𝛽

1

≥ 0,

𝛼

1

+ 𝛾

1

≥ 0

This in research , dynamic o ʻ variability clear forecast can popular , popular ARIMA model with together in

vibration effective working ARCH from models used without under study of enterprises action prices forecast This
was done . models together use forecast accuracy to increase help gives . Forecast done increase for , 7 under
study stock ownership societies from January 1, 2017 August 1, 2024 until daily action grades received .

Table 4.9

Test results for forecasting

KWTS

QZSM

KUMZ

UZMC

AGMK

TNGK

KYEZ

Dickey-Fuller test (p-value)

0
difference

0.0839

0.5252

0.0056

0.0864

0.0012

0.0000

0.0000

Difference
I

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

Difference
II

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

Phillips-Perron test (p-value)

0
difference

0.4239

0.6995

0.3960

0.0224

0.0347

0.0000

0.0006

Difference
I

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

Difference
II

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

Lags outside the confidence interval

AR 0

difference

9

4

9

17

5

3

6

2

Bollerslev , T. (1986). Generalized autoregressive conditional heteroskedasticity . Journal of Econometrics, 31(3), 307–327.

3

Engle, RF, & Bollerslev , T. (1986). Modeling the persistence of conditional variances. Econometric Reviews, 5(1), 1–50

4

Nelson, D. B. (1991). Conditional heteroskedasticity in asset returns: A new approach. Econometrica , 59(2), 347–370.

5

Glosten , LR, Jagannathan , R., & Runkle , DE (1993b). On the relationship between the expected value and the volatility of the nominal

excess return on stocks. The Journal of Finance, 48(5), 1779–1801

6

Zakoian , JM 1994. Threshold heteroskedastic models. Journal of Economic Dynamics and Control 18: 931-955.

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International Journal of Management and Economics Fundamental (ISSN: 2771-2257)

Difference
I

8

7

8

20

8

2

12

Difference
II

20

23

21

23

22

28

27

I

Lag

1

1

1

2

1

1

1

MA Difference

I

7

4

7

14

8

2

15

Difference
II

6

2

6

10

7

7

15

ARCH effect test

0

chi2

1785.89

1837.81

1798.81

1754.98

1686.9

1689.9

1298.9

p-value

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

I

chi2

21,272

88,191

37,511

2.856

37,511

0.003

0.367

p-value

0.0000

0.0000

0.0000

0.0910

0.0000

0.9593

0.5446

II

chi2

360,554

385,533

466,675

426,650

466,675

459,632

476,590

p-value

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

Above table from the information to see possibly
Dickey - Fuller test to the results see KVTS , QZSM and
UZMC action prices first level stationary , remaining all
under study of enterprises action prices stationary
Phillips - Perron test to the results see but only TNGK
action prices first level stationary to be , to remain
enterprises shares are 2nd level stationary Therefore ,
the value of I in the ARIMA forecast is 1 in KVTS, QZSM
and UZMK

, and 0 in the rest .

In order to evaluate the GARCH model in combination
with ARIMA in stock price estimation, it is necessary to
have volatility in stock prices. For this, an ARCH test was
conducted. According to it, at the level of difference 0,
the p-value is equal to 0 in all enterprises, which means
that the H 0 null hypothesis is rejected and the
alternative hypothesis is accepted. This means that
stock prices have volatility and they have an ARCH
effect. Accordingly, it was considered appropriate to
use the ARCH and GARCH models in stock price

forecasting. However, although UZMK achieved
stationarity at level I, the ARCH effect at this level has
not been proven. The remaining enterprises are
forecasted at level I. Because the MA lags are outside
the confidence interval at level 0, this is a sign of non-
stationarity.

In the AR indicator, all enterprises except UZMK accept
the results of 0 difference. That is, in KVTS the AR value
is from 1 to 9, in QZSM it is from 1 to 4, etc. The results
of the MA value can also be described in the same way.
Based on the above, a total of 848 ARIMA combinations
were formed, of which 56 for KVTS, 28 for QZSM, 56 for
KUMZ, 460 for UZMK, 64 for AGMK and 180 for KYEZ.
From the formed ARIMA combination models of each
joint-stock company, the most optimal model with the
minimum number of statistically significant indicators,
logarithmic probability, AIC and BIC indicators was
selected. The indicators of these models are given in
the table below. (See Table 4.10)

Table 4.10

The most optimal models

AJ

ARIMA

Paramet

er

Log

likelihoo

d

AIC

BIC

Hair

L/l

AIC

BIC

KWTS

(7,1,5)

16(15)

-12195.1

24422.3

24510.8

+

+

QZSM

(1,1,1)

6(5)

-10741.2

21494.5

21527.6

+

+

KUMZ

(1,1,1)

6(5)

-9055.93

18123.9

18157.1

+

+

+

UZMC

(18,1,9)

31(30)

-15099.2

30260.4

30431.9

+

+

AGMK

(1,1,1)

6(5)

-14989.3

29990.6

30023.8

+

+

+

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KYEZ

(6,1,7)

17(16)

-11031.4

22096.8

22190.6

+

From this table, it can be seen that out of the 56 models
calculated for KVTS, the most optimal was the value of
ARIMA (7,1,5). In this model, out of 16 indicators, 15
were found to be statistically significant, and the AIC

indicator was lower than in other models. Therefore,
this model was selected for forecasting. For the QZSM
enterprise, out of the 28 models formed, ARIMA (1,1,1)
was selected for its superiority in terms of logarithmic
likelihood and BIC indicator.

Table 4.11

Regression results of GARCH and ARIMA models

VARIABLES

KWTS

QZSM

KUM Z

UZMC

AGMK

KYEZ

L.ar

0.621***

-0.00861

0.145***

0.115***

0.296***

-0.569***

(0.0622)

(0.0740)

(0.0514)

(0.0343)

(0.0554)

(0.0335)

L2.ar

-0.685***

0.414***

0.546***

(0.0784)

(0.0265)

(0.0432)

L3.ar

-0.927***

-0.223***

1.023***

(0.0955)

(0.0245)

(0.0329)

L4.ar

0.488***

0.199***

0.767***

(0.0847)

(0.0238)

(0.0315)

L5.ar

-0.666***

0.00201

-0.323***

(0.0625)

(0.0166)

(0.0375)

L6.ar

-0.208***

-0.0259

-0.815***

(0.0483)

(0.0166)

(0.0259)

L7.ar

-0.0845**

0.353***

(0.0376)

(0.0184)

L8.ar

0.245***

(0.0195)

L9.ar

-0.320***

(0.0211)

L10.ar

-0.0795***

(0.0229)

L11.ar

-0.179***

(0.0170)

L12.ar

0.0624***

(0.0118)

L13.ar

-0.117***

(0.0122)

L14.ar

0.00229

(0.0126)

L15.ar

0.153***

(0.0137)

L16.ar

-0.135***

(0.0128)

L17.ar

-0.00103

(0.0139)

L18.ar

-0.0846***

(0.0118)

L.ma

-0.959*** -0.401*** -0.676***

-0.623***

0.225***

(0.0506)

(0.0657)

(0.0355)

(0.0420)

(0.0522)

L2.ma

0.850***

-0.824***

(0.0683)

(0.0379)

L3.ma

0.770***

-0.919***

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(0.0860)

(0.0426)

L4.ma

-0.894***

-0.440***

(0.0631)

(0.0627)

L5.ma

0.878***

0.601***

(0.0436)

(0.0458)

L6.ma

0.327***

0.765***

(0.0117)

(0.0279)

L7.ma

-0.360***

-0.293***

(0.0147)

(0.0438)

L8.ma

-0.386***

(0.0109)

L9.ma

0.582***

(0.0177)

L.arch

0.191***

0.206***

0.125***

3.460***

0.0477***

0.127***

(0.0311)

(0.00889) (0.00936)

(0.187)

(0.00232)

(0.0118)

L. though

0.365***

0.832***

0.821***

0.190***

0.951***

0.853***

(0.0815)

(0.00490) (0.00871)

(0.0123)

(0.00163)

(0.00930)

Constant ARCH

15,144*** 78.23***

77.46***

3,167***

2,975***

576.7***

(1,936)

(4.152)

(3.785)

(589.3)

(225.6)

(37.42)

Constant

0.523

0.206***

-0.0858

-20.97***

-0.837

0.309

(4.280)

(0.00889)

(0.315)

(0.801)

(8.153)

(0.759)

Observations

1,867

1867

1867

1867

1867

1867

Standard errors in parentheses

*** p<0.01, ** p<0.05, * p<0.1

The model selected for the KUMZ enterprise
outperformed the calculated models in 3 indicators,
namely, the number of statistically significant
indicators, the probability of the graph, and the
minimum value of the BIC indicator. The presence of
volatility in the share prices of enterprises indicated the
possibility of using ARCH and GARCH models with 1 lag.
In addition, in all calculated models, ARCH and GARCH
indicators were found to be statistically significant. The
table above shows the regression results of the

selected models. According to it, most of the indicators
are statistically significant, which means that it is
possible to forecast using these models. The positive
correlation between the ARCH and GARCH indicators
indicates that the share prices are positively correlated
with their volatility.

Using these models, daily stock price forecasts were
made from 1.08.2024 to 29.12.2027. The results are
presented in the following figure: (See Figure 4.10)

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Figure 4.10. Forecast results (daily)

above , it can be seen that the share prices of the
studied enterprises were stable. The main reason for
this was that the share prices of the enterprises were
relatively stable in the period after 2022, compared to
the periods before. Also, the fact that the forecast in
this figure consists of only a straight line is due to the
sharp increases in UZMK shares in previous periods. If
each joint-stock company were taken separately with
their forecast indicators, or if the data of the UZMK
enterprise were removed from this figure, the real pace

would be shown. - The annexes present the forecasts
of individual prices and volatility of each of the studied
enterprises.

To assess the level of error in the forecast, it is
necessary to examine the forecast standard error, that
is, how much the forecast indicators differ from the
actual indicator. The following table presents the
analytical statistics of the forecast standard error. (See
Table 4.12)

Table 4.12

Standard error analytical statistics

Descriptive Statistics

Variable

Obs.

Mean

Std. Dev.

Min

Max

errorkvts

1867

-1.479

175,141

-1158.586

4984.217

errorqzsm

1867

2.479

108,865

-965.373

732,833

errorkumz

1867

-.576

36,291

-180.927

410,226

erroruzmk

1867

47,519

3566.179

-64869.043

85363.547

erroragmk

1867

9.161

911,087

-7579.332

4424.676

errorkeyz

1846

-1.002

142,334

-1691.111

2781.728

This from the table to see maybe , what is being studied of enterprises action prices every one enterprise for 1867
from information consists of was if only KYEZ enterprise 1846 information Because this enterprise 2024 August
from the month starting information presented not yet , maybe his/her shares Tashkent Republic fund from the
stock exchange delisting done increased to be possible . In general when received , all in enterprises standard

error big not , only UZMC in the enterprise o ʻ average 47.5 units organization This is relatively high indicator , this
of the enterprise action prices high o ʻ to variability has that with is characterized .

0

20000

40000

60000

80000

100000

120000

140000

160000

180000

200000

03.01.2017

05.04.2017

06.07.2017

06.10.2017

24

.01.2

01

8

24.05.2018

26.08.2018

27

.11.2

01

8

27.02.2019

03.06.2019

05

.09.2

01

9

04.12.2019

04.03.2020

03.06.2020

04.09.2020

04.12.2020

10.03.2021

10.06.2021

14.09.2021

21.12.2021

29.03.2022

01.07.2022

04.10.2022

05.01.2023

06.04.2023

13.07.2023

13.10.2023

15.01.2024

22.04.2024

24 07 24

24.10.2024

25.01.2025

01.05.2025

03.08.2025

03.11.2025

06.02.2026

12.05.2026

15.08.2026

15.11.2026

16.02.2027

24.05.2027

25.08.2027

27.11.2027

kvts

fkvts

qzsm

fqzsm

kumz

fkumz

uzmk

fuzmk

agmk

fagmk

kyez

fkyez

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Table 4.13

T

est for “white noise”

KWTS

QZSM

KUMZ

UZMC

AGMK

KYEZ

Portmanteau (Q) statistic

57,845

145,656 118,754 1166.91 66.2425 250,212

Prob>Chi2(40)

0.0336

0.0000

0.0000

0.0000

0.0056

0.0000

Bartlett's (B) statistic

1.55

3.58

3.17

10.54

1.21

3.80

Probe > B

0.0162

0.0000

0.0000

0.0000

0.1080

0.0000

This test to the results Therefore , H 0 - 0 hypothesis refusal mature , alternative hypothesis acceptance These
residuals are not stationary, they do not contain white noise, but they indicate the presence of serial
autocorrelation. Because the model has an ARCH effect .

The following table checks whether the forecast is normally distributed. (See Table 4.14)

Table 4.14

Normal distribution test

Shapiro-Wilk W test for normal data

Variable

Obs.

W

V

z

Prob>z

erkwts

1,867

0.525

529,480

15,914

0.000

erqzsm

1,867

0.832

187,456

13,279

0.000

my dear

1,867

0.844

174,002

13,090

0.000

eruzmk

1,867

0.353

720,698

16,696

0.000

eragmk

1,867

0.853

164,280

12,944

0.000

old man

1,846

0.566

478,979

15,653

0.000

From the results of this test , it can be seen that the
residuals are not normally distributed. Therefore, the
null hypothesis H 0 - 0 is rejected and the alternative
hypothesis is accepted.

In conclusion, many scientific studies have been
conducted to forecast stock prices and profitability, and
these forecasts are mainly carried out using time series
forecasting models. These models can be conditionally
divided into 2 groups: linear and nonlinear models.
Linear models include statistical (AR, MA, ARMA,
ARIMA) and stochastic (Geometric Brownian motion)
models, and nonlinear models include statistical (ARCH,
GARCH, etc.) and neutral network (ANN, CNN, RNN,
LSTM, etc.) models. Since the shares of joint-stock
companies are volatile, ARIMA models based on the
GARCH model were used for forecasting. 848 models of
ARIMA models were created to forecast the share
prices of 6 joint-stock companies. The most optimal
models were selected. The presence of the ARCH effect
on the share prices of the studied enterprises was
assessed, and since the test result was positive, ARIMA
and GARCH regression analysis was conducted. Since
these generated models were found to be statistically
significant, a daily medium-term forecast was
implemented from August 2024 to December 2027.

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