A Three-State Markov Approach to Predicting Asset Returns

Maruf A. Raheem and Patrick O Ezepue

Statistics and Information Modelling Research Group, Department of Engineering and Mathematics, Sheffield

Hallam University, Sheffield S1 1WB, United Kingdom

*Tel: +44(0)70867262705; Email: Maruf.A.Raheem@student.shu.ac.uk

Tel: +44(0)1142256708; Email: p.ezepue@shu.ac.uk

Abstract

We present in this work, alternative approach to determining and predicting the fluctuations in the stock returns

of a company at the stock market. A three- state Markov is proposed to estimate the expected length of an asset

return to remain in a state, which may be, rising(positive) state(RK), falling(negative) state(Rm) or stable(zero)

state (RL). Daily closing prices of stocks of a major and first generation bank in Nigeria are studied. The results

show that for the 5 years, encompassing the period of post banking reform of 2004 and period of global

financial crisis of 2008, no significant asymmetric and leverage effect on the returns of this bank. Rather, the

bank’s asset prices remain stable; thereby given rise to making little or no gain, and at the same time the loss

was kept at bay. It is optimistic that adopting this method, investors are better guided in their choice of future

investment.

Keywords: Markov model, predictability, stock returns, probability transition matrix, Trading cycle

1.0 INTRODUCTION

In recent times, reports on daily basis, from the print media as well as the radio and television, with

respect to the happenings in the financial markets, such as latest stock market index values, currency

exchange rates, electricity prices, and interest rates create awareness to the general public; and in

particular to the stakeholders in the financial markets. Curiosity generated by the price behaviour,

among the private and corporate investors, businessmen as well as the individuals involving in

international trade; has made the field of financial time series become popular. The window of

opportunities offered by studying and understanding price behaviour, to the relevant practitioners in

this market has over the years helped many traders to deal with the risks associated with fluctuations

in prices. These risks can often be summarised by the variances of future returns.

To the financial analysts, understanding how price behaves is of great importance; and also to use the

knowledge of price behaviour to reduce risk or take better and well informed decisions about the

future states of the price. The price tomorrow remains uncertain, hence must be described by a

probability distribution. This implies that for price to be investigated, statistical techniques such as

building a model, which gives a detailed description of how successive prices would be, might be

desirable. Analysing financial data has also for sometimes now been subject of keen interest among

various stakeholders such as financial experts (or engineers) in banks and other financial institutions;

as an empirical discipline as well as the scientists and academicians, as a theoretical field with the aim

of drawing inferences and making forecasts. Furthermore, the inherent uncertainty associated with

financial time series, given the complexity involved in its application has made its study and analysis

subject of concern to statisticians and economists(Tsay, 2005) as well as the physicists( Shinha et al.,

2010 and Chakabarti et al., 2006).

Predicting financial series like asset returns has however been a challenging task (Lendasse, et al.,

2008). Studying and understanding how stock prices behave has also widely been a subject of

research in finance. Fama(1965) identified the highly stochastic nature of stock price behaviour.

Bachelier(1914) proposed the theory of random walk to characterise the fluctuations in stock prices

overtime. Fama(1965) confirmed the empirical evidence of stock prices to satisfy the principle of

random hypothesis; that a series of price changes has no memory, indicating past price dynamics

cannot be used in forecasting the future price. According to efficient market hypothesis(EMH),

security price changes can only be explained by the arrival of new information, which is quite

challenging to predict(Lendasse et al., 2008).

In the meantime, empirical evidence on the stochastic behaviour of stock returns has led to identifying

some important stylized facts. Fama(1965) Mandelbrot (1963) and Nelson(1991) observed that the

distribution of stock returns appears to be leptokurtic. Engle(1982) and Bollerslev(1986) also used

ARCH-type models to study volatility clustering of short term returns. Black (1976), Christie (1982)

and Bekaert and Wu (2000) submitted that changes in stock prices tend to be inversely related to

changes in Volatility. Another vital concept about asset returns is that of asymmetry in volatility

which has its origin in the works of Black (1976), French et al.(1987) and Nelson(1991). In most of

these studies, it has been argued that negative returns cause volatility to rise significantly compared to

the positive returns of equal magnitude. It is to be noted that the presence of asymmetric volatility is

greatly pronounced during stock market crashes. At this point, a sharp drop in stock price is

associated with significant rise in market volatility (Wu.G, 2001). Black(1976) and Christie(1982)

also identify leverage effect in stock returns, whereby it was found that a negative return, due to

falling prices leads to increase in financial leverage, thereby making stock to be very risky and thus

increases the volatility.

In this research however, rather than determining the extent of volatility of the series, we concentrate

on the persistence of the three possible scenarios( called states) of a given return series, with a view to

obtaining the expected length of each of the scenarios. Consequently, this would afford us opportunity

to determine what in this research, we refer to as ‘Return ( or Trading) Cycle’. The decision to adopt

this approach, using a Markov model to describe the price behaviour was born out of the fact that,

according to Bachellier(1914) and Fama(1965) as well as a couple of researches, stock price follow

the theory of random walk; and that the possible states (k- positive, l-zero and m-negative) are distinct

and non-overlapping. In addition, the price behaviour could be likened to the rainfall pattern and

Markov model have been applied extensively to study the pattern of occurrence of dry, wet and rainy

spells for (daily, weekly and monthly) rainfall data( see Weiss,1964; Green, 1965, 1970; Purohit et

al., 2008; Garg and Singh, 2010 and Raheem et al., 2012).

2.0 Methods and Methodology

Many variables such as asset returns undergo episodes in which the series behaviour appears to

change dramatically. Similar dramatic fluctuations are common to almost any macroeconomic or

financial time series for unspecified length of time. The reasons for these changes mostly include

wars, financial panics, or significant changes in government policies. Meanwhile, if a process has

changed in the past, it could likely change again in the future, and this forms the basis for prediction.

The change in regime therefore should be seen as a random variable, which is governed by a

probability law.

It then indicates that such process (series) might be influenced by an unobserved random variable, ,

known as the state or regime at which such process was(is) at date ‘t’ . Thus in this work, three states

(regimes), defined as have been identified with regime,

= k, called positive (+

=1) state; zero (0=2)or stable state and

, negative( - = 3) state. Since takes on only

discrete values, and the simplest time series model for a discrete-valued random system(series) is a

Markov Chain, we therefore take the daily stock returns used in this research as random variables

possibly falling in any of the stated discrete-valued regimes.

Daily closing prices of stock (security)of a bank ( First bank Nigeria ) for the period starting from 1st

August, 2005 to 1st August, 2012 were used in generating simple daily returns( ), which is assumed

to fall in any of the regimes defined above. Thus, is said to be in k state( = k),

at time ‘t’

when it takes on positive value; Rt is in l-state = l),

at time ‘t’ when it assumes zero( 0-value)

and in m-state, ( = m),

when it takes on negative value. This indicates that in forming the

possible states, the ‘signs’ are considered rather than using the actual value of a return.

2.1 Markov Chain

Let Rt be a random variable that can assume an integer value {1, 2, 3…, N}. Suppose the probability

that Rt equals some certain values depends on the past only through the most recent value . Thus,

Pr( ǀ , ,……………………….) = Pr ( ǀ ) = ; [ . Such

a process is described as attaining N- state; with N=3, for . The transition

probability, gives the probability that state " " will be followed by state " ". Also note that +

+ + ……………. + = 1. Hence for this work we have that + + = = 1;

The data observed as the daily returns are taken as three-state Markov chain with state space, S =

{ }. The current daily return was expected to depend only on that of the preceding day; thus, the

observed frequency and the transition probability matrix are given as:

Table 1. Observed Frequency Table

Current Day

Total Positive( ) Zero ) Negative )

Previous

Day

Positive( )

Zero )

Negative )

Where

(i, j = ) are the number of observed returns falling in row i and column j

Daily number of positive returns preceded by, positive previous day’s returns.

: Number of negative returns preceded by positive one in the previous day

: Number of zero returns preceded by positive one in the previous day

: Number of positive returns preceded by negative one in the previous day

: Number of negative returns preceded by a negative one in the previous day; and so on.

; total number of positive returns

i.e total number of zero returns

; i.e. total number of negative returns

The maximum likelihood estimators of (i, j = ) are given by

where i, j =

The transition probability matrix is defined as

P = ( ) = P (j/i) where i, j S

and is given in the table below :

Table2: Transition Probability Matrix

Current Day

Positive(k) Zero(l) Negative(m)

Previous

Day

Positive(k)

Zero(l)

Negative(m)

Where

= P (k/k): Probability of a dry day preceded by a dry day

= P (l/k): Probability of a wet day preceded by a dry day

= P (m/l): Probability of a rainy day preceded by a wet day and so on.

Subject to the condition that the sum of probabilities of each row is one (1 ) i.e.

= 1

+ = 1

+ + 1

2.1.1 ASSUMPTIONS OF MARKOV CHAIN MODEL

For any system to be modeled by the Markov chain model, it must satisfy the following assumptions

viz:

1. The present state of the system (process) depends only on the immediate past state.

2. Transition probability matrices are the result of processes that are stationary in time or space;

the transition probability does not change with time or space.

2.2 TEST OF GOODNESS OF FIT

Our task in this section is to validate the use of a three-state Markov Chain with a view to ascertaining the

suitability of this method to the set assumption that the current day’s return depends on the return of the

previous day. To realize this, two methods have been adopted; these are: the conventional test for independence

via chi-square statistic and WS test statistic that was proposed by Wang and Maritz(1990) for the purpose of

testing the goodness-of-fit of the Markov model.

Hence, we set the hypotheses:

H0: Simple Returns on consecutive days are independent

H1: Asset returns on consecutive days are not independent

a.) Chi-square statistic: = ∑( )

) )

Where is the expected number of returns, which is computed using the formula:

b.) WS statistic is given as:

√ )

→ )

Where ) represents the variance of the maximum likelihood estimator given by

) ) [

]

Represent the stationary probabilities calculated as follows:

) )

[ )

] (

)

(

) [

)]

(

)

The critical region for the WS test statistic is given by ) > at ‘ ’ level of significance. That is the

null hypothesis ( ) can be rejected if │WS│ where is the 100(1- ) lower percentage point of a

standard normal distribution.

2.4 Expected Length of Different Trading Runs and Trading Cycle (TC)

(i) A positive run (k) represents the sequence of consecutive daily positive returns preceded and

followed by either zero or negative returns. Thus the probability of a sequence of ‘k’ positive

days is given by

P (k) = ) (1- )

The expected length of positive runs is given by

E (K) =

)

Where k is the number of positive returns preceded and followed by zero or negative returns.

(1- ) is the probability of a return being either zero or negative.

(ii) A zero runs (l) stands for the sequence of consecutive daily zero returns preceded and

followed by positive or negative daily returns. The probability of a sequence of ‘l’ is given by

P(l) = ) (1 - )

The expected length of zero run is given by

E (L) =

)

Where ‘l’ is the number of zero daily returns preceded by either positive or negative daily

returns, while (1- ) is the probability of a return being positive or negative.

(iii) Finally, for negative runs (m) stands for the probability of a sequence of daily negative returns,

and is given as:

P(m) = ) (1 - ) ; with the expected length of rainy spell obtained as:

E(M) =

)

Where ‘m’ represents the number of negative returns preceded by either zero or positive days;

while (1 - ) is the probability of a return being either zero or positive.

(iv) Trading Cycle (TC): The Returns (trading) cycle is given by

E(TC) = E(K) + E(L) + E(M).

Where ;

E(TC) is the expected length of TRADING cycle; that is, the length of time it will take the

series (returns) to be found in each of the three regimes (positive, zero and negative); and go

back to a particular state after leaving the regime.

E(K) is the expected length of daily positive returns

E(L) is the expected length of zero returns

E(M) is the expected length of negative returns.

(v.) The number of days (N) after which equilibrium state is achieved represents the number of times the

probability transition matrix is powered till the elements of the rows of the matrix becomes the same. Thus

for a 3x 3 matrix, we expect the equilibrium point to be attained when we have the probability transition matrix

to be powered until we have:

=

3.0 Results and Discussion

Having analyzed the data on asset returns, as discussed earlier the results are have been summarized the the

tables below for ease of access. For instance, the tables 1-12 below give the probability transition matrices for

the various months considered in this work. It is to be noted that for the purpose of this research, we only

considered the possible signs the returns can take on. Thus, we see the return series to possibly assume positive

alue (regime-k), negative value (regime m) and zero value (regime l). With this consideration, the actual value

of the return is of no relevance. We actually looked at the daily simple returns of a bank, which were computed

from the daily closing prices of the stocks of the bank for five years, encompassing the period of post 2004

Nigerian banking reform to include periods of 2008, global financial crisis whose effects greatly impacted

majority of the banks, other financial institutions as well as the economy in general. The approach adopted in

this research serves as a mean to an end in itself and not the end in the real sense. Rather than looking at the

distributional properties of the returns, or the stylized facts about return series ,as contained in many of the

existing works in this field, our intent is to determine the extent of persistence of the extremes( positive-

negative) of an asset returns; since these extremes represent the parameter for determining the asymmetry as

well as the leverage effects of stock returns.

Having obtained the transition matrices for the series, we first tested how fit is the Markov to the data; to

achieve this we used both traditional chi-square method and WS statistics, proposed by Wang and Martiz

(1990). According to the result, the test was significant for all months except for February in the case of Chi-

square. Whereas, for WS, it was only the month of June was in significant, as this can be confirmed from table

13. Having ascertained the fitness of the model, we computed the transition matrix for each month and

correspondingly, the equilibrium probabilities were obtained. Subsequently, we proceeded on to computing

expected length of experiencing each of the regimes within a month of trading. It would be recalled that

virtually in every stock market, 22 days of trading are the minimum that could be found in a given month. Form

the table, it could be observed that the return of the bank, though not been drifted by both positive and negative

returns, one can see that, the asset prices were more stable, given the suspicion of no pronounced change. This

indicates, that price was a bit stable. Take for instance, for the months of May, July and Octobers for the five

years this research covers; there have been little or no change in the daily closing prices remain the same, which

consequently led to more ‘Zero’ returns (see Figures 1 & 2). It was also discovered that the stock of the bank in

question seems to be less influenced by external variations in the months of February, January, August, June and

September based on the time taken for the transition probability to assume equilibrium in the ranking order as

listed above. Also from our results as shown in table 14, we found that the months of June and October were

characterized with more instability in the returns subject to the length of time it took the transition matrix to

arrive at equilibrium point. Another notable discovery made in this research was that going by table14 still, at

long run, the expected number of trading days of having positive, negative and zero returns in say, January are

10, 9 and 2 days respectively. Whereas in December, expected of positive, negative and zero returns are

respectively 6 days, 11days and 5days.

Table1:Prob. Transition Matrix for January

K L M

K 0.7 0.025 0.275

L 0.2 0.4 0.4

m 0.3095 0.071 .019

Table 2: Prob. Transition for February

K L M

K 0.5152 0.0303 0.4545

L 0.2 0.0 0.8

m 0.3095 0.071 .019

Table 3: Prob. Transition for March

K L M

K 0.6154 0.0513 0.3333

L 0.2222 0.7778 0.0

m 0.3514 0.0274 .6216

Table 4: Prob. Transition for April

K L M

K 0.5385 0.0 .4615

L 0.0769 0.8077 .1154

m 0.3611 0.1389 .5

Table5: Prob. Transition Matrix for May

K L M

K 0.5484 0.0 0.4545

L 0.0870 0.913 0.0

m 0.3529 0.0588 .5882

Table6: Prob. Transition Matrix for June

K L m

K 0.4828 0.0345 0.4828

L 0.2 0.6 0.2

m 0.4483 0.0690 0.4828

Table7: Probability transition for July

K L M

K 0.6667 0.0 0.3333

L 0.1364 0.8636 0.0

m 0.3333 0.0333 .6333

Table8: Prob. Transition for August

K L M

K 0.5472 0.0377 0.4151

L 0.0 0.5 0.5

M 0.3818 0.0182 .6

Table9: Prob. Transition Matrix for September

K L M

K 0.4444 0.1111 0.4444

L 0.1579 0.6316 0.2105

m 0.4038 0.0192 0.5769

Table 10: Transition Prob. Matrix for October

K L M

K 0.533 0.0222 0.4444

L 0.0869 0.913 0.0

m 0.439 0.0244 0.5366

Table11: Prob. Transition Matrix for November

K L M

K 0.6512 0.093 0.2558

L 0.087 0.7826 0.1304

m 0.35 0.025 0.625

Table12: Prob. Tran. Matrix for December

K L M

K 0.5333 0.0333 0.4333

L 0.1 0.8333 0.0667

0.2245 0.0612 0.7143

Table 13: Test of Goodness for Markov Model

Months Chi-square result WS-statistic

Jan 21.856 (significant) 8.447(significant)

Feb 4.932(Not significant) 3.2916(significant)

March 49.481(significant) 56.58(significant)

April 49.481(significant) 5.89(significant)

May 71.606(significant) 21.73((significant)

June 24.308(significant) 0.64( Not significant)

July 76.816(significant) 37.05(significant)

August 16.688(significant) 21.948(significant)

Sept 40.655((significant 10.40(significant)

October 116.187(significant) 7.78(significant)

November 68.08(significant) 40.45(significant)

December 78.258(significant) 30.56(significant)

Table14: Equilibrium state Probabilities, Expected length of different Regimes Runs, Trading

Cycles and length of time for equilibrium attainment

Months Positive

runs

Zero

Runs

Negative

Runs

Trading

cycle

N- length

of time it

takes to

reach

equilibrium

Jan .49 .o7 .43 4 2 3 9 10

Feb .38 0.05 0.57 3 1 3 7 6

March .45 .15 0.4 3 5 3 11 28

April 0.34 0.28 0.38 3 6 2 11 32

May 0.35 0.26 0.39 3 12 3 18 45

June 0.44 0.12 0.45 2 3 2 7 15

July 0.47 0.10 0.43 4 8 3 15 33

Aug 0.43 0.05 0.52 2 3 3 8 10

Sept 0.38 0.14 0.48 2 3 3 8 15

Oct 0.40 0.22 0.38 3 12 3 18 45

Nov 0.42 0.22 0.36 3 5 3 11 19

Dec 0.28 0.24 0.48 3 6 4 13 23

4.0 Conclusion

From our findings in this research, this approach would be of benefit to determine the riskiness of an asset of a

company; because it serves as a pointer to predicting the future of an asset return. Aside this it would help guide

the planners or market participants on the future stance of their investment based on the current performance of

such asset (or portfolio) in a given company they intend to trade with. It would also serve as a search light to the

business owners or managers about the period of the years (months) their investment has been yielding realistic

returns. For instance, going by our findings on the bank considered in this work, for the five years(2005-

2009),the runs of both positive and negative returns are almost the same, meaning neither there was significant

gains on the returns nor loss on their assets despite the challenges faced due to financial crisis of 2008 across

2009. The months of May and October have the longest length of trading cycles (see fig 3)

Fig 1: Bar Graph Showing the Distribution of Runs of the Three Possible Regimes

Fig 2: Line Plot for the Distribution of Runs for the Three Regimes

0

5

10

15

Positive runs

Zero Runs

Negative Runs

0

2

4

6

8

10

12

14

Positive runs

Zero Runs

Negative Runs

0

2

4

6

8

10

12

14

16

18

20

Trading cycle

Trading cycle

Fig 3: Bar Graph for the Distribution of the Trading Cycle per Month

Fig 4: Price Series (2005-2012)

Fig 5: Returns Series (2005-2012)

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