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This paper investigates the universal financial dynamics in two dominant stock markets in Sub-Saharan Africa, through an in-depth analysis of the cross-correlation matrix of price returns in Nigerian Stock Market (NSM) and Johannesburg Stock Exchange (JSE), for the period 2009 to 2013. The strength of correlations between stocks is known to be higher in JSE than that of the NSM. The stock price dynamics in the NSM is important for modeling Nigerian derivatives in the future, and in the light of this, it is pertinent to note that the interactions of other stocks with the oil sector are weak, whereas the banking stocks have strong positive interactions with the other sectors in the stock exchange. For the JSE, it is the oil sector and beverages that have greater sectorial correlations, instead of the banks which have a weaker one in relation to other assets in the stock exchange.

This paper investigates and compares the spectral properties of correlation matrices of price fluctuations in Nigerian and South African Stock Markets, using the Random Matrix Theory (RMT). Alternative approaches, namely factor and principal components analysis for measuring the extent of correlations could be found as presented in [

Pafka and Kondor [

The interesting question that concerned investors need to answer is how volatility, which is a measure of market fluctuations, affects the dynamics of the market or vice versa. It is, therefore, expedient to explore the relationship between volatility and the coupling of stocks with one another, using correlation matrices [

V a r ( ρ ) = ∑ i , j , k , l w i w l Λ i j Λ l k C j k v j v k σ j σ k (1)

where w i are the weights in the portfolio, C i j is the correlation matrix for the implied volatility for the underlying assets and the Vega matrix Λ i j is defined as

Λ i j = ∂ p i ∂ v j (2)

with p i as the price of option i, v j is the implied volatility of asset underlying option j and σ i is the standard deviation of the implied volatility v i .

Similarly, for investors using derivatives products as a hedge on the underlying assets and for risk management, it is advisable that such investors should buy call and put options respectively for assets whose returns move in opposing directions, as may be witnessed from the calculated empirical correlation matrix. Furthermore, an accurate quantification of correlations between the returns of various stocks is practically important in quantifying risks of stock portfolios, pricing options, and forecasting. [

Theoretically, the comparative analyses of asset price fluctuations (hence correlation structures) between the JSE and NSM will enable us to calibrate suitable derivative models to be proposed for adoption in the NSM for portfolio optimization and risk management. This is because from the research visit embarked upon by the researchers to the Nigerian Stock Exchange (NSE) in 2014; policy makers in the NSE are taking a clue from the JSE in their proposed introduction of some pioneer derivative products and subsequently an appropriate pricing and valuation of such products in the NSM. The research into the correlations between price changes of different stocks is not only necessary for quantifying the risk in a given portfolio, but it is also of scientific interest to researchers in Economics and Financial Mathematics [

Similarly, for financial time series in a stock exchange, the nature of interactions among constituent stock are unknown, hence the need to adopt the RMT approach in exploring these interactions between individual pairs of stocks, for use in portfolio optimization and risk management. The estimation of risk and expected returns based on variance and expected returns in a given portfolio constitutes Markowitz’s model [

Szilard and Kondor [

on the ratio of stocks to the period considered, given by r = N T , where N is the

number of stocks considered and T the length of the available time series. They note that for the ratio r = 0.6 and above, there will be a pronounced effect of noise on the empirical analysis as was discovered by [

In our case for NSM r = 82 1018 = 0.08 and that of JSE we have r = 35 1147 = 0.03

thus both lying in the admissible region in the values of r. When this is done, if the eigenvalues of the empirical correlation matrix and that of the Wishart matrix lie in the same region without any significant deviations, then the stocks are said to be uncorrelated and therefore no information or deduction can be made about the nature of the market, since it is the deviations of the eigenvalues of the correlation matrix from that of the Wishart matrix that carries information about the entire market. However, if there exists at least one eigenvalue lying outside the theoretical predicted bound of the eigenvalues in the empirical correlation matrix obtained from the stock market returns, then the deviating eigenvalue(s) is(are) known to carry information about the market under consideration.

To the best of our knowledge, no such work on the comparison of stock market correlations has been carried out on African emerging markets, especially JSE and NSM which are major emerging markets in the Sub-Saharan Africa. Most of the work on such comparison has been carried out for developed markets or developed versus emerging markets, see, for instance: [

In some sense, the JSE is gradually approaching a developed market whereas the NSM is an ideal African emerging market with no known trades on derivative products currently existing in the market, unlike the JSE where trade on derivatives has been in existence for over two decades. Option contracts were introduced in JSE in October 1992, agricultural commodity futures in 1995 and a fully automated trading system in May 1996, whereas in the NSM trade in derivative products are still at the formative stage, with a recently approved derivative trade on foreign exchange future under the auspices of Financial Market Derivative Quotations (FMDQ) in 2016. As the policy makers in the NSM are benchmarking themselves on the relevant trade on derivatives in JSE towards an effective take off of derivative trade in the NSM, it is pertinent to compare the asset return correlations between the two markets, to understand the similarities and differences in the statistical properties using random matrix theory.

The data set consists of the daily closing prices of 82 stocks listed in the Nigerian Stock Market, NSM from 3^{rd} August 2009 to 26^{th} August 2013, giving a total of 1019 daily closing returns after removing (a) assets that were delisted, (b) those that did not trade at all or (c) are partially in business for the period under review. The stocks considered for NSM are drawn from the Agriculture, Oil and Gas, Real Estates/Construction, Consumer Goods and Services, Health care, ICT, Financial Services, Conglomerates, Industrial Goods, and Natural Resources. For the JSE, we have a total in 35 stocks selected from Top 40 shares in the Industrial Metals and Mining, Banking, Insurance, Health care, Mobil Telecommunications, Oil and Gas, Financial services, Food and Drugs, Tobacco, Forestry and Paper, Real Estate, Media, Personal Goods and Beverages, covering the period 2^{nd} January 2009 to 01^{st} August 2013 covering a similar period as that of NSM (This period was chosen for the research because that was the period when we could get the complete market information for the two stock exchanges being considered).

For the values of the daily asset prices to be continuous and to minimize the effect of thin trading, we remove the public holidays in the period under consideration and to reduce noise in the analysis, market data for the present day is assumed to be the same with the previous day for cases where there are no information on trade for any particular asset on a given date. Also, we eliminate stocks that infrequently traded within the period under review. Let P i ( t ) be the closing price on a given day t, for stock i and define the natural logarithmic return of the index as

r i ( t ) = ln P i ( t + 1 ) P i ( t ) (3)

where r i ( t ) is the number of observations in the two stock exchanges, NSM and JSE.

We calculate the price changes of assets in the two markets over a time scale Δ t which is equivalent to one day and denote the price of i at a time t as s i ( t ) with the corresponding price change or logarithmic returns G i ( t ) over time scale Δ t as

G i ( t ) = ln [ S i ( t + Δ t ) ] − ln [ S i ( t ) ] (4)

We quantify the volatility in the respective asset return as a local average of the absolute value of daily returns of indices in an appropriate time window of T days as

v = ∑ t = 1 T − 1 | G i ( t ) | T − 1 (5)

To standardize the values of obtained from Equation (4) above for all values of i, we normalize G ( t ) i as follows

g ( t ) i = ≺ G ( t ) i − G ( t ) i σ i (6)

where σ i = G ( t ) i 2 − G ( t ) i 2 and … represents the average in the period studied.

From real time series data of the implied volatility surface, we can calculate the element of N × N correlation matrix C as follows

C i j = 〈 g i ( t ) g j ( t ) 〉 = 〈 [ G i ( t ) − 〈 G i 〉 ] [ G j ( t ) − 〈 G j 〉 ] 〉 [ 〈 G i 2 〉 − 〈 G i 2 〉 ] [ 〈 G j 2 〉 − 〈 G j 2 〉 ] (7)

C i j lies in the range of the closed interval − 1 ≤ C i j ≤ 1 , with C i j = 0 means there is no correlation, C i j = − 1 implies anti-correlation and C i j = 1 means perfect correlation for the empirical correlation matrix.

As stated earlier, our aim is to extract information about the cross-correlation from the empirical correlation matrix C. To this end, we are going to compare the properties of C with those of a random matrix; see, [

C = 1 L G G T (8)

where G is the normalized N × L matrix and G^{T} is the transpose of G. This empirical matrix will be compared with a random Wishart matrix R given by:

R = 1 L A A T (9)

to classify the information and noise in the system [

In our bid to use the random matrix theory in portfolio optimization and (derivative) assets risk management, we should be conversant with the universal properties of random matrices. Wilcox et al. [

N → ∞ , and L → ∞ , we have that Q = L N ( ≥ 1 ) is fixed. The probability func-

tion P r m ( λ ) of eigenvalues λ of the random correlation matrix R is given by

P ( λ ) = Q 2 π σ 2 ( λ max − λ ) ( λ − λ min ) λ (10)

for λ such that λ min ≤ λ ≤ λ max , where σ 2 is the variance of the elements of A. Here σ 2 = 1 and λ min and λ max satisfy

λ max / min = σ 2 ( 1 + 1 Q ∓ 2 1 / Q ) (11)

The values of lambda from Equation (10) that satisfy (11) and (12) are called the Wishart distribution of eigenvalues from the correlation matrix. These values of lambda obtained from Equation (11) as stated before determine the bounds of theoretical eigenvalue distribution. When the eigenvalues of empirical correlation matrix C are beyond these bounds, they are said to deviate from the random matrix bounds and are therefore supposed to carry some useful information about the market, [

The distribution of eigenvalue spacing was introduced as the required test for the case when there are not significant deviations of the empirical eigenvalue distribution to that of the random matrix prediction Wilcox et al. [

P ( s ) = s 2 π exp ( − s π 2 4 ) , (12)

where ( λ i + 1 − λ i ) / d and d denotes the average of the differences λ i + 1 − λ i as i varies.

The concept that low lying eigenvalues are really random can also be verified by studying the statistical structure of the corresponding eigenvectors. The jth component of the eigenvector corresponding to each eigenvalue λ α will be denoted by, v α , j and then normalized such that ∑ j = 1 N v α , j 2 = N . Plerou et al. [

p ( u ) = 1 2 π exp ( − u 2 2 ) (13)

In line with the assumption of pure randomness and independence, the distribution of the components, u a ( l ) for l = 1 , 2 , 3 , ⋯ , N of an eigenvector u a of a random correlation matrix, R should obey the standard normal distribution with zero mean and unit variance, [

Guhr, T. et al. [

I α = ∑ j = 1 N ( v α j ) 4 (14)

where N is the number of the time series (the number of implied volatility considered) and hence the number of eigenvalue components and v α j is the j th component of the eigenvector, v α . There are two limiting cases of I α ( i ) ; If an

eigenvector v α has an identical component, v α ( j ) = 1 N , then I α = 1 N and

(ii) For the case when the eigenvector v α has one element with v α ( j ) = 1 and the remaining components zero, then I α = 1. Therefore, the IPR can be illustrated as the inverse of the number of elements of an eigenvector that are different from zero that contribute significantly to the value of the eigenvector. [

〈 I α 〉 = N ∫ − ∞ ∞ [ v α ( j ) ] 4 1 2 π N exp ( − [ v α ( j ) ] 2 2 N ) d v α ( j ) = 3 N (15)

since the kurtosis (extreme deviations) for a distribution of eigenvector components s 3.

We took a sample study of eighty-two (N = 82) stocks from the Nigerian stock exchange which gave rise to L = 1019 daily closing prices. For the Johannesburg stock exchange, JSE we had a sample study of thirty-five (N' = 35) stocks with a total of L' = 1148. The theoretical eigenvalue bounds in the NSM are respectively

λ_{−} = 0.51 and λ_{+} = 1.65 as minimum and maximum values with Q = L N = 12.41 .

Further from the calculation, the market value shows that the largest eigenvalue λ_{1} = 4.87 which is approximately three times larger than the predicted RMT of value (1.64). Similarly for the JSE, the theoretical eigenvalue bounds of the correlation matrix are λ_{−} = 0.21 and λ_{+} = 2.37 as minimum and maximum eigenval-

ues respectively, with Q ′ = L ′ N ′ = 32.77. A high percentage of the eigenvalues

obtained from the empirical correlation matrix of stock market price returns lie below λ min ( λ − ) , just as obtained by [_{1} = 11.86 which is five times larger than the predicted RMT value of 2.37 above. If there were no correlations between the stocks in NSM and JSE, the eigenvalues derived from the market data would have been bounded between λ_{−} = 0.51 and λ_{+} = 1.65 for NSM and λ_{−} = 0.21 and λ_{+} = 2.37 for JSE respectively. In NSM 7.3% of the eigenvalue lie outside the theoretical value and therefore contain information about the market whereas in JSE 8.57% of the total eigenvalue carry information about the entire market (see

The average 〈 C i j 〉 of the elements of the market correlation matrix for the NSM is 0.041, and that of the JSE is 0.168, showing that even though the two markets are both emerging the JSE is about four times more correlated than that of the NSM. Thus, this shows that the Johannesburg market is much more emerging than the Nigerian market, [

The comparable informative indices (7.3% and 8.6%) for NSM and JSE, respectively, suggest a similarity between the market microstructures in the system.

The overwhelming non-informativeness of the remaining 92.7% and 91.4% of the overall markets, further suggests typical random behaviour of the two markets. Typically, the distribution of the first three eigenvectors indicates the key features (mean, standard deviation and kurtosis) of a market. A look at these first three distributions for the NSM shows compared to the normal distribution, they are skewed and leptokurtic in mean and standard deviations, but fairly symmetric in kurtosis. The JSE versions portray similar non-symmetric behaviours, but fairly symmetric in kurtosis. The NSM distributions would seem to follow a beta-gamma family of distribution while the JSE ones are mostly negatively skewed, as opposed to the first two NSM distributions which are positively skewed. In general, higher-order distributions are examined for a more detailed understanding of market-dynamics, for example, market microstructure.

These distributions present the same profiles as the first three distributions in the two markets, which suggest persistence of market features and the driving economic forces. Given the fact the distributions reveal the presence of market information outside the noisy RMT range; the results suggest potential market inefficiency and ability to make money from the markets. We cannot, however, say more that this regarding the stylised fats and market features, without a detailed examination of the key financial economics features typically explored in empirical finance, namely market efficiency, volatility, bubbles, anomalies, valuations and predictability.

V34 and V35 represent the eigenvectors corresponding to an eigenvalue outside the region predicted by RMT which contain the information about the market. The other eigenvectors correspond to the eigenvalues due to noise as they lie in the region predicted by RMT.

The key interest in this paper is to assess how similar the NSM and JSE are, to facilitate future modelling of as yet non-existent derivative prices in the NSM using available information on existing derivative prices in the JSE. For this, a comparative look at the two sets of eigenvector distributions suggest a flipping over or reverse dynamics in the JSE in comparison with the NSM. For example, the U2 and U_{3}(NSM) versus V2 and V_{3}(JSE) eigenvalue distributions are mirror reflections of each other. The practical implication of this reveals that different market forces seem to drive the NSM and JSE. This result is intuitively meaningful because the NSM is an oil-dependent and erratic in its price dynamics and market microstructure unlike the JSE which is mining dependent, and is therefore relatively stable in nature. Consequently, attempts to model, say, non-existent derivative prices in Nigeria using existing prices in the JSE have to be taken cautiously. That said, the flipping-over features suggest that including NSM and JSE stocks in an African Emerging Markets portfolio would achieve reasonable portfolio diversification and corresponding Markowitz-style mean-variance portfolio optimization. These insights reveal the power of statistical physics tools such as RMT in peering through complex market dynamics which may not manifest with traditional mathematical finance techniques.

The inverse participation ratio (IPR) is the multiplicative inverse of the number of eigenvector components that contribute significantly to the eigenmode, [

The average IPR value is around 3/82 for NSM & 1/35 for JSE respectively larger than would be expected 1/N = 1/82 = 0.01 for NSM & 1/35 = 0.03 for JSE, if all components contributed to each eigenvector, [

The lower end of JSE and the higher end of the eigenvalues for both exchanges (NSM and JSE) show deviations suggesting the existence of localized modes. It is noticeable from

This paper stems from a doctoral research which aims to model yet non-existing derivative prices in the NSM, using existing prices in the JSE. The underpinning heuristics (not developed in detail in this paper) is to backtrack from measures of similarity or dissimilarity between the stylized facts and other empirical correlates of the two market dynamics, one of which is the random matrix correlation structures. The paper, therefore, is novel in foregrounding the modeling of non-existing financial derivatives for the first time known to the authors.

It would have been preferable to use up to date data (2009-2016) for the two markets to accommodate the recent impact of oil price fluctuation on the market dynamics. This was not possible since for the NSM available data from the Nigerian Stock Exchange when this research was being carried out range from 2009-2013. The authors therefore, used this range that was available for the analysis. Strictly speaking from the point of using the results in derivative pricing, this limitation is not severe as one can forecast parts of the data that are not available or simulate alternative impact scenarios for the revealed price paths of crude oil between 2013 and 2016, for example.

The analysis of the correlation and structure of stock market returns for the two most dominant markets in the Sub-Saharan Africa, NSM and JSE, was carried out in this paper using RMT. Marcenko-Pastur eigenvalue distribution predicted that the theoretical eigenvalues should be in the range of 0.52 and 1.65 for NSM and 0.21 and 2.37 for JSE respectively. While for NSM it was observed that 6 out of 82 stocks considered that have their corresponding eigenvalues lie outside this theoretical bound of eigenvalues, in JSE 3 out of the 35 stocks has their eigenvalues outside the predicted eigenvalue bounds. Therefore, 89% of the information from the return distributions is purely random thereby leaving us with the alternative hypothesis of the RMT which states that the information on the market lies on the deviating eigenvalues which imply then that for NSM the true market characteristic lies with only 11% of the assets examined. Similarly, for JSE, only 9% of the stocks considered have information about the market which can be used in constructing portfolios with better stable returns and optimal risk management. As stated earlier, these correlation matrices contain some relevant information for option pricing and hedging [

We noted earlier in the literature review that random matrix theory could be very useful in options trading, hedging and in the management of risks associated with a portfolio of investment. In this regard, we intend to use the RMT results in this paper to construct suitable investment portfolios from overall market and sector-based results, for given weights and implied volatilities of the stocks in the respective portfolios under consideration. As Nigeria is yet to commence trade on derivative products, we will carry out heuristic analyses of the option price data for NSM, and execute same for JSE using obtained data from the Johannesburg Stock Exchange, to adjudicate the relative performances of different derivative pricing models in the two markets.

Urama, T.C., Ezepue, P.O. and Nnanwa, C.P. (2017) Analysis of Cross-Correlations in Emerging Markets Using Random Matrix Theory. Journal of Mathematical Finance, 7, 291- 307. https://doi.org/10.4236/jmf.2017.72015