Market Efficiency and Cointegration: A Post-demutualization Analysis of Canadian Life Insurance Stocks Gilles Bernier1 and Chaouki Mouelhi2, 3
Abstract: This paper characterizes the (weak-form) efficiency of the Toronto Stock Exchange (TSX) with respect to the life insurance sector in a post-demutualization context, using a methodology called cointegration analysis. The major conclusion that can be drawn from this analysis is that the Canadian stock market appears to have been inefficient in pricing the securities of the three newly demutualized life insurance firms that became part of the S&P/TSX index of Canada’s financial sector. Indeed, it appears that it would have been possible to predict the future price behavior of these life insurance stocks by relying on past information following their demutualization. [Key words: market efficiency, demutualization, cointegration.]
key question concerning capital markets is their informational efficiency. Indeed, this concept is important in a wide range of applied topics, such as accounting information, new issues of securities, and portfolio management (Copeland, Weston, and Shastri, 2005). The simplest definition of market efficiency is that prices already reflect all available information (past, public, private) and thus buying or selling stocks should,
A
1 Gilles Bernier is Professor of Finance and Insurance in the Faculty of Business Administration at Laval University, Department of Finance and Insurance, Faculty of Business Administration, Laval University, 2325 de la Terrasse Street, Quebec City, G1V 0A6, Canada,
[email protected]. 2 Chaouki Mouelhi, PhD, is a Lecturer in the Department of Finance and Economics at Université du Québec à Trois-Rivières, and a Researcher with the Research Laboratory on Business Valuation at Laval University. 3 The authors express their appreciation for the helpful comments of anonymous reviewers who contributed to significant improvements of this article. The authors would also like to thank participants at the 2008 annual meeting of the American Risk and Insurance Association held in Portland, Oregon.
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Journal of Insurance Issues, 2009, 32 (2): 107–132. Copyright © 2009 by the Western Risk and Insurance Association. All rights reserved.
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on average, provide only a normal risk-adjusted return (net of transaction costs). Weak-form informational efficiency refers to the case where the information set includes only securities’ past prices and returns (Fama, 1970). If a market is found to be weak-form inefficient, this means that it would be possible to predict the future price behavior of stocks traded on that market by relying on past information and, by doing so, potentially earn an abnormal positive return. Likewise, this would be an indication that asset prices do not properly reflect all available relevant information about firms. This would also be a violation of the so-called “random walk hypothesis” (RWH). Technical analysts (also known as chartists) strongly believe they can predict future price movements on the basis of historical prices in order to beat the market. Prior research (mostly US) has resulted in the rejection of RWH, confirming that technical analysis can potentially be a worthwhile investment approach (Lo and MacKinlay, 1988). The purpose of this paper is to study the (weak-form) efficiency of the Toronto Stock Exchange (TSX) with respect to the Canadian life insurance sector in a post-demutualization context, using a methodology called cointegration analysis. Indeed, as new insurance firms became listed on the market (following demutualization), questions can be raised about whether their stock behavior is inter-connected or cointegrated and whether the actual order of demutualization had an impact on this behavior. These questions are of particular interest, given that the demutualization of Canada’s life insurance industry has been described by many as the most significant event to have taken place in its financial services industry in the last decade (CIBC World Markets, 1999). Indeed, over CDN$25 billion worth of stock was brought into the publicly traded equity markets during this 1999–2000 demutualization wave. Babin and Bernier (2001) find some evidence of significant IPO underpricing (average of 9%) associated with this wave. Their results show that a secondary market investor would have outperformed the market from day one up to two years after the IPOs, a result which is consistent with market inefficiency. Our cointegration analysis also provides evidence in support of market inefficiency (weakform) with respect to the pricing of Canadian life insurance firms during the post-demutualization period. The paper is organized as follows. First, we provide an overview of the Canadian life insurance industry. Second, we discuss the statistical concept of cointegration pioneered by Granger (1983), Granger and Weiss (1983), and Engle and Granger (1987). Then, we specifically explain the methodology we intend to apply for testing purposes. Following that, we describe our data set and the time periods covered by our study and
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provide a rationale for the different hypotheses to be tested. Next, our empirical results are shown and explained, and we provide a discussion of whether our hypotheses are supported by those results. Finally, we conclude.
THE CANADIAN LIFE INSURANCE INDUSTRY Canada’s financial services sector is significantly integrated, as different players offer similar services and financial groups or conglomerates offer a variety of financial products and services that cut across what was known as the four pillars (banks, trust companies, insurers, and securities dealers). This integration trend is particularly prevalent in the banking and life and health insurance sectors, where companies have established specialized subsidiaries to provide many different financial service products (Bernier and Nathan, 2007). In the life insurance industry, annuity products (including a variety of tax-sheltered retirement vehicles) are currently the key premium generators, followed by health insurance and then life insurance. In terms of total life insurance premium volumes (in US dollars) sold during 2007, Canada ranked tenth, with 1.91 percent of the world market (Swiss Re, 2008). The big players in the Canadian life sector are mostly insurance and financial groups of the stock form, with domestic ownership and with a multi-channel hybrid distribution system (agency and brokerage) for the group as a whole. But the individual firms within each group typically focus on only one distribution channel (Bernier and Nathan, 2007). Currently, four life insurance firms are listed on the Toronto stock exchange (TSX): Great-West Lifeco Inc., which has always been a public stock company, and Manulife, Sun Life, and Industrial-Alliance, which have been listed on the TSX since their demutualization through IPOs over the period 1999–2000. Manulife Financial has been the most active firm in the international arena, expanding its operations in Asia through joint ventures with Chinese and Indian firms and in the United States through a merger with Boston-based John Hancock Financial Services Inc. The four listed companies are among the so-called Top 5 life insurers in Canada, along with Desjardins Financial Security, which is a subsidiary of the Desjardins Movement, a large cooperative provider of financial services, based in Quebec.
COINTEGRATION ANALYSIS Cointegration is a property that some nonstationary time series data may possess.4 Engle and Granger (1987) first developed a two-step estimation technique in order to analyze long-term equilibrium (cointegration)
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relationships in time series data. This technique has been used by many other researchers to study exchange rate market efficiency (Hakkio and Rush, 1989, 1991; Copeland, 1991) and underwriting margins in P&C insurance (Haley, 1993, 2007). According to Engle and Granger’s methodology, two nonstationary variables (e.g., Xt and Yt) are cointegrated even if the variables evolve independently from each other over time and even if they do not fluctuate around a mean value over time, as long as there is an economic linear relationship between them that remains stable over time. Such linear combination represents the long-term relationship between the variables, a relationship that may be regarded as equilibrium, or an attractor (step 1). In such a two-variable case, the linear combination forms a line that connects the long-term equilibrium pairs of values of the two variables. The deviations from this line, which represent the short-term movements around equilibrium, must be stationary and statistically and significantly related with the first differences of at least one of the original variables (step 2). The initial determination of nonstationarity of the individual variables (e.g., time series of stock prices of listed life insurers) is a pre-test to cointegration analysis (Haley, 2007). It consists in determining whether the variables being considered are indeed integrated of order 1 ( Xt ~ I(1) and Yt~ I(1)). An I(1) variable will contain one unit root and will need to be differenced once to become stationary, that is, I(0). This test attempts to conclude whether the individual variables consistently fluctuate around a fixed mean. For that purpose, we perform a unit root test on each variable, namely an Augmented Dickey-Fuller test ADF (K*) per Dickey and Fuller (1981). To determine the optimal lag order K*, we use the following three criteria of information: Akaike (AIC), Schwartz (SC), and Hann-Quin (HC). If this nonstationarity condition is met, then we can apply Engle and Granger’s two-step procedure as follows: Step one requires an ordinary least square (OLS) estimation of the relationship between the stock prices of each pair of securities as twovariable cases: Y t = α + βX t + ε t ,
(1)
where 4 The concept of cointegration is used to test the weak-form efficiency of markets because the error-correction model of two cointegrated variables implies that the spread between the two variables possesses a predictive power on the future variations of these two variables.
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-Yt is stock price of security Y on day t - Xt is stock price of security X on day t - εt is the error term, and - α and β are two regression OLS parameters to be estimated. The purpose here is to extract the error terms ( εˆ t ) of the regression ( εˆ t = Y t + αˆ + βˆ X t ). Afterward, we test whether the error terms εˆ t are indeed stationary, using the same ADF (K*) test as in pre-testing. Here again, the optimal lag order, K*, will be determined by the three criteria of information identified previously. Hence, if the error terms εˆ are found to t
be stationary [ εˆ t ∼ I ( 0 ) ] , then we can move to step two. Step two requires an ordinary least square (OLS) estimation of two error-correction models (ECM), one for each I(1) variable (each firm’s stock price series) under consideration. The tight connection between cointegration and error correction models stems from the Granger representation theorem.5 Thus, a representation type ECM is given by the following two equations: p
ΔY t = μ 1 + λ 1 ε t – 1 +
ΔX t = μ 2 + λ 2 ε t – 1 +
∑
p
δ i ΔX t – i +
∑ ηi ΔYt – 1 + ut
i=1
i=1
p
p
∑ δi ΔXt – i + ∑ ηi ΔXt – 1 + u't
i=1
(2)
i=1
where p, the number of lags, is chosen arbitrarily (p = 1 in this paper), ΔY t and ΔX t are the changes in the cointegrated variables, and ( u t ,u' t ) is bivariate white noise. The two coefficients λ 1 and λ 2 reflect the speed of adjustment to the long-run equilibrium. For a cointegration relationship to exist at least one of the two lambda coefficients ( λ 1 or λ 2 ) has to be significantly different from zero. If such a statistically significant errorcorrection is not found in step two of the estimation procedure, then the 5
According to this theorem, two or more integrated time series that are cointegrated have an error correction representation, and two or more time series that are error correcting are cointegrated.
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analyst must conclude it is a likely case of spurious correlation (Haley, 2007). Engle and Granger’s procedure has been found to be asymptotically valid and more appropriate when the number of observations is reduced. Nevertheless, it has a few weaknesses on its own, as discussed by Hall (1994) and Ng and Perron (1995), particularly when it comes to the performance of stationary ADF tests, which have been shown to be very sensitive to the chosen optimal lag.6 An alternative to Engle and Granger’s technique is the one put forward by Johansen (1988, 1991). However, Johansen’s procedure aims particularly at testing for multivariate cointegration. Several authors have also noticed some problems with this procedure, particularly in finite samples.
DATA, SAMPLE, AND PERIOD OF STUDY In explaining the statistical meaning of cointegration when applied to two traded securities, Chan (2006: 1) argues that, if cointegrated, then “the two price series cannot wander off in opposite directions for very long without coming back to a mean distance eventually. But it doesn’t mean that on a daily basis the two prices have to move in synchrony at all.” In order to further distinguish between cointegration and correlation, Chan (2006: 1) adds: “Cointegration is the foundation upon which pair trading (‘statistical arbitrage’) is built. If two stocks simply move in a correlated manner, there may never be any widening of the spread. Without a temporary widening of the spread in either direction, there is no opportunity to short (or buy) the spread, and no reason to expect the spread to revert to the mean either.” Our cointegration tests will be carried out for the following four securities: Great-West Lifeco Inc. (GWO), Industrial Alliance Group (IAG), Manulife Financial Corporation (MFC), and Sun Life Financial Inc. (SLF). The data we use comprise the daily closing prices of these four securities. Prices have been obtained on Datastream. The overall period covered by our study goes from March 23, 2000 to December 31, 2007. Hence, we have obtained 2028 observations for every stock. For testing whether the cointegration relationships appear to be stable over time, we will also split the entire period into two equal sub-periods. The first one goes from March 23, 2000 to February 10, 2004 (1014 observations for every stock) 6
This explains why, in this paper, we use the main criteria of information found in the econometrics literature [Akaike (AIC), Schwartz (SC), and Hann-Quin (HC)] in order to choose the optimal lag in ADF tests.
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while the second sub-period goes from February 11, 2004 to December 31, 2007 (again, 1014 observations for every stock).
RESEARCH HYPOTHESES Given that GWO has always been a listed company, it is our assertion that the three newly listed life insurers that opted in favor of demutualization (IAG, MFC, and SLF) are likely to show significant cointegration in terms of their stock price behavior relative to GWO. In our view, this would be the case for the entire period of investigation, but perhaps even more so for the first sub-period, during which GWO was used by market participants as a benchmark security to assess and price the other three life insurers. Hence, our first two hypotheses can be stated as follows: Hypothesis 1 (H1): For the overall period of study, we expect that IAG, MFC, and SLF all are significantly cointegrated with GWO.7 Hypothesis 2 (H2): Cointegration relationships between IAG, MFC, SLF, and GWO are also stronger (statistically) during the first sub-period compared to the second sub-period. Moreover, we consider that it is also very likely that the three newly listed life insurers (IAG, MFC, and SLF) will exhibit strongly significant statistical cointegration relationships among themselves during the entire period of study, simply due to the fact that they were considered by market participants as the new kids on the block. In addition, we feel that the intensity of these cointegration relationships is likely to be directly linked to the actual order of demutualization, particularly during the first subperiod.8 Indeed, we think that as time went by, market participants learned more and more about the specifics of each firm so that the intensity of these cointegration relationships became somewhat independent of the actual order of demutualization. 7
The order of demutualization of the three firms is as follows: (1) MFC (September 1999), (2) IAG (February 2000), and (3) SLF (March 2000). 8 The meaning we give to the word “intensity” has to do with whether the empirical results based on the different tests required by Engle and Granger’s methodology indicate either a weak or a strong statistical significance for these cointegration relationships. Here, we follow a practice similar to the one used by Su et al. (2007) in reporting their panel data cointegration results on the relationship between stock prices and dividends on the Taiwan stock market.
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Therefore, our last three hypotheses are stated as follows: Hypothesis 3 (H3): For the overall period of study, IAG, MFC, and SLF are significantly cointegrated among themselves. Hypothesis 4 (H4): For the first sub-period, the intensity of these cointegration relationships is directly linked to the actual order of demutualization. Hypothesis 5 (H5): For the second sub-period, the intensity of these cointegration relationships is independent of the actual order of demutualization.
PRE-TESTING ANALYSIS FOR THE ENTIRE PERIOD Recall that pre-testing is about determining whether the variables being considered are indeed distributed I(1). These variables are each firm’s stock price series over the entire period (March 23, 2000 to December 31, 2007). To test for unit roots, an ADF(K*) test was performed on each variable. Here, the optimal lag order (K*) is defined as the minimum lag order value (ranging between 1 and 25) for each of the three information criteria identified above. Table 1 shows the results of the process leading to the determination of the optimal lag order for each stock in our sample, according to the ADF(K*) tests: Table 1. Optimal Lag Order (K*) for Each Security According to the Three Criteria of Information
Minimum Optimal Minimum under lag under HC order SC
Optimal lag order Optimal according to the three lag criteria order
Stock
Minimum under AIC
Optimal lag order
GWO
0.380828
20
0.399277
1
0.394043
1
1 and 20
IAG
0.511117
4
0.520660
1
0.515426
1
1 and 4
MFC
0.699425
7
0.709767
1
0.704533
1
1 and 7
SLF
1.624572
16
1.640151
1
1.634917
1
1 and 16
The third column represents the lag order corresponding to a minimum under AIC, the fifth column represents the lag order corresponding to a minimum under SC, and the seventh column represents the lag order corresponding to a minimum under HC. The eighth column represents the lag order according to three criteria (AIC, SC, and HC).
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Table 2. Augmented Dickey-Fuller Tests for Unit Roots Securities
Optimal lag order (K*)
Levels
First differences
GWO
1 20
–1.699892 –1.354216
–33.15854*** –11.72769***
IAG
1 4
–1.281011 –1.157362
–33.66443*** –21.28854***
MFC
1 7
–1.516335 –1.398765
–32.63254*** –17.79966***
SLF
1 16
–2.512866 –2.215506
–32.19429*** –11.62204***
The null hypothesis of these tests is that variables contain a unit root (implying nonstationarity), while the alternative is that the variables are I(0). The statistics of the ADF (K*) tests marked with (***), (**), and (*) reject the null hypothesis at a significance level of 1%, 5%, and 10%, respectively. The 1%, 5%, and 10% critical values for the ADF test are –3.43, –2.86, and –2.56, respectively (see MacKinnon, 1996).
After determining the optimal lag order for each security, we then applied the ADF(K*) test to both levels and first differences of the price series for each firm. Results of these tests are presented in Table 2. In interpreting the content of Table 2, let us focus on the case of the Industrial Alliance Group (IAG). As we can see, the ADF(1) and ADF(4) statistics for the stock’s level series (–1.281011 and –1.157362, respectively) are not even significant at 10%; thus we can infer that the level series is not stationary because of the presence of a unit root. With respect to ADF(1) and ADF(4) tests on first differences (–33.66443 and –21.28854, respectively) we can reject the null hypothesis, and therefore accept stationarity. This result confirms that IAG is indeed integrated of order 1 [IAG ~ I(1)]. Results for the other three securities (GWO, MFC, and SLF) can be interpreted in the same way. So we can argue that the four selected securities are indeed candidates for cointegration, given that they are distributed I(1). Morever, the individual stock price time series of levels and first differences for the four securities shown in Figures 1 to 8 (see Appendix 1) also confirm this result. Indeed, these figures provide very strong visual evidence of mean nonstationarity in levels and mean stationarity in the first differences of IAG, GWO, MFC, and SLF. This means the following six relations of cointegration can now be tested using Engle and Granger’s two-step method: (IAG-GWO), (MFCGWO), (SLF-GWO), (IAG-MFC), (IAG-SLF), and (MFC-SLF).
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COINTEGRATION TESTING RESULTS FOR THE ENTIRE PERIOD First step : To test the above six pairs of cointegration relations, we estimate through OLS the following cointegration equations : Equation 1: IAG t = α 1 + β 1 GWO t + ε 1 ,t Equation 2: MFC t = α 2 + β 2 GWO t + ε 2 ,t Equation 3: SLF t = α 3 + β 3 GWO t + ε 3 ,t Equation 4: IAG t = α 4 + β 4 MFC t + ε 4 ,t Equation 5: IAG t = α 5 + β 5 SLF t + ε 5 ,t Equation 6: MFC t = α 6 + β 6 SLF t + ε 6 ,t OLS results are shown in Table 3. Table 3. OLS Results of Cointegration Equations Intercept
GWO
MFC
SLF
2 · R – adj.
IAG
0.6589*** (0.000)
1.0700*** (0.000)
–
–
91.31%
MFC
–1.7481*** (0.000)
1.1801*** (0.000)
–
–
93.08%
SLF
9.4959*** (0.000)
1.1832*** (0.000)
–
–
86.50%
IAG
1.2574*** (0.000)
–
0.8941*** (0.000)
IAG
–6.9183*** (0.000)
–
–
0.8425*** (0.000)
91.63%
MFC
–8.8870*** (0.000)
–
–
0.9355*** (0.000)
94.67%
95.40%
(***), (**), (*) statistically significant at 1%, 5%, and 10%, respectively. Numbers in parentheses represent p-value.
As mentioned above, our purpose here is to extract the error terms ( εˆ t ) of the six cointegration regression equations ( εˆ = Y + αˆ + βˆ X ). Thus, our t
t
t
intent is to test whether these error terms εˆ t are indeed stationary, using
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CANADIAN LIFE INSURANCE STOCKS
again an ADF (K*) test similar to the one used before in order to choose the optimal lag order K*. Hence, if the error terms εˆ are found to be stationary t
[ εˆ t ~ I(0)], then such a result would allow us to continue on with the analysis. Table 4 shows the results of the process leading to the determination of the optimal lag order of the error terms ( ε 1 ,t , ε 2 ,t , ε 3 ,t , ε 4 ,t , ε 5 ,t ,
and ε 6 ,t ) for the six cointegration equations, according to the ADF(K*) tests. Table 4. Optimal Lag Order of Error Terms for Each Cointegration Equation According to the Three Criteria of Information
Minimum Optimal Minimum Optimal Minimum Optimal lag under lag under lag E r r o r under order HC order SC order AIC terms
Optimal lag order according to the three criteria
ε 1 ,t
0.933111
11
0.948159
2
0.940880
4
2, 4, and 11
ε 2 ,t
0.994804
7
1.012563
1
1.003923
7
1 and 7
ε 3 ,t
1.639723
7
1.659419
1
1.650183
6
1, 6, and 7
ε 4 ,t
0.846696
1
0.854961
1
0.849728
1
1
ε 5 ,t
1.360373
20
1.374339
1
1.369017
2
1, 2, and 20
ε 6 ,t
1.104618
25
1.123780
1
1.115328
6
1, 6, and 25
The third column represents the lag order corresponding to a minimum under AIC, the fifth column represents the lag order corresponding to a minimum under SC, and the seventh column represents the lag order corresponding to a minimum under HC. The eighth column represents the lag order according to three criteria (AIC, SC, and HC).
After determining the optimal lag order, we then applied the ADF(K*) test on the error term series for the six cointegration equations. Results of these tests are presented in Table 5.9 Overall, these results lead us to proceed with the second step of our cointegration analysis for all pairs of stocks except (IAG–GWO), for which ADF tests results do not clearly indicate statistical stationarity of the error terms. Indeed, in this particular case, ADF(2) shows stationarity of the error 9
Graphs of error term behavior (levels) for the six cointegration equations are shown in Appendix 2.
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Table 5. Augmented Dickey-Fuller Tests for Unit Roots on Error Terms for Each Cointegration Equation Error terms
Optimal lag order (K*)
Levels
ε 1 ,t
2 4 11
–2.994209* –2.676929 –2.127456
ε 2 ,t
1 7
–3.641929** –2.875892
ε 3 ,t
1 6 7
–3.997793*** –3.137292* –3.180047*
ε 4 ,t
1
–4.392254***
ε 5 ,t
1 2 20
–4.832929*** –4.604904*** –3.438917**
ε 6 ,t
1 6 25
–5.367039*** –4.360487*** –4.027118***
The null hypothesis of these tests is that variables contain a unit root (implying nonstationarity), while the alternative is that the variables are I(0). The statistics of the ADF (K*) tests marked with (***), (**) and (*), reject the null hypothesis at a significance level of 1%, 5%, and 10%, respectively. The 1%, 5%, and 10% critical values for the ADF test are –3.78, –3.25, and –2.98, respectively (see Engle and Yoo, 1987).
term ( ε 1 ,t ) only at a 10% significance level. This allows us to conclude that these two securities were clearly not cointegrated during the entire period covered by our analysis. Second step: Table 6 shows the estimation results of Error-Correction Models (ECM) of the different cointegration relationships found for the overall period. The results indicate that for each pair of Error-Correction Models at least one of the lambdas ( λ 1 or λ 2 ) is statistically and significantly different from zero. The significance of at least one of the lambdas means that one life insurer’s returns are adjusting to the other one. As an example, let us look at the case of (MFC–SLF). The first ECM representation has ΔMFC t as the dependent variable, while the other has ΔSLF t as the dependent variable (see the last two lines at the bottom of Table 6). The value of the coefficient λ 2 indicates the speed of adjustment of any disequilibrium towards a long-run equilibrium, and its statistical significance means that the size of ε 6 ,t – 1 influences ΔSLF t . Overall, these results mean that the different cointegration relationships found in step one of our cointegration tests are validated by the ECM representations.
0.011* (0.076) 0.016 (0.170) 0.011* (0.068) 0.014** (0.033) 0.012* (0.097) 0.014** (0.033) 0.016 (0.163) 0.013* (0.086) 0.016 (0.173)
ΔGWO t = μ 2 + λ 2 ε 2 ,t – 1 + δ 2 ΔMFC t – 1 + η 2 ΔGWO t – 1 + u' t
ΔSLF t = μ 1 + λ 1 ε 3 ,t – 1 + δ 1 ΔGWO t – 1 + η 1 ΔSLF t – 1 + u t
ΔGWO t = μ 2 + λ 2 ε 3 ,t – 1 + δ 2 ΔMFC t – 1 + η 2 ΔGWO t – 1 + u' t
ΔIAG t = μ 1 + λ 1 ε 4 ,t – 1 + δ 1 ΔMFC t – 1 + η 1 ΔIAG t – 1 + u t
ΔMFC t = μ 2 + λ 2 ε 4 ,t – 1 + δ 2 ΔIAG t – 1 + η 2 ΔMFC t – 1 + u' t
ΔIAG t = μ 1 + λ 1 ε 5 ,t – 1 + δ 1 ΔSLF t – 1 + η 1 ΔIAG t – 1 + u t
ΔSLF t = μ 2 + λ 2 ε 5 ,t – 1 + δ 2 ΔIAG t – 1 + η 2 ΔSLF t – 1 + u' t
ΔMFC t = μ 1 + λ 1 ε 6 ,t – 1 + δ 1 ΔSLF t – 1 + η 1 ΔMFC t – 1 + u t
ΔSLF t = μ 2 + λ 2 ε 6 ,t – 1 + δ 2 ΔMFC t – 1 + η 2 ΔSLF t – 1 + u' t
0.025*** (0.000)
–0.001 (0.672)
0.015*** (0.003)
–0.009*** (0.001)
0.003 (0.427)
–0.018*** (0.000)
0.001 (0.440)
–0.013*** (0.000)
0.006** (0.024)
–0.006** (0.049)
λˆ
δˆ
0.016 (0.706)
0.034** (0.040)
–0.034 (0.398)
0.019 (0.138)
0.050** (0.046)
0.030 (0.148)
0.046*** (0.000)
–0.006 (0.878)
0.097*** (0.000)
–0.011 (0.684)
(***), (**), (*) statistically significant at 1 %, 5 %, and 10 %, respectively. Numbers in parentheses represent p-values.
0.013* (0.081)
μˆ
ΔMFC t = μ 1 + λ 1 ε 2 ,t – 1 + δ 1 ΔGWO t – 1 + η 1 ΔMFC t – 1 + u t
ECM
Table 6. Estimation Results of Error-Correction Models (ECM) for the Overall Period
0.014 (0.593)
–0.056** (0.036)
0.025 (0.269)
–0.058** (0.011)
–0.035 (0.123)
–0.055** (0.015)
–0.114*** (0.000)
0.019 (0.406)
–0.123*** (0.000)
–0.019 (0.193)
ηˆ
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Table 7. Augmented Dickey-Fuller Tests for Unit Roots Securities
Optimal lag order (K*)
Levels
First differences
Panel A : First sub-period GWO
1 20
–1.851499 –0.836856
–24.17710*** –8.897728***
IAG
1 5
–2.683420 –2.453487
–23.84780*** –14.16083***
MFC
1 6
–2.900069 –2.652559
–22.54566*** –13.16146***
SLF
1 16
0.533961 0.771041
–22.33684*** –8.507132***
–1.366581 –1.447257
–22.77986*** –19.06575***
Panel B : Second sub-period GWO
1 2
IAG
1
–1.114532
–23.75187***
MFC
1 5
–1.687786 –1.681201
–23.92561*** –13.99395***
SLF
1
–1.714315
–23.41099***
The null hypothesis of these tests is that variables contain a unit root (implying nonstationarity), while the alternative is that the variables are I(0). The statistics of the ADF (K*) tests marked with (***), (**), and (*) reject the null hypothesis at a significance level of 1%, 5%, and 10%, respectively. The 1%, 5%, and 10% critical values for the ADF test are –3.43, –2.86, and –2.56, respectively (see MacKinnon, 1996).
PRE-TESTING ANALYSIS FOR BOTH SUB-PERIODS We now pre-test for two sub-periods. The first sub-period goes from March 23, 2000 to February 10, 2004, whereas the second sub-period extends from February 11, 2004 to December 31, 2007. In each sub-period, our sample includes 1014 observations for every stock. Having determined the optimal lag order, we then applied ADF(K*) tests to both levels and first differences of the price series for every security during both sub-periods. Results for unit root tests are shown in Table 7. Results for both sub-periods show that the four securities are integrated of order 1. This means that the prerequisite condition for cointegration analysis is met. This also means that we can go on with Engle and Granger’s two-step method in order to test for cointegration in both subperiods.
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CANADIAN LIFE INSURANCE STOCKS
Table 8. OLS Results of Cointegration Equations Intercept
2
GWO
MFC
SLF
R – adj.
Panel A: First sub-period IAG
3.5061*** (0.000)
0.8491*** (0.000)
–
–
51.75%
MFC
4.8082*** (0.000)
0.8088*** (0.000)
–
–
54.36%
SLF
14.586*** (0.000)
0.9019*** (0.000)
–
–
27.86%
IAG
–0.1528 (0.576)
–
0.9774*** (0.000)
IAG
1.7746*** (0.000)
–
–
0.5493*** (0.000)
63.09%
MFC
1.8625*** (0.000)
–
–
0.5656*** (0.000)
77.44%
82.51%
Panel B: Second sub-period IAG
–10.247*** (0.000)
1.3826*** (0.000)
–
–
87.95%
MFC
–6.1970*** (0.000)
1.3301*** (0.000)
–
–
84.46%
SLF
2.8881*** (0.000)
1.4010*** (0.000)
–
–
85.68%
IAG
–1.1232*** (0.002)
–
0.9595*** (0.000)
IAG
–9.4076*** (0.000)
–
–
0.9044*** (0.000)
86,21%
MFC
–7.6523*** (0.000)
–
–
0.9206*** (0.000)
92.69%
88.73%
(***), (**), (*) statistically significant at 1%, 5%, and 10%, respectively. Numbers in parentheses represent p-value.
COINTEGRATION TESTING RESULTS FOR BOTH SUB-PERIODS First step : For each of the following pairs of securities [(IAG-GWO), (MFCGWO), (SLF-GWO), (IAG-MFC), (IAG-SLF), and (MFC-SLF)], and for both sub-periods, cointegration regression equations are estimated using OLS in order to extract the error terms. These OLS regression results are shown in Table 8.
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Table 9. Augmented Dickey-Fuller Tests for Unit Roots on Error Terms for First Sub-period Error terms
Optimal lag order (K*)
Levels
ε 1 ,t'
1 4 10
–2.287658 –1.834357 –1.702635
ε 2 ,t'
1 6 7
–3.006814* –2.331830 –2.239465
ε 3 ,t'
1 7
–3.089465* –2.598523
ε 4 ,t'
1
–4.590546***
ε 5 ,t'
1 11
–3.204155* –1.932528
ε 6 ,t'
4 6
–3.251848** –2.786480
The null hypothesis of these tests is that variables contain a unit root (implying nonstationarity), while the alternative is that the variables are I(0). The statistics of the ADF (K*) tests marked with (***), (**), and (*) reject the null hypothesis at a significance level of 1%, 5%, and 10%, respectively. The 1%, 5%, and 10% critical values for the ADF test are –3,78, –3,25, and –2.98, respectively (see Engle and Yoo, 1987).
After determining the optimal lag order for both sub-periods, we then apply the ADF(K*) test to error term series for the six cointegration equations. Results for the first sub-period are shown in Table 9, while those of the second sub-period appear in Table 10. Results shown in Table 9 suggests we should proceed with the second step of our cointegration analysis for all pairs of stocks, except for (IAGGWO) and (MFC-GWO), both of which do not clearly indicate statistical stationarity of their error terms in the first sub-period. Likewise, Table 10 suggests continuing on with the second step of our cointegration analysis for all pairs of stocks, except for (MFC-GWO) and (IAG-MFC), again because they do not meet the requirement of statistical stationarity of their error terms in the second sub-period. From the above analysis, it appears that Great-West (GWO) and Manulife (MFC) lose their property of being cointegrated once we split our entire period of study in two sub-periods. Second step: Table 11 shows the estimation results of Error-Correction Models (ECM) of the different cointegration relationships found for the first sub-period (Panel A) and the second one (Panel B). The results indicate
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CANADIAN LIFE INSURANCE STOCKS
Table 10. Augmented Dickey-Fuller Tests for Unit Roots on Error Terms for Second Sub-period Error terms
Optimal lag order (K*)
Levels
ε 1 ,t''
2 3
–3.009551* –2.811106
ε 2 ,t''
1 3
–2.809267 –2.653370
ε 3 ,t''
1 3
–3.423097** –3.106024*
ε 4 ,t''
1
–2.544838
ε 2 ,t''
1 2
–3.065004* –2.789475
ε 2 ,t''
1 3
–3.423097** –3.106024*
The null hypothesis of these tests is that variables contain a unit root (implying nonstationarity), while the alternative is that the variables are I(0). The statistics of the ADF (K*) tests marked with (***), (**), and (*) reject the null hypothesis at a significance level of 1%, 5%, and 10%, respectively. The 1%, 5%, and 10% critical values for the ADF test are –3.78, –3.25, and –2.98, respectively (see Engle and Yoo, 1987).
that for each pair of Error-Correction Models at least one of the lambdas ( λ 1 or λ 2 ) is statistically and significantly different from zero. This means that, for both sub-periods, the different cointegration relationships found in step one are validated by the ECM representations.
DISCUSSION OF RESULTS IN LIGHT OF OUR RESEARCH HYPOTHESES From our previous empirical cointegration tests, we observe that the results of the second step of Engle and Granger’s procedure, for both the entire period of study and the two sub-periods, do confirm those of step one. Hence, we can rely on the results of step one in order to validate whether our five research hypotheses are met. Based on our interpretation of the results shown in Tables 5, 9, and 10 (step one of Engle and Granger’s procedure), Table 12 below provides a synthesis of the statistical intensity of the cointegration relationships among the different pairs of securities.
0.021 (0.264) 0.017* (0.057) 0.013 (0.134) 0.011 (0.262) 0.013 (0.132) 0.021 (0.269) 0.012 (0.243) 0.020 (0.291)
ΔGWO t' = μ 2 + λ 2 ε 2 ,t' – 1 + δ 2 ΔMFC t' – 1 + η 2 ΔGWO t' – 1 + u' t'
ΔIAG t' = μ 1 + λ 1 ε 4 ,t' – 1 + δ 1 ΔMFC t' – 1 + η 1 ΔIAG t' – 1 + u t'
ΔMFC t' = μ 2 + λ 2 ε 4 ,t' – 1 + δ 2 ΔIAG t' – 1 + η 2 ΔMFC t' – 1 + u' t'
ΔIAG t' = μ 1 + λ 1 ε 5 ,t' – 1 + δ 1 ΔSLF t' – 1 + η 1 ΔIAG t' – 1 + u t'
ΔSLF t' = μ 2 + λ 2 ε 5 ,t' – 1 + δ 2 ΔIAG t' – 1 + η 2 ΔSLF t' – 1 + u' t'
ΔMFC t' = μ 1 + λ 1 ε 6 ,t' – 1 + δ 1 ΔSLF t' – 1 + η 1 ΔMFC t' – 1 + u t'
ΔSLF t' = μ 2 + λ 2 ε 6 ,t' – 1 + δ 2 ΔMFC t' – 1 + η 2 ΔSLF t' – 1 + u' t'
μˆ
ΔSLF t' = μ 1 + λ 1 ε 3 ,t' – 1 + δ 1 ΔGWO t' – 1 + η 1 ΔSLF t' – 1 + u t'
Panel A: First sub-period
ECM
0.007 (0.581)
–0.020*** (0.009)
0.005 (0.601)
–0.017*** (0.000)
0.018** (0.020)
–0.023*** (0.000)
0.000 (0.817)
–0.014*** (0.003)
λˆ
Table 11. Estimation Results of Error-Correction Models (ECM) for the Two Sub-periods
0.077 (0.266)
0,023 (0.262)
–0.044 (0.508)
0.013 (0.404)
0.051 (0.167)
0.019 (0.488)
0.036** (0.017)
–0.044 (0.511)
δˆ
0.015 (0.682)
0,057 (0.132)
0.046 (0.153)
–0.032 (0.309)
0.077** (0.018)
–0.026 (0.417)
–0.143*** (0.000)
0.049 (0.126)
ηˆ
124 BERNIER AND MOUELHI
0.006 (0.508) 0.012 (0.406) 0.006 (0.483) 0.015 (0.121) 0.012 (0.395) 0.013 (0.198) 0.012 (0.396)
ΔGWO t' = μ 2 + λ 2 ε 1 ,t'' – 1 + δ 2 ΔIAG t'' – 1 + η 2 ΔGWO t'' – 1 + u' t''
ΔSLF t'' = μ 1 + λ 1 ε 3 ,t'' – 1 + δ 1 ΔGWO t'' – 1 + η 1 ΔSLF t'' – 1 + u t''
ΔGWO t'' = μ 2 + λ 2 ε 3 ,t'' – 1 + δ 2 ΔSLF t'' – 1 + η 2 ΔGWO t'' – 1 + u' t''
ΔIAG t'' = μ 1 + λ 1 ε 5 ,t'' – 1 + δ 1 ΔSLF t'' – 1 + η 1 ΔIAG t'' – 1 + u t''
ΔSLF t'' = μ 2 + λ 2 ε 5 ,t'' – 1 + δ 2 ΔIAG t'' – 1 + η 2 ΔSLF t'' – 1 + u' t''
ΔMFC t'' = μ 1 + λ 1 ε 6 ,t'' – 1 + δ 1 ΔSLF t'' – 1 + η 1 ΔMFC t'' – 1 + u t''
ΔSLF t'' = μ 2 + λ 2 ε 6 ,t'' – 1 + δ 2 ΔMFC t'' – 1 + η 2 ΔSLF t'' – 1 + u' t''
0.030*** (0.002)
–0.002 (0.763)
0.012** (0.041)
–0.009* (0.056)
0.004 (0.247)
–0.017*** (0.009)
0.008* (0.068)
–0.012** (0.015)
λˆ
–0,026 (0.613)
0.020 (0.466)
–0.016 (0.731)
0.024 (0.281)
0.064*** (0.002)
0.043 (0.412)
0.064** (0.028)
–0.019 (0.583)
δˆ
(***), (**), (*) statistically and significant at 1%, 5%, and 10%, respectively. Numbers in parentheses represent p-values.
0.016 (0.115)
μˆ
ΔIAG t'' = μ 1 + λ 1 ε 1 ,t'' – 1 + δ 1 ΔGWO t'' – 1 + η 1 ΔIAG t'' – 1 + u t''
Panel B: Second sub-period
ECM
–0.013 (0.730)
–0.141*** (0.000)
–0.021 (0.514)
–0.079** (0.016)
–0.091*** (0.006)
–0.035 (0.295)
–0.068** (0.034)
–0.063* (0.051)
ηˆ
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Table 12. Statistical Intensity of Cointegration Relationships: A Synthesis Pair of securities
Overall period
First sub-period
Second sub-period
IAG-GWO
No
No
Weak
MFC-GWO
Weak
No
No
SLF-GWO
Strong
Weak
Strong
IAG-MFC
Very strong
Very strong
No
IAG-SLF
Very strong
Weak
Weak
MFC-SLF
Very strong
Weak
Strong
As shown in Table 5, it is noticeable that, over the entire period of study, the cointegration relationship of (SLF, GWO) was strong while the one between (MFC, GWO) was weak. Indeed, for the model (SLF-GWO), the ADF(1) test indicates stationarity of the error terms obtained from equation 3 at a 1% significance level. Also, both the ADF(6) and ADF(7) test results for equation 3 also confirm stationarity of the error terms at a 10% significance level. Accordingly, it is fair to claim the existence of a strong cointegration relationship between Great-West (GWO) and Sun Life (SLF). Moreover, for the model (MFC-GWO), the ADF(1) test indicates stationarity of the error terms extracted from equation 2 at a 5% significance level. However, ADF(7) test indicates no stationarity. These results are supportive of a rather weak cointegration relationship between Great-West (GWO) and Manulife (MFC). The results of Table 5 also show that there is no cointegration relationship between IAG and GWO. This might be because IAG was much smaller and less diversified than the other three firms when it demutualized itself. Hence, these results allow us to accept our first hypothesis (H1). Results shown in Tables 9 and 10 are not supportive of hypothesis 2 (H2) since among the three demutualized firms, during the first subperiod, only SLF was cointegrated with GWO. However, during the second sub-period, we found significant cointegration relationships for the following pairs: (IAG-GWO) and (SLF-GWO). Indeed, the cointegration relationship of SLF with GWO appear to be even more powerful during the second sub-period but not during the first one, as hypothesized. The latter is indicated by our ADF(K*) tests results found in Table 10, which reject the null hypothesis of non stationarity of the error terms at the 5% and 10% confidence levels. On the other hand, the ADF(1) test result shown in Table 9 indicates stationary error terms at the 10% significance level. However, the ADF(7) test result shows no stationarity. Hence, this can be seen as evidence that, contrary to our expectations, market participants did not
CANADIAN LIFE INSURANCE STOCKS
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rely on GWO as a benckmark security right after the demutualization wave of 1999–2000. Moreover, for the entire period of study, pairs of securities that include only the three demutualized life insurers [(IAG-MFC), (IAG-SLF), and (MFC-SLF)] do indicate very strong cointegration relationships. Indeed, for these pairs of securities, ADF tests results in Table 5 clearly indicate stationary error terms at a 1% significance level, independently of the lag order chosen for testing purposes.10 This evidence is in support of hypothesis 3 (H3). This means that these securities do share a long-term linear relationship that remains stable over time and which may be regarded as an attractor. Deviations from the line represent the short-term (temporary) movements around equilibrium. In addition, we can notice that the intensity of the cointegration relationships among the three demutualized securities during the first subperiod seems to depend upon the order of demutualization of these securities.11 Indeed, cointegration between IAG (second to demutualize) and MFC (first to demutualize) is very strong given that the null hypothesis of non stationarity is rejected at the 1% significance level as shown in Table 9. On the contrary, the relationship between IAG (2nd to demutualize) and SLF (third to demutualize) is weak. In effect, the ADF(1) test indicates stationary error terms generated from equation 5 at a 10% significance level. However, the ADF(11) test result shows no stationarity. At the same time, the cointegration relationship between MFC (first to demutualize) and SLF (third to demutualize) is found to be weak. For the latter, the ADF(4) test indicates stationary error terms at a 5% significance level. However, the ADF(6) test result indicates no stationarity. Overall, these results are in support of hypothesis 4 (H4), to the effect that, for the first sub-period, cointegration relationships depend upon the order of demutualization of the three life insurers that adopted such a strategy. However, as shown in Table 10, the previous result is not verified during the second sub-period since IAG (second to demutualize) and MFC (first to demutualize) are not cointegrated. In addition, we can see that IAG (second to demutualize) and SLF (third to demutualize) show a weak cointegration relationship. Indeed, the ADF(1) test indicates stationarity at a 10% significance level. However, the ADF(2) test result shows no stationarity of the error terms. At the same time, the cointegration relationship 10
This holds true except for the model (IAG-SLF) in the case of the ADF(20) test, which shows stationarity of the error terms at a 5% significance level. 11 We repeat that the order of demutualization of the three firms is as follows: (1) MFC (September 1999), (2) IAG (February 2000) and (3) SLF (March 2000).
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between MFC (first to demutualize) and SLF (third to demutualize) is found to be strong, as indicated by the ADF(1) and the ADF(3) tests results rejecting the null hypothesis of nonstationarity at a 5% and a 10% significance level, respectively. This is supportive of hypothesis 5 (H5) to the effect that, during the second sub-period, cointegration relationships among the securities of recently demutualized life insurers are independent of the actual order of demutualization. This is consistent with the fact that investors stopped considering MFC, IAG, and SLF as new comers to the market, four years after demutualization.
CONCLUSION The major conclusion that can be drawn from this cointegration analysis is that, during the post-demutualization period, the Canadian stock market appears to have been inefficient (weak-form) in pricing the securities of the four life insurance firms that are part of the S&P/TSX index of Canada’s financial sector. It is particularly interesting to notice that, over the entire period of our study (3/2000 to 12/2007), the three newly demutualized life insurers have shown very strong cointegration relationships among themselves. Moreover, during the first half of this time interval, the statistical intensity of these cointegration relationships has been shown to depend upon the actual order of demutualization. Overall, this means that these securities did exhibit a long-term comovement behavior that remained stable during the investigation period. Short-term deviations from this equilibrium relationship were temporary in nature, so that mean-reversion was always prevalent. This also means that, during this time period, it would have been possible to predict the future price behavior of these life insurance stocks by relying on past information in order to potentially “beat the market” in the sense of earning abnormal (excess) returns. Future research could attempt to test whether a “statistical arbitrage” strategy using pairs of Canadian life insurance stocks could indeed lead to abnormal returns. Likewise, one could also consider running a crosscointegration analysis between the four Canadian life insurance shares and a group of listed securities representative of the US Life and Health insurance sector.
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REFERENCES Babin, M and G Bernier (2001) Démutualisation des sociétés canadiennes d’assurance de personnes: une caractérisation de leurs premiers appels publics à l’épargne, Insurance and Risk Management Review 2: 229–258. Bernier, G and A Nathan (2007) A Descriptive Analysis of Canadian Insurance Markets, Chapter 8: 403–453; as printed in JD Cummins and B Venard (Eds.) Handbook of International Insurance and Financial Markets: Between Global Dynamics and Local Contingencies, New York: Springer. Chan, EP (2006) Cointegration Is Not the Same as Correlation (www.tradingmarkets.com/.site/stocks/commentary/quantitative_trading/Cointegration-isnot-the-same-as-correlation.cfm) CIBC World Markets (1999) Shifting Sands: The Transformation of the Canadian Life Insurance Industry, CEO Seminar/Financial Services: Canadian Imperial Bank of Commerce. Copeland, L (1991) Cointegration Tests with Daily Exchange Rate Data, Oxford Bulletin of Economics and Statistics 53 #2: 185–198. Copeland, TE, JF Weston, and K Shastri (2005) Financial Theory and Corporate Policy (4th), Boston: Pearson Addison Wesley. Dickey, D and W Fuller (1981) Likelihood Ratio Statistics for Autoregressive Time Series with a Unit Root, Econometrica 49: 1057–1072. Engle, R and CW Granger (1987) Co-integration and Error Correction: Representation, Estimation and Testing, Econometrica 55: 251–276. Engle, R and BS Yoo (1987) Forecasting and Testing in Cointegrated Systems, Journal of Econometrics 35: 143–159. Fama, EF (1970) Efficient Capital Markets: A Review of Theory and Empirical Work, Journal of Finance 25: 383–417. Granger, CW (1983) Cointegrating Variables and Error Correcting Models, Discussion Paper 83-13, San Diego: University of California, San Diego. Granger, CWJ and AA Weiss (1983) Time Series Analysis of Error Correction Models, in S Karlin, T Amemiya, and LA Goodman (Eds.) Studies in Econometrics, Time Series and Multivariate Statistics, New York: Academic Press. Hakkio, CS and M Rush (1989) Market Efficiency and Cointegration: An Application to the Sterling and Deutschemark Exchange Markets, Journal of International Money and Finance 8: 75–88. Hakkio, CS and M Rush (1991) Cointegration: How Short Is the Long Run?, Journal of International Money and Finance 10: 571–581. Haley, JD (1993) A Cointegration Analysis of the Relationship between Underwriting Margins and Interest Rates: 1930–1989, Journal of Risk and Insurance 60 (3): 480–493. Haley, JD (2007) Further Considerations of Underwriting Margins, Interest Rates, Stability, Stationarity, Cointegration, and Time Trends, Journal of Insurance Issues 30 (1) (Spring): 62–75. Hall, A (1994) Testing for Unit Root in Time Series with Pretest Data-Based Model Selection, Journal of Business and Economics Statistics 12: 461–470.
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Johansen, S (1988) Statistical Analysis of Cointegration Vectors, Journal of Economic Dynamics and Control 12: 231–254. Johansen, S (1991) Estimation and Hypothesis Testing of Cointegration Vectors in Gaussian Vector Autoregressive Models, Econometrica 59: 1551–1580. Lo, AW and AC MacKinlay (1988) Stock Market Prices Do Not Follow Random Walks: Evidence from a Simple Specification Test, Review of Financial Studies 1 (1): 41–66. MacKinnon, JG (1994) Approximate Asymptotic Distribution Functions for Unit Root and Cointegration Tests, Journal of Business and Economic Statistics 12: 167– 176. MacKinnon, JG (1996) Numerical Distribution Functions for Unit Root and Cointegration Tests, Journal of Applied Econometrics 11: 601–618. Ng, S and P Perron (1995) Unit-Root Tests in ARMA Models with Data-dependent Methods for the Selection of the Truncation Lag, Journal of the American Statistical Association 90: 268–281. Su, C-W, H Chang, T Chang, and C-C Wei (2007) Re-examining the Relationship between Stock Prices and Dividends: Evidence Based on Taiwan Panel Data Investigation, The Business Review 7 (1): 137–142. Swiss Re (2008) World Insurance in 2007: Emerging Markets Leading the Way, sigma No. 3/2008.
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APPENDIX 1 Stock Price Behavior (Levels and First-Differences) of the Four Canadian Life Insurers: Overall Period
GWO
D1GWO
40
4
36
3
32
2
28 1
24 0
20
-1
16
-2
12
-3
8 250
500
750
1000
1250
1500
1750
250
2000
Fig. 1. Stock price behavior of GWO
500
750
1000
1250
1500
1750
2000
Fig. 2. First differences for GWO
IAG
D1IAG
44
3
40 2
36 32
1
28 24
0
20 -1
16 12
-2
8 250
500
750
1000
1250
1500
1750
250
2000
Fig. 3. Stock price behavior of IAG
500
750
1000
1250
1500
1750
2000
Fig. 4. First differences for IAG
D1MFC
MFC 1.6
45
1.2
40
0.8 35
0.4
30
0.0
25
-0.4 -0.8
20
-1.2 15
-1.6
10
-2.0 -2.4
5 250
500
750
1000
1250
1500
1750
250
2000
Fig. 5. Stock price behavior of MFC
500
750
1000
1250
1500
1750
2000
Fig. 6. First differences for MFC
SLF
D1SLF
60
3 2
50
1
40 0
30 -1
20
-2 -3
10 250
500
750
1000
1250
1500
1750
2000
Fig. 7. Stock price behavior of SLF
250
500
750
1000
1250
1500
1750
2000
Fig. 8. First differences for SLF
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APPENDIX 2 Error Term Behavior (Levels) of the Six Cointegration Equations: Overall Period RESDIAGGWO
RESDMFCGWO
8
6
6
4
4
2 2
0 0
-2
-2
-4
-4 -6 250
500
750
1000
1250
1500
1750
2000
Fig. 9. Errors of IAG-GWO regression
-6 250
500
750
1000
1250
1500
1750
2000
Fig. 10. Errors of MFC-GWO regression
RESDSLFGWO
RESDIAGMFC
12
8 6
8
4 4
2 0
0
-2 -4
-4 -8 250
500
750
1000
1250
1500
1750
2000
Fig. 11. Errors of SLF-GWO regression
-6 250
500
750
1000
1250
1500
1750
2000
Fig. 12. Errors of IAG-MFC regression RESDMFCSLF
RESDIAGSLF 8
6
6
4
4
2
2 0
0 -2
-2 -4
-4
-6
-6
-8 250
500
750
1000
1250
1500
1750
2000
Fig. 13. Errors of IAG-SLF regression
-8 250
500
750
1000
1250
1500
1750
2000
Fig. 14. Errors of MFC-SLF regression
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