Portfolio Construction Strategies Using Cointegration M.Sc. Student: IONESCU GABRIEL Supervisor: Professor MOISA ALTAR BUCHAREST, JUNE 2002 ACADEMY OF ECONOMIC STUDIES DOCTORAL SCHOOL OF FINANCE AND BANKING

2 1.Introduction  Traditional models seek portfolio weights so as to minimize the variance of the portfolio for a given level of return.  Portfolio variance is measured using a covariance matrix which is not only difficult to estimate, but also very unstable in time.  Additionally, the mean-variance criterion has nothing to ensure that portfolio deviations (errors) relative to a benchmark are stationary, in the majority of cases being a random walk.  As a consequence the portfolio will drift virtually anywhere away from the benchmark unless is frequently rebalanced.  =>transaction costs=>negative influence on performances !

3 The Problem  Root of the problem: MV analysis is based on returns (I(0)) rather than prices (I(1)).  The difference:  So when we move from prices to returns we actually lose valuable information! What can be done?... the prices are highly autocorrelated, sharing long-term trends relative to spreads and relative market direction; return analysis is based on low autocorrelated market information, having less stable, short-lived portfolios, and little long term predictive value, limited trend information

4  Cointegration enables us to avoid this drawback because it measures how the prices, and not the returns, are moving together in the long run, having in contradiction to the classical correlation concept the advantage of using the entire set of information from the price levels.  If the spreads are mean-reverting, asset prices are tied together in the long run by a common stochastic trend => the prices are cointegrated  Cointegration tells us that when found, stable co-relationships between groups of assets will remain stable for some period of time as a result of prevailing market factors. COINTEGRATION

5 Cointegration in portfolio management  Lucas (1997)  Alexander (1999)  DiBartolomeo (1999)  Alexander and Weddington (2001)  Alexander and Dimitriu (2002)

6 Portfolio Construction Strategies Using Cointegration 2. Portfolio Construction Strategies Using Cointegration  Cloning strategies: aim to construct a portfolio, that clones a given benchmark, in terms of return and volatility, and preferably with the use of a small number of assets. Cloning portfolio will be strong correlated with the market.  Cointegration method: Engle -Granger (1987). Reasons: 1.we know a priori that we have a single cointegrating relation (portfolio weights) 2. Its simplicity; 3. For portfolio management the criterion of minimizing the variance is far more important than Johansen’s criterion of maximizing stationarity.  Once we ensured that the candidate asset price series are non- stationary, we will estimate a cointegrating regression, having as dependent variable the price series of the benchmark, and as independent variables the candidate clone portfolio components. Estimation will be made using a prespecified window of data, called calibration period.

7 More formally, we will estimate by OLS the following equation: where: P benchmark is the time series of (daily) benchmark price; P Ai is the time series of asset “i”;  i are the estimated coefficients from the above regression, coefficients that after normalization will play the role of portfolio weights; and ε is residual series, which is nothing but the tracking error.

A simple algorithm of optimization  To fully benefit of the common stochastic trend followed by the asset prices that will compose the clone portfolio, it is paramount to select from the candidate assets, the basket that is the most cointegrated with the benchmark.

9 Data: MSCI equity indices for Eurozone countries

10  We will estimate a cointegrating regression using all candidate assets as independent variables, and as dependent variable EURO MSCI index plus 2% p.a.  We will try to find the most cointegrated portfolio eliminating successively variables from the regression, and testing the stationarity of the resulting residuals.

11 Optimization rounds (eliminating FI, NE, SP, BE)

12 Suboptinal round  a further attempt to optimize the portfolio composition will end up in obtaining a suboptimal portfolio, because eliminating the Austrian equity index from portfolio will lead to an error less stationary (ADF t-stat of –7.0568) comparing to the previous round (ADF t-stat of –7.2266). We will conclude that previous round gives us the most cointegrated portfolio.

13 Once we found the composition, we determine the weights...

14 Testing the residual...

15 BACK-TESTING THE MODEL .1. Rolling window Engle-Granger cointegration tests .2. Differential return between the cloning strategy and the benchmark .3. Information ratio .4. Turnover index and transaction costs .5. Volatility of cloning portfolio returns .6. Correlation between benchmark return and cloning strategy return .7. Distributional properties of the tracking error

16 Figure 1. Monthly EG rolling cointegration tests for the unmanaged portfolio 1. Rolling window Engle-Granger cointegration tests

17 If we rebalance the portfolio... Figure 3. Monthly EG rolling cointegration tests for the rebalanced portfolio

18 2. Differential return between the cloning strategy and the benchmark

19 Cumulated returns Figure 5. Cumulated returns during testing period of the two substrategies

20 4. Turnover index and transaction costs

21 5. Volatility of cloning portfolio returns Figure 6. a) Historical 30 day volatility b) conditional EWMA volatilities with =0.94 Figure 7. Historical and EWMA volatilities of the excess return for the two sub-strategies

Figure 8. Historical and EWMA correlations between a) rebalanced portfolio and market b) unmanaged portfolio and market Figure 9. Historical and EWMA correlations between a) market and unmanaged residual b) market and rebalanced residual 6. Correlations

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24 Arbitrage strategies  This type of strategies aims to construct a self-financing portfolio, which will generate positive returns irrespective of market direction, with a low volatility and in conditions of zero correlation with the market. To ensure the self-financing of the strategy, we construct two cointegrating portfolios: a long portfolio, which clone a benchmark plus a spread, and a short portfolio, which clone a benchmark minus a spread. The arbitrage portfolio will be given by the difference of the above portfolios, and will earn approximately the sum of the absolute values of the two spreads.

25  We need to construct the short portfolio, which clones MSCI EURO minus 2%. Using the optimization algorithm we obtain:

26 EG cointegrating regression for short portfolio

27 1. Rolling window Engle-Granger cointegration tests

28 2. Arbitrage returns Figure 11. Cumulated returns of the arbitrage strategy with monthly rebalancing

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30 3. Turnover index and transaction costs

31 4. Volatility Figure 12. Historical and EWMA volatilities of the arbitrage returns for the two sub-strategies

32 5. Correlation benchmark return - the arbitrage portfolio return Figure 13. Correlation between MSCI EURO returns and arbitrage returns

33 3.Conclusions  we succeeded to find a cloning portfolio that systematically over- performed the benchmark in terms of returns, had a smaller volatility, and moreover was composed of a smaller number of assets than the original benchmark.  cloning strategy remained cointegrated with the benchmark during the entire testing period, even if the portfolio was left unmanaged  monthly rebalanced portfolio was more cointegrated than in the first case; also with a greater excess return and a reduced risk.  The performances of the model persisted even after accounting for brokerage fees.  The arbitrage strategy aimed to produce a positive return in all states of the nature. The enhanced stationarity of the tracking errors, gained by rebalancing, made it possible for the arbitrage portfolio to generate positive risk-free returns after deducting the corresponding transaction costs.