Portfolio Construction StrategiesPortfolio Construction Strategies Using Cointegration 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 !

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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

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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)

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2. Portfolio Construction Strategies Using CointegrationPortfolio 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.

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n

ittAiitbenchmark PcP

1,, )log()log(

More formally, we will estimate by OLS the following equation:

where: Pbenchmark is the time series of (daily) benchmark price; PAi 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.

Definition 1. We will call cointegrating portfolio the linear combination

m

iAii PPC

1

)log( , m <= n,

with the property that log(Pbenchmark) is cointegrated with PC, n being the number of available assets.

D e f i n i t i o n 2 . W e s a y t h a t c o i n t e g r a t i n g p o r t f o l i o

1

11 )log(

m

iAii PPC i s m o r e c o i n t e g r a t e d ( i n E n g l e

G r a n g e r s e n s e ) t h a n p o r t f o l i o

2

12 )log(

m

iAii PPC , i f n o t i n g 2,1,)log( jPCP jbenchmarkj

t h e n t - s t a t (ε 1 ) < t - s t a t (ε 2 ) , w h e r e t - s t a t i s t a k e n f r o m t h e u n i t r o o t t e s t a p p l i e d t o r e s i d u a l ε j .

Definition 3 (optimality) . Let

!)!(

!|

kkn

njPCj be the set of all cointegrating portfolios that

can be formed with the n assets, using the same calibration period. Let

!)!(

!...2,1,)log(|

kkn

njPCP jbenchmarkjj be the set of residual series

corresponding to the above portfolios. We say that PCk is optimal cointegrating portfolio if and only if t-stat(ε k) < t-stat(ε h) for kh .

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Data: MSCI equity indices for Eurozone countries

Series name Description

1 LAUSTRIA MSCI Austria Equity Index

2 LBELGIUM MSCI Belgium Equity index

3 LFINLAND MSCI Finland Equity index

4 LFRANCE MSCI France Equity index

5 LGERMANY MSCI Germany Equity index

6 LGREECE MSCI Greece Equity index

7 LIRELAND MSCI Ireland Equity index

8 LITALY MSCI Italy Equity index

9 LNETHERLANDS MSCI Netherlands Equity index

10 LPORTUGAL MSCI Portugal Equity index

11 LSPANIA MSCI Spain Equity index

12 LEURO MSCI EURO Equity index

13 LEUROPLUS MSCI EURO Equity index plus a spread of 2% p.a. uniformly distributed

14 LEUROMINUS MSCI EURO Equity index minus a spread of 2% p.a. uniformly distributed

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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)

SERIES NAME ADF H0: I(1) vs I(0)RESIDAU -4.7471RESIDBE -4.7094RESIDFI -6.6327RESIDFR -6.1154RESIDGE -4.7469RESIDGR -5.2723RESIDIR -4.6777RESIDIT -4.9688RESIDNE -5.4707RESIDPO -3.9057RESIDSP -6.3582

SERIES NAME ADF H0: I(1) vs I(0)RESID01AU -7.1659RESID01BE -6.4796RESID01FR -6.5213RESID01GE -5.0125RESID01GR -6.318RESID01IR -6.4678RESID01IT -5.8177RESID01NE -7.3124RESID01PO -5.4401RESID01SP -6.8202

SERIES NAME ADF H0: I(1) vs I(0)RESID02AU -7.0592RESID02BE -7.1881RESID02FR -6.3224RESID02GE -5.2677RESID02GR -6.9923RESID02IR -7.0693RESID02IT -6.0594RESID02PO -6.2361RESID02SP -7.3568

SERIES NAME ADF H0: I(1) vs I(0)RESID03AU -7.1127RESID03BE -7.2266RESID03FR -5.5149RESID03GE -5.8952RESID03GR -7.0518RESID03IR -6.8181RESID03IT -6.893RESID03PO -6.9268

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Suboptinal round

SERIES NAME ADF H0: I(1) vs I(0)RESID04AU -7.0568RESID04FR -5.525RESID04GE -4.8763RESID04GR -6.4989RESID04IR -6.7358RESID04IT -6.8774RESID04PO -6.0192

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.

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Once we found the composition, we determine the weights...Dependent Variable: LEUROPLUSMethod: Least SquaresSample: 4/30/1997 5/02/2001Included observations: 1046Variable Coefficient Std. Error t-Statistic Prob.

LAUSTRIA -0.04919 0.003945 -12.4691 0.0000LFRANCE 0.586314 0.00798 73.47163 0.0000LGERMANY 0.317436 0.006915 45.90446 0.0000LGREECE -0.02272 0.001998 -11.3686 0.0000LIRELAND 0.051311 0.005324 9.637921 0.0000LITALY 0.211019 0.006652 31.72453 0.0000LPORTUGAL -0.08221 0.005488 -14.9816 0.0000C -0.25137 0.033517 -7.49981 0.0000

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Testing the residual...

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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

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-7.5

-7.0

-6.5

-6.0

-5.5

-5.0

-4.5

-4.0

1 2 3 4 5 6 7 8 9 10 11 12 13

EG ADF t-statistic1% critical value

5% critical value10% critical value

4-year rol l ing w indow ADF testmonthly frequency

Figure 1. Monthly EG rolling cointegration tests for the unmanaged portfolio

1. Rolling window Engle-Granger cointegration tests

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If we rebalance the portfolio...

-6.0

-5.6

-5.2

-4.8

-4.4

1 2 3 4 5 6 7 8 9 10 11 12 13

EG ADF t-statistic1% critical value

5% critical value10% critical value

4-year rol l ing w indow ADF testmonthly frequency

Figure 3. Monthly EG rolling cointegration tests for the rebalanced portfolio

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2. Differential return between the cloning strategy and the benchmark

RMSCI EURO RPClone RPClone Rebal RMSCI EURO

cumulated

RPClone

cumulated

RPClone Rebal

cumulated

May 2001 -7.6830% -6.8367% -6.7542% -7.6830% -6.8367% -6.7542%Jun 2001 -4.0890% -2.6386% -2.8392% -11.7720% -9.4753% -9.5935%Jul 2001 0.1812% 0.0639% 0.0944% -11.5907% -9.4114% -9.4991%Aug 2001 -5.1310% -4.5884% -4.9981% -16.7217% -13.9998% -14.4972%Sep 2001 -13.2248% -15.7097% -14.4758% -29.9465% -29.7094% -28.9729%Oct 2001 4.1796% 4.8391% 4.8078% -25.7670% -24.8703% -24.1652%Nov 2001 4.7549% 4.7061% 4.2784% -21.0121% -20.1642% -19.8867%Dec 2001 2.9018% 3.3577% 3.1018% -18.1102% -16.8065% -16.7850%Jan 2002 -6.2934% -6.2711% -5.9570% -24.4036% -23.0777% -22.7420%Feb 2002 -0.6254% -0.1915% -0.0479% -25.0290% -23.2692% -22.7899%Mar 2002 5.4747% 6.3870% 5.8837% -19.5542% -16.8822% -16.9062%Apr 2002 -2.2441% -3.1133% -2.2029% -21.7983% -19.9955% -19.1091%

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Cumulated returns

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

5/02 7/11 9/19 11/28 2/06 4/17

EUROMSCI rand cumulatPclona rand cumulatPclona (reb) rand cumulat

Randamentul cumulat pe perioada de testare

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

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4. Turnover index and transaction costs

Month TO(%) Long TO(%) Short TO(%) ArbitrageMay-01 4.06 3.22 6.40Jun-01 1.45 5.02 6.47Jul-01 1.31 3.93 5.25

Aug-01 0.78 2.27 3.05Sep-01 3.36 2.42 5.60Oct-01 1.45 3.44 4.89

Nov-01 2.50 3.37 5.87Dec-01 1.31 1.10 2.42Jan-02 0.28 0.52 0.65Feb-02 4.13 1.95 5.81Mar-02 1.31 2.53 3.83Apr-02 1.13 3.91 4.98

Table 14.The effects of transaction costs on the returns generated by the rebalanced cloning strategy

a=0.1% a=0.2% a=0.5%Net

rebalancedcloningreturn

Cumulated netrebalanced

cloning return

Netrebalanced

cloningreturn

Cumulated netrebalanced

cloning return

Netrebalanced

cloningreturn

Cumulated netrebalanced

cloning return

-6.8065% -6.8065% -6.8587% -6.8587% -7.0155% -7.0155%-2.8562% -9.6627% -2.8733% -9.7320% -2.9243% -9.9397%0.0792% -9.5836% 0.0639% -9.6680% 0.0183% -9.9215%

-5.0072% -14.5907% -5.0163% -14.6843% -5.0436% -14.9650%-14.5178% -29.1085% -14.5598% -29.2441% -14.6858% -29.6508%

4.7897% -24.3188% 4.7716% -24.4725% 4.7173% -24.9335%4.2476% -20.0713% 4.2167% -20.2558% 4.1241% -20.8094%3.0859% -16.9854% 3.0699% -17.1859% 3.0222% -17.7872%

-5.9603% -22.9457% -5.9637% -23.1495% -5.9736% -23.7608%-0.0983% -23.0441% -0.1488% -23.2983% -0.3000% -24.0608%5.8675% -17.1766% 5.8513% -17.4470% 5.8026% -18.2582%

-2.2164% -19.3930% -2.2299% -19.6769% -2.2703% -20.5285%

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5. Volatility of cloning portfolio returns

Figure 6. a) Historical 30 day volatility b) conditional EWMA volatilities with =0.94

10

20

30

40

50

4/30/98 2/04/99 11/11/99 8/17/00 5/24/01 2/28/02

Vol hist 30 Rand EURO MSCIVol hist 30 Rand PclonaVol hist 30 Rand Pclona rebal

0

10

20

30

40

50

4/30/98 2/04/99 11/11/99 8/17/00 5/24/01 2/28/02

VOL EWMA 30 PClona rebVOL EWMA 30 MSCI EUROVOL EWMA 30 PClona

0

10

20

30

40

4/30/98 3/30/00 2/28/02

V OL E W MA 30 eroare1V OL hist 30 eroare1

0

10

20

30

40

4/30/98 3/30/00 2/28/02

V OL E W MA 30 resV OL hist 30 res

Figure 7. Historical and EWMA volatilities of the excess return for the two sub-strategies

0.94

0.95

0.96

0.97

0.98

0.99

1.00

5/01 7/10 9/18 11/27 2/05 4/16

CORR EWMA 30 PC rebalCORR HIST 30 PC rebal

0.94

0.95

0.96

0.97

0.98

0.99

1.00

5/01 7/10 9/18 11/27 2/05 4/16

CORR EWMA PClona EUROCORR hist PClona EURO

Figure 8. Historical and EWMA correlations between

a) rebalanced portfolio and market b) unmanaged portfolio and market

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

5/01 7/10 9/18 11/27 2/05 4/16

CORR HIST eroare1 euro CORR EWMA eroare1 euro

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

5/01 7/10 9/18 11/27 2/05 4/16

CORR EWMA res euro CORR hist res euro

Figure 9. Historical and EWMA correlations between

a) market and unmanaged residual b) market and rebalanced residual

6. Correlations

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7. Distributional properties of cloning errors

Error Long P Long rbl P Short P Short rbl P Mean 0.010147 0.001207 0.006660 0.002074 Median 0.011981 0.002274 -0.006811 -0.005947 Maximum 0.032453 0.014586 0.080644 0.049457 Minimum -0.020394 -0.020299 -0.051729 -0.026488 Std. Dev. 0.011858 0.007568 0.034781 0.021043 Skewness -0.358671 -0.449998 0.601555 0.721445 Kurtosis 2.232754 2.301240 1.976044 2.103339

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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 .

n

ittAiitplusbenchmark PcP

1,1,_ )log()log( (2)

n

ittAiitusbenchmark uPcP

1,2,min_ )log()log( (3)

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

Table 16. Short portfolio optimization summary

Residual Series ADF t-statround 1 r_ge -5.8476round 2 r_01fr -6.2491round 3 r_02au -6.2563round 4 r_03it -6.2609round 5 r_04sp -6.4368round 6 r_05ne -6.2523

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EG cointegrating regression for short portfolio

Dependent Variable:LEUROMINUSSample: 4/30/1997 5/02/2001Included observations: 1046

Variable Coefficient Std. Error t-Statistic Prob.LBELGIUM 0.048556 0.005851 8.299036 0.0000LFINLAND 0.193238 0.001956 98.81191 0.0000LGREECE 0.020864 0.003528 5.913643 0.0000LIRELAND -0.076175 0.007751 -9.828342 0.0000

LNETHERLANDS 0.404709 0.012000 33.72456 0.0000LPORTUGAL 0.354806 0.006464 54.88697 0.0000

C 0.741652 0.068012 10.90479 0.0000

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1. Rolling window Engle-Granger cointegration tests

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2. Arbitrage returns

-60

-40

-20

0

20

40

60

5/01 7/10 9/18 11/27 2/05 4/16

MSCI EURO cum returnLONG port cum returnSHORT port cum returnARBITAGE port cum return

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

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RLONG RSHORT RARBITRAGE RLONG_

REBALANCED

RSHORT_

REBALANCED

RARBITRAGE

_REBALANCED

May 2001 -6.8367% 9.0058% 2.1691% -6.7542% 7.1222% 0.3680%Jun 2001 -9.4753% 17.0103% 7.5349% -9.5935% 10.2975% 0.7040%Jul 2001 -9.4114% 16.0273% 6.6159% -9.4991% 10.5551% 1.0560%Aug 2001 -13.9998% 22.4745% 8.4747% -14.4972% 15.9052% 1.4080%Sep 2001 -29.7094% 28.4680% -1.2414% -28.9729% 30.7009% 1.7280%Oct 2001 -24.8703% 22.8646% -2.0057% -24.1652% 26.2612% 2.0960%Nov 2001 -20.1642% 19.7057% -0.4585% -19.8867% 22.3347% 2.4480%Dec 2001 -16.8065% 17.4388% 0.6323% -16.7850% 19.5690% 2.7840%Jan 2002 -23.0777% 22.3647% -0.7130% -22.7420% 25.8940% 3.1520%Feb 2002 -23.2692% 23.5181% 0.2489% -22.7899% 26.2619% 3.4720%Mar 2002 -16.8822% 18.8532% 1.9710% -16.9062% 20.7142% 3.8080%Apr 2002 -19.9955% 22.9627% 2.9671% -19.1091% 23.2691% 4.1600%

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3. Turnover index and transaction costs

Table 21.The effects of transaction costs on the returns generated by the rebalanced arbitrage strategy

a=0.1% a=0.2% a=0.5%Net

rebalancedarbitrage

return

Cumulated netrebalanced

arbitrage return

Netrebalancedarbitrage

return

Cumulated netrebalanced

arbitrage return

Netrebalancedarbitrage

return

Cumulated netrebalanced

arbitrage return

0.2885% 0.2885% 0.2090% 0.2090% -0.0296% -0.0296%0.2630% 0.5515% 0.1900% 0.3990% -0.0290% -0.0586%0.2906% 0.8420% 0.2291% 0.6281% 0.0448% -0.0138%0.3159% 1.1580% 0.2798% 0.9079% 0.1716% 0.1578%0.2486% 1.4066% 0.1773% 1.0852% -0.0369% 0.1209%0.3040% 1.7106% 0.2401% 1.3252% 0.0482% 0.1691%0.2759% 1.9866% 0.1999% 1.5251% -0.0283% 0.1409%0.3055% 2.2921% 0.2750% 1.8001% 0.1834% 0.3243%0.3595% 2.6516% 0.3511% 2.1512% 0.3257% 0.6500%0.2474% 2.8990% 0.1747% 2.3259% -0.0432% 0.6068%0.2886% 3.1875% 0.2411% 2.5670% 0.0989% 0.7056%0.2900% 3.4775% 0.2279% 2.7950% 0.0419% 0.7475%

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4. Volatility

0

5

10

15

20

25

30

35

6/01 8/10 10/19 12/28 3/08

VOL EWMA30 PArbitrage(PA)VOL EWMA 30 PA RBLVOL Hist 30 PAVOL Hist 30 PA RBL

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

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5. Correlation benchmark return - the arbitrage portfolio return

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

6/01 8/10 10/19 12/28 3/08

COR_PARBL_EURO_EWMACOR_PARBL_EURO_HIST

COR_PA_EURO_EWMACOR_PA_EURO_HIST

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.