Transcript Portfolio Construction Strategies Using Cointegration

ACADEMY OF ECONOMIC STUDIES
DOCTORAL SCHOOL OF FINANCE AND BANKING
Portfolio Construction Strategies
Using Cointegration
M.Sc. Student: IONESCU GABRIEL
Supervisor: Professor MOISA ALTAR
BUCHAREST, JUNE 2002
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 !
2
The Problem
 Root of the problem: MV analysis is based on returns (I(0)) rather
than prices (I(1)).
 The difference:
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
 So when we move from prices to returns we actually lose valuable
information! What can be done?...
3
COINTEGRATION
 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.
4
Cointegration in portfolio management
 Lucas (1997)
 Alexander (1999)
 DiBartolomeo (1999)
 Alexander and Weddington (2001)
 Alexander and Dimitriu (2002)
5
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.
6
More formally, we will estimate by OLS the following equation:
n
log(Pbenchmark , t )  c    i  log(PAi, t )   t
i 1
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.
7
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 PC 
m

i 1
i
 log( PAi ) , m <= n,
with the property that log(Pbenchmark) is cointegrated with PC, n being the number of available assets.
Definition 2. We say that cointegrating portfolio PC1 
Granger sense) than portfolio PC 2 
m2

i 1
i
m1

i 1
i
 log( PAi ) is more cointegrated (in Engle
 log( PAi ) , if noting  j  log( Pbenchmark )  PC j , j  1,2
then t-stat (ε1) < t-stat (ε2), where t-stat is taken from the unit root test applied to residual εj.


n!
Definition 3 (optimality) . Let    PC j | j 
 be the set of all cointegrating portfolios that
(
n

k
)!
k
!


can be formed with the n assets, using the same calibration period. Let


n!
   j |  j  log( Pbenchmark )  PC j , j  1,2...
 be the set of residual series
( n  k )! k!

corresponding to the above portfolios. We say that PCk is optimal cointegrating portfolio if and only if tstat(εk) < t-stat(εh) for  h  k .
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
9
 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.
10
Optimization rounds (eliminating FI, NE, SP, BE)
SERIES NAME ADF H0: I(1) vs I(0)
RESIDAU
-4.7471
RESIDBE
-4.7094
RESIDFI
-6.6327
RESIDFR
-6.1154
RESIDGE
-4.7469
RESIDGR
-5.2723
RESIDIR
-4.6777
RESIDIT
-4.9688
RESIDNE
-5.4707
RESIDPO
-3.9057
RESIDSP
-6.3582
SERIES NAME ADF H0: I(1) vs I(0)
RESID02AU
-7.0592
RESID02BE
-7.1881
RESID02FR
-6.3224
RESID02GE
-5.2677
RESID02GR
-6.9923
RESID02IR
-7.0693
RESID02IT
-6.0594
RESID02PO
-6.2361
RESID02SP
-7.3568
SERIES NAME ADF H0: I(1) vs I(0)
RESID01AU
-7.1659
RESID01BE
-6.4796
RESID01FR
-6.5213
RESID01GE
-5.0125
RESID01GR
-6.318
RESID01IR
-6.4678
RESID01IT
-5.8177
RESID01NE
-7.3124
RESID01PO
-5.4401
RESID01SP
-6.8202
SERIES NAME ADF H0: I(1) vs I(0)
RESID03AU
-7.1127
RESID03BE
-7.2266
RESID03FR
-5.5149
RESID03GE
-5.8952
RESID03GR
-7.0518
RESID03IR
-6.8181
RESID03IT
-6.893
RESID03PO
-6.9268
11
Suboptinal round
 a further attempt to optimize the
portfolio composition will end up
SERIES NAME ADF H0: I(1) vs I(0)
RESID04AU
-7.0568
RESID04FR
-5.525
RESID04GE
-4.8763
RESID04GR
-6.4989
RESID04IR
-6.7358
RESID04IT
-6.8774
RESID04PO
-6.0192
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.
12
Once we found the composition, we determine the weights...
Dependent Variable: LEUROPLUS
Method: Least Squares
Sample: 4/30/1997 5/02/2001
Included observations: 1046
Variable
Coefficient Std. Error
LAUSTRIA
LFRANCE
LGERMANY
LGREECE
LIRELAND
LITALY
LPORTUGAL
C
-0.04919
0.586314
0.317436
-0.02272
0.051311
0.211019
-0.08221
-0.25137
0.003945
0.00798
0.006915
0.001998
0.005324
0.006652
0.005488
0.033517
t-Statistic
-12.4691
73.47163
45.90446
-11.3686
9.637921
31.72453
-14.9816
-7.49981
Prob.
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
13
Testing the residual...
14
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
15
1. Rolling window Engle-Granger cointegration tests
4-year rolling w indow ADF test
monthly frequency
-4.0
-4.5
-5.0
-5.5
-6.0
-6.5
-7.0
-7.5
1
2
3
4
5
6
7
EG ADF t-statistic
1% critical value
8
9
10
11
12
13
5% critical value
10% critical value
Figure 1. Monthly EG rolling cointegration tests for the unmanaged portfolio
16
If we rebalance the portfolio...
4-year rolling w indow ADF test
monthly frequency
-4.4
-4.8
-5.2
-5.6
-6.0
1
2
3
4
5
6
7
EG ADF t-statistic
1% critical value
8
9
10 11 12 13
5% critical value
10% critical value
Figure 3. Monthly EG rolling cointegration tests for the rebalanced portfolio
17
2. Differential return between the cloning strategy and the benchmark
RMSCI EURO
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
Jan
Feb
Mar
Apr
2001
2001
2001
2001
2001
2001
2001
2001
2002
2002
2002
2002
RPClone
-7.6830% -6.8367%
-4.0890% -2.6386%
0.1812%
0.0639%
-5.1310% -4.5884%
-13.2248% -15.7097%
4.1796%
4.8391%
4.7549%
4.7061%
2.9018%
3.3577%
-6.2934% -6.2711%
-0.6254% -0.1915%
5.4747%
6.3870%
-2.2441% -3.1133%
RPClone Rebal
-6.7542%
-2.8392%
0.0944%
-4.9981%
-14.4758%
4.8078%
4.2784%
3.1018%
-5.9570%
-0.0479%
5.8837%
-2.2029%
RMSCI EURO
RPClone
RPClone Rebal
cumulated
cumulated
cumulated
-7.6830%
-11.7720%
-11.5907%
-16.7217%
-29.9465%
-25.7670%
-21.0121%
-18.1102%
-24.4036%
-25.0290%
-19.5542%
-21.7983%
-6.8367%
-9.4753%
-9.4114%
-13.9998%
-29.7094%
-24.8703%
-20.1642%
-16.8065%
-23.0777%
-23.2692%
-16.8822%
-19.9955%
-6.7542%
-9.5935%
-9.4991%
-14.4972%
-28.9729%
-24.1652%
-19.8867%
-16.7850%
-22.7420%
-22.7899%
-16.9062%
-19.1091%
18
Cumulated returns
Randamentul cumulat pe perioada de testare
0.0
-0.1
-0.2
-0.3
-0.4
-0.5
5/02
7/11
9/19
11/28
2/06
4/17
EUROMSCI rand cumulat
Pclona rand cumulat
Pclona (reb) rand cumulat
Figure 5. Cumulated returns during testing period of the two substrategies
19
4. Turnover index and transaction costs
Month TO(%) Long
May-01
4.06
Jun-01
1.45
Jul-01
1.31
Aug-01
0.78
Sep-01
3.36
Oct-01
1.45
Nov-01
2.50
Dec-01
1.31
Jan-02
0.28
Feb-02
4.13
Mar-02
1.31
Apr-02
1.13
TO(%) Short
3.22
5.02
3.93
2.27
2.42
3.44
3.37
1.10
0.52
1.95
2.53
3.91
TO(%) Arbitrage
6.40
6.47
5.25
3.05
5.60
4.89
5.87
2.42
0.65
5.81
3.83
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
Cumulated net
Net
Cumulated net
Net
Cumulated net
rebalanced
rebalanced
rebalanced
rebalanced
rebalanced
rebalanced
cloning
cloning return
cloning
cloning return
cloning
cloning return
return
return
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%
20
5. Volatility of cloning portfolio returns
50
50
40
40
30
30
20
20
10
4/30/98
10
2/04/99
11/11/99
8/17/00
5/24/01
2/28/02
0
4/30/98
2/04/99
V ol hist 30 Rand E URO MS CI
V ol hist 30 Rand P clona
V ol hist 30 Rand P clona rebal
Figure 6. a) Historical 30 day volatility
30
30
20
20
10
10
V OL E W MA 30 eroare1
V OL hist 30 eroare1
5/24/01
2/28/02
b) conditional EWMA volatilities with =0.94
40
3/30/00
8/17/00
V OL E WMA 30 P Clona reb
V OL E W MA 30 MS CI E URO
V OL E W MA 30 P Clona
40
0
4/30/98
11/11/99
2/28/02
0
4/30/98
3/30/00
2/28/02
V OL E W MA 30 res
V OL hist 30 res
21
Figure 7. Historical and EWMA volatilities of the excess return for the two sub-strategies
6. Correlations
1.00
1.00
0.99
0.99
0.98
0.98
0.97
0.97
0.96
0.96
0.95
0.95
0.94
5/01
7/10
9/18
11/27
2/05
4/16
0.94
5/01
CORR EWMA 30 PC rebal
CORR HIST 30 PC rebal
7/10
9/18
11/27
2/05
4/16
CORR EWMA PClona EURO
CORR hist PClona EURO
Figure 8. Historical and EWMA correlations between
a) rebalanced portfolio and market
b) unmanaged portfolio and market
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.0
0.0
-0.2
-0.2
-0.4
-0.6
5/01
7/10
9/18
CORR HIST eroare1 euro
11/27
2/05
4/16
CORR EWMA eroare1 euro
-0.4
5/01
7/10
9/18
CORR EWMA res euro
11/27
2/05
CORR hist res euro
Figure 9. Historical and EWMA correlations between
a) market and unmanaged residual
4/16
b) market and rebalanced residual
7. Distributional properties of cloning errors
Error
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
Long P Long rbl P Short P Short rbl P
0.010147 0.001207 0.006660 0.002074
0.011981 0.002274 -0.006811 -0.005947
0.032453 0.014586 0.080644 0.049457
-0.020394 -0.020299 -0.051729 -0.026488
0.011858 0.007568 0.034781 0.021043
-0.358671 -0.449998 0.601555 0.721445
2.232754 2.301240 1.976044 2.103339
23
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
log(Pbenchmark _ plus , t )  c1    i  log(PAi, t )   t
i 1
n
log(Pbenchmark _ minus , t )  c2    i  log(PAi, t )  ut
(3)
i 1
24
 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
round 1
round 2
round 3
round 4
round 5
round 6
Residual Series
r_ge
r_01fr
r_02au
r_03it
r_04sp
r_05ne
ADF t-stat
-5.8476
-6.2491
-6.2563
-6.2609
-6.4368
-6.2523
25
EG cointegrating regression for short portfolio
Dependent Variable:LEUROMINUS
Sample: 4/30/1997 5/02/2001
Included observations: 1046
Variable
LBELGIUM
LFINLAND
LGREECE
LIRELAND
LNETHERLANDS
LPORTUGAL
C
Coefficient
0.048556
0.193238
0.020864
-0.076175
0.404709
0.354806
0.741652
Std. Error t-Statistic
0.005851 8.299036
0.001956 98.81191
0.003528 5.913643
0.007751 -9.828342
0.012000 33.72456
0.006464 54.88697
0.068012 10.90479
Prob.
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
26
1. Rolling window Engle-Granger cointegration tests
27
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 return
LONG port cum return
SHORT port cum return
ARBITAGE port cum return
Figure 11. Cumulated returns of the arbitrage strategy with monthly rebalancing
28
RLONG
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
Jan
Feb
Mar
Apr
2001
2001
2001
2001
2001
2001
2001
2001
2002
2002
2002
2002
-6.8367%
-9.4753%
-9.4114%
-13.9998%
-29.7094%
-24.8703%
-20.1642%
-16.8065%
-23.0777%
-23.2692%
-16.8822%
-19.9955%
RSHORT
9.0058%
17.0103%
16.0273%
22.4745%
28.4680%
22.8646%
19.7057%
17.4388%
22.3647%
23.5181%
18.8532%
22.9627%
RARBITRAGE
2.1691%
7.5349%
6.6159%
8.4747%
-1.2414%
-2.0057%
-0.4585%
0.6323%
-0.7130%
0.2489%
1.9710%
2.9671%
RLONG_
RSHORT_
RARBITRAGE
REBALANCED
REBALANCED
_REBALANCED
-6.7542%
-9.5935%
-9.4991%
-14.4972%
-28.9729%
-24.1652%
-19.8867%
-16.7850%
-22.7420%
-22.7899%
-16.9062%
-19.1091%
7.1222%
10.2975%
10.5551%
15.9052%
30.7009%
26.2612%
22.3347%
19.5690%
25.8940%
26.2619%
20.7142%
23.2691%
0.3680%
0.7040%
1.0560%
1.4080%
1.7280%
2.0960%
2.4480%
2.7840%
3.1520%
3.4720%
3.8080%
4.1600%
29
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
Cumulated net
Net
Cumulated net
Net
Cumulated net
rebalanced
rebalanced
rebalanced
rebalanced
rebalanced
rebalanced
arbitrage arbitrage return arbitrage arbitrage return arbitrage arbitrage return
return
return
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%
30
4. Volatility
35
30
25
20
15
10
5
0
6/01
8/10
10/19
12/28
3/08
VOL EWMA30 PArbitrage(PA)
VOL EWMA 30 PA RBL
VOL Hist 30 PA
VOL Hist 30 PA RBL
Figure 12. Historical and EWMA volatilities of the arbitrage returns for the two sub-strategies
31
5. Correlation benchmark return - the arbitrage portfolio return
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
6/01
8/10
10/19
COR_PARBL_EURO_EWMA
COR_PARBL_EURO_HIST
12/28
3/08
COR_PA_EURO_EWMA
COR_PA_EURO_HIST
Figure 13. Correlation between MSCI EURO returns and arbitrage returns
32
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.
33