SemiVariance and SemiCorrelation

8/7/2019 SemiVariance and SemiCorrelation

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S EM I - VARIANCE AND S EM I -C ORRELATIONS FOR F INANCIAL I NVESTMENTS

GIAMPAOLO GABBI Financial Management Department, University of Siena

Banking and Finance Area, SDA Bocconi - [email protected]

Many studies show that international correlations have changed through time.This phenomenon changed many portfolio managers practices, which are nowstrictly linked with sectors behaviours. In order to give reason for thismanagement style, we provide some new evidences for correlation dynamicsamong geographic areas and business sectors.Nevertheless some researches offer theoretical basis for semi-varianceoptimisation, fewer authors analyse its contribution to asset allocation. Here weapply the concept to compare whether it applies efficiently to sectors andcountries.The paper is aimed at answering to the following questions:1) is short-term correlation useful for tactical asset allocation?2) is it possible to estimate less volatile indexes of correlation, so to ease theforecasting process?3) how to obtain a good coherence with frameworks incorporating downsiderisk as measure of risk?4) can portfolios outperform those allocated with the usual correlation index?

Conclusions are that:1) short-term correlation, if adequately modelled, can be fundamental toimplement a successful tactical asset allocation;2) correlations lower volatile thanks to the application of semi-correlationmeasure;3) neural networks applied to benchmark returns generate good results,especially in terms of direction, which is, in fact, the necessary input for theapplication of semi-correlation to the asset allocation process;4) finally, gap ratios among returns and volatility grow for both the groups andextreme values of the frontiers and the upward of risk/return ratio is still betterfor geographical diversification, especially in the minimum values of expectedreturns.

1. INTRODUCTION Many studies show that international correlations have changed throughtime 1. During the last years, globalisation has accelerated the process of towards increasing positive values.

1 Solnik-Bouorelle-Le Fur (1996); Groenen-Franses (2000).


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This phenomenon changed many portfolio managers practices, which arenow strictly linked with sectors behaviours. In order to give reason for thismanagement style, we provide some new evidences for correlation dynamicsamong geographic areas and business sectors.Many studies offer theoretical basis for semi-variance optimisation 2. Fewerauthors analyse the contribution of semi-correlation to asset allocation 3. Herewe try to apply the concept to compare whether it applies more efficiently tosectors or to countries.The paper is aimed at answering to the following questions:5) is short-term correlation useful for tactical asset allocation?6) is it possible to estimate less volatile indexes of correlation, so to ease

the forecasting process?7) how to obtain a good coherence with frameworks incorporating

downside risk as measure of risk [Bramante Gabbi (2001)]?8) can portfolios outperform those allocated with the usual correlation

index [Harlow Rao (1989); Harlow (1991)]?

2. SHORT-TERMCORRELATION ANDTACTICALASSETALLOCATION

In our paper, we consider correlation computed on a short-term basis:periods are six months long, in order to evaluate the opportunity toimplement a tactical asset allocation.

Moreover, portfolio managers who apply sector benchmarks to theirdecisions must face the problem of time series shortage, which affects thedataset.In our empirical study we compare two large groups of equity benchmarks,respectively observed for sectors and for countries, as shown in Table 1 , forthe period January 1994 January 2000.

2 Harlow (1991).3 Erb, Harvey, Viskanta (1994).


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Table 1 Morgan Stanley Indexes for Sectors and CountriesSECTORS COUNTRIESENERGY AUSTRALIAMATERIALS AUSTRIACAPITAL GOODS BELGIUMCOMML SVC & SUPPL CANADATRANSPORTATION DENMARKAUTO & COMPONENTS FINLANDCONS DUR & APPAREL FRANCEHOTELS REST & LEISURE GERMANYMEDIA HONG KONGRETAILING IRELANDFOOD & DRUG RETL ITALYFOOD BEV & TOBACCO JAPANHOUSE & PERS PROD NETHERLANDSH CARE EQUIP & SVC NEW ZEALAND

PHARMA & BIOTECH NORWAYBANKS PORTUGALDIVERS FINANCIAL SINGAPOREINSURANCE SPAINREAL ESTATE SWEDENSOFTWARE & SERVICES SWITZERLANDTECH HARD & EQUIPMENT UNITED KINGDOMTELECOM SVC USAUTILITIES WORLD

The analysis of correlation volatility through short-term rolling periodsshows that volatility correlation is, on average, 12,99 per cent for countries(with MSCI World Index), and 10,19 per cent for sectors.

Figure 1 Short-term rolling correlation of MSCI World Index with MSCIsectors

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CONS DUR & APPAREL HOTELS REST & LEIS MEDIAR ETAI LI NG F OO D & D RU G R ET L F OO D B EV & T OB AC CO


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Figure 1 (for sectors) and Figure 2 (for countries) show how correlation,

usually considered substantially stable in the long run, can be considered animportant variable for tactical asset allocation, if satisfactorily forecast.

Figure 2 Short-term rolling correlation of MSCI World Index with MSCIcountries

Figure 3 and Figure 4 show a candlestick representation of globalcorrelation. The evidence is that sectors correlation is, on balance,decreasing (black candlesticks), while countries correlation is increasing,even though less intensively.

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Figure 3 Global correlation candlestick (MSCI World vs. sectors)

Figure 4 Global correlation candlestick (MSCI World vs. countries)

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Applying historical returns to a six months holding period, we derive theefficient frontiers reproduced in Appendix A (sectors) and Appendix B(countries). We observe how volatile portfolio composition is even rollingtime only by six months (Table 2).

Table 2 Extreme values of efficient frontiersSECTORS COUNTRIES

MIN MAX MIN MAXPERIODS E[r] E[r] E[r] E[r] 1 0,02 4,36 9,94 7,65 -3,06 3,73 7,59 16,362 -2,70 3,59 6,63 5,69 -0,73 2,09 6,23 10,523 5,59 0,87 10,27 2,52 9,69 1,06 10,95 15,174 6,42 1,83 11,15 7,83 5,05 0,21 14,31 8,645 -0,50 1,68 5,15 6,34 2,19 0,40 10,58 8,756 0,97 2,57 8,77 7,27 4,34 1,28 17,97 10,847 3.02 1,81 14,20 8,29 9,73 2,18 22,11 9,338 -3,93 3,61 9,26 12,12 -3,90 6,02 15,98 15,949 6,64 3,25 17,96 7,63 7,54 2,23 29,74 13,5210 3,83 6,84 13,57 12,69 4,20 7,91 15,97 24,4311 -1,08 1,56 15,08 17,05 1,90 0,96 28,64 16,53

AVERAGE 1.66 2.91 11.09 8.64 3.36 2.55 16.37 13.64ST. DEV. 3.50 1.61 3.63 3.76 4.44 2.32 7.46 4.52RATIO 0.57 1.28 1.32 1.20

Here we can view how range is, on average, 9,43 for sectors ( =3,41) and

13,01 for countries ( =6,99). The ratio computed in the last row is simply

)(avg)]r(E[avg

=r [1]

This means that, whether some portfolio managers wish to implementtactical asset allocation should be able to forecast not only returns andvolatilities, but also correlation indexes, so to determine optimal assetweights.

3. SEMI-CORRELATIONCORRELATION Recently different correlation formulas have been proposed, especially inorder to support financial decision processes. Accepted the idea that the lastcorrelation matrix hides some traps, because of static view of variance-covariance, it should be better analyse the time series of these correlations,as shown in Figure 1 and Figure 2.Nevertheless, the implementation of the expected correlation values could becomplicated by short term volatility. Here we apply an already known


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measure of correlation to verify whether it is useful for sector and countrydatasets.According to Erb, Harvey and Viskanta [1994], semi-correlation helps theanalyst to find out the evolution process of two variables through time, so todecide whenever introduce them into a portfolio in order to diversifyfinancial assets.Computed in the same way as the downside risk, the semi-correlationprovides a measure of equity comovements in common up, common down,and mixed markets 4.

We define up-up semi-correlation equation [2]

( ) ( )( ) ( )

==

= =

=

h

ijj

k

iii

k

i

h

jjiij

yx

yynxxn

yyxxn

1

2.

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1 1, [2]

solved for all x i > avg(x) and y i > avg(y).

Down-down correlation is the [2] solved for solved for all x i < avg(x) and y i < avg(y).

While, mixed correlation is the [2] when x i > avg(x) and y i < avg(y) or whenwhen x i < avg(x) and y i > avg(y).

Possible benefits of this way to calculate correlation are:1) it should be possible to estimate a less volatile indexes of

correlation, so to ease the forecasting process;2) coherence with frameworks incorporating downside risk as measure

of risk [Bramante Gabbi (2001)];3) portfolios could be outperforming with respect those allocated with

the usual correlation index [Harlow Rao (1989); Harlow (1991)].

Four different measures of correlation (global; up-up; down-down and up-down) have been computed through 120 rolling periods. Table 3 shows theaverage and volatility values for all the sectors and the world index.

4 Erb, Harvey and Viskanta [1994, 32]


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Table 3 Average Correlations (MSCI WORLD vs. MSCI sectors)

TOT CORR UP-UP CORRDOWN-DOWN

CORR MIXED CORRWORLD

AVG VOL AVG VOL AVG VOL AVG VOL

ENERGY 47.65 27.19 71.58 8.88 79.03 10.39 -54.49 6.38

MATERIALS 65.01 21.47 80.01 8.40 84.93 8.57 -53.61 9.95

CAPITAL GOODS 86.20 8.01 88.79 5.38 93.23 4.18 -43.44 11.04

COMML SVC & SUPPL 70.94 10.99 81.82 4.37 81.57 11.16 -55.92 10.23

TRANSPORT 66.27 19.93 82.86 6.37 84.94 6.70 -56.64 10.88

AUTO COMPONENTS 58.49 15.90 74.17 7.29 80.59 9.23 -55.48 6.59

CONS DUR & APPAREL 58.74 13.81 73.61 8.08 80.26 8.05 -53.96 9.17

HOTELS REST & LEISURE 67.00 12.46 78.40 6.38 83.74 7.44 -54.62 8.42

MEDIA 79.34 9.46 82.34 8.66 90.17 4.94 -54.72 9.65

RETAILING 70.06 18.14 80.61 9.23 85.72 6.05 -51.34 13.63

FOOD & DRUG RETL 60.79 19.79 76.06 7.80 82.46 9.04 -56.01 8.55

FOOD BEV & TOBACCO 60.69 32.01 79.19 9.76 81.95 13.65 -54.28 11.30

HOUSE & PERS PROD 54.22 23.10 77.25 8.02 79.85 12.76 -54.95 11.27

H CARE EQUIP & SVC 60.54 15.10 77.75 8.24 80.61 9.00 -62.63 8.10

PHARMA & BIOTECH 68.02 24.85 80.11 10.47 86.19 10.18 -53.13 11.91

BANKS 81.84 11.48 87.01 5.88 89.30 6.59 -46.86 16.92

DIVERS FINANC 82.69 9.34 86.62 5.36 92.74 3.96 -45.48 8.84

INSURANCE 77.43 12.92 84.74 6.99 87.53 6.90 -49.27 13.62

REAL ESTATE 47.76 18.09 67.43 8.10 77.77 8.14 -52.73 6.86

SOFTWARE & SERVICES 67.24 9.41 79.90 6.52 83.71 5.56 -47.23 9.85

TECH HARD & EQUIP 78.61 7.88 83.37 5.06 88.36 5.06 -52.68 11.58

TELECOM SVC 77.66 7.10 82.59 6.65 89.15 4.86 -45.41 11.27

UTILITIES 64.35 17.91 79.05 8.65 83.64 8.25 -53.35 8.43


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figures, which offer a technical view of direction and range for everybenchmark in respect with MSCI World Index.

The best result comes from the sector benchmarks, both in sense of decreasing volatility and in sense of higher negative correlation with mixedmarkets.

Nevertheless, our results can be applied only when indexes directions can besatisfactorily forecast. Next section will show some results obtained with aneural network for all the dataset.

4. FORECASTINGRETURNS 4.1. Chaotic and non linear dynamicsWe know that time series generated by a chaotic process, if studied throughconventional statistical methods like autocorrelation function or spectralanalysis, come into view apparently random. Brock and al. [1987] proposeda methodology useful to distinguish stochastic and deterministic processesthrough a statistics able to verify the hypothesis of a series identically andindependently distributed (i.i.d). Ashley and Patterson [1989] and Hsieh[1991] demonstrate that the independence of a variable from its past valuesdoes not necessarily imply a white noise process. The alternative reason for

the i.i.d. are: chaos, non-stationarity and conditional heteroskedasticity.Therefore, we adopted opportune tests [Barnett and Chen (1986); Frank andStengos (1988a/b)] like the dimension of correlation, the BDS test and theLyapunov exponent, in order to evaluate possible chaotic behaviours.The dimension of correlation, independently by the time frequency,increases linearly with m, and this suggests that the underlying data processgeneration are primarily stochastic. Besides, the dimension assumesrelatively low values, between 1 and 2 and they are always below therandom model values.The BDS tests [Brock, Dechert e Scheinkman, 1987] allow to verify whethertime series are identically and independently distributed, both for seriesproduced by chaotic systems and for non linear stochastic systems.The high BDS values point out that it is not possible to accept the nullhypothesis of i.i.d. data, while they suggest that the generating process is nonlinear. Besides, the BDS test disconnects the random model N(0,1) from thechaotic one. The empirical evidences bring us to conclude that time seriesare non linear even though not necessarily chaotic.Peculiar characteristic of the chaotic systems is the dependence from thestarting conditions, with trajectories that diverge exponentially, despite verysimilar initial values. The most important tool to quantify the dependence


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from the initial conditions in a dynamic system is the Lyapunov exponents[McCafferty et al., 1992 and Dechert and Gencay, 1993].Our results are consistent with those previously obtained through thedimension of correlation and seem to exclude the presence of a chaoticregime. In fact, being the Lyapunov exponents negative for all the currenciesexamined and all the frequencies, this is indicative of a stable generatingprocess.All the empirical results show a strong evidence of the existence of linearand non linear dependencies for all the examined financial time series, eventhough deterministic chaos is not an explanation. These considerations arecoherent with the implementation of econometric models and neuralnetworks, in order to fit the linear and non linear components here observed.Data properties authorise some conclusions:

1) time series are asymmetrical and leptokurtic, therefore non normaldistribution is a coherent result with that traditionally obtained for dailyand weekly observations;2) dependencies found in data are not linked with a white noisegenerating process; however, as well underlined by Hsieh [1991], it isopportune to treat this conclusion with extreme caution, since the higherthe frequency the greater the probability of false dependencies, linked tothe market microstructure;3) the possibility to describe the patterns through a little dimensionchaotic model has been clearly refused. This result is in contradictionwith a large part of financial literature which found a strong chaoticcomponent for daily and weekly time series.

4.2. Forecasting methodologiesHere we apply a back-propagation neural network characterised by threelayers, a logistic activation function, and constant leaning rate (0,1),momentum (0,1) and initial weight (0,1). Our inputs are technical indicatorsoptimised on the one-period lead calculated for the target variable.Technical indicators applied are:1. Lower and Upper Bolling Bands 5;

5

Bollinger Band High creates a band above and below price. Because the band iscreated by adding the standard deviation of price to the moving average of price, theband width is determined by the fluctuation in price over the last n time periods(advantage of quick band width reaction to large movements in the market). Thusthe band widens when price fluctuates wildly and narrows when price shows verylittle fluctuation. Adjustment of the time periods provides a means of focusing onanything from short term trends to long term trends. Adjustment of the standarddeviation multiplier controls the relative width of the band and thus determines thestrength of the price movement needed for a price breakout (movement above theband). A higher standard deviation multiplier increases the relative band width


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2. Linear weighted moving average difference;3. Linear extrapolation;4. Linear regression;5. Linear regression slope;6. Linear weighted moving average;7. Differential moving average;8. Exponential moving average;9. Lagged Exponential moving average;10. Wilders RSI 6.Optimisation methods depend on: a) time periods used by the majority of indicators (simple moving average, lag, change, etc.). b) exponential movingaverage factor is used by all indicators based upon one or more exponentialmoving average calculations; c) standard deviation time periods are used byall indicators which calculate a standard deviation; d) linear regression timeperiods are used by all indicators which are based upon linear regression.Three datasets have been extracted with the following rule:

a) training set: 50 per cent;b) test set: 20 per cent;c) generalization set (out-of-sample): 30 per cent.

Since time series are built through rolling periods over the period 1994-2001,we train the neural networks on 61 periods; we test them on 24 periods,generalizing the results on 36 periods of six months each.The pattern selection has been random, although it does not guarantee thatevery pattern will be chosen an equal number of times. The weight updatesnot only included the change dictated by learning rate, but also a portion of the last weight change as well. Like momentum in physics, a highmomentum term will keep the network generally going in the direction it hasbeen going. Weight fluctuations will tend to be dampened by a highmomentum term 7.

(stronger price movement needed for a breakout), while a lower standard deviationmultiplier decreases the relative band width (weaker price movement needed for abreakout).6 WilderRSI = 100 - ( 100 / (1 + RS))

where:RS = Up / Down (equals zero if down=0).InitiallyUp = Sum of increases over last n time periods divided by nDown = Sum of decreases over last n time periods divided by nRemainder of timeUp = [ Previous Up * (n-1) + increase ] / nDown = [ Previous Down * (n-1) + decrease ] / n7 The same parameters have been used for correlations forecasts, showed in the nextsection of the paper.


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4.3. Forecasting results

The application of neural networks to benchmarks returns can besynthetically reported in the following figures.

Table 5 contains error measures for every neural network computed in thegeneralisation set (out-of-sample) for all the sector indexes (Appendix Dshows all the figures with actual and estimated data).

Table 5 Out-of-sample forecast performance (MSCI sectors)

R squared r squared MSE MAE MIN AE MAX AECORRCOEFF

ENERGY 0.7325 0.7407 0.003 0.042 0.001 0.104 0.8606

MATERIALS 0.8985 0.9044 0.001 0.026 0.001 0.080 0.951

CAPITAL GOODS 0.8052 0.8193 0.001 0.030 0.000 0.087 0.9051

COMML SVC & SUPPL 0.4404 0.5933 0.002 0.031 0.000 0.107 0.7703

TRANSPORT 0.7922 0.7924 0.001 0.028 0.001 0.079 0.8902

AUTO COMPONENTS 0.832 0.8472 0.001 0.030 0.000 0.076 0.9205

CONS DUR & APPAREL 0.8104 0.8148 0.002 0.037 0.001 0.110 0.9027

HOTELS REST & LEISURE 0.7887 0.8136 0.001 0.028 0.000 0.100 0.9020

MEDIA 0.8225 0.8226 0.004 0.043 0.001 0.180 0.9069

RETAILING 0.7745 0.7756 0.003 0.045 0.001 0.120 0.8807

FOOD & DRUG RETL 0.8734 0.9317 0.002 0.038 0.000 0.104 0.9653

FOOD BEV & TOBACCO 0.8429 0.8467 0.002 0.040 0.004 0.081 0.9202

HOUSE & PERS PROD 0.8014 0.8154 0.003 0.044 0.001 0.146 0.9030

H CARE EQUIP & SVC 0.8235 0.8313 0.003 0.042 0.002 0.124 0.9117

PHARMA & BIOTECH 0.7656 0.7697 0.003 0.037 0.000 0.116 0.8774

BANKS 0.7156 0.7185 0.002 0.041 0.001 0.101 0.8477

DIVERS FINANC 0.4748 0.4997 0.003 0.047 0.002 0.116 0.7069

INSURANCE 0.7859 0.8101 0.002 0.040 0.005 0.108 0.9001

REAL ESTATE 0.7899 0.8088 0.003 0.047 0.000 0.149 0.8993

SOFTWARE & SERVICES 0.7723 0.7834 0.013 0.089 0.006 0.237 0.8851

TECH HARD & EQUIP 0.9041 0.9076 0.005 0.053 0.003 0.160 0.9527

TELECOM SVC 0.8682 0.8815 0.003 0.04 0.004 0.208 0.9389

UTILITIES 0.7666 0.8001 0.001 0.027 0.002 0.097 0.8945

Table 6 contains the same error measures for the country indexes (AppendixE shows the figures with actual and estimated values).


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Table 6 Out-of-sample forecast performance (MSCI countries)R squared r squared MSE MAE MIN AE MAX AE

CORRCOEFF

AUSTRALIA 0.4363 0.5063 0.002 0.032 0.001 0.105 0.7115

AUSTRIA 0.763 0.7668 0.003 0.042 0.002 0.193 0.8757

BELGIUM 0.8379 0.8559 0.002 0.039 0.004 0.156 0.9252

CANADA 0.7435 0.7488 0.005 0.046 0.003 0.273 0.8653

DENMARK 0.5557 0.6063 0.004 0.051 0.002 0.145 0.7786

FINLAND 0.7887 0.7889 0.027 0.110 0.001 0.567 0.8882

FRANCE 0.8029 0.8058 0.004 0.051 0.003 0.154 0.8977

GERMANY 0.7480 0.7591 0.007 0.067 0.005 0.209 0.8712

HONG KONG 0.7600 0.7860 0.010 0.076 0.005 0.245 0.8865

IRELAND 0.7918 0.7967 0.004 0.046 0.004 0.153 0.8926

ITALY 0.7624 0.7733 0.007 0.066 0.005 0.256 0.8794

JAPAN 0.7640 0.8033 0.004 0.045 0.001 0.161 0.8962

NETHERLANDS 0.7232 0.7468 0.004 0.053 0.003 0.169 0.8642

NEW ZEALAND 0.6201 0.6795 0.002 0.042 0.002 0.096 0.8243

NORWAY 0.7263 0.7555 0.003 0.048 0.002 0.187 0.8692

PORTUGAL 0.8421 0.843 0.006 0.064 0.000 0.159 0.9181

SINGAPORE 0.8913 0.8949 0.006 0.057 0.005 0.240 0.9460

SPAIN 0.7657 0.7664 0.007 0.065 0.005 0.220 0.8754

SWEDEN 0.8461 0.8508 0.008 0.063 0.000 0.330 0.9224

SWITZERLAND 0.6891 0.7114 0.005 0.054 0.002 0.161 0.8435

UNITED KINGDOM 0.6705 0.6747 0.002 0.036 0.001 0.094 0.8214

USA 0.6232 0.6624 0.003 0.040 0.004 0.188 0.8139

WORLD 0.6260 0.6638 0.002 0.039 0.003 0.124 0.8148

On average, statistical error measures award sector forecasts, rather thangeographical ones (Table 7).

Table 7 Average out-of-sample forecast performance

MSCI SECTORS MSCI COUNTRIES

R squared 0.777 0.729r squared 0.797 0.750Mean squared error 0.003 0.006Mean absolute error 0.040 0.054Min. absolute error 0.002 0.003Max. absolute error 0.121 0.199Correlation coefficient 0.891 0.864


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Determination coefficient, MAE and MSE can be misshapen indexes whenerrors are in terms of direction. In fact, semi-correlation needs a goodforecast of co-movements and mixed ones.If we measure reliability of estimates by path accuracy from one period tothe following one (Table 8), the results show that neural networks forecastbetter countries indexes than sectors (92,17% vs. 88,45%).

Table 8 Correct direction in out-of-sample data (per cent values)SECTORS % COUNTRIES %ENERGY 94.29 AUSTRALIA 85.71MATERIALS 88.57 AUSTRIA 91.43CAPITAL GOODS 88.57 BELGIUM 97.14

COMML SVC & SUPPL 77.14 CANADA 97.14TRANSPORTATION 82.86 DENMARK 94.29AUTO & COMPONENTS 94.29 FINLAND 97.14CONS DUR & APPAREL 77.14 FRANCE 100.00HOTELS REST & LEISURE 71.43 GERMANY 97.14MEDIA 91.43 HONG KONG 91.43RETAILING 88.57 IRELAND 91.43FOOD & DRUG RETL 97.14 ITALY 82.86FOOD BEV & TOBACCO 88.57 JAPAN 94.29HOUSE & PERS PROD 85.71 NETHERLANDS 97.14H CARE EQUIP & SVC 94.29 NEW ZEALAND 82.86PHARMA & BIOTECH 91.43 NORWAY 82.86BANKS 85.71 PORTUGAL 97.14DIVERS FINANCIAL 85.71 SINGAPORE 91.43INSURANCE 94.29 SPAIN 97.14REAL ESTATE 85.71 SWEDEN 94.29SOFTWARE & SERVICES 94.29 SWITZERLAND 91.43

TECH HARD & EQUIPMENT 97.14 UNITED KINGDOM 82.86TELECOM SVC 94.29 USA 91.43UTILITIES 85.71 WORLD 91.43

AVERAGE 88.45 AVERAGE 92.17

5. FORECASTINGCORRELATIONS

Once determined expected returns, we must compare forecastingperformances over correlations time series, both in global terms, and thethree semi-correlation outcomes.We apply the same forecasting rules to global and semi-correlation time

series: back-propagation neural network with three layers and logisticactivation function 8.We trained neural networks for each benchmark in respect with MSCI WorldIndex and each side of semi-correlations (up-up down-down, and mixed) soto compare various outcomes for every group of variables.Table 9 shows how sector global correlations forecasts are more consistentthan countries. This is coherent with the evidence of Figure 3 and Figure 4.

8 For every technical detail, see section 4.


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The interesting outcomes here are the lower predictability of semi-correlation for sectors, especially down-down and mixed ones.

Table 9 Global and Semi-Correlations Forecasts (MSCI WORLD vs. MSCIsectors)

GLOBAL UP-UP DOWN-DOWN MIXEDAVG VOL AVG VOL AVG VOL AVG VOL

R squared 0.788 0.130 0.808 0.052 0.619 0.200 0.564 0.082r squared 0.807 0.119 0.832 0.040 0.660 0.187 0.621 0.082Mean squared error 0.006 0.006 0.001 0.001 0.004 0.003 0.004 0.002Mean absolute error 0.041 0.014 0.025 0.005 0.026 0.013 0.041 0.016Min. absolute error 0.001 0.001 0.001 0.001 0.000 0.000 0.001 0.001Max. absolute error 0.296 0.160 0.094 0.035 0.174 0.108 0.172 0.088Correlation coefficient 0.895 0.069 0.921 0.032 0.804 0.113 0.786 0.052

Moreover, the purpose of decreasing volatility is failed (mainly when bothmarkets are bearish). Similar outcomes can be seen in Table 10, but herecountries data benefits of semi-correlation fitting. This conclusion isespecially true in mixed markets phases, when diversification works better.

Table 10 Global and Semi-Correlations Forecasts (MSCI WORLD vs. MSCIcountries)

GLOBAL UP-UP DOWN-DOWN MIXEDAVG VOL AVG VOL AVG VOL AVG VOL

R squared 0.502 0.264 0.588 0.232 0.545 0.253 0.635 0.241r squared 0.586 0.185 0.681 0.103 0.591 0.229 0.693 0.202Mean squared error 0.008 0.010 0.002 0.002 0.002 0.002 0.004 0.002Mean absolute error 0.043 0.022 0.029 0.013 0.024 0.010 0.042 0.008Min. absolute error 0.001 0.001 0.001 0.001 0.001 0.001 0.002 0.002Max. absolute error 0.358 0.212 0.094 0.033 0.163 0.075 0.190 0.081Correlation coefficient 0.754 0.130 0.823 0.064 0.750 0.170 0.823 0.127

This result is consistent with the ability to forecast the correct direction of benchmark returns, as seen in Table 8, in particular for countries indexes.


17/35

17

6. EFFICIENTFRONTIERS The last section of the paper is aimed at verifying whether portfoliooptimised using semi-correlation measures can actually, according Harlow Rao (1989) and Harlow (1991), outperform global correlation.We implemented return forecasts with neural networks 9. For all the 11periods we got the direction forecast and then, we applied the suitable semi-correlation. Then, we compared expected with definite results (alreadyshowed in Table 2), so to evaluate whether and when this measureconcretely works.Table 11 contains the minimum and maximum values of the efficientfrontiers

Table 11 Extreme values of efficient frontiers computed with semi-correlations

SECTORS COUNTRIESMIN MAX MIN MAXPERIODS E[r] E[r] E[r] E[r]

1 0.36 4.27 10.50 6.70 -3.05 2.91 8.85 15.272 -2.84 3.17 7.31 5.11 0.06 1.61 6.63 9.303 5.94 0.68 10.40 1.81 9.72 0.33 11.23 14.564 7.22 1.48 12.01 7.78 5.22 -0.58 15.86 7.465 0.43 1.64 5.82 6.21 1.54 0.05 10.85 7.876 1.92 2.17 9.02 7.25 4.69 1.17 18.78 10.407 3.14 1.20 14.32 7.79 8.41 1.28 23.55 9.078 -3.47 3.35 9.47 11.77 -3.59 5.18 17.69 15.229 7.44 2.88 18.27 6.81 8.08 1.37 30.58 12.7510 3.88 6.24 14.55 11.70 3.90 6.95 16.82 23.2711 -0.39 1.07 15.22 16.26 2.28 0.53 30.02 15.51

AVERAGE 2.15 2.56 11.54 8.11 3.39 1.89 17.35 12.79ST. DEV. 3.58 1.58 3.56 3.69 4.25 2.19 7.66 4.44RATIO 0.84 1.42 1.79 1.36

Table 12 shows the differences in average between extreme values of theefficient frontiers and the relative ratio, computed as already seen in equation[1].

9 Pagnoni Gabbi (2001) find out a comparison of other forecasting methods, suchas historical, capital asset pricing, building block, applied to equity and bondmarkets.


18/35

18

Table 12 Return and volatility gaps between global and semi-correlationefficient frontiersSECTORS COUNTRIES

MIN MAX MIN MAXE[r] E[r] E[r] E[r]

RETURN GAP 0.49 -0.35 0.45 -0.53 0.03 -0.66 0.98 -0.85

VOL GAP 0.08 -0.03 -0.07 -0.07 -0.19 -0.13 0.20 -0.08

RATIO 0.27 0.14 0.47 0.16

For all the expected returns we have an increasing value, especially for themaximum values. This is coupled with a lower level of volatility, due to thebetter definition of correlation in the optimising process.Figure 5 and Figure 6 report all the spreads for every forecasting period,respectively for sectors and countries.

Figure 5 Return spreads (MSCI sectors)

-0.20 0.00 0.20 0.40 0.60 0.80 1.00

1

2

3

4

5

6

7

8

9

10

11

min max


19/35

19

Figure 6 Return spreads (MSCI countries)

Despite in periods 5 and 7 we note a negative spread, gap values grant

countries, coherently with outcomes in terms of ability to forecast returns(section 4) and correlations (section 5).

7. CONCLUSIONS Our study shows that in periods of high volatility in correlation coefficients,the application of the phase indicator can facilitate the process of optimisation in building financial portfolios.The answers offered by the study are:

5) short-term correlation, if adequately modelled, can be fundamentalto implement a successful tactical asset allocation. Decisive can bethe existence of numerous cycle inversions in correlation.

6) We obtained correlations lower volatile thanks to the application of semi-correlation measure. The evidence is that the up-up correlationis 7,1% for sectors and 6,7% for countries, while down-downcorrelation is 7,5% and 7,2%. Mixed markets outcomes arerelatively unstable.

7) Neural networks applied to benchmark returns generate good results,especially in terms of direction: this is, in fact, the necessary input

-1.50 -1.00 -0.50 0.00 0.50 1.00 1.50 2.00

1

2

3

4

5

6

7

8

9

10

11

min max


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20

for the application of semi-correlation to the asset allocation process.Outcomes confirm that if applied to countries, the model worksbetter: direction residuals are 11,55% for sectors and 7,83% forcountries.

8) Finally, expected returns and semi-correlations have been used tobuild the efficient frontiers. Results that is worth quoting are:

a. Gap ratios among returns and volatility grow for both thegroups and extreme values of the frontiers;

b. The upward of risk/return ratio is still better forgeographical diversification, especially in the minimumvalues of expected returns.

REFERENCES

Ashley R.A., Patterson D.M., 1989, Linear versus Nonlinear Macroeconomics: A StatisticalTest International Economic Review, 30 (3), 685-704.

Barnett W., Chen, P., 1986, The Aggregation-Theoretic Monetary Aggregates are Chaoticand have Strange Attractions, in W. Barnett, E. Berndt and H. R White (eds),Dynamic Econometric Modelling. Cambridge: Cambridge University Press.

Bramante, Colombo, Gabbi, 1998, Are Neural Network and Econometric Forecasts Good for Trading? Stochastic Variance Model as a Filter Rule , in A.-P. N. Refenes A. N.Burgess J. E. Moody (eds), Decision Technologies for Computational Management

Science , Kluwer Academic Publishers, Boston.Bramante, Gabbi, 2001, Optimal Portfolio Choice Under Changing Risk , Forecasting

Financial Markets 2001, London, May-June.Brock W.A., Dechert W.D., Scheinkman J.A., 1987, A Test for Independence Based on the

Correlation Dimension , Working Paper, University of Houston and University of Chicago.

Dechert W.D., Gencay R., 1993, Lyapunov Exponents as a Nonparametric Diagnostic for Stability Analysis , Journal of Applied Econometrics, 7, S41-S61.

Diebold F.X., 1988, Lecture Notes in Economics and Mathematical System , Springer-Verlag,New York.

Erb, Harvey and Viskanta, 1994, Forecasting International Equity Correlations, FinancialAnalysts Journal, November-December.

Frank M.Z., Stengos T., 1988a, Chaotic Dynamics in Economic Time Series , Journal of

Economic Surveys 2, 103-133.Frank M.Z. Stengos T., 1988b, Some Evidence Concerning Macroeconomic Chaos , Journal of Monetary Economics 22, 423-438.

Gabbi, Colombo, Bramante, Viola, De Vito, Tumietto, 2000, Predicting the Exchange Rate: AComparison of Econometric Models, Neural Networks, and Trading Systems, ITFAJournal, 2000 edition.

Groenen, Franses, 2000, Visualizing time-varying correlations across stock markets , Journalof Empirical Finance, vol. 7.

Harlow, 1991, Asset Allocation in a Downside Risk Framework , Financial Analysts Journal,vol. 47, n. 5.


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Harlow, Rao, 1989, Asset Pricing in a Generalized Mean-Lower Partial Moment Framework ,Journal of Financial and Quantitative Analysis, vol. 24, n. 3.

Hsieh D. A., 1991, Chaos and Nonlinear Dynamics: Application to Financial Market , TheJournal of Finance, No.5.

McCafferty D.F., Ellner S. Gallant A.R., Nychka D.W., 1992, Estimating the LyapunovExponent of a Chaotic System with Nonparametric Regression , Journal of AmericanStatistical Association, 87.

Pagnoni, Gabbi, 2001, Tactical Asset Allocation and Time Diversification for Bond and Equity Markets , Forecasting Financial Markets 2001, London, May-June.

Solnik, Boucrelle, Le Fur, 1996, International Correlation and Volatility, FinancialAnalysts Journal , September/October.

Zhang G., Patuwo B. E., Hu M. Y., 1998, Forecasting with Artificial Neural Networks: TheState of the Art , International Journal of Forecasting, vol. 14.


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Appendix A

Short-term (6 months) holding period efficient frontiers(MSCI sectors)

Appendix BShort-term (6 months) holding period efficient frontiers(MSCI countries)

Appendix CCorrelation Candlestick

Appendix DGeneralization sets MSCI sectors

Appendix DGeneralization sets MSCI countries


23/35

Appendix A Short-term (6 months) holding period efficient frontiers (MSCIsectors)

Standard Deviation (Risk)

Expected Return

0.0 16.01.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0

-10.0

10.0

-9.0

-8.0

-7.0

-6.0

-5.0

-4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0 MSCI Automobiles TR

MSCI Banking TR

MSCI Broadcast & Publish TR

MSCI Building Mat & Construct TR

MSCI Business & Public Serv TR

MSCI Chemicals TR MSCI Construct & Housing TR

MSCI Electrical & Electronics TRMSCI Energy Sources TR

MSCI Financial Services TR

MSCI Food & Hshold Prod TR

MSCI Forest Prod & Paper TR

MSCI Health & Personal Care TR

MSCI Indust Components TR

MSCI Insurance TR MSCI Leisure & Tourism TR

MSCI Machinery & Engineering TR

MSCI Misc Materials&Commod TR

MSCI Real Estate TR

MSCI Recreation, Cons Goods TR

MSCI Telecomm TR

MSCI Transport - Road&Rail TR

MSCI Utilities - Elec&Gas TR


Expected Return

0.0 10.00 .4 0 .8 1 .2 1 .6 2 .0 2 .4 2 .8 3 .2 3 .6 4 .0 4 .4 4 .8 5 .2 5 .6 6 .0 6 .4 6 .8 7 .2 7 .6 8 .0 8 .4 8 .8 9 .2 9 .6

-7.0

7.0

-6.5

-6.0

-5.5

-5.0

-4.5

-4.0

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

6.5

MSCI Automobiles TRMSCI Banking TR




MSCI Chemicals TR

MSCI Construct & Housing TR

MSCI Electrical & Electronics TR

MSCI Energy Sources TR






MSCI Insurance TR

MSCI Leisure & Tourism TRMSCI Machinery & Engineering TR


MSCI Real Estate TR

MSCI Recreation, Cons Goods TRMSCI Telecomm TR




Expected Return

0.0 13.00 .5 1 .0 1 .5 2 .0 2 .5 3 .0 3 .5 4 .0 4 .5 5 .0 5 .5 6 .0 6 .5 7 .0 7 .5 8 .0 8 .5 9 .0 9 .5 1 0. 0 10 .5 1 1. 0 1 1. 5 12 .0 1 2. 5

-1.0

11.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

6.5

7.0

7.5

8.0

8.5

9.0

9.5

10.0

10.5

MSCI Automobiles TR

MSCI Banking TR




MSCI Chemicals TR



MSCI Energy Sources TRMSCI Financial Services TR





MSCI Insurance TR

MSCI Leisure & Tourism TR



MSCI Real Estate TR


MSCI Telecomm TRMSCI Transport - Road&Rail TR



Expected Return

0.0 10.00 .4 0 .8 1 .2 1 .6 2 .0 2 .4 2 .8 3 .2 3 .6 4 .0 4 .4 4 .8 5 .2 5 .6 6 .0 6 .4 6 .8 7 .2 7 .6 8 .0 8 .4 8 .8 9 .2 9 .6

-5.0

12.0

-4.5-4.0-3.5-3.0

-2.5-2.0-1.5-1.0-0.5

0.00.51.01.5

2.02.5

3.03.54.04.5

5.05.5

6.06.57.0

7.58.08.59.0

9.510.0

10.511.011.5

MSCI Automobiles TR

MSCI Banking TRMSCI Broadcast & Publish TR



MSCI Chemicals TR









MSCI Insurance TR




MSCI Real Estate TR


MSCI Telecomm TR




Expected Return

0.0 9.00 .3 0 .6 0 . 9 1 . 2 1 . 5 1 . 8 2 . 1 2 .4 2 .7 3 .0 3 .3 3 . 6 3 . 9 4 . 2 4 . 5 4 . 8 5 .1 5 .4 5 .7 6 .0 6 . 3 6 . 6 6 . 9 7 . 2 7 . 5 7 . 8 8 .1 8 .4 8 .7

-3.0

6.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

MSCI Automobiles TR

MSCI Banking TR

MSCI Broadcast & Publish TRMSCI Building Mat & Construct TR

MSCI Business & Public Serv TR MSCI Chemicals TR





MSCI Food & Hshold Prod TRMSCI Forest Prod & Paper TR



MSCI Insurance TR




MSCI Real Estate TR


MSCI Telecomm TR




Expected Return

0.0 9.00 .3 0 .6 0 .9 1 .2 1 .5 1 .8 2 .1 2 .4 2 .7 3 .0 3 .3 3 .6 3 .9 4 .2 4 .5 4 .8 5 .1 5 .4 5 .7 6 .0 6 .3 6 .6 6 .9 7 .2 7 .5 7 .8 8 .1 8 .4 8 .7

-14.0

9.0

-13.0

-12.0

-11.0

-10.0

-9.0

-8.0

-7.0

-6.0

-5.0

-4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

MSCI Automobiles TR

MSCI Banking TRMSCI Broadcast & Publish TR



MSCI Chemicals TR


MSCI Electrical & Electronics TRMSCI Energy Sources TR MSCI Financial Services TR





MSCI Insurance TR




MSCI Real Estate TR

MSCI Recreation, Cons Goods TRMSCI Telecomm TR




Expected Return

0.0 16.01.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0

-20.0

10.0

-19.0

-18.0

-17.0

-16.0

-15.0

-14.0

-13.0

-12.0

-11.0

-10.0

-9.0

-8.0

-7.0

-6.0

-5.0

-4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

4.0

5.06.0

7.0

8.0

9.0

MSCI Automobiles TR

MSCI Banking TR




MSCI Chemicals TR









MSCI Insurance TR




MSCI Real Estate TR


MSCI Telecomm TR




Expected Return

0.0 11.00 .4 0 .8 1 .2 1 .6 2 .0 2 .4 2 .8 3 .2 3 .6 4 .0 4 .4 4 .8 5 .2 5 .6 6 .0 6 .4 6 .8 7 .2 7 .6 8 .0 8 .4 8 .8 9 .2 9 .6 10.0 10 .4

-9.0

15.0

-8.0

-7.0

-6.0

-5.0

-4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

10.0

11.0

12.0

13.0

14.0

MSCI Automobiles TR

MSCI Banking TR




MSCI Chemicals TR





MSCI Food & Hshold Prod TR MSCI Forest Prod & Paper TR



MSCI Insurance TR




MSCI Real Estate TRMSCI Recreation, Cons Goods TR

MSCI Telecomm TR




24/35

24


Expected Return

0.0 14.00 .5 1 .0 1 .5 2 .0 2 .5 3 .0 3 .5 4 .0 4 .5 5 .0 5 .5 6 .0 6 .5 7 .0 7 .5 8 .0 8 .5 9 .0 9 .5 10 .0 10 .5 11 .0 11.5 12 .0 12 .5 13 .0 13 .5

- 0.0

20.0

-18.0

-16.0

-14.0

-12.0

-10.0

-8.0

-6.0

-4.0

-2.0

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

18.0

MSCI Automobiles TR MSCI Banking TR




MSCI Chemicals TRMSCI Construct & Housing TR








MSCI Insurance TR




MSCI Real Estate TR


MSCI Telecomm TR




Expected Return

0.0 23.01 .0 2 .0 3 .0 4 .0 5 .0 6 .0 7 .0 8 .0 9 .0 1 0. 0 1 1 .0 1 2. 0 1 3 .0 1 4. 0 1 5 .0 1 6. 0 1 7 .0 1 8. 0 1 9 .0 2 0. 0 2 1 .0 2 2. 0

-11.0

14.0

-10.0

-9.0

-8.0

-7.0

-6.0

-5.0

-4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

10.0

11.0

12.0

13.0

MSCI Automobiles TR

MSCI Banking TR




MSCI Chemicals TR









MSCI Insurance TR


MSCI Machinery & Engineering TRMSCI Misc Materials&Commod TR

MSCI Real Estate TR


MSCI Telecomm TR




Expected Return

0.0 19.01.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0-7.0

16.0

-6.0

-5.0

-4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

10.0

11.0

12.0

13.0

14.0

15.0

MSCI Automobiles TR

MSCI Banking TR


MSCI Building Mat & Construct TRMSCI Business & Public Serv TR

MSCI Chemicals TR









MSCI Insurance TR




MSCI Real Estate TRMSCI Recreation, Cons Goods TR

MSCI Telecomm TR




25/35

25

Appendix B Short-term (6 months) holding period efficient frontiers (MSCIcountries)


Expected Return

0.0 17.01.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0

-10.0

8.0

-9.5-9.0-8.5-8.0-7.5-7.0-6.5-6.0-5.5-5.0-4.5-4.0-3.5-3.0-2.5-2.0-1.5-1.0-0.50.00.51.01.52.02.53.03.54.04.55.05.56.06.5

7.07.5

MSCI Australia TR

MSCI Austria TR

MSCI Belgium TR MSCI Canada TR

MSCI Denmark TR

MSCI Finland TR

MSCI France TRMSCI Germany TR

MSCI Hong Kong TR

MSCI Ireland TR

MSCI Italy TR

MSCI Japan TR

MSCI Netherlands TRMSCI New Zealand TR

MSCI Norway TR

MSCI Portugal TR

MSCI Singapore TR

MSCI Spain TR

MSCI Sweden TR

MSCI Switzerland TR

MSCI U.K. TR

MSCI U.S. TR

MSCI World TR


Expected Return

0.0 13.00 .5 1 .0 1 .5 2 .0 2 .5 3 .0 3 .5 4 .0 4 .5 5 .0 5 .5 6 .0 6 .5 7 .0 7 .5 8 .0 8 .5 9 .0 9 .5 1 0. 0 10 .5 1 1. 0 11 .5 1 2. 0 12 .5

-10.0

7.0

-9.5-9.0-8.5-8.0-7.5-7.0-6.5-6.0-5.5-5.0-4.5-4.0-3.5-3.0-2.5-2.0-1.5-1.0-0.50.00.51.01.52.02.53.03.54.04.55.05.56.06.5

MSCI Australia TR

MSCI Austria TR

MSCI Belgium TR

MSCI Canada TR

MSCI Denmark TR

MSCI Finland TR

MSCI France TR

MSCI Germany TR

MSCI Hong Kong TR

MSCI Ireland TR

MSCI Italy TR

MSCI Japan TR

MSCI Netherlands TR

MSCI New Zealand TR

MSCI Norway TR

MSCI Portugal TR

MSCI Singapore TR

MSCI Spain TR

MSCI Sweden TR

MSCI Switzerland TR

MSCI U.K. TR

MSCI U.S. TR

MSCI World TR


Expected Return

0.0 18.01.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0

-8.0

11.0

-7.0

-6.0

-5.0

-4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

10.0

MSCI Australia TR

MSCI Austria TR

MSCI Belgium TR

MSCI Canada TR

MSCI Denmark TR

MSCI Finland TR

MSCI France TR

MSCI Germany TR

MSCI Hong Kong TR

MSCI Ireland TR

MSCI Italy TR

MSCI Japan TR

MSCI Netherlands TR

MSCI New Zealand TR

MSCI Norway TR

MSCI Portugal TR

MSCI Singapore TR

MSCI Spain TR

MSCI Sweden TR

MSCI Switzerland TR

MSCI U.K. TR

MSCI U.S. TR

MSCI World TR


Expected Return

0.0 16.01.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0

-9.0

15.0

-8.0

-7.0

-6.0

-5.0

-4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

10.0

11.0

12.0

13.0

14.0

MSCI Australia TR

MSCI Austria TR

MSCI Belgium TR

MSCI Canada TR

MSCI Denmark TR

MSCI Finland TR

MSCI France TR

MSCI Germany TR

MSCI Hong Kong TR

MSCI Ireland TR

MSCI Italy TR

MSCI Japan TR

MSCI Netherlands TR

MSCI New Zealand TR

MSCI Norway TR

MSCI Portugal TR

MSCI Singapore TR

MSCI Spain TR

MSCI Sweden TR

MSCI Switzerland TRMSCI U.K. TR

MSCI U.S. TR

MSCI World TR


Expected Return

0.0 13.00 .5 1 .0 1 .5 2 .0 2 .5 3 .0 3 .5 4 .0 4 .5 5 .0 5 .5 6 .0 6 .5 7 .0 7 .5 8 .0 8. 5 9 .0 9. 5 1 0. 0 10 .5 1 1. 0 11 .5 1 2. 0 12 .5

-7.0

11.0

-6.5-6.0

-5.5-5.0-4.5-4.0-3.5-3.0-2.5-2.0-1.5-1.0-0.50.00.51.01.52.02.53.03.54.04.55.05.56.06.57.07.58.08.59.09.5

10.010.5

MSCI Australia TR

MSCI Austria TR

MSCI Belgium TR

MSCI Canada TR

MSCI Denmark TRMSCI Finland TR

MSCI France TR

MSCI Germany TR

MSCI Hong Kong TR

MSCI Ireland TR

MSCI Italy TRMSCI Japan TR

MSCI Netherlands TR

MSCI New Zealand TR

MSCI Norway TR

MSCI Portugal TR

MSCI Singapore TR

MSCI Spain TR

MSCI Sweden TR

MSCI Switzerland TR

MSCI U.K. TR

MSCI U.S. TRMSCI World TR


Expected Return

0.0 17.01.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0

-20.0

20.0

-18.0

-16.0

-14.0

-12.0

-10.0

-8.0

-6.0

-4.0

-2.0

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

18.0

MSCI Australia TR

MSCI Austria TR

MSCI Belgium TRMSCI Canada TR

MSCI Denmark TR

MSCI Finland TR

MSCI France TRMSCI Germany TR

MSCI Hong Kong TRMSCI Ireland TR MSCI Italy TR

MSCI Japan TR

MSCI Netherlands TR

MSCI New Zealand TR

MSCI Norway TRMSCI Portugal TR

MSCI Singapore TR

MSCI Spain TRMSCI Sweden TR

MSCI Switzerland TR

MSCI U.K. TRMSCI U.S. TR

MSCI World TR


Expected Return

0.0 16.01.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0

-10.0

30.0

-8.0

-6.0

-4.0

-2.0

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

18.0

20.0

22.0

24.0

26.0

28.0

MSCI Australia TR

MSCI Austria TR

MSCI Belgium TR

MSCI Canada TR

MSCI Denmark TR

MSCI Finland TR

MSCI France TR

MSCI Germany TR

MSCI Hong Kong TR

MSCI Ireland TR

MSCI Italy TR

MSCI Japan TR

MSCI Netherlands TR

MSCI New Zealand TR

MSCI Norway TR

MSCI Portugal TR

MSCI Singapore TR

MSCI Spain TR

MSCI Sweden TR

MSCI Switzerland TR

MSCI U.K. TR

MSCI U.S. TR

MSCI World TR


Expected Return

0.0 21.01 .0 2 .0 3 .0 4 .0 5 .0 6 .0 7 .0 8 .0 9 .0 1 0. 0 1 1 .0 1 2. 0 1 3 .0 1 4. 0 1 5. 0 1 6 .0 1 7. 0 1 8. 0 1 9. 0 2 0 .0

-30.0

20.0

-28.0

-26.0

-24.0

-22.0

-20.0

-18.0

-16.0

-14.0

-12.0

-10.0

-8.0

-6.0

-4.0

-2.0

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

18.0

MSCI Australia TR

MSCI Austria TRMSCI Belgium TR

MSCI Canada TR

MSCI Denmark TR

MSCI Finland TRMSCI France TR MSCI Germany TR

MSCI Hong Kong TR

MSCI Ireland TR

MSCI Italy TR

MSCI Japan TR

MSCI Netherlands TR

MSCI New Zealand TRMSCI Norway TR

MSCI Portugal TR

MSCI Singapore TR

MSCI Spain TR

MSCI Sweden TR

MSCI Switzerland TRMSCI U.K. TR

MSCI U.S. TR

MSCI World TR


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Expected Return

0.0 20.01.0 2.0 3.0 4.0 5.0 6 .0 7 .0 8 .0 9 .0 10 .0 11 .0 1 2. 0 1 3.0 1 4.0 1 5.0 1 6.0 17 .0 18 .0 19 .0

-20.0

30.0

-18.0

-16.0

-14.0

-12.0

-10.0

-8.0

-6.0

-4.0

-2.0

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

18.0

20.0

22.0

24.0

26.0

28.0

MSCI Australia TR

MSCI Austria TR

MSCI Belgium TR

MSCI Canada TR

MSCI Denmark TR

MSCI Finland TR

MSCI France TR

MSCI Germany TR

MSCI Hong Kong TR

MSCI Ireland TR

MSCI Italy TR

MSCI Japan TR

MSCI Netherlands TR

MSCI New Zealand TRMSCI Norway TR

MSCI Portugal TR

MSCI Singapore TR

MSCI Spain TR

MSCI Sweden TR

MSCI Switzerland TR

MSCI U.K. TR

MSCI U.S. TR

MSCI World TR


Expected Return

0.0 29.02.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0 22.0 24.0 26.0

-20.0

20.0

-18.0

-16.0

-14.0

-12.0

-10.0

-8.0

-6.0

-4.0

-2.0

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

18.0

MSCI Australia TR

MSCI Austria TR

MSCI Belgium TR

MSCI Canada TR

MSCI Denmark TR

MSCI Finland TR

MSCI France TR

MSCI Germany TR

MSCI Hong Kong TR

MSCI Ireland TR MSCI Italy TR

MSCI Japan TRMSCI Netherlands TR

MSCI New Zealand TR

MSCI Norway TR

MSCI Portugal TR

MSCI Singapore TR

MSCI Spain TR

MSCI Sweden TRMSCI Switzerland TR

MSCI U.K. TR

MSCI U.S. TR

MSCI World TR


Expected Return

0.0 23.01 .0 2 .0 3 .0 4 .0 5 .0 6 .0 7 .0 8 .0 9 .0 1 0. 0 1 1 .0 1 2. 0 1 3 .0 1 4. 0 1 5 .0 1 6. 0 1 7 .0 1 8. 0 1 9 .0 2 0. 0 2 1. 0 2 2 .0

-10.0

30.0

-8.0

-6.0

-4.0

-2.0

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

18.0

20.0

22.0

24.0

26.0

28.0

MSCI Australia TR

MSCI Austria TR

MSCI Belgium TR

MSCI Canada TR

MSCI Denmark TR

MSCI Finland TR

MSCI France TR

MSCI Germany TR

MSCI Hong Kong TR

MSCI Ireland TRMSCI Italy TR

MSCI Japan TR

MSCI Netherlands TR

MSCI New Zealand TR

MSCI Norway TR

MSCI Portugal TR

MSCI Singapore TR

MSCI Spain TR

MSCI Sweden TR

MSCI Switzerland TR

MSCI U.K. TR

MSCI U.S. TR

MSCI World TR


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27

Appendix C Correlation Candlestick

Figure 7 Up-up correlation candlestick (MSCI countries)

Figure 8 Down-down correlation candlestick (MSCI countries)

-100.00%

-80.00%

-60.00%

-40.00%

-20.00%

0.00%

20.00%

40.00%

60.00%

80.00%

100.00%

AUS AUT BEL CAN D EN FIN FRA G ER H OK I RE ITA JAP NE T N EZ NOR P OR SI N SPA S WE S WI UK USA

-100.00%

-80.00%

-60.00%

-40.00%

-20.00%

0.00%

20.00%

40.00%

60.00%

80.00%

100.00%

AUS AUT BEL CAN DEN FIN FRA GER HOK IRE ITA JAP NET NEZ NOR POR SIN SPA SWE SWI UK USA


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Figure 9 Mixed correlation candlestick (MSCI countries)

Figure 10 Up-up correlation candlestick (MSCI sectors)

-100.00%

-80.00%

-60.00%

-40.00%

-20.00%

0.00%

20.00%

40.00%

60.00%

80.00%

100.00%

AUS AUT BEL CAN DEN FIN FRA GER HOK IRE ITA JAP NET NEZ NOR POR SIN SPA SWE SWI UK USA

-100.00%

-80.00%

-60.00%

-40.00%

-20.00%

0.00%

20.00%

40.00%

60.00%

80.00%

100.00%



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Figure 11 Down-down correlation candlestick (MSCI sectors)

Figure 12 Mixed correlation candlestick (MSCI sectors)

-100.00%

-80.00%

-60.00%

-40.00%

-20.00%

0.00%

20.00%

40.00%

60.00%

80.00%

100.00%


-100.00%

-80.00%

-60.00%

-40.00%

-20.00%

0.00%

20.00%

40.00%

60.00%

80.00%

100.00%



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Appendix D Generalization sets MSCI sectors

Out-of-sample returns - MSCI ENERGY

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

1 2 3 4 5 6 7 8 9 1 0 11 1 2 13 1 4 15 1 6 17 1 8 19 2 0 21 2 2 2 3 24 2 5 26 2 7 28 2 9 30 3 1 32 3 3 34 3 5 36

ENERGY NEURAL NET

Out-of-sample returns - MSCI MATERIAL

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

1 2 3 4 5 6 7 8 9 1 0 11 1 2 13 1 4 15 1 6 17 1 8 19 2 0 21 2 2 23 24 2 5 26 2 7 28 2 9 30 31 3 2 33 3 4 35 3 6

MATER IAL NE UR AL NET

Out-of-sample returns - MSCI CAPITAL GOODS

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

1 2 3 4 5 6 7 8 9 1 0 11 1 2 13 1 4 15 1 6 17 1 8 19 2 0 21 2 2 23 2 4 25 2 6 27 2 8 29 3 0 31 3 2 33 3 4 35 3 6

C AP ITAL G OO DS N EU RA L N ET

Out-of-sample returns - MSCI COMML SVC & SUPPL

-0.1

-0.05

0

0.05

0.1

0.15

0.2

1 2 3 4 5 6 7 8 9 1 0 11 1 2 13 1 4 15 16 1 7 18 1 9 20 2 1 22 2 3 24 25 2 6 27 2 8 29 3 0 31 3 2 33 3 4 35 3 6

C OM ML S VC & S UP PL N EU RA L N ET

Out-of-sample returns - MSCI TRANSPORTATION

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

1 2 3 4 5 6 7 8 9 1 0 11 1 2 13 1 4 15 1 6 17 1 8 19 2 0 21 2 2 23 2 4 25 2 6 27 2 8 29 3 0 31 3 2 33 3 4 35 3 6

T RA NS PO RTAT IO N N EU RA L N ET

Out-of-sample returns - MSCI AUTO COMPONENTS

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

1 2 3 4 5 6 7 8 9 1 0 11 1 2 13 1 4 15 16 1 7 18 1 9 20 2 1 22 2 3 24 25 2 6 27 2 8 29 3 0 31 3 2 33 3 4 35 3 6

A UTO & C OM PO NE N TS N EU RA L N ET

Out-of-sample returns - MSCI CONS DUR & APPAREL

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

1 2 3 4 5 6 7 8 9 1 0 11 1 2 13 1 4 15 1 6 17 1 8 19 2 0 21 2 2 23 2 4 25 2 6 27 2 8 29 3 0 31 3 2 33 3 4 35 3 6

C ON S D U R & A PPAR EL N EU R AL N ET

Out-of-sample returns - MSCI HOTELS RESTAURANTS AND LEISURE

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

1 2 3 4 5 6 7 8 9 1 0 11 1 2 13 1 4 15 1 6 1 7 18 1 9 20 2 1 22 2 3 24 2 5 26 2 7 2 8 29 3 0 31 3 2 33 3 4 35 3 6

H OT EL S R ES T & L EI S N EU RA L N ET


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Out-of-sample returns - MSCI MEDIA

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

1 2 3 4 5 6 7 8 9 1 0 11 1 2 13 1 4 15 1 6 17 1 8 19 2 0 21 2 2 23 2 4 25 2 6 27 2 8 29 3 0 31 3 2 33 3 4 35 3 6

MEDIA NEURAL NET

Out-of-sample returns - MSCI RETAILING

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

1 2 3 4 5 6 7 8 9 1 0 11 1 2 13 1 4 15 1 6 17 1 8 19 2 0 21 2 2 23 2 4 25 2 6 27 2 8 29 3 0 31 3 2 33 3 4 35 3 6

RE TAIL ING N EUR AL N ET

Out-of-sample returns - MSCI FOOD & DRUG

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

1 2 3 4 5 6 7 8 9 1 0 11 1 2 13 1 4 15 1 6 17 1 8 19 2 0 21 2 2 23 2 4 25 2 6 27 2 8 29 3 0 31 3 2 3 3 34 3 5 36

FOOD & DRUG RETL NEURAL NET

Out-of-sample returns - MSCI FOOD BEV & TOBACCO

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

1 2 3 4 5 6 7 8 9 1 0 11 1 2 13 1 4 15 1 6 17 1 8 19 2 0 21 2 2 23 2 4 25 2 6 27 2 8 29 3 0 31 3 2 33 3 4 35 3 6

F OO D B EV & TOB AC C O N EU R AL N E T

Out-of-sample returns - MSCI HOUSE

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

1 2 3 4 5 6 7 8 9 1 0 11 1 2 13 1 4 15 1 6 17 1 8 19 2 0 21 2 2 23 2 4 25 2 6 2 7 28 2 9 30 3 1 32 3 3 34 3 5 36

HOUSE & PERS PROD NEURAL NET

Out-of-sample returns - MSCI HEALTH CARE

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

1 2 3 4 5 6 7 8 9 1 0 11 1 2 13 1 4 15 1 6 17 1 8 19 2 0 21 2 2 23 2 4 25 2 6 27 2 8 29 3 0 31 3 2 33 3 4 35 3 6

H C AR E E QU IP & S VC N EU RA L N ET

Out-of-sample returns - MSCI PHARMA & BIOTECH

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

1 2 3 4 5 6 7 8 9 1 0 11 1 2 13 1 4 15 1 6 17 1 8 19 2 0 21 2 2 23 2 4 25 2 6 27 2 8 29 3 0 31 3 2 33 3 4 35 3 6

PHARMA & BIOTECH NEURAL NET

Out-of-sample returns - MSCI BANKS

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

1 2 3 4 5 6 7 8 9 1 0 11 1 2 13 1 4 15 1 6 17 1 8 19 2 0 21 2 2 23 2 4 25 2 6 27 2 8 29 3 0 31 3 2 33 3 4 35 3 6

BANKS NEURAL NET


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Out-of-sample returns - MSCI DIVERS FINANCIAL

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

1 2 3 4 5 6 7 8 9 1 0 11 1 2 13 1 4 15 1 6 17 1 8 19 2 0 21 2 2 23 2 4 25 2 6 27 2 8 29 3 0 31 3 2 33 3 4 35 3 6

D IV ER S F INA NC N EU RA L N ET

Out-of-sample returns - MSCI INSURANCE

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

1 2 3 4 5 6 7 8 9 1 0 11 1 2 13 1 4 15 1 6 17 1 8 19 20 2 1 22 2 3 24 2 5 26 2 7 28 2 9 30 31 3 2 33 3 4 35 3 6

INSURAN CE NEURAL NET

Out-of-sample returns - MSCI SOFTWARE

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1 2 3 4 5 6 7 8 9 1 0 11 1 2 13 1 4 15 1 6 17 1 8 19 2 0 21 2 2 23 2 4 25 2 6 2 7 28 2 9 30 3 1 32 3 3 34 3 5 36

SOFTWARE & SERVICES NEURAL NET

Out-of-sample returns - MSCI TECH HARD & EQUIPMENT

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1 2 3 4 5 6 7 8 9 1 0 11 1 2 13 1 4 15 1 6 17 18 1 9 20 2 1 22 2 3 24 2 5 26 2 7 28 2 9 30 3 1 32 3 3 34 3 5 36

T EC H H AR D & E QU IP N EU RA L N ET

Out-of-sample returns - MSCI TELECOM

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

1 2 3 4 5 6 7 8 9 1 0 11 1 2 13 1 4 15 1 6 17 1 8 19 2 0 21 2 2 23 2 4 25 2 6 2 7 28 2 9 30 3 1 32 3 3 34 3 5 36

TE LE CO M S VC N EU RA L N ET

Out-of-sample returns - MSCI UTILITIES

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

1 2 3 4 5 6 7 8 9 1 0 1 1 12 1 3 14 1 5 16 1 7 18 1 9 20 2 1 22 2 3 24 2 5 26 2 7 28 2 9 30 3 1 32 3 3 34 3 5 36

UTILITIES NEURAL NET


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Appendix E Generalization sets MSCI countries

Out-of-sample returns - MSCI AUSTRALIA

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

1 2 3 4 5 6 7 8 9 1 0 11 1 2 13 1 4 15 1 6 17 1 8 19 2 0 21 2 2 23 2 4 25 2 6 2 7 28 2 9 30 3 1 32 3 3 34 3 5

A USTRAL IA NE URAL NET

Out-of-sample returns - MSCI AUSTRIA

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

1 2 3 4 5 6 7 8 9 1 0 11 1 2 13 1 4 15 1 6 17 1 8 19 2 0 21 2 2 23 2 4 25 2 6 27 2 8 29 3 0 31 3 2 33 3 4 35

AUSTRIA NEURAL NET

Out-of-sample returns - MSCI BELGIUM

-0.2

-0.1

0

0.1

0.2

0.3

0.4

1 2 3 4 5 6 7 8 9 10 1 1 12 1 3 14 1 5 16 1 7 18 1 9 20 2 1 22 2 3 24 2 5 26 2 7 28 2 9 30 3 1 32 3 3 34 3 5

BELGIUM NEURAL NET

Out-of-sample returns - MSCI CANADA

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

1 2 3 4 5 6 7 8 9 1 0 11 1 2 13 1 4 15 1 6 17 1 8 19 2 0 21 2 2 23 2 4 25 2 6 27 2 8 29 3 0 31 3 2 33 3 4 35

CANADA NEURAL NET

Out-of-sample returns - MSCI DENMARK

-0.2

-0.1

0

0.1

0.2

0.3

0.4

1 2 3 4 5 6 7 8 9 1 0 11 1 2 13 1 4 15 1 6 17 1 8 19 2 0 21 2 2 23 2 4 25 2 6 27 2 8 29 3 0 31 3 2 33 3 4 35

DENMARK NEURAL NET

Out-of-sample returns - MSCI FINLAND

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1 2 3 4 5 6 7 8 9 1 0 11 1 2 13 1 4 15 1 6 17 1 8 19 2 0 21 2 2 23 2 4 25 2 6 27 2 8 29 3 0 31 3 2 33 3 4 35

FINLAND NEURAL NET

Out-of-sample returns - MSCI FRANCE

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

1 2 3 4 5 6 7 8 9 1 0 11 1 2 13 1 4 15 1 6 17 1 8 19 2 0 21 2 2 23 2 4 25 2 6 27 2 8 29 3 0 31 3 2 33 3 4 35

FRANCE NEURAL NET

Out-of-sample returns - MSCI GERMANY

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

1 2 3 4 5 6 7 8 9 1 0 11 12 1 3 1 4 1 5 16 1 7 18 1 9 20 2 1 22 2 3 24 2 5 26 2 7 28 2 9 30 3 1 32 3 3 34 3 5

GERMANY NEURAL NET


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Out-of-sample returns - MSCI HONG KONG

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

1 2 3 4 5 6 7 8 9 1 0 11 1 2 13 1 4 15 1 6 17 1 8 19 2 0 21 2 2 23 2 4 25 2 6 27 2 8 29 3 0 31 3 2 33 3 4 35

HONG K ONG NE URA L NET

Out-of-sample returns - MSCI IRELAND

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

1 2 3 4 5 6 7 8 9 1 0 11 1 2 13 1 4 15 1 6 17 1 8 19 2 0 21 2 2 23 2 4 25 2 6 27 2 8 29 3 0 31 3 2 33 3 4 35

IRELAND NEURAL NET

Out-of-sample returns - MSCI ITALY

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

1 2 3 4 5 6 7 8 9 1 0 11 1 2 13 1 4 15 1 6 17 1 8 19 2 0 21 2 2 23 2 4 25 2 6 27 2 8 29 3 0 31 3 2 33 3 4 35

ITALY NEURAL NET

Out-of-sample returns - MSCI JAPAN

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

1 2 3 4 5 6 7 8 9 1 0 11 1 2 13 1 4 15 1 6 17 1 8 19 2 0 21 2 2 23 2 4 25 2 6 27 2 8 29 3 0 31 3 2 33 3 4 35

JAPAN NEURAL NET

Out-of-sample returns - MSCI NETHERLANDS

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

1 2 3 4 5 6 7 8 9 1 0 11 1 2 13 1 4 15 1 6 17 1 8 19 2 0 21 2 2 23 2 4 25 2 6 27 2 8 29 3 0 31 3 2 33 3 4 35

NE TH ER LA ND S N EU RA L N ET

Out-of-sample returns - MSCI NEW ZEALAND

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

1 2 3 4 5 6 7 8 9 1 0 11 12 1 3 14 15 1 6 17 18 1 9 20 2 1 22 23 2 4 25 26 2 7 28 2 9 30 31 3 2 33 34 3 5

N EW Z EA LA ND N EU RA L N ET

Out-of-sample returns - MSCI NORWAY

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

1 2 3 4 5 6 7 8 9 1 0 11 1 2 13 1 4 15 1 6 17 1 8 19 2 0 21 2 2 23 2 4 25 2 6 27 2 8 29 3 0 31 3 2 33 3 4 35

NORWAY NEURAL NET

Out-of-sample returns - MSCI PORTUGAL

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

1 2 3 4 5 6 7 8 9 1 0 11 1 2 13 1 4 15 1 6 17 1 8 19 2 0 21 2 2 23 2 4 25 2 6 27 2 8 29 3 0 31 3 2 33 3 4 35

POR TUGA L NEU RAL NET


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Out-of-sample returns - MSCI SINGAPORE

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

1 2 3 4 5 6 7 8 9 1 0 11 1 2 13 1 4 15 1 6 17 18 1 9 20 2 1 22 2 3 24 2 5 26 2 7 28 2 9 30 3 1 32 3 3 34 3 5

S INGA POR E NEURA L NE T

Out-of-sample returns - MSCI SPAIN

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

1 2 3 4 5 6 7 8 9 1 0 11 1 2 13 1 4 15 1 6 17 1 8 19 2 0 21 2 2 23 2 4 25 2 6 27 2 8 29 3 0 31 3 2 3 3 3 4 35

SPAIN NEURAL NET

Out-of-sample returns - MSCI SWITZERLAND

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

1 2 3 4 5 6 7 8 9 1 0 11 1 2 13 1 4 15 1 6 17 1 8 19 2 0 21 2 2 2 3 24 2 5 26 2 7 28 2 9 30 3 1 32 3 3 34 3 5

S WI TZ ERL AND N EUR AL NE T

Out-of-sample returns - MSCI UNITED KINGDOM

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

1 2 3 4 5 6 7 8 9 1 0 11 12 1 3 14 15 1 6 17 1 8 19 20 2 1 22 2 3 24 2 5 26 2 7 28 2 9 30 31 3 2 33 3 4 35

UNITED KINGDOM NEURAL NET

Out-of-sample returns - MSCI USA

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

1 2 3 4 5 6 7 8 9 1 0 11 1 2 13 1 4 15 1 6 17 1 8 19 2 0 21 2 2 23 2 4 25 2 6 27 2 8 29 3 0 31 3 2 33 3 4 35

USA NEURAL NET

Out-of-sample returns - MSCI WORLD

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

1 2 3 4 5 6 7 8 9 1 0 11 1 2 13 1 4 15 1 6 17 1 8 19 20 2 1 22 2 3 24 25 2 6 27 2 8 29 30 3 1 32 3 3 34 3 5

WORLD NEURAL NET

SemiVariance and SemiCorrelation

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Transcript of SemiVariance and SemiCorrelation