mean-variance portfolio model modiﬁed by nonlinear bagging

Mean-Variance Portfolio Model Modified by

Nonlinear Bagging Predictors

Tomoya Suzuki and Kiyoharu Tanaka

Department of Intelligent Systems Engineering, College of Engineering, Ibaraki University4-12-1 Nakanarusawa-cho, Hitachi, Ibaraki 316-8511, Japan

E-mail: [email protected]

Abstract In Markowitz’s mean-variance portfolio model, the probability distribution of a future return

is composed of recent historical prices, and the future return and future risk are estimated as the mean

and standard deviation of the distribution, respectively. Namely, the future return is predicted by a simple

moving average, and the risk is simply the historical fluctuation. In this study, to improve the prediction

accuracy of the future return, we apply a nonlinear prediction method following local spatial dynamics,

and to estimate the future risk, we produce a probability distribution by aggregating predicted values by

the bagging algorithm. Then, each risk is reduced by making a portfolio, that is, we apply the portfolio

effect. Namely, our method attempts to simultaneously improve the prediction accuracy and reduce the

risk of its prediction error. To confirm the validity of our method, we performed investment simulations.

As a result, we could realize higher profit and lower risk in investment than by the conventional method.

Keywords: portfolio theory, nonlinear prediction, bagging algorithm, financial markets

1. Introduction

Markowitz’s portfolio theory [1] is useful for de-ciding the allocation of stocks for investment and canreduce its risk by the portfolio effect. For this theory,it is necessary to calculate the future return and riskof each stock, but these are completely unknown. Forthis reason, the conventional method estimates themas the mean and standard deviation of recent histor-ical data, respectively. Namely, because this estima-tion of the future return corresponds to the moving-average prediction, it might be insufficient to predictreal financial systems, which are typical examples ofcomplex systems.

In the present study, we apply a nonlinear predic-tion model and the bagging algorithm [2] to the con-ventional portfolio theory. First, a nonlinear predic-tion is used to improve the prediction accuracy of fu-ture return rates because it can model the relationshipbetween the past and the future, that is, the temporalevolution of financial systems, even if this relationshipis nonlinear. Moreover, the bagging algorithm is usedfor ensemble learning to estimate the probability dis-tribution of future return rate. In particular, in Ref.[3], the mean of an ensemble distribution composed

of nonlinear predictors is used as the predicted value,and this ensemble learning can improve the predictionaccuracy compared with a single nonlinear prediction.Furthermore, in Ref. [4] it was reported that the stan-dard deviation of this ensemble distribution is relatedto the difficulty of prediction. Namely, we considerthis standard deviation as the risk and attempt toreduce the risk by making a portfolio in the presentstudy.

In Sect. 2, we introduce the conventional methodused to make a portfolio. In Sect. 3, we introduce anonlinear prediction and the bagging algorithm, andthen propose a new portfolio model based on thesetechniques. In Sect. 4, we perform some investmentsimulations with real stock prices to confirm the va-lidity of our proposed method. Then, we discuss howto select better stocks to compose a multistock portfo-lio and evaluate its investment performance from theviewpoints of profitability and safety. In Sect. 5, weconclude our research.

Journal of Signal Processing, Vol.18, No.6, pp.283-290, November 2014

PAPER

Journal of Signal Processing, Vol. 18, No. 6, November 2014 283

2. Conventional Portfolio Models

2.1 Mean-variance portfolio model

If we denote xi as the price of the ith stock (i =1, 2, · · · , N) at time t, the return rate ri(t) is given by

ri(t) =xi(t) − xi(t − 1)

xi(t − 1)(1)

In the mean-variance portfolio model, the future re-turn and future risk are respectively given by the meanand standard deviation of the future probability dis-tribution. However, because this distribution is com-pletely unknown, an empirical distribution made fromthe recent historical data is used to estimate the re-turn rate r̂i(t + 1) and risk σ̂i(t + 1), which are givenby

r̂i(t + 1) = r̄i(t)

=1T

T−1∑a=0

ri(t − a) (2)

σ̂i(t + 1) = σi(t)

=

√√√√ 1T

T−1∑a=0

[ri(t − a) − r̄i(t)]2 (3)

where T is the length of the historical data.Then, in the case of making a portfolio with N

stocks, the expected return rate r̂p(t+1) and expectedrisk σ̂p(t+1) of the portfolio are respectively given by

r̂p(t + 1) =N∑

i=1

cir̂i(t + 1) (4)

σ̂p(t + 1) =

√√√√N∑

i=1

N∑j=1

cicj σ̂ij(t + 1) (5)

where ci is the allocation rate andN∑

i=1

ci = 1. Then,

σ̂ij(t + 1) is defined by

σ̂ij(t+1) = σij(t)

=1T

T−1∑a=0

[ri(t−a)−r̄i(t)]·[rj(t−a)−r̄j(t)] (6)

In the portfolio effect [1], the risk of a portfolioσ̂p(t + 1) can be reduced by increasing the number ofstocks in the portfolio, N , and reducing the correlationamong them. Here, if all N stocks have no correlation,σ̂ij(t + 1) = 0 (i ̸= j). Thus, Eq. (5) is rewritten as

σ̂2p(t + 1) =

N∑j=1

c2j σ̂

2j (t + 1) (7)

Here, the upper limit of σ̂2j (t+1) is set as σ̂2

j (t+1) ≤ P ,

σ̂2p(t + 1) ≤

(c21 + · · · + c2

N

)P (8)

If N stocks are allocated uniformly, i.e., ci = 1/N , Eq.(8) is rewritten as

0 ≤ σ̂2p(t + 1) ≤ P

N(9)

Therefore, σ̂2p(t + 1) → 0 as N → ∞. This is the

portfolio effect.Next, if we allocate {ci} so as to maxmize r̂p(t+1)

and minimize σ̂p(t + 1), this investment can be rea-sonable. From this viewpoint, the Sharpe ratio

Sr(t) =r̂p(t + 1) − rf

σ̂p(t + 1)(10)

has been proposed [5]. In the mean-variance portfoliomodel, we maximize Sr to optimize the allocation of{ci}. Here, we used the interior point method becauseSr is a convex function. After this optimization, ifr̂p(t + 1) < 0, we take short (sell) positions for allstocks composing a portfolio to make its future returna positive value. Then, rf is the risk-free return, butwe set rf = 0 because the short-term interest rate hasbeen nearly zero in Japan.

2.2 Risk-parity portfolio model

As another portfolio model, the risk-parity port-folio model [6, 7] has been recently applied to assetallocation because the mean-variance portfolio oftenbiases the risk contributions of each asset to a few as-sets. Here, the risk contribution of the ith asset isdefined as

φi =ci

∑Nj=1 cj σ̂ij(t + 1)σ̂p(t + 1)

= ciβi(t + 1)σ̂p(t + 1) (11)

where βi(t+1) = σ̂ip(t+1)/σ̂2p(t+1). If these risk con-

tributions are biased to a few assets, this portfolio ispractically dangerous even if its Sharpe ratio is large.For this reason, the risk-parity portfolio optimizes theallocation rates of each asset {ci} so as to equalizetheir risk contributions {φi}. Namely, we maximizethe following entropy:

E = exp

(−

N∑i=1

φi log φi

)(12)

Therefore, the risk-parity portfolio does not needany future return rates for asset allocation because itis not easy to predict future return rates. However,in deciding whether to take long (buy) or short (sell)positions to compose a portfolio, we use the predictedfuture return rates for Eq. (4). After calculating r̂p(t+1) by using {ci} optimized by Eq. (12), if r̂p(t+1) ≥ 0,we take long positions; if r̂p(t + 1) < 0, we take shortpositions for a portfolio, similarly to that in the caseof the mean-variance model.

284 Journal of Signal Processing, Vol. 18, No. 6, November 2014

3. Our Portfolio Model

3.1 Nonlinear prediction for dynamical systems

First, to reproduce the background dynamics thatderives the time-series data ri(t), we reconstruct amultidimensional attractor vi(t) from ri(t) by theTakens embedding method [8]:

vi(t) = {ri(t), ri(t − τ), . . . , ri(t − τ(d − 1))} (13)

where τ is the delay time and d is the embedding di-mension. Then, we merge all the {vi(t)} into an at-tractor:

V (t) = {v1(t), v2(t), . . . , vN (t)} (14)

Next, we predict the future state of V (t) by the lo-cal linear approximation method [9, 10] as a nonlinearprediction. Here, some local neighbors V (tk) wherek = 1 ∼ K are selected from all of the historical at-tractors V (t), that is, tk < t. Then, by averagingthe next states of the neighbors, we can obtain thepredicted value of V (t + 1) as

V̂ (t + 1) =1K

K∑k=1

V (tk + 1) (15)

3.2 Bagging predictors for the probability distribu-tion of a future return

As mentioned in Sect. 3.1, because the local lin-ear prediction uses only local data whose length is notlong enough, ensemble learning is useful to improvethe prediction accuracy [3] and to estimate the pos-sibility of its prediction error [4]. Here, the ensemblelearning applied for prediction is called the bootstrapaggregating (bagging) predictors [2]. In our portfo-lio model, these bagging predictors are applied to im-prove the prediction accuracy of future returns and toestimate the risk of each prediction.

First, we randomly sample K neighbors from{V (tk)} with replacement to obtain a new set of nearneighbors. Then, we apply the nonlinear prediction ofEq. (15) to the new neighbors and obtain another pre-dicted value V̂b(t + 1). After repeating this procedureB times, we can estimate the possible distribution ofthe future value as {V̂b(t + 1)} where b = 1 ∼ B. Inthe present study, we set B = 1000. In the field of fi-nancial engineering, because the expected return andrisk correspond to the mean and the standard devi-ation of the distribution of the possible return rates,respectively, we estimate the final predicted value by

V̂ (t + 1) =1B

B∑b=1

V̂b(t + 1) (16)

The predicted value of Eq. (16) can be more accuratethan that of Eq. (15) due to the effect of the ensemble

learning [3]. Moreover, because the predicted futurereturns {r̂i(t + 1)} are included in V̂ (t + 1) of Eq.(15), if each future return of V̂b(t + 1) is denoted asr̂i,b(t + 1), we can rewrite Eq. (16) as

r̂i(t + 1) =1B

B∑b=1

r̂i,b(t + 1) (17)

By substituting Eq. (17) into Eq. (4), we can calcu-late the expected return rate of a portfolio.

Next, because the risk of a portfolio is consideredas the standard deviation of the possible return rates,we estimate the risk by

σ̂i(t + 1) =

√√√√ 1B

B∑b=1

[r̂i,b(t + 1) − r̂i(t + 1)]2 (18)

Similarly, the covariance σ̂ij(t + 1) is given by

σ̂ij(t + 1) =1B

B∑b=1

[ri,b(t+1) − r̂i(t+1)]

· [rj,b(t+1) − r̂j(t+1)] (19)

By substituting Eqs. (18) and (19) into Eq. (5), wecan calculate the expected risk of a portfolio σ̂p(t+1).Then, we maximize the Sharpe ratio Sr to optimizethe allocation rate {ci} similarly to that in the caseof the mean-variance portfolio model. Namely, ourmethod aims to simultaneously improve the predictionaccuracy and reduce the risk of its prediction error.

4. Investment Simulations

In this section, to confirm the validity of our pro-posed method, we perform some investment simula-tions with two sets of real stock data. Details ofthe data are shown in Table 1. In the first term,the Japanese stock market is relatively stable; thisperiod starts some years after the collapse of theJapanese bubble economy in 1991. On the other hand,the second term is unstable because it includes theglobal financial crisis caused by the Lehman Broth-ers bankruptcy in September 2008. In both terms, weonly used daily data because it is difficult to predictlarger time scale data such as weekly and monthlydata [11, 12]. This seems to be because a large timescale, that is, a large sampling interval, destroys theoriginal dynamics of financial markets, and thereforeprediction models cannot work well [13].

Before starting the investment, we have to optimizethe parameters of each portfolio model. For the con-ventional portfolio models mentioned in Sect. 2, thelength of the historical data T should be optimized,and for our proposed model outlined in Sect. 3.1, d,τ , and K should be optimized.

First, the delay time τ should be determined so asto make each axis of vi in Eq. (13) uncorrelated [8]. As


Table 1 Details of real financial data used for investment simulations

First term Second termLearning period 1995/2 – 2000/4 (about 5 years) 2003/4 – 2008/6 (about 5 years)Investment period 2000/5 – 2005/6 (about 5 years) 2008/7 – 2013/8 (about 5 years)Market Tokyo Stock Exchange (TSE)Target stocks 50 stocks randomly selected from each sectorData source Yahoo! Finance Japan [14]

a result, τ was set to one in every case. For the otherparameters, we optimized them so as to maximize theprediction accuracy during the learning period. Here,we repeated one-step predictions by Eq. (2) or Eq.(17) through the learning period, and calculated thecorrelation coefficient as the prediction accuracy as

ηi =∑

t(ri(t) − ⟨ri⟩)(r̂i(t) − ⟨r̂i⟩)√∑t(ri(t) − ⟨ri⟩)2

√∑t(r̂i(t) − ⟨r̂i⟩)2

(20)

where r̂i(t) is a predicted value, ri(t) is its true value,and ⟨·⟩ is the mean. If N ≥ 2, we maximize

⟨η⟩ =∑N

i=1 ηi

N(21)

to optimize T , d, and K.Next, in the investment period, the daily invest-

ment process is carried out as follows:

Step 1 Estimation of the future return and riskAfter obtaining new opening prices of each stock{xi(t)} in every morning, we predict the futurereturns {r̂i(t + 1)} by Eq. (2) or Eq. (17), es-timate the future risks {σ̂i(t + 1)} by Eq. (3)or Eq. (18), and also estimate the covariances{σ̂ij(t + 1)} by Eq. (6) or Eq. (19).

Step 2 Optimization of allocation rates for a portfo-lioBy substituting these predicted values into Eqs.(4) and (5), we can obtain the portfolio’s futurereturn r̂p(t + 1) and future risk σ̂p(t + 1). Then,we optimize the allocation rates {ci(t)} so thatthe Sharpe ratio Sr [Eq. (10)] or the entropy E[Eq. (12)] becomes maximum.

Step 3 Taking portfolio positionsIf this is the first investment, we compose a newportfolio based on the optimized allocation rates{c∗i (1)} with the initial asset A(1). Otherwise,we rebalance the portfolio so that the currenttotal asset A(t) is reallocated by {c∗i (t)}, whereA(t) can be calculated by converting all of ourpositions to cash. Moreover, if r̂p(t + 1) > 0, wetake long (buy) positions for all stocks compos-ing the portfolio. If r̂p(t + 1) < 0, we take short

(sell) positions for all stocks to make r̂p(t + 1) apositive return. Then, we do nothing until thenext morning.

After daily investments for five years, the invest-ment performance is evaluated by the following mea-sures: the asset growth rate M , the maximum draw-down rate Rd, the profit factor P , and the winningrate Rw. These are commonly used to evaluate trad-ing performance. First, the asset growth rate is cal-culated by

M(t) =A(t)A(1)

(22)

Of course, a larger M(t) is better, and we make aprofit if M(t) > 1. The drawdown rate is defined by

rd(t) =A(t)

max1≤t′≤t

{A(t′)}× 100[%] (23)

Therefore, the maximum drawdown rate is calculatedby

Rd(t) = max1≤t′≤t

{rd(t′)} (24)

This indicates the value lost on our asset so far,that is, the degree of possible danger of an investmentstrategy. Namely, a smaller rd(t) is safer. The profitfactor is calculated by

P (t) =

t∑t′=1

{∆A(t′)|∆A(t′) ≥ 0}

t∑t′=1

{∆A(t′)|∆A(t′) < 0}(25)

where ∆A(t) = A(t) − A(t − 1) = rp(t)M(t − 1).If P (t) > 1, the total profit is larger than the to-tal loss, which means that this investment strategy isprofitable. Finally, the winning rate is calculated by

Rw(t) =

t∑t′=1

U(∆A(t′))

t′× 100[%] (26)

where U is the unit step function:

U(x) ={

1 if x ≥ 00 if x < 0


Table 2 Results of investment simulations for the first term with two stocks composing each portfolio: Eachstatistic is calculated from different 50C2 portfolios. “MV” refers to the mean-variance portfolio model mentionedin Sect. 2.1, “RP” refers to the risk-parity portfolio model mentioned in Sect. 2.2, and “Ours” refers to ourportfolio model proposed in Sect. 3. Moreover, “SD” stands for the standard deviation. Each bold numberindicates the best performance of these portfolio models. Finally, α is the significance level if the superiority ofour portfolio model over the mean-variance or risk-parity portfolio model can be confirmed by a one-sided pairedt-test.

M Rw P Rd

MV RP Ours MV RP Ours MV RP Ours MV RP OursMean 0.602 0.583 2.666 48.740 48.938 51.056 0.929 0.923 1.045 67.820 70.451 56.102Median 0.458 0.396 1.670 48.905 49.139 50.783 0.921 0.917 1.045 65.876 72.941 56.444SD 0.433 0.497 3.548 1.218 1.179 1.616 0.065 0.067 0.094 11.995 13.020 16.453Maximum 2.331 2.262 22.69 51.017 51.565 55.086 1.091 1.078 1.360 93.449 91.849 88.109Minimum 0.072 0.112 0.285 46.166 45.853 47.809 0.738 0.805 0.893 42.917 31.968 21.206α 0.01 0.01 – 0.01 0.01 – 0.01 0.01 – 0.01 0.01 –

Table 3 The same as Table 2 but the results for the second term

M Rw P Rd

MV RP Ours MV RP Ours MV RP Ours MV RP OursMean 0.457 0.592 1.369 48.783 49.007 50.022 0.899 0.990 0.991 73.180 70.414 61.709Median 0.400 0.519 1.009 48.983 49.061 50.000 0.902 0.932 1.001 73.194 71.667 61.792SD 0.235 0.395 1.660 1.050 1.312 1.396 0.056 0.064 0.110 9.411 13.247 13.563Maximum 1.337 1.605 10.760 50.7040 51.252 53.834 1.038 1.055 1.321 89.324 92.269 93.162Minimum 0.122 0.085 0.074 46.479 44.992 46.479 0.758 0.815 0.682 41.937 43.501 34.646α 0.01 0.01 – 0.01 0.01 – 0.01 0.01 – 0.01 0.01 –

In a sense, the winning rate corresponds to theprediction accuracy of r̂p(t+1). If Rw(t) > 50[%], thisprediction model works well. We do not consider anytrading commission in this study because the purposeof our portfolio model is not to reduce the tradingcommission, similarly to other portfolio models.

4.1 Composing a portfolio with two stocks

In the first simulation, we compose each portfoliowith only two stocks to statistically compare the per-formance between each conventional portfolio modeland our proposed model. Here, the two stocks in eachportfolio do not change during the investment period,and thus we can obtain the results of 50C2 portfolios.

The results are shown in Tables 2 and 3. We canconfirm that our model improves M , Rw, P , and Rw.In particular, every one-sided paired t-test can be re-jected at a significance level of α = 0.01, which meansthat our portfolio model is more profitable and saferthan the conventional models. The results are almostthe same in the first and second terms. However, it ismore difficult to make profits in the second term, inwhich there is a larger drawdown, that is, this termis more dangerous not only using our proposed modelbut also using the conventional models. This might be

because the second term includes the global financialcrisis caused by the Lehman Brothers bankruptcy inSeptember 2008.

4.2 Composing a multi-stock portfolio with N stocks

As shown in Sect. 2.1, we can theoretically reducethe total risk of a portfolio by including many stocksin the portfolio. To confirm this portfolio effect, wecompose a multistock portfolio with more than twostocks. Here, we hope to select profitable stocks whoseprediction accuracy is high, but the prediction accu-racy is unknown. However, if there is some correla-tion between the likelihood of a prediction model {ηi}during the learning period and its prediction accuracy{ξi} during the investment period, we can estimatethe unknown prediction accuracy ξi before startingthe investment by using the already known likelihoodηi, and can select profitable stocks beforehand.

To confirm this possibility, Figs. 1 and 2 show thecorrelations between {ηi} and {ξi}. In the first term(Fig. 1), we can see the following. Figure 1(a) shows aslight correlation, and therefore the conventional port-folio models might be able to select profitable stocksto some degree. However, the overall prediction accu-racy {ξi} for the moving-average prediction is small.


-0.1 -0.05 0 0.05 0.1 0.15-0.15

-0.1

-0.05

0

0.05

0.1

0 0.1 0.2-0.1

0

0.1

0.2

ηi

ξi

(a) (b)0.3

ηi

ξi

ξξ

Fig. 1 Correlations between the likelihood of eachprediction model during the learning period {ηi} andits prediction accuracy during the investment period{ξi}: (a) Moving-average prediction using the con-ventional portfolio models: the mean value of {ξi}, ξ̄,is −0.03 and the correlation between {ηi} and {ξi} is0.25, (b) Nonlinear bagging prediction using our port-folio model: ξ̄ = 0.03 and the correlation is 0.72.

0 0.05 0.1-0.2

-0.1

0

0.1

0 0.05 0.1

-0.05

0

0.05

0.1

(a) (b)

ηi

ξi

ηi

ξiξ ξ

Fig. 2 The same as Fig. 1 but the results for thesecond term: (a) Moving-average prediction using theconventional portfolio models: ξ̄ = −0.02 and the cor-relation between {ηi} and {ξi} is 0.51, (b) Nonlinearbagging prediction using our portfolio model: ξ̄ = 0.01and the correlation is 0.33.

2 10 20 30 40 500

4

8

2 10 20 30 40 500

50

100

2 10 20 30 40 50

0.8

1

1.2

2 10 20 30 40 50

46

50

54

N

N

N

N

M

Rd

P

Rw

Our proposed portfolio

1

[%]

[%]

Mean-variance portfolio

Risk-parity portfolio

Fig. 3 Results of investment simulations for the firstterm with N stocks composing each portfolio

2 10 20 30 40 500

5

10

2 10 20 30 40 5020

50

80

100

2 10 20 30 40 500.6

1

1.4

2 10 20 30 40 5045

50

55

N

N

N

N

M

Rd

P

Rw

1

[%]

[%]




Fig. 4 The same as Fig.3 but the results for thesecond term

In particular, ξ̄ is a negative value. Figure 1(b) showsa strong correlation, and thus our portfolio model caneffectively select profitable stocks. Moreover, the over-all prediction accuracy for the bagging prediction isbetter than that in Fig. 1(a).

In the second term (Fig. 2), we can see the fol-lowing. Figure 2(a) shows a good correlation, but theprediction accuracy ξ̄ is as poor as that in Fig. 1(a).Figure 2(b) shows a small correlation, but the predic-tion accuracy ξ̄ is better than that for the conventionalportfolio models in Figs. 1(a) and 2(a). Overall, wecan expect that our proposed model will be most ef-fect for selecting profitable stocks and for composinga multistock portfolio.

Next, Figs. 3 and 4 show the performance of mul-

tistock portfolios with N stocks, which were selectedfrom the viewpoint of the likelihood ηi. In the firstterm (Fig. 3), we can see the following. In the mean-variance model, the asset growth rate M increaseswith the number of stocks N , which is caused by theportfolio effect. However, in our proposed model, theportfolio effect is not confirmed. If N is larger, theportfolio has to also include unfavorable stocks whoseprediction accuracy is lower, and therefore the perfor-mance of our portfolio might decrease. In addition,because the total of embedding dimensions is Nd inEq. (14), this becomes too large to find suitable localneighbors for the nonlinear prediction if N is larger.However, in most cases, our proposed model showshigher performance than the other portfolio models,


0 200 400 600 800 1000 12000

2

4

6

8

0 200 400 600 800 1000 12000

1

2

3

0 200 400 600 800 1000 1200

1

2

0 200 400 600 800 1000 12000

1

2

0 200 400 600 800 1000 12000

2

4

1

1

3

t

t

t

t

t

M

M

M

M

M

(a)

(b)

(c)

(d)

(e)

0
















Fig. 5 Temporal movement of the asset growth rateM(t) during the investment period of the first term:Each result is obtained for a portfolio composed ofN stocks: (a) N = 3, (b) N = 10, (c) N = 20, (d)N = 35, and (e) N = 50

t

t

t

t

t

M

M

M

M

M

(a)

(b)

(c)

(d)

(e)

0 200 400 600 800 1000 12000

1

3

1

0 200 400 600 800 1000 12000

1

3

0 200 400 600 800 1000 12000

5

10

0 200 400 600 800 1000 12000

0.5

1

1.5

0 200 400 600 800 1000 12000.2

0.6

1




2

2

1.4













Fig. 6 The same as Fig. 5 but the results for thesecond term: The drawdown at t ≅ 100 is caused bythe global financial crisis in September 2008.

that is, the asset growth rate M and profit factor Pare larger. This means that our proposed model ismore profitable. Also, the maximum draw downrateRd is smaller, which means that our proposed model issafer. Moreover, the winning rate Rw is larger, whichmeans that our nonlinear bagging prediction is moreaccurate than the conventional moving-average pre-diction.

The second term in Fig. 4 shows almost the sameresults as the first term. However, the reduction inM is larger in our proposed portfolio. This is becausethe correlation shown in Fig. 2(b) is small, and there-fore we cannot avoid unprofitable stocks in the stockselection based on the likelihood ηi. In addition, it isbecause the financial crisis in the second term madeit difficult to make profits.

However, in both terms, it is better to use a smallnumber of stocks for a portfolio, such as N = 3. Insuch a case, the investment performance is better thanthose in Tables 2 and 3 in terms of every measure: M ,Rw, P , and Rd. Moreover, if we send orders manuallyor take account of the trading commission, a smallernumber of stocks is more favorable. However, in any

case, we have to discuss how to optimize the num-ber of stocks in the portfolio as a future work, whichcurrently appears to be difficult.

Finally, for more details, Figs. 5 and 6 show tem-poral movements of the asset growth rate M(t) duringthe investment period. Similarly to those in Figs. 3and 4, although the investment performance is betterwhen N is smaller, especially in the second term, it isclearer that our proposed portfolio is superior to theconventional models in every case.

5. Conclusions

We proposed a means of improving the mean-variance portfolio model by using the bagging algo-rithm with local linear (nonlinear) prediction. First,the nonlinear prediction improved the prediction accu-racy of future returns because the prediction methodof the mean-variance model is a simple moving averageof the recent historical data. Then, because the bag-ging algorithm can estimate the possible distributionof a future return, its mean and standard deviationare the expected future return and risk of trading a


single stock, respectively. Moreover, by applying theseexpected values to the mean-variance portfolio model,the risk is reduced because of the portfolio effect ifthe number of stocks in the portfolio is not very large.Through investment simulations with real stock data,we confirmed that our method could simultaneouslyrealize higher profit and lower risk, even during thehuge financial crisis caused by the Lehman Brothersbankruptcy.

However, in our portfolio model, we could not seethe portfolio effect very clearly, although it was possi-ble to select profitable stocks by referring to the likeli-hood obtained by the bagging prediction. One of thereasons for this might be the embedding dimension be-coming very large relative to the learning data lengthas we increase the number of stocks in the portfolio.Therefore, nonlinear prediction cannot find suitablelocal neighbors. To solve this issue, we might be ableto use principal component analysis to reduce the em-bedding dimension, which we hope to discuss in an-other paper. In addition, we hope to discuss how tooptimize the number of stocks in a portfolio as a futurework.

Acknowledgement

This research was partially supported by a Grant-in-Aid for Scientific Research (C) (No. 25330280) fromthe Ministry of Education, Culture, Sports, Scienceand Technology of Japan.

References

[1] H. M. Markowitz: Portfolio selection, J. Finance, Vol. 7,

No. 1, pp. 77–91, 1952.

[2] L. Breiman: Bagging predictors, Mach. Learn., Vol. 24,

pp. 123–140, 1996.

[3] D. Haraki, T. Suzuki, H. Hashiguchi and T. Ikeguchi:

Bootstrap nonlinear prediction, Phys. Rev. E, Vol. 75,

056212-1–9, 2007.

[4] T. Suzuki and K. Nakata: Risk reduction for nonlinear

prediction and its application to the surrogate data test,

Physica D, Vol. 266, No. 1, pp. 1–12, 2014.

[5] W. F. Sharpe: Capital asset prices: A theory of market

equilibrium under conditions of risk, J. Finance, Vol. 19,

No. 3, pp. 425–442, 1964.

[6] E. Qian: Risk parity portfolios, Research Paper,

PanAgora, 2005.

[7] E. Qian: Risk parity and diversification, J. Investing, Vol.

20, No. 1, pp. 119–127, 2011.

[8] F. Takens: Detecting strange attractors in turbulence,

Lecture Notes in Mathematics, Vol. 898, pp. 366–381,

Springer-Verlag, 1981.

[9] E. N. Lorenz: Atmospheric predictability as revealed by

naturally occurring analogues, J. Atmos. Sci., Vol. 26, pp.

636–646, 1969.

[10] J. D. Farmer and J. J. Sidorowich: Predicting chaotic time

series, Phys. Rev. Lett., Vol. 59, pp. 845–848, 1987.

[11] K. Tanaka and T. Suzuki: Dynamical portfolio theory by

nonlinear bagging predictors, Proc. NOLTA’12, pp. 527–

530, 2012.

[12] S. Inose and T. Suzuki: Portfolio selection based on non-

linear time series prediction, IEICE Trans. Fundam., Vol.

J96-A, No. 7, pp. 410–422, 2013. (in Japanese)

[13] T. Suzuki: Appropriate time scales for nonlinear analyses

of deterministic jump systems, Phys. Rev. E, Vol. 83, No.

6, 066203-1–9, 2011.

[14] Yahoo! Finance Japan: http://finance.yahoo.co.jp

Tomoya Suzuki received hisB.S., M.S., and Ph.D. degrees inphysics from Tokyo University ofScience, Tokyo, Japan, in 2000,2002, and 2005, respectively. Hejoined Tokyo Denki University as anAssistant in 2005, and was a Lecturein Doshisha University from 2006 to2009. Since 2009, he has been anAssociate Professor in Ibaraki Uni-versity, Japan. His research inter-ests are to analyze and predict com-plex systems such as financial mar-

kets by using nonlinear time series analysis, machine learning,data mining, etc. He is a member of IEICE, IPSJ, the Physi-cal Society of Japan (JPS), and the Nippon Technical AnalystsAssociation (NTAA).

Kiyoharu Tanaka received hisB.E. and M.E. degrees in engineer-ing from Ibaraki University, Ibaraki,Japan, in 2012 and 2014, respec-tively. His research interest is theimprovement of financial engineer-ing by using prediction models basedon the nonlinear dynamical theoryand advanced statistical methods.

(Received May 15, 2014; revised August 8, 2014)


mean-variance portfolio model modiﬁed by nonlinear bagging

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