Complex Valued Risk Diversification
aa r X i v : . [ q -f i n . P M ] O c t Complex Valued Risk Diversification
Yusuke Uchiyama a , Takanori Kadoya a , Kei Nakakagawa b a MAZIN, Inc., 1-60-20 Minami Otsuka, Toshima-ku, Tokyo, Japan b Nomura Asset Management Co., Ltd. 1-21-1, Nihonbashi, Chuo-ku, Tokyo, Japan
Abstract
Risk diversification is one of the dominant concerns for portfolio managers.Various portfolio constructions have been proposed to minimize the risk ofthe portfolio under some constrains including expected returns. We propose aportfolio construction method that incorporates the complex valued principalcomponent analysis into the risk diversification portfolio construction. Theproposed method is verified to outperform the conventional risk parity andrisk diversification portfolio constructions.
Keywords:
Portfolio management, Risk diversification, Hilbert transform,Principal component analysis
1. Introduction
Both individual and institutional investors are concerned with risk di-versification for portfolio construction. Portfolio managers have employedappropriate mathematical techniques to minize the risk of the portfolios,formulated as constrained nonlinear optimization problems. Indeed, as thepioneer of quantitative finance, Markowitz proposed the mean-variance (MV)portfolio construction [1]. In the framework of the mathematical portfolioconstruction, the return and risk of the portfolios are defined by the meanand variance respectively, then the MV portfolio construction is determinedby the minimization of the risk of the portfolio constrained with the expectedreturn. However, it was pointed out that the risk allocations of the MV port-folio construction are often biased [2]. In other words, the weight levels ofparticular assets are much higher than others in the MV portfolio.In general, risk biased portfolios seem to be vulnerable to asset pricechange. The MV portofolio construction is thus undesireble from the pointof view of risk diversification. The risk parity (RP) portfolio construction
Preprint submitted to arXiv October 11, 2018 as designed to allocate market risk equally across asset classes, includingstocks, bonds, commodities, and so on [3]. Subsequntly, a return weightedsum of assets is introduced to the RP portfolio construction for improving itsperformance [4]. Some variations of the RP portfolio construction have beenproposed and verified to outperform the MV portfolio construction [5, 6, 7].Nevertheless, the RP portfolio construction cannot fully disperse the originof risk because almost all parts of the world mutually interact in modernsociety, causing entanglement of different asset classes.In the field of data science and multivariate analysis, the principal compo-nent analysis (PCA) has been developed to decompose mutually correlateddata subspaces [8]. The maximum risk diversification (MRD) portfolio con-struction utilizes the PCA to decompose and allocate the risk contributionof assets [9]. Then the constrained optimization of the MRD portfolio con-struction is expected to design a risk allocated portfolio. The MRD portfolioconstruction is also confirmed to outperform the MV portfolio constructionand to be able to allocate the risk contribution of assets [9, 10]On the other hand, in the filed of the atmospheric physics, the PCAhas been utilized and extended to capture principal modes of spatiotemporaldynamics, which are known as empirical orthogonal functions (EOFs) [11].In practice, it is extremely difficult to investigate all the degrees of freedom ofglobal atmospheric changes. Thus, the method of EOFs has been employedto extract essential dynamics [12, 13, 14].The conventional portfolio constructions have not considered the tempo-ral dynamics of the portfolios despite the importance of the temporal fluctua-tion of assets. In this research we incorporate dynamic effects into the MRDportfolio construction by using the method of EOFs. The Hilbert trans-form is utilized to generate analytical signals from the prices of the assets.In addition the corresponding optimization problem is presented to equalizedynamic risk allocations and then is verified to outperform the conventionalmethods.
2. Related works
Markowitz first introduced the MV portfolio as a sophisticated method inmodern portfolio theory. In this theory, the risk of the asset is defined as thestandard deviation of the return. With this setup, a portfolio is presentedby the weighted sum of the assets considered.2iven the sequence of m -th asset prices { p ( m ) t } ≤ t ≤ T (1 ≤ m ≤ M ), the re-turn of the asset is defined by r ( m ) t = p ( m ) t +1 − p ( m ) t p ( m ) t . (1)Subsequently, the return of the portfolio is obtained as R t = M X m =1 w m r ( m ) t , (2)where { w m } ≤ m ≤ M is the set of weight coefficients. The risk of the portfoliois defined by the standard deviation of the return in Eq. (2). In generalrisk averse investors tend to minimize the risk of their portfolios under ex-pected returns. This strategy is mathematically formalized by constrainedquadratic programming with respect to the covariance matrix of the returnof the portfolio.The expected return of the portfolio in Eq. (2) is expressed by the weightedsum of the expected return of each asset asE[ R t ] = M X m =1 w m E[ r ( m ) t ] , (3)where E[ · ] denotes the expectation for a random variable. The covariancematrix of the return of the portfolio is defined byΣ = E (cid:2) ( r t − E[ r t ])( r t − E[ r t ]) T (cid:3) , (4)where components of r t are the return of each asset and ( · ) T denotes thetranspose of a vector. With the use of the covariance matrix in Eq. (4), thevariance of the portfolio is obtained as σ = w T Σ w (5)with w being a weight coefficient vector. The MV optimized portfolio withexpected return µ is realized as the solution of the minimization for σ inEq. (5) subject to w T r t = µ . Also, constraints for the weight coefficients canbe added to the objective function as a Lagrangian form with multipliers.3 .2. Risk parity portfolio It has been pointed out that the asset classes of the MV portfolio are notfully allocated. To disperse the risk contributions of portfolios, risk parity(RP) portfolio constructions have been proposed. Based on the idea of theRP portfolio construction, a measure of risk contribution was introduced.The risk contribution of the m -th asset is derived from the variance ofthe RP portfolio as follows: σ m = w m ∂σ∂w m (6)= (Σ w ) m √ w T Σ w , (7)where (Σ w ) m denotes the m -th component of Σ w . Equal risk contributionfor the RP portfolio requires that all the risk contributions have the samevalue, whereby the weight coefficients of the portfolio are determined byoptimization as follows:arg min w M X m =1 (cid:20) w m − σ (Σ w ) m M (cid:21) (8)This portfolio construction enables one to obtain equally allocated assets.In addition various subclasses of the RP portfolio construction have beenproposed. For instance, the return weighted RP portfolio construction wasdeveloped to improve the performance of the equally risk allocated RP port-folio [4]. In general, the origin of the risk of assets seems to be entangled. Namely,the covariance matrix of the return of portfolios contains non-diagonal com-ponents and thus the pair of assets exhibit a linear correlation. To unravelthe entangled risks, the PCA has been incorporated into the portfolio con-structions [9].The covariance matrix of the return of portfolios can be transformedinto a diagonal matrix by an appropriate orthogonal matrix since all of theeigenspaces are mutually independent. The eigenvalues of the covariancematrix introduce a probability distribution of risk contribution. Thus theentropy with respect to the probability distribution is defined and is employed4s the objective function of the MRD portfolio construction. The origin ofrisk is expected to be decomposed on the principal axes of the covariancematrix.
3. Complex valued risk diversification portfolio construction
As has been reviewed in the previous section, almost all of the portfolioconstruction methods utilize the covariance matrix to estimate the risk ofthe portfolios as the objective functions of the optimization. The covariancematrix of a random vector contains autocorrelations of pairs of vector com-ponents. Thus one can extract stationary information of the random vectorfrom the corresponding covariance matrix. However, in general, the price ofan asset exhibits non-stationary random fluctuations. Hence it is necessaryto utilize dynamic information of the fluctuations of the assets to accuratelyestimate the risk of the portfolios.In order to incorporate the dynamics of the price of the assets into theportfolio constructions, we apply the method of EOFs to the timeseries ofthe return of the assets. This portfolio construction method utilizes a com-plex valued timeseries and the corresponding covariance matrix whereby wename this method a complex valued risk diversification (CVRD) portfolioconstruction.The Hilbert transform of a timeseries x ( t ) on t ∈ [0 , ∞ ) is defined by H [ x ( t )] = 1 π Z ∞ x ( τ ) t − τ dτ, (9)where the improper integral is understood in the sense of principal value[15].In practice, empirical timeseries are recoded at a certain sampling rate ∆ t ,which introduces discrete time t n = n ∆ t with n being integer. The Hilberttransform for a discrete timeseries is given by H D [ x k ] = − i sgn (cid:18) k − N (cid:19) N − X n =0 x n e i πnN , (10)where sgn( · ) is the sign function [16]. Here we apply the Hilbert transform inEq. (10) to the return of the portfolio in Eq. (2) and then obtain the analyticsignal as z t = r t + i H D [ r t ] . (11)5s with the PCA for real valued time series, the analytic signal, z t (0 ≤ t ≤ T ),provides a complex valued covariance matrix defined as C z = 1 T + 1 T X t =0 z t z ∗ t (12)with z ∗ t being the transjugate of z t . Since C z in Eq. (12) is a positive defi-nite Hermitian matrix, the corresponding eigenvalues are positive real valuesincluding zeros. An unitary matrix U , which consists of eigenvectors of C z ,transforms C z as U C z U ∗ = Λ , (13)where Λ is the orthogonal matrix with respect to the set of eigenvalues { λ n } of C z , which is arranged in descending order. The weight coefficient vector w , at the same time, is transformed into ˜ w = U w . The contribution of theeigenvector is introduced as v m = ˜ w m λ m (14)with ˜ w m being the m -th component of the transformed weight coefficient ˜w ,and the probability distribution for v m is defined by p m = v m P Mm =1 v m . (15)From the probability distribution in Eq. (15), the corresponding entropy canbe introduced as H = − M X m =1 p m log p m . (16)In general, weight coefficients of portfolios are constrained based on tradingstrategies. Thus we construct a Lagrangian function with the aid of theentropy in Eq. (16) and constraint functions for weight coefficients as L = H − L X l =1 µ l g l ( ˜w ) , (17)where µ l is a Lagrange multiplier and g l ( · ) is a constraint function. Opti-mizing L in Eq. (17) with respect to w gives the weight coefficients of theexpected CVRD portfolio. 6 . Result In this section, we test the performance of the CVRD portfolio construc-tion by comparing it with the RP and MRD portfolio constructions. As atest dataset, we selected bonds, commodities, indexes and swaps during May2000 to April 2017. The descriptive statistics of the dataset are shown intable 1.We use the annual return, risk and Sharpe ratio as measures of the per-formance of the portfolio constructions. Each portfolio is rebalanced everymonth by previous data from a year without transaction costs. Table 2 showsthe annual return, risk and Sharpe ratio of the RP, MRD, and CVRD port-folio constructions. The CVRD portfolio construction outperforms the RDportfolio construction with respect to all of the measures. The risk of the RPportfolio construction is the lowest because the RP portfolio mainly consistsof bonds, which means that the risk contributions of the RP portfolio arestrongly biased toward the bonds and thus is not fully dispersed, as is seenin fig. 1. On the other hand, the risk of the assets in the CVRD portfolioconstruction are well allocated as is shown in fig. 2.Figure 3 shows the time sequence of the annual return of the RP, MRD,and CVRD portfolio constructions. The return of the CVRD portfolio con-struction is confirmed to outperform that of the RP and MRD portfolioconstructions during almost all the periods. This result seem to be realizedby capturing the dynamics of the return of the portfolio with the use of thecomplex valued PCA. In other words, the CVRD portfolio construction canappropriately time the rebalancing of the portfolio based on the dynamicproperties of the assets.
5. Conclusion
Risk diversification for portfolio management is of great interest for bothindividual and institutional investors. Indeed, various portfolio constructionmethods have been developed and employed in both individual and industrialtrades. Nevertheless, almost all of the portfolio constructions fail to accountfor the dynamic property of the assets.To utilize the dynamic property of the assets, we introduce the method ofEOFs into portfolio constructions. The Hilbert transform is used to producethe imaginary part of the analytic signal, from which the complex valuedcovariance matrix obtained. The PCA for the covariance matrix enables one7 igure 1: The allocation of the assets in the RP portfolio construction.Figure 2: The allocation of the assets in the CVRD portfolio construction. able 1: Descriptive statistics of the dataset. Mean Std. Dev. Skewness KurtosisTY1 Comdty 0.00661 0.465 -0.284 4.56XM1 Comdty 0.000876 0.0621 -0.140 1.81CN1 Comdty 0.00935 0.444 -0.222 2.87RX1 Comdty 0.01288 0.464 -0.821 6.95G1 Comdty 0.00334 0.532 -7.35 214JB1 Comdty 0.00414 0.283 -0.571 5.78SP1 Index 0.210 14.7 -0.250 5.37XP1 Index 0.624 46.9 -0.289 5.60PT1 Index 0.0776 7.42 -0.628 7.42GX1 Index 1.10 90.8 -0.227 4.01Z1 Index 0.166 61.3 -0.223 3.61NK1 Index 0.165 194 -0.302 5.44AUD Curncy 0.000037 0.00641 -0.361 6.11CAD Curncy -0.000026 0.00686 0.0726 2.92EUR Curncy 0.0000400 0.00772 0.0346 1.85GBP Curncy -0.0000600 0.00939 -0.665 7.55JPY Curncy 0.000695 0.675 -0.147 3.19to estimate the contribution of each principal axis and to obtain the entropyof the risk contribution distribution. Appropriately constrained optimizationmethods with respect to the entropy yields the CVRD portfolio construction.The performance of the CVRD portfolio construction was compared withthat of the RP and MRD portfolio constructions. It was confirmed that theannual return of the CVRD portfolio construction outperformed that of theothers. In addition, the risk of the assets in the CVRD portfolio constructionwas well allocated. This result verified that the CVRD portfolio constructionsucceeded in diversifying the risk of the portfolio.In practice, the time window of estimating the covariance matrix varieson the investors’ policy. Also, the accessible test period depends on theresources of the institution where investor belongs. Hence the performanceof the CVRD portfolio construction seems to vary depending on the sizeof the time windows and test periods. Comprehensive investigation for theeffect of the time window and the test period is our future work.9 able 2: Annual return, risk and, Sharpe ratio of the RP, RD and CVRD portfolio con-structions. Each portfolio is rebalanced every month by previous data for a year.
RP RD CVRDReturn 1.340 1.621 3.816Risk 1.728 4.219 6.152Sharpe Ratio 0.7756 0.384 0.620
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