Key-Words: - Investment portfolio, Transaction costs, Serially correlated returns

Size: px

Start display at page:

Download "Key-Words: - Investment portfolio, Transaction costs, Serially correlated returns"

Clarence Neal
5 years ago
Views:

1 Portfolio Optimizatio i the Fiacial Maret with Correlated Returs uder Costraits, rasactio Costs ad Differet Rates for Borrowig ad Ledig VLADIMIR DOMBROVSKII, AYANA OBEDKO Departmet of Ecoomics oms State Uiversity Leia Street 36, oms RUSSIAN FEDERAION dombrovs@ef.tsu.ru, tatyaa.obedo@mail.ru Abstract: - I this wor, we cosider the optimal portfolio selectio problem uder hard costraits o tradig amouts, trasactio costs ad differet rates for borrowig ad ledig whe the risy asset returs are serially correlated. No assumptios about the correlatio structure betwee differet time poits or about the distributio of the asset returs are eeded. he problem is stated as a dyamic tracig problem of a referece portfolio with desired retur. Our approach is tested o a set of a real data from Russia Stoc Exchage MICEX. Key-Words: - Ivestmet portfolio, rasactio costs, Serially correlated returs Itroductio he ivestmet portfolio (IP) maagemet is a area of both theoretical iterest ad practical importace. he foudatio for moder portfolio selectio theory is the sigle-period mea-variace approach suggested by Marowitz [3] ad the Merto s [4] IP model i cotiuous time. At preset, there exists a variety of models ad approaches to the solutio of the IP optimizatio problem, but most of them are the complicatios ad extesios of the Marowitz ad Merto approaches to various versios of stochastic models of the prices of risy ad ris-free assets ad utility fuctios. Most studies assume the time idepedecy of the retur vector, with a few exceptios. he vast majority of the existig literature o dyamic portfolio selectio is based o usig dyamic programg approach for deterig the solutio. However, that approach leads to the wellow curse of dimesioality, which hiders desig of the decisio strategies uder costraits. herefore, the most of the results preseted i the literature are limited to the cases without tradig costraits ad trasactio costs. Also, the rates of borrowig ad ledig are assumed to be the same. However, it's well-ow that realistic ivestmet models must iclude these features []. I this paper, we cosider the dyamic ivestmet portfolio selectio problem subject to hard costraits o tradig amouts (a borrowig limit o the total wealth ivested i the risy assets, ad log- ad shortsale restrictios o all risy assets), taig ito accout the presece of quadratic trasactio costs. Other realistic feature we icorporate is that i our model the rates of borrowig ad ledig are differet (the rate of borrowig is greater tha that of ledig). Empirical evidece shows that the returs of the risy assets always exhibit certai degree of depedecy amog time periods, e.g., see [8,9] ad referece therei. We assume also that the risy asset returs are serially correlated. he oly coditios imposed o the distributios of the asset returs are the existeces of the coditioal mea vectors ad of the coditioal secod-order momets. No assumptios about the correlatio structure betwee differet time poits or about the distributio of the asset returs are eeded. he problem is stated as a dyamic tracig problem of a referece portfolio with desired retur. he ivestor s objective is to choose the dyamic tradig strategy to imize the coditioal measquare error betwee the ivestmet portfolio value ad a referece (bechmar) portfolio, pealized for the trasactio costs assosiated with tradig. We cosider quadratic trasactio costs. he atural iterpretatio of a quadratic cost is that price impact is liear i the trade size, resultig i a quadratic cost [5].

2 I this wor, we use the model predictive cotrol (also ow as recedig horizo cotrol) method i order to solve the problem. he major attractio of such techique lies i the fact that it ca hadle hard costraits o the iputs (maipulated variables) ad states/outputs of a process ad allows to avoid the curse of dimesioality [6-8,,8-2]. here are may examples of the MPC i fiace applicatios. Some recet wors ca be foud i [,4-7,9,6,7]. I [4-7,9] ivestmet portfolio optimizatio with costraits usig MPC is cosidered. Dyamic optio hedgig usig MPC is preseted i [6] ad i []. I all of these papers, authors assume the hypothesis of serially idepedet returs ad/or cosider the explicit form of the model describig the price process of the risy assets (e.g., geometric Browia motio, etc.). Related results i multi-period portfolio optimizatio ca be foud i [2-3] where a multistage optimizatio model is developed. I a developed model portfolio, diversity costraits are imposed i expectatio (soft costraits). Calafiore [2-3] proposed a approximated techique to solve the problem via stochastic simulatios of the retur series that ca be used i practice whe a full stochastic model for retur dyamics is available. he purpose of the preset paper is to provide umerically tractable algorithm for practical applicatios. We wat to demostrate the performace of our model uder real maret coditios. We pay a particular attetio to testig of our approach o a set of a real data from the Russia Stoc Exchage MICEX. his wor is orgaized as follows. Sectio 2 presets portfolio model ad the optimizatio problem formulatio. he mai results of this article are preseted i Sectio 3 where we desig the optimal ivestmet strategy for the problem uder cosideratio. I Sectio 4 the umerical modelig results are preseted. his paper is cocluded i Sectio 5 with some fial remars. 2 Portfolio Model ad Optimizatio Problem 2. he Proposed Portfolio Model Let us cosider the ivestmet portfolio of risy assets ad oe ris-free asset (e.g. a ba accout or a govermet bod). Let u i(), (i=,,2,,) deote the amout of moey ivested i the ith asset at time ; u () is the amout ivested i a ris-free asset. Ivestor also ca borrow the capital i case of eed. he volume of the borrowig of the ris-free asset is equal to u +(). If u i()<, (i=,2,,), the we use short positio with the amout of shortig u i(). he wealth process V() satisfies i () i V u u u. Let P i() deote the maret value of the ith risy asset at time, ad η i(+) deote the correspodig retur per period [,+], defied as Pi( ) Pi i ( ). P i ( ) It is a stochastic value uobserved at time. We cosider self-fiacig portfolio. Selffiacig meas that the whole wealth obtaied at the tradig period will be exactly reivested at the tradig period +. By cosiderig the self-fiace strategies, the wealth dyamics are give by i i (2) V ( ) ( ) u r u i [ r2] u, with iitial value V(), where r is the risless ledig rate, r 2 is the risless borrowig rate (r <r 2). Usig (), the dyamics (2) ca be rewritte as follows V ( ) r V ( ) r ui (3) i i [ r r ] u, 2 here u V ui u is the amout i ivested i a ris-free asset. We impose the followig costraits o the decisio variables (a borrowig limit o the total wealth ivested i the risy assets, ad log- ad short-sale restrictios o all risy assets) u u u,( i, ), (4) i i i i i V u u u, (5) u u. (6) If u i ()<, (i=,2,,), so we suppose that the amouts of the short-sale are restricted by u i () ; if the short-sellig is prohibited the u i (), (i=,2,,). he amouts of log-sale are restricted by u i (), (i=,2,,); u () defied the imum amout of moey we ca ivest i the ris-free asset; the borrowig amout is restricted by u. Note, that values u i (), (i=,,,), u i (), (i=,,,+) are ofte deped o commo wealth of portfolio i practice. So that we ca write u i ()=β iv(), u i ()=γ iv(), where β i, γ i are costat parameters. Let =( F ) be the complete filtratio with σ- field F geerated by the {η(s): s=,, 2,,} that

3 models the flow of iformatio about asset returs to time. Let us assume that the vectors of risy asset returs η()=[η () η 2() η ()], =,,, form a serially correlated o-statioary discrete-time multivariate process with fiite coditioal momets / E ( i) / F ( i), E ( i) ( j) F, ( i, j, l;,,2,). herefore, the lead-lag relatioships betwee compoet series η t(+i) ad η f(+j) are described by the matrices of the secod-order ij coditioal momets. hroughout the paper, we use the followig otatios. For ay matrix ψ[η(+i),+i], depedet o η(+i), ( i) E [ ( i), i] / F, without idicatig the explicit depedece of matrices o η(+i). Oe motivatio for such a model is the fact that a large umber of empirical aalyses of assets price dyamics show that there exists saliet serial correlatios i the returs of fiacial assets [8,9]. 2.2 Optimizatio Problem (Ris Fuctio) Our objective is to cotrol the ivestmet portfolio, via dyamics asset allocatio amog the stocs ad the ris-free asset, as closely as possible tracig the deteristic bechmar V ( ) [ ] V, (7) where μ is a give parameter represetig the growth factor, the iitial state is V ()=V(). We use the MPC methodology i order to defie the optimal cotrol portfolio strategy. For the give predictio horizo m, a sequece of predictive cotrols (tradig amouts) u(/), u(+/),, u(+m /) depedig o the portfolio wealth at the curret time ad all the iformatio about asset returs to time is calculated at each step. his sequece optimizes the criterio chose by the ivestor for the predictio horizo. At the time, u()=u(/) is assumed to be cotrol u(). o obtai the cotrol at the ext step +, the procedure is repeated, ad the cotrol horizo is oe step shifted. We cosider the followig objective with recedig horizo (ris fuctio) m 2 J ( m / ) E V ( i) V ( i) i (8) (, i) V ( i) V ( i) V, F / ij m i E u( i / ) u( i / ) R(, i) u i u i / V F ( / ) ( / ) ( ),, where m is the predictio horizo, u(+i/)=[u (+i/),,u +(+i/)] is the predictive cotrol vector, i, m,,,2, ; u(- /)=u(-) is the optimal cotrol vector obtaied o the previous step, u(-/)=; R(,i)> is a positivedefiite symmetric matrix mesurig the level of trasactio costs, ρ(,i)> is a positive weight E a / b is the coditioal expectatio coefficiet; of a with respect to b. Notice that variable V () is ow for all time istat ad may be cosidered as a pre-chose parameter. Let us explai the terms i the objective fuctio (8). he first term represets the coditioal mea-square error betwee the ivestmet portfolio value ad a referece (bechmar) portfolio, the secod term pealizes wealth values that less tha the desired value. he third term peelizes for trasactio costs assosiated with tradig amout u( i / ) u( i / ). A importat advatage of tracig a referece portfolio approach uder quadratic criterio (8) is its capability to predict the trajectory of growth portfolio wealth, which would follow close to the deteristic (give by the ivestor) bechmar or beat it. It maes possible to obtai a smooth curve of the growth of the portfolio wealth o the etire ivestmet horizo. It is oe of the basic requiremets for the tradig strategies of ivestors i fiacial marets. he growth factor μ is selected by ivestor, based o the aalysis of the fiacial maret. 3 he Proposed Ivestmet Strategy Desig he problem of imizig the criterio (8) is equivalet to the quadratic cotrol problem with criterio m i m 2 ( / ) { ( ) J m E V i i R (, i) V ( i) V, F } E u( i / ) u( i / ) R(, i) u i u i / V F ( / ) ( / ) ( ),, / (9) where we eliated the term that is idepedet of cotrol variables, R (+i)=2v (+i)+ρ(,i). We have the followig theorem.

4 heorem. Let the wealth dyamics is give by (3) uder costraits (4)-(6). he the MPC policy with recedig horizo m, such that it imizes the objective (9), for each istat is defied by the equatio u I U, () where I is (+)-dimesioal idetity matrix; is (+)-dimesioal zero matrix; U()=[u (/),,u (+m-/)] is the set of predictive cotrols defied from the solvig of quadratic programg problem with criterio Y( m / ) 2 V G F U () U H R U uder costraits (elemet-wise iequality) U SU U, (2) where U [ u, 2,, 2], U [ u, 2,, 2], u u u2 u2 u, u, u ( ) u V ( ) u V u S diag S, 2,, 2, S, R(,) R(,) R(,) R(,) R(,) R(,2) R ( ), R(, m ) R(, m) R(, m) R(, m) R(, m) H(), G(), F() are the bloc matrices H Htf, G Gt, F Ft, (3) ( t, f, m), ad the blocs satisfy the followig recursive equatios H E{ b [ ( t), t] tt Q ( m t) b[ ( t), t] / F }, f t (4) Htf A E{ b [ ( t), t] (5) Q ( m f ) b[ ( f ), f ] / F }, t f, ft H H, t f, (6) tf t G A Q ( m t) b( t), (7) t F Q ( m t) b( t) 2 R(,) u( ), (8) t 2 2 Q ( t) A Q ( t ), Q (), Q2 ( t) AQ2 ( t ) R (, m t), Q2() R(, m), R (,t)=2v (+t)+ρ(,t), ( t, m), A r, b, r r r r. 2 A brief proof of this theorem is reported i the Appedix. 3 A real data umerical example I this sectio, we preset several umerical examples demostratig the applicatio of our approach to portfolio of a real stocs. We wat to assess the performace of our model uder real maret coditios by computig the portfolio wealth over a log period of time. he data used for these examples are tae from the Russia Stoc Exchage MICEX ( hey iclude the daily stoc prices of the largest Russia compaies such as Sberba, Gazprom, VB, LUKOIL, NorNicel, Roseft, ad Sibeft. he portfolio was composed of five risy assets. Perforg umerical modellig, we looed over all of the possible combiatios of the five assets. We cosider the situatio of a ivestor who has to allocate oe uit of wealth over the ivestmet horizo of approximately 5 tradig days (about four years) amog risy assets ad oe ris-free asset. he ris-free asset is cosidered here as a ba accout with r =., r 2=.2. he updatig of the portfolio based o the MPC is executed oce every tradig day. We set the tracig target to retur.5% per day (μ =.5). We assumed a iitial portfolio wealth of V()=V ()=. he matrix measurig the level of trasactio costs is set as R(,i)=diag( -4,, -4 ) for all,i, the weight coefficiet ρ(,i)=. for all,i. We impose costraits o the tracig portfolio problem with parameters β i=-.6, (i=,,), γ i=3, (i=,,+). herefore we allow borrowig ad short sellig. For the o-lie fiite horizo MPC problems, we used a horizo of m=, ad umerically solved it i MALAB by usig the quadprog.m fuctio.

At each time, the optimizatio problem requires as iput parameters the predicted returs ad predicted secod momets of returs over the predictive horizo m.

5 At each time, the optimizatio problem requires as iput parameters the predicted returs ad predicted secod momets of returs over the predictive horizo m. hese parameters ca be estimated usig differet model specificatios describig the retur asset evolutio. Examples iclude usig autoregressive models, coditioal heteroscedastic models, factor models, complex oparametric methods ad others (see, for istace, [2,9]). As a simple example, we assume that the multivariate process of risy asset returs follows the VAR(2) model (vector autoregressive model of order 2) [2] ( ) A A2 ( ) ( ), where A, A 2 are the coefficiet matrices, ( I A A2) is a vector of itercept terms, E{ ( )}; ad ( ) is a -dimesioal white oise, that is, E{ } ; E{ ( ) ( )} ; E{ ( i) ( j)}, i j. he covariace matrix is assumed to be osigular. We estimated parameters of this model by the ordiary least squares method usig the observed historical data based o the past 2 tradig days prior to the tracig period. hese parameters were cosidered costat alog the etire period uder study ad equal to the iitial empirical estimates, based o bacwards data. We calculated the predicted coditioal secod momets based o this VAR(2) model ad substituted them ito equatios (4)-(5). I practice, time series of risy asset returs have a tredig behaviour which is ot compatible with the assumptios of the classical VAR model. I order to capture short-ru treds of risy asset returs, we use the followig modificatio of the forecastig procedure based o the VAR(2) model. We calculate the sample meas of returs ˆ( ) usig 2-day widows of past historical retur data ad icorporate these estimates i the VAR(2) - predictor Eˆ{ ( h) / } ˆ Aˆ Eˆ{ ( h ) / } Aˆ 2E ˆ{ ( h 2) / ( )}, where the true coefficiets, A are replaced by estimators ˆ, Aˆ ˆ ˆ ; ˆ ( I A A2 ) ˆ ; h,2,, m. his formula was used for recursively computig the h-step predictors startig with h=. his predictor is used to predict the expected returs over the predictive horizo m at each decisio time i equatios (7) ad (8). Whe a ew measuremet becomes available, the oldest measuremet is discarded ad the ew measuremet is added. So, we use the adjusted procedure, updatig the estimates of mea returs at each time. Oe motivatio for such a heuristic approach is that we have o restrictios to costruct ay type of predictors i order to obtai the best asset allocatio strategies. However, forecastig is too large a topic to address adequately i this wor ad the ivestigatio of the sesitivity of optimizatio results to the estimated parameters is outside the scope of this wor. We preset the typical results of the experimets o fig. -3. I the pictures below, the portfolio was composed of five risy assets: LUKOIL, Gazprom, Sberba, Roseftj, ad NorNicel. Ivestmet period is from to.9.24 (approximately 6 years). Fig. plots the tracig portfolio ad a referece portfolio values. I fig. 2, we have ivestmets i the risy asset Gazprom. Fig. 3 plots risy asset returs for asset Gazprom. Several isights ca be gathered from the examples illustrated above. Fig. shows that the tracig a referece portfolio strategy allows us to obtai a smooth curve of growth. he advatage of the cotrol accordig to the quadratic criterio is that it is possible to predict the trajectory of the growth of portfolio wealth, which should follow as close as possible to the deteristic bechmar give by the ivestor. Fig.. racig performace (V real portfolio, V referece portfolio).

We preseted the umerical modelig results, based o a set of real data from the Russia Stoc Exchage MICEX. We fid that o actual data the proposed approach is reasoable.

6 We preseted the umerical modelig results, based o a set of real data from the Russia Stoc Exchage MICEX. We fid that o actual data the proposed approach is reasoable. he value of the portfolio follows the value of the reverece portfolio, beatig it most of the time ad the costraits are satisfied. Fig.2. Asset allocatio decisio (u is the amout ivested i Gazprom). Fig.3. Risy asset returs (Gazprom). It is importat to acowledge that i our experimets, where we use a rather simple model for parameters estimatio, the performace of proposed strategies appears to be rather efficiet. So, our approach allows us to desig strategies which are desesitized, i.e., robustified, to parameters estimatio. It is clear that oe ca use more sophisticated estimatio schemes to improve the precisio of parameters estimatio. 4 Coclusio ad future wor I this paper, we studied a discrete-time portfolio selectio problem with serially correlated returs, for which oly the first ad the secod coditioal momets are ow. he owledge of the statistical distributios of the returs is ot assumed. We proposed to use the MPC methodology i order to solve the problem. he optimal portfolio cotrol strategy was derived uder hard costraits o tradig amouts, trasactio costs ad differet rates for borrowig ad ledig. he advatage of usig a recedig horizo implemetatio is that at each decisio stage we ca profit from observatios of actual maret behavior durig the precedig period ad use iformatio to feed fresh estimates to the model. Appedix Proof of the heorem : he portfolio dyamics (3) ca be rewritte i the form V( ) r V b[ ( ), ] u, where η()=[η () η 2() η ()] is the vector of risy asset returs, u [ u u2 u ( )] is the vector of iput (maipulated) variables, ad b, r r r r. 2 Costraits(4)-(6) ca be rewritte i matrix form (elemet-wise iequality): u Su u, (9) where I, S E, E u u u2 u2. u ( ) u V ( ) u V u u, u he objective (9) ca be writte i the form J( m / ) E X ( ) X ( ) (2) ( ) X ( ) U R U 2 u( / ) R(,) u( ) u ( ) R(,) u( ) / V, F, subject to X( ) V [ ( ), ] U, (2) where V ( ) A 2 V ( 2) A X ( ),, A r, m V ( m) A U u ( / ), u ( / ),, u ( m / ), ( ) ( 2) ( ), ( m)

7 [ ( ), ] b[ ( ), ] Ab[ ( ), ] b[ ( 2), 2] m m2 A b A b [ ( ), ] [ ( 2), 2], b[ ( m), m] ( ) R (,), R (,2),, R (, m), R(,) R(,) R(,) R(,) R(,) R(,2) R ( ). R(, m ) R(, m) R(, m) R(, m) R(, m) Usig (2), we ca rewrite (2) as follows 2 J ( m / ) V V (22) 2 V ( ) F U E ( ), ( ), / F U E ( ), L U u ( ) R(,) u( ), where L 2 R(,) u( ). Deote the matrices F F H E ( ), ( ), / F, G E ( ),, F ( ) E ( ), L. We have that the imizatio of the criterio (9) uder costraits (4)-(6) is equivalet to the quadratic programg problem with criterio Y( m / ) 2 V G F U U [ H R] U uder costraits (9). Straightforward calculatios lead to the expressios (4)-(8) for the matrices H(), G(), F(). his completes the proof. Refereces: [] A. Bemporad, L. Puglia,. Gabbriellii, A stochastic model predictive cotrol approach to dyamic optio hedgig with trasactio costs, Proc. America Cotrol Coferece. Sa Fracisco, CA, USA, Jue 29-July, 2, pp [2] G.C. Calafiore, Multi-period portfolio optimizatio with liear cotrol policies, Automatica, Vol.44, 28, pp [3] G.C. Calafiore, A affie cotrol method for optimal dyamic asset allocatio with trasactio costs, SIAM J. Cotrol. Optim, Vol.48, No.4, 29, pp [4] V.V. Dombrovsii, D.V. Dombrovsii, E.A. Lyasheo, Ivestmet portfolio optimizatio with trasactio costs ad costraits usig model predictive cotrol, IEEE Proc. 8th Korea-Russia It. Sympos. Sci. echology, KORUS, oms, Russia, 24, pp [5] V.V. Dombrovsii, D.V. Dombrovsii, E.A. Lyasheo, Predictive cotrol of radomparameter systems with multiplicative oise. Applicatio to ivestmet portfolio optimizatio, Automatio ad remote cotrol, Vol.66, No.4, 25, pp [6] V.V. Dombrovsii, D.V. Dombrovsii, E.A. Lyasheo, Model predictive cotrol of systems with radom depedet parameter uder costraits ad It s applicatio to the ivestmet portfolio optimizatio, Automatio ad remote cotrol, Vol.67, No.2, 26, pp [7] V.V. Dombrovsii,.Yu. Ob edo, Predictive cotrol of systems with Marovia jumps uder costraits ad it s applicatio to the ivestmet portfolio optimizatio, Automatio ad remote cotrol, Vol.72, No.5, 2, pp [8] E. Fama, K. Frech, Permaet ad temporary compoets of stoc prices, Joural of Political ecoomy, Vol.96, 988, pp [9] F. Herzog, G. Dodi, H.P. Geerig, Stochastic model predictive cotrol ad portfolio optimizatio, Iteratioal Joural of heoretical ad Applied Fiace, Vol., No.2, 27, pp [] Y. Hu, X.Y. Zhou, Costraied stochastic LQ cotrol with radom coefficiets, ad applicatio to portfolio selectio, SIAM J.

8 Cotrol Optim., Vol.44, No.2, 25, pp [] P.N. Kolm, R. ütücü, F.J. Fabozzi, 6 Years of portfolio optimizatio: Practical challeges ad curret treds, Europea Joural of Operatioal Research, Vol.234, 24, pp [2] H. Lütepohl, New itroductio to multiple time series aalyses, Spriger Verlag Berli Heidelberg, 25. [3] H.M. Marcowitz, Portfolio selectio, J. Fiace, Vol.7, No., 952, pp [4] R.C. Merto, Cotiuous-time fiace, Cambridge: Blacwell, 99. [5] N. Gârleau, L.H. Pederse, Dyamic tradig with predictable returs ad trasactio costs, he Joural of Fiace, Vol.LXVIII, No.6, 23, pp [6] J.A. Primbs, Dyamic hedgig of baset optios uder proportioal trasactio costs usig recedig horizo cotrol, It. J. of Cotrol, Vol.82, No., 29, pp [7] J.A. Primbs, C.H. Sug, A stochastic recedig horizo cotrol approach to costraied idex tracig, Asia-Pacific Fia Marets, Vol.5, 28, pp [8] J. Rawligs, utorial: Model Predictive Cotrol echology, I Proc. Amer. Cotrol Cof., 999, pp [9] R.S. say, Aalysis of fiacial time series, A Wiley-Itersciece Publicatio, Joh Wiley ad Sos. Ic., 22.

Subject CT1 Financial Mathematics Core Technical Syllabus

Subject CT1 Financial Mathematics Core Technical Syllabus Subject CT1 Fiacial Mathematics Core Techical Syllabus for the 2018 exams 1 Jue 2017 Subject CT1 Fiacial Mathematics Core Techical Aim The aim of the Fiacial Mathematics subject is to provide a groudig