FINVEX WHITE PAPER ON ASSET ALLOCATION WITH RISK FACTORS

Transcription

1 FINVEX WHITE PAPER ON AET ALLOCATION WITH RIK FACTOR By Dr Kris Boud PhD Professor of Finance & Research Parner a Finvex Group Benedic Peeers Co-Founder Finvex Group July 3

2 Execuive ummary In his paper, we will presen in a simple way a new approach o asse allocaion. The idea is o uild porfolios ha offer diversificaion over so called risk facors and his wihin a minimum variance porfolio consrucion framework. We elieve his approach is an imporan advancemen compared o radiional asse allocaion as i achieves a higher level of rue risk diversificaion, aking ino accoun he common and unique risk facors ha each asse class is exposed o. In a previous working paper on ond-equiy allocaion, we have sressed he shif in he invesmen paradigm from reurn-ased o risk-ased porfolio allocaion. Especially he equal risk conriuion (ERC) approach is very popular in asse allocaion: porfolio weighs are dynamically se such ha he asse classes conriue equally o he porfolio risk. As such, he porfolio loads auomaically less on he more risky asse and, y diversifying across asse classes, porfolio drawdowns are reduced. A poenial weakness of he ERC approach is ha i may no guaranee sufficien diversificaion across risk facors, especially in he case where asse classes are srongly correlaed. Consider e.g. an ERC porfolio invesed in a ond, equiy and converile ond index. Clearly, he hree asse classes have common risk exposures. Depending on he ond/equiy exposure of he converile ond index, an ERC consrain a he asse class level may lead o a concenraion of porfolio risk ino he underlying ond or equiy risk. We argue ha, for asse allocaion, no only he risk conriuion of he asses, u also he risk conriuions of he facors are imporan elemens o consider in he porfolio decision. To idenify such sources, we propose he use of a linear facor model ha decomposes he risk of an asse class ino he exposure o direcly oservale risk facors explaining he comovemen across asse classes and he idiosyncraic asse-specific risk facors ha can e idenified in he reurn series of he asse classes. We are also he firs o emphasize he imporance of shrinkage esimaors for hese applicaions where he numer of parameers o esimae is large compared o he numer of availale oservaions We provide wo applicaions of he proposed risk facor conriuion mehodology for a porfolio invesed in EU governmen onds, EU corporae onds, EU high yield onds and European equiy. The firs applicaion consiss of an ex-pos facor risk conriuion analysis where we decompose he porfolio risk ino he risk associaed o he economic aciviy, inflaion, ineres rae, exchange rae, credi risk and marke risk. In he second applicaion we consruc minimum variance porfolios ha saisfy ex-ane consrains on he facor risk conriuions. Copyrigh : Finvex Group, 3 P a g e

3 Inroducion Risk-ased porfolio soluions are porfolio allocaion echniques ha do no require explici modelling of expeced reurns. In asse allocaion, he equal risk conriuion (ERC) approach is very popular: porfolio weighs are dynamically se such ha he asse classes conriue equally o he porfolio risk. As such, he porfolio loads auomaically less on he more risky asse and, y diversifying across asse classes, porfolio drawdowns are reduced. An imporan cavea when consrucing ERC porfolios is ha he power of diversificaion of he equal risk conriuion consrain depends on he underlying asses. When hose asses are very dependen on underlying common risk facors, he porfolio risk may effecively e very concenraed in a few underlying risk facors. We propose o unravel his hidden risk concenraion hrough he use of linear facor models, in which he asse reurns are rewrien as a linear cominaion of reurns coming from exposure o common risk facors and he componen ha is specific o he asse class. We assume hose risk facors o e oservale and ha all relevan risk facors are included. The same parameric seup is aken in he relaed work of Roncalli and Weisang () on risk pariy porfolios wih risk facors. In order o calculae he facor risk conriuions, we hen need o rewrie he porfolio reurn as an exac linear cominaion of facors. We follow Zivo () in defining he se of facors of he join se of risk facors specified y he modeller and he unexplained variaion in he sock reurns (called idiosyncraic risk facors). The common risk facors can of course e correlaed, while he idiosyncraic risk facors are assumed o e uncorrelaed. Roncalli and Weisang () follow Meucci (7) in calculaing he facor risk conriuions in one sep. In our view, his comes a he price of racailiy. Zivo () proposes a wo-sep approach of firs esimaing he exposures y ordinary leas squares. Given hose exposure, one can hen calculae he facor risk conriuions in an analogous way as he asse risk conriuions are calculaed. As such, more racale analyical expressions are oained for he facor risk conriuions. Imporanly, his wo-sep framework also allows more flexiiliy in he choice of esimaion mehods. In fac, our applicaion is on asse allocaion for which macroeconomic facors are crucial deerminans of he asse reurns. Many of hese facors can only e oserved a a quarerly frequency. Because of he relaively large numer of parameers o esimae compared o he numer of oservaions, we recommend no o use he ordinary leas squares and sample covariance esimaors, u so-called shrinkage esimaors in which he esimaes are de-noised. We focus on porfolio sandard deviaion as a risk measure, u, under he assumpion of ellipical symmery or using simulaion mehods, he approach can e exended o downside risk measures such as value-a-risk and expeced shorfall. Copyrigh : Finvex Group, 3 3 P a g e

4 In wha follows, we firs review he mehodology of facor risk analysis. We hen illusrae he differences eween asse risk conriuions and facor risk conriuions for a porfolio invesed in EU governmen onds, EU corporae onds, EU high yield onds and European equiy. We furher illusrae he effec on he porfolio weigh allocaion of consraining he percenage risk conriuion of he underlying risk facors. Finally, for all porfolios considered, an ou-of-sample reurn analysis is performed o illusrae he effecs of he imposed risk diversificaion on he porfolio reurns. Copyrigh : Finvex Group, 3 4 P a g e

5 Proposed Mehodology uppose we have N asses wih covariance marix Σ and a porfolio vecor w=(w,, w N ). The porfolio sandard deviaion is given y: ( w) w w. For porfolio risk managemen purposes, i is imporan o disenangle he differen sources of porfolio risk. Ariuion of porfolio volailiy o he porfolio asses In a firs sep we use he Euler decomposiion o reak down he porfolio volailiy ino he volailiy conriuions of each asse and represen hese componen risk measures as a percenage of oal porfolio risk. The percenage volailiy risk conriuion of he ih asse in he porfolio is given y: % ARC i ( w) ( w) w i w ( w) i w w i (see e.g. Boud, Carl and Peerson, ). Ariuion of porfolio volailiy o he underlying risk facors The percenage asse conriuions o porfolio volailiy provide insigh in he disriuion of risk across he porfolio asses, u, in some regards, i is sill superficial as i does no reveal any economic insigh in how risk facors drive he porfolio risk. In fac, a common view is o consider ha he variaion in asse reurns is driven y muliple macroeconomic facors and idiosyncraic facors ha are specific o each asse. The use of such facor models is now widespread. I has een used y Ross (976) o derive expeced reurns under no arirage assumpions (he so-called Arirage Pricing Theory). Very ofen, i is used wihou any furher assumpions, as a descripive ool o inspec he exposures of an invesmen syle. To inroduce he mehodology, we firs enumerae he assumpions of he linear facor model and he implicaions i has for he srucure of he covariance marix and he porfolio variance. We hen derive he percenage facor risk conriuions and discuss some of he esimaion issues ha arise in he pracical implemenaion. Copyrigh : Finvex Group, 3 5 P a g e

6 Copyrigh : Finvex Group, 3 6 P a g e The linear facor model uppose ha K oservale facors are idenified as eing influenial for he porfolio variailiy. A a given frequency (e.g. monhly or quarerly), he asse reurns r =(r,, r N ) and he facors f =(f,, f K ) are recorded. The asse reurns are assumed o depend linearly on he facors, wherey he variaion in he asse reurns ha is no explained y he facors, is assumed o e uncorrelaed wih each of he facors and also o e uncorrelaed across asses. Explain ha his is ok alhough simplified. The linear approach leads o he following sysem of equaions:. N N K NK N N K K N N e e e f f f a a a r r r The covariance marix of e =(e,, e N ) is he ideniy marix and e has mean zero. In marix noaion, he sysem is given y:. De Bf a r Le e he KxK covariance marix of he K facors. Because of he assumpion ha he unexplained asse reurn variaion e is uncorrelaed wih he facors, we can rewrie Σ (he NxN covariance marix of he N asse reurns) as: D. BB Our ineres is in explaining he porfolio reurn. Premuliplying he asse reurn y he porfolio weigh vecor gives us he porfolio reurn:, e f w De w Bf w a w r wih β=w B he Kx row-marix of exposure of he porfolio reurn o each of he facors and δ=w D he he Kx row-marix of exposure of he porfolio reurn o each of he asse-specific facors. Following Zivo (), we join hese wo exposures in he vecor γ of size K+N: e f e f r w. The join covariance marix of f and e is

7 Copyrigh : Finvex Group, 3 7 P a g e. KK K K K K Percenage facor risk conriuions under he linear facor model Under he linear facor model he porfolio volailiy can hus e wrien as. ) ( In an analogous way o he percenage volailiy conriuion of he ih asse, he percenage volailiy risk conriuion of he ih facor is given y:. ) ( ) ( ) ( % i i i FRC i Because of he one-homogeneiy of σ(γ) (i.e. he propery ha σ(kγ)=k σ(γ)), he percenage facor risk conriuions add up o one. Implemenaion For he effecive calculaion of he percenage risk conriuions a he level of he individual asses and facors, we need o esimae he covariance marix of he asse reurns (Σ), he covariance marix of he facors () and he facor exposures γ. The radiional approach is o use he sample covariance esimaor and he ordinary leas squares esimae. Bu, as menioned in he inroducion, in our seup we ypically have a large numer of parameers o esimae and a small numer of oservaions. In such cases, he sample covariance marix and leas squares esimaes are known o e unreliale and shrinkage esimaors perform eer. everal shrinkage mehods exis. We will use he covariance shrinkage esimaor of Ledoi and Wolf (3) and ase all our esimaes on he Ledoi-Wolff shrinkage esimae of he covariance marix of he asse reurn and facors, joinly. The facor exposures implied y his esimaed covariance marix are oained following Engle ()

8 Applicaions in risk monioring and porfolio allocaions The proposed mehodology has imporan applicaions in monioring he porfolio risk as well as in he design of opimal porfolios. Nex we illusrae oh applicaions for he universe of European governmen onds, corporae onds, high yield onds and equiy. The daa source is Bloomerg. As some of he macro-facors ha we will consider are only availale a a quarerly frequency, we consider quarerly realancing of he porfolio. The cumulaive reurn evoluion of each of he asse classes over he period 999- is shown in Figure. Besides he differences in volailiy and reurn over he period, he graph clearly shows he diversificaion poenial across he differen invesmen syles. The shaded area in Figure corresponds o he ou-of-sample evaluaion period used o compare he differen porfolio allocaion sraegies. Panel of Tale summarizes he reurn performance of he four asse classes over his period. Over his period, he wors performing asse class in all aspecs is EU equiy wih an average annualized reurn of.8%, a volailiy of 6.8% and a maximum drawdown of 54%. The corporae and governmen ond indices perform similarly wih reurns of 5% and a sandard deviaion eween 4 %and 5%. The high yield ond index has over he period he highes reurn (9.3%) wih an annualized volailiy of 5%. We consider 3 economic facors ha we have grouped ino six caegories:. Aciviy: EU GDP growh, indusrial producion growh and he economic senimen index as pulished y he European Commission;. Inflaion: consumer prices and commodiy prices; 3. Ineres rae: real ineres rae and slope of he yield curve; 4. Currency: percenage changes real effecive exchange rae; 5. Credi risk: EU corporae Baa-AA onds spread, U corporae Baa-AA onds spread, TED spread; 6. Marke risk: Implied Volailiy &P 5 (VIX) and DAX (VDAX). The firs four caegories are also considered in he risk facor analysis of Roncalli and Weisang (). The daa frequency used is quarerly. For he asse classes, he index names are Bloomerg/EFFA Bond Indices (EUGATR), IBOXX CRP OA TR (EU governmen onds), Pan-European High Yield and MCI EUROPE NR. Copyrigh : Finvex Group, 3 8 P a g e

9 .5 EU governmen onds EU high yield onds EU corporae onds EU equiy.5.5 3//998 3// 3// 3//4 3//6 3//8 3// 3// Figure Quarerly cumulaive value of EU governmen ond, corporae ond, high yield ond and equiy index low risk, value and equal risk conriuion low risk-value porfolio versus he marke porfolio for a gloal universe over he period Decemer 998-Decemer. The grey area indicaes he ou-of-sample evaluaion period. Tale Monhly reurns analysis of single-asse sraegies and dynamically realanced asse allocaion porfolios over he period Augus 6 Decemer. Annualized reurn Annualized sandard deviaion harpe raio (RF=) Max drawdown Panel : ingle-asse class sraegies Governmen onds 4.8% 4.3%.4-6.7% Corporae onds 5.% 4.7%.58-8.% High yield onds 9.3% 5.% % Equiy.8% 6.8%.6-54.% Panel : andard dynamically realanced porfolios Equally weighed 5.% 8.% % Equal Risk Conriuion 4.5% 4.7% % Minimum Variance 4.6% 4.% % Panel 3: Risk facor consrained minimum variance porfolios RFCP 3.5% 5.9% % RFCP 5.% 4.3% % RFCP3 4.8% 4.7%.4-8.6% Copyrigh : Finvex Group, 3 9 P a g e

10 Applicaion : Risk monioring and ex-pos porfolio risk analysis The firs applicaion of facor risk udges is he ex-pos analysis of he risk concenraion of he porfolio. The risk analysis is done for he following hree porfolios: The equally weighed porfolio: each of he four asses is ariued a 5% porfolio weigh; The equal risk conriuion porfolio: each asse conriues o 5% of he porfolio risk; The minimum variance porfolio: porfolio weighs are such ha he porfolio variance is minimal, under he consrain of full invesmen and no shor sales. The covariance marix is esimaed on rolling samples of 4 (quarerly) oservaions. The ou-of-sample analysis period corresponds o Augus 6-Decemer and porfolios are realanced on a quarerly frequency. The average porfolio weighs over his period are repored in Tale. Noe ha he equal risk conriuion and minimum variance porfolios are concenraed in governmen onds and have a relaively lower allocaion o he more risky high yield onds and he equiy index. Equally Weighed Equal Risk Conriuion Minimum Variance Governmen onds Corporae onds High yield onds.5.. Equiy Tale Average quarerly weigh allocaion of equally weighed, equal risk conriuion and minimum variance porfolio invesed in EU governmen onds, corporae onds, high yield onds and equiy over he period Augus 6-Decemer. Porfolios are realanced quarerly. Panel of Tale shows he porfolio performance of hese hree dynamic asse allocaion sraegies, compared o he single-asse alernaives in panel of Tale. The differences in porfolio weighs ranslae direcly in he porfolio volailiy, wherey he (annualized) volailiy of he equal risk conriuion porfolio (4.7%) is in eween he volailiy of he minimum variance porfolio (4.%) and he equally weighed porfolio (8.%). The annualized reurn of he hree porfolios is around 5%. Copyrigh : Finvex Group, 3 P a g e

11 Le us now focus on he key quesion, namely he ariuion of he porfolio volailiy o he percenage volailiy caused y he asses (yielding he percenage asse risk conriuions) and he disriuion of he percenage volailiy across he differen risk facors (yielding he percenage facor risk conriuions). The percenage volailiy conriuions of each asse are repored in Tale 3. Comparing he equal risk conriuion and minimum variance porfolios, we see ha increasing he weigh of he high yield onds from % o % and of equiy from 3% o 8% has a large impac on he percenage volailiy conriuions of hose asses, which increase sharply from and 3% o 5%, respecively. Equally Weighed Equal Risk Conriuion Minimum Variance Governmen onds Corporae onds High yield onds.4.5. Equiy Tale 3 Average quarerly asse-ased risk allocaion of equally weighed, equal risk conriuion and minimum variance porfolio invesed in EU governmen onds, corporae onds, high yield onds and equiy over he period Augus 6- Decemer. Porfolios are realanced quarerly. The percenage volailiy conriuion of each facor is shown in Tale 4. Equally Weighed Equal Risk Conriuion Minimum Variance Aciviy..9. Inflaion...9 Ineres rae..4. Currency... Credi risk Marke risk..3. Percenage risk conriuion y all macrofinancial risk facors 69% 57% 48% Idiosyncraic governmen..8.7 onds facor Idiosyncraic corporae onds..8. facor Idiosyncraic high yield onds.3.8. facor Idiosyncraic equiy facor.6.9. Percenage risk conriuion y all idiosyncraic risk facors 3% 43% 5% Tale 4 Average quarerly facor-ased risk allocaion of equally weighed, equal risk conriuion and minimum variance porfolio invesed in EU governmen onds, corporae onds, high yield onds and equiy over he period Augus 6- Decemer. Porfolios are realanced quarerly. Copyrigh : Finvex Group, 3 P a g e

12 As could e inuiively expeced, he volailiy of he equally weighed porfolio eing 5% invesed in all asses is explained y all facors, excep currency. The idiosyncraic governmen and corporae onds facors have negligile impac, in conras wih he idiosyncraic high yield onds and equiy facors ha explain 3% and 6% of he volailiy of he equally weighed porfolio. The equal risk conriuion and he minimum variance porfolio have a much lower exposure o he European high yield ond and equiy asse classes. As a consequence, hese porfolios have a higher exposure o he idiosyncraic governmen and corporae ond facors. In paricular, for he equal risk conriuion porfolio 8% and 8% of porfolio volailiy are explained y he governmen and corporae onds facors, and for he minimum variance porfolios, hese facors explain 7% and % respecively. For all porfolios, he economic aciviy, inflaion and ineres rae facors are he hree mos imporan macro-economic conriuors o he porfolio volailiy. Joinly, he macro-economic facors explain 69% of he volailiy of he equally weighed porfolio, 57% of he volailiy of he equal risk conriuion porfolio and 48% of he minimum variance porfolio volailiy. Applicaion : Ex-ane facor risk consrains in porfolio allocaion An imporan shorcoming of he minimum variance porfolio in his applicaion is ha only hree facors are responsile for 68% of he oal porfolio volailiy: inflaion (9%), idiosyncraic governmen onds facor (7%) and he idiosyncraic corporae onds facor (%). We now invesigae he ineresing applicaion of implemen facor risk udges ha resric ex-ane he risk conriuions of he differen facors. To illusrae his, we consider following risk facor consrained minimum variance porfolios: [RFCP] Minimum variance porfolio under he consrain ha he maximum percenage facor risk conriuion is less han %. [RFCP] Minimum variance porfolio under he consrain ha he maximum percenage idiosyncraic facor risk conriuion is less han %. [RFCP3] Minimum variance porfolio under he consrain ha he maximum asse reurn percenage conriuion is less han 3% and he maximum percenage facor risk conriuion is less han %. Tales 5-7 show he corresponding weigh and risk allocaions, while he ou-of-sample reurn performance is in Panel 3 of Tale. Copyrigh : Finvex Group, 3 P a g e

13 RFCP RFCP RFCP3 Governmen onds Corporae onds High yield onds Equiy Tale 5 Average quarerly weigh allocaion of risk facor consrained porfolios invesed in EU governmen onds, corporae onds, high yield onds and equiy over he period Augus 6-Decemer. Porfolios are realanced quarerly. RFCP RFCP RFCP3 Governmen onds Corporae onds High yield onds.4..3 Equiy Tale 6 Average asse-ased risk allocaion of risk facor consrained porfolios invesed in EU governmen onds, corporae onds, high yield onds and equiy over he period Augus 6-Decemer. Porfolios are realanced quarerly. RFCP RFCP RFCP3 Aciviy Inflaion..5. Ineres rae...4 Currency... Credi risk..9.9 Marke risk.6..3 Percenage risk conriuion y all macrofinancial risk facors 56% 53% 57% Idiosyncraic governmen.3..7 onds facor Idiosyncraic corporae onds.3.. facor Idiosyncraic high yield onds facor Idiosyncraic equiy facor.3.4. Percenage risk conriuion y all idiosyncraic risk facors 44% 47% 43% Tale 7 Average facor-ased risk allocaion of risk facor consrained minimum variance porfolios invesed in EU governmen onds, corporae onds, high yield onds and equiy over he period Augus 6-Decemer. Porfolios are realanced quarerly. Copyrigh : Finvex Group, 3 3 P a g e

14 Le s firs consider he RFCP porfolio. Rememer ha he minimum variance porfolio wihou risk facor consrains is 5% invesed in governmen onds and 44% in corporae onds. By imposing ha he maximum percenage facor risk conriuion should e less han %, an imporan diversificaion is achieved, a he porfolio weigh level (44% is invesed in corporae onds and 33% in governmen onds), he asse risk conriuion level (he maximum risk conriuion drops from 5% o 44%) and he facor risk conriuion, where he upper ound of % is clearly inding. This increase in diversificaion is accompanied y a decrease in reurn (4.6% o 3.5%), an increase in volailiy (4.% o 5.9%) and an imporan increase in he maximum drawdown (-5.4% o -7.8%). For he RFCP porfolio, he maximum % consrain on all risk facors is clearly oo resricive and does no srike a alance eween he ojecives of high reurn, low risk and high diversificaion across oh asses and risk facors. In he RFCP porfolio, he maximum % percenage facor risk conriuion consrain is only imposed on he idiosyncraic facors and he resuling porfolio weigh and risk allocaion is in eween he unconsrained minimum variance porfolio and he RFCP porfolio. The maximum % ound consrain is inding for he idiosyncraic governmen and corporae ond facors. The RFCP porfolio has a higher harpe raio (.83) compared o he harpe raio of he minimum variance porfolio (.37) and a comparale maximum drawdown. The RFCP porfolio risk is well diversified across all risk facors, u a he asse class level, he RFCP porfolio risk is concenraed in he corporae (46%) and governmen onds (35%). Imposing only consrains on he risk conriuions of he risk facors herefore fails in guaraneeing a sufficien risk diversificaion across he asse classes. We herefore consider as a final design he RFCP3 porfolio ha has consrains on he risk conriuion of he asses and he risk facors. More precisely, he RFCP3 porfolio comines a maximum % percenage facor risk conriuion consrain on he idiosyncraic facors wih a maximum 3% percenage risk conriuion on each of he asses. This porfolio loads significanly less on he corporae ond (porfolio weigh reduces from 44% o 35%) and more on high yield onds (8% insead of 4%) and equiy (9% insead of 5%). Is performance in erms of reurn and risk is comparale (and even slighly eer) o he performance of he equal risk conriuion porfolio. Copyrigh : Finvex Group, 3 4 P a g e

15 Conclusion A mere analysis of he componen risk conriuions of he porfolio asses is insufficien o uncover he risk facor concenraions of he porfolio. The decision on he risk he porfolio manager is willing o ake should e done a differen aggregaion levels, among which he level of he asse and facor percenage risk conriuions. Based on Meucci (7) and Zivo () we consider a mehodology o do so in a compuaionally simple and ransparen way. The proposed wo sep approach comines shrinkage esimaion of he exposures and covariance marices, wih he usual percenage risk calculaion ased on he Euler expansion. We apply he mehodology o he case of a diversified asse allocaion porfolio invesed in EU governmen onds, EU corporae onds, EU high yield onds and European equiy. We firs illusrae he usefulness of he ool for ex-pos risk analysis and hen analyse he impac of ex-ane risk facor consrains on he porfolio allocaion. In our applicaion, i is possile o achieve simulaneously a high diversificaion a he asse level and risk facor level, while sill offering comparale (and even slighly eer) performance han he equal risk conriuion approach ha allocaes he porfolio risk equally across asse classes. Copyrigh : Finvex Group, 3 5 P a g e

16 Acknowledgemens The auhors would like o hank David Ardia, François Berrand, Peer Carl, Jorn De Boeck, Brian Peerson and Eric Zivo. REFERENCE Boud Kris, Carl, Peer and Brian Peerson. 3. Asse allocaion wih Condiional Value-a-Risk Budges. Journal of Risk 5, Engle, R.. Dynamic condiional ea. NYU working paper. Ledoi, Oliver and Michael Wolf. 3. Improved esimaion of he covariance marix of sock reurns wih an applicaion o porfolio selecion. Journal of Empirical Finance, Meucci, Ailio. 7. Risk conriuions from generic user-defined facors. Risk, June, Roncalli, Thierry and Guillaume Weisang.. Risk pariy porfolios wih risk facors. Lyxor working paper. Ross, ephen The arirage heory of capial asse pricing. Journal of Economic Theory 3, Zivo, Eric.. Facor model risk analysis. Presenaion a R/Finance. FINVEX WHITE PAPER ERIE June 3 Asse allocaion wih risk facors. January 3 mar harvesing of equiy syle premia. epemer Dynamic risk ased asse allocaion. April - Risk opimised invesing in equiy markes. Novemer Risk opimized invesmen. Availale a: hp:// Copyrigh : Finvex Group, 3 6 P a g e