Thomas G. Stephan*, Raimond Maurer** and Martin Dürr* *) Deutscher Investment Trust (DIT),Frankfurt/Main

Transcription

1 A Muliple Facor Model for European Socks Thomas G. Sephan*, Raimond Maurer** and Marin Dürr* *) Deuscher Invesmen Trus (DIT),Frankfur/Main **) Johann Wolfgang Goehe Universiy of Frankfur/Main, Chair for Invesmen, Porfolio Managemen and Pension Sysems 1

2 2 General Srucure of Muliple Facor Models General Srucure of Muliple Facor Models > r i : reurn o securiy i in period wih covariance marix Σ > f j : reurn o facor j in period wih covariance marix Φ > ε i : specific reurn o securiy i in period wih covariance marix Ω > under sandard assumpions follows: ε ε ε δ δ δ δ + + f r f f r r n 1 k 1 nk n1 1k 11 n 1 M M M M M Ω + Φ T

3 The Role of Muliple Facor Models in Invesmen > muliple facor models describe asse reurns and heir covariance marix as a funcion of a limied number of risik aribues > main applicaions of muliple facor models: > analysis and forecas of risk > performance and risk aribuion > porfolio consrucion (risk reurn opimizaion) > main advanages of muliple facor models: > inuiive descripion of porfolio characerisics > comparaively small number of risk dimensions > number of parameers o be esimaed dramaically reduced: > unresriced covariance marix has ½n(n + 1) parameers > facor model requires esimaion of ½(k² + k) + n parameers > e.g.: for 500 asses and 5 facors he number of parameers is reduced from 125,250 o 3,015 (2,4% of he original number) 3

4 General Esimaion Mehodology > We use cross secion analysis in he following way: > he facor exposures or loadings are prespecified and serve as regressors > he esimaed parameer vecor can be inerpreed as facor reurn values > he cross secion analysis has o be execued for each poin in ime in order o creae a ime series of facor reurns ha forms he basis for he facor covariance marix > he maxrix of facor loadings akes he following form: δ 11 δ 1u l11 l1v b11 b1w M M M M M M n1 nk ln1 lnv bn1 b nw 14 δ δ > δ i1,...,δ iu sand for he risk indices (see nex slide), l i1,...,l iv for he counry dummy variables and b i1,..., b iv for he indusry dummy variables for sock i. The dummy variables ake he form: 1 Sock i is from counry j l ij and 0 oherwise 1 Sock i belongs o indusry j b ij. 0 oherwise 4

5 Risk Descripors and Risk Indices > for each sock we calculae risk descripors on he basis of fundamenal aribues > he following risk indices are calculaed as he arihmeic mean wihin groups of he (sandardized) descripors: Risk Index Size Success Value Variabiliy Yield Profi Blue Chip Descripor(s) Naural log of marke capialisaion 12 monh railing reurn, relaive srengh index (evy) over 6 monhs Book o price, earnings o price, price o earnings o growh, price o book o reurn o equiy Hisorical specific variance, 1-monh-earnings revision, coefficien of variaion of analyss esimaes for he curren financial year as dividend divided by curren marke cap Reurn on equiy, cash flow reurn on equiy Dummy variable: member of he larges European companies (ha represen 10% of oal marke cap) 5

6 Esimaion of Facor Reurns > Esimaion of facor reurns on he basis of weighed leas squares: ) f ( T W 1 ) T W r subjec o: k f i i u + v+1 0 > W is he diagonal marix of weighs which are he roos of he marke caps of he individual securiies (he variance of he reurns decreases empirically wih he size of he company) > he resricion on he indusry facors assures ha general marke movemens are solely accouned for by he counry facors (oherwise we face he problem of mulicollineariy) 6

7 7 Esimaion of Covariance Marix Esimaion of Covariance Marix > Sep One: using an exponenial smoohing facor (in order o give less weigh o he observaions of pas periods), each elemen ϕ ij of he facor covariance marix Φ is esimaed as follows: > Sep Two: Esimaion of securiy covariance marix: Λ 1 0 u j j u, i u,i u ij ) ˆ ( ) ˆ ˆ ( 1 1 ˆ f f f f λ ϕ The parameer Λ is used o sandardise he weighs in a way so ha hey sum up o. Λ is calculaed as follows: Λ -1 0 i i -1 1 λ T ˆ ˆ ˆ Ω + Φ Σ

8 Daa base and research design Objecive is he esimaion of a fundamenal muliple facor model for a universe of European socks using cross-secion analysis. This mehodology was preferred o he saisical facor analysis because he resuls are more easily inerpreable and because he consancy of he sensiiviies is no required. In addiion, he sudy examines he explanaory power of he model over ime. Daabase consiss of a selecion of 656 European socks from 12 EU member counries as well as Norway and Swizerland. These socks, due o heir marke capialisaion and liquidiy, form a sandard working universe for European invesmen funds. Time series for he weekly reurns were calculaed in local currency from o

9 Descripors - The descripors examined herein are variables ha play a cenral role in fundamenal equiy research. - The selecion of he descripors is based on he lis of descripors ha has already proven is worh in he BARRA model. The following descripors have been seleced: - Daabase is cleaned for exreme values using he skipped Huber mehod. - The median of he observaions plus/minus 5.2 imes he median of he deviaions serve as limis. Exreme values are hen referred o he respecive limis. 9

10 is of Descripors SIZE: The naural logarihm of he marke capialisaion (marke cap number of shares muliplied by he curren sock price). 2. SUCCESS: The naural logarihm of las year s reurn. 3. BTOP (book o price): The equiy value saed in he las balance shee of he company divided by he curren marke capialisaion. 4. ETOP (earnings o price): Prediced earnings of he curren financial year prediced by analyss divided by he curren marke capialisaion. 5. VIM (variabiliy in markes): The hisorical specific variance, i.e. a measure for he variabiliy in he pas. 6. YIED (dividend yield): as dividend divided by curren marke capialisaion. 7. PEG (price o earnings o growh): The price earnings raio (he reciprocal of ETOP) divided by earnings growh of he las four years. 8. PBROE (price o book o reurn on equiy): The quoien of marke cap o book value (he reciprocal of BTOP) is divided by he reurn on equiy over he las four years. 9. RSI6M (relaive srengh index, 6 monh): Relaive srengh index by EVY. 10. REV1M: Revision of earnings on a one-monh basis. 11. REV3M: Revision of earnings on a hree-monh basis. 12. ROE (reurn on equiy): Raio of he earnings of he curren financial year o he equiy capial of he firm. 13. CROE (cash flow reurn on equiy): Raio of he cash flow of he curren financial year o he equiy capial of he firm. 14. F1CV: The coefficien of variaion of analyss earnings esimaes for he curren financial year. 15. F2CV: The coefficien of variaion of analyss earnings esimaes for he following financial year.

11 Aggregaion of Descripors Filered and sandardised descripors are aggregaed o so-called risk indices like in he BARRA model (based on fundamenal crieria) Risk Index SIZE SUCCESS VAUE VIM YIED PROFIT Descripor(s) SIZE SUCCESS, RSI6M BTOP, ETOP, PEG, PBROE VIM, REV1M, REV2M, F1CV, F2CV YIED ROE, CROE The index BUECHIP was added o hese six risk indices so ha here were seven risk indices plus 14 counry dummy variables, 10 coninenal indusry dummy variables and 10 Briish indusry variables 11

12 Hisogram of he values of R² A Hisogram of R² A Frequency % 10% 15% 20% 25% 30% 35% 40% 45% 50% 55% 60% 65% 70% 75% R ² A 12

13 R² over ime A R² A over ime 70% 60% 50% 40% R ² A 30% 20% 10% 0% Period 13

14 Moving averages for R² A Moving average for R ² A 50% 40% Average of R ² A 30% 20% 10% 0% Period 14

15 Significan facors a a 5% level 15 o. Facor Frequen cy No. Facor Frequency 1 Unied Kingdom 70,6% 22 UK Uiliies / Telecom 25,4% 2 Ialy 68,2% 23 Oil / Energy Equipmen 22,1% 3 France 67,1% 24 Norway 21,6% 4 Germany 62,0% 25 Ireland 21,4% 5 Spain 61,6% 26 UK Machine Consrucion / 21,4% Elecrical Goods 6 Neherlands 52,5% 27 YIED 18,6% 7 Swizerland 44,6% 28 UK Commercial Services 18,1% 8 Belgium 43,3% 29 Uiliies / Telecom 15,9% 9 Sweden 41,0% 30 VAUE 14,8% 10 SUCCESS 36,6% 31 Machine Consr. / Elecr. Goods 13,9% 11 Ausria 36,4% 32 UK-Media / eisure / Sofware 13,7% 12 UK Healhcare 32,7% 33 UK - Muli Indusry 13,7% 13 Denmark 32,5% 34 Consumer Goods and Reail 13,2% 14 Finland 32,2% 35 Raw Maerials 12,4% 15 Finance 29,3% 36 PROFIT 12,1% 16 UK Raw Maerials 28,5% 37 BUECHIP 11,0% 17 UK Finance 27,4% 38 Healhcare 10,8% 18 UK Consumer Goods and 27,4% 39 Commercial Services 10,4% Reail 19 VIM 26,0% 40 Media / eisure / Sofware 7,3% 20 UK Oil / Energy 25,8% 41 Muli Indusry 6,0% Equipmen 21 SIZE 25,4%

16 Conclusion Muliple facor models are a powerful ool for he saisical formulaion of reurn generaing processes and can hus make deailed risk analysis and prognosis easier For reasons of more accurae inerpreaion we chose he cross-secional approach was chosen for his sudy The resuls of he esimaions can be considered very saisfacory, as he muliple facor model explains more han 30% of he variance on average Possible exensions of his sudy are he enlargemen of he sock universe or he inegraion of bond porfolio analysis 16