Principal components of stock market dynamics. Methodology and applications in brief (to be updated ) Andrei Bouzaev, bouzaev@ya.

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1 Principal componens of sock marke dynamics Mehodology and applicaions in brief o be updaed Andrei Bouzaev, bouzaev@ya.ru

2 Why principal componens are needed Objecives undersand he evidence of more han one facor driving he sock marke, marke facor should be supplemened wih ohers, possibly less significan idenify se of facor componens driving sock marke and is basic segmens se up muli-facor model ha is beer han onefacor for monioring of marke dynamics, hedge, pricing, risk-valuaion 1-2

significan idenify se of facor componens driving sock marke and is basic segmens se up

Sensiiviies of zero-coupon yields o he hree facors Similar o modeling he shape of he yield curve Modelling he shape of he yield curve very similar process evidence of more han one

3 Sensiiviies of zero-coupon yields o he hree facors Similar o modeling he shape of he yield curve Modelling he shape of he yield curve very similar process evidence of more han one facor driving he erm srucure principal componens of dynamics are needed o hedge he shape of he yield curve in pracice The folowing hree facors are used: level, slope, curvaure 1-3

han one facor driving he erm srucure principal componens of dynamics are needed o hedge

4 In realiy, do we have only one facor? Correlaion beween USA sock marke porfolios formed on size. Daa: Fama-French decile porfolios reurns formed on marke equiy, based on monhly reurns, LO10 DEC2 DEC3 DEC4 DEC5 DEC6 DEC7 DEC8 DEC9 HI10 LO10 1,00 0,96 0,93 0,91 0,88 0,86 0,85 0,81 0,79 0,70 DEC2 1,00 0,97 0,97 0,95 0,92 0,91 0,88 0,86 0,78 DEC3 1,00 0,98 0,97 0,96 0,94 0,92 0,90 0,82 DEC4 1,00 0,98 0,97 0,96 0,94 0,92 0,84 DEC5 1,00 0,98 0,97 0,96 0,94 0,87 DEC6 1,00 0,98 0,97 0,96 0,90 DEC7 1,00 0,98 0,97 0,91 DEC8 1,00 0,98 0,93 DEC9 1,00 0,95 HI10 1,00 Correlaion marix indicaes ha difference in reurn dynamics is increasing by he difference in marke equiy. Are here underlying facors ha influence differen porfolios in differen manner? 1-4

0,81 0,79 0,70 DEC2 1,00 0,97 0,97 0,95 0,92 0,91 0,88 0,86 0,78 DEC3 1,00 0,98 0,97 0,96 0,94 0,92 0,90 0,82 DEC4 1,00 0,98 0,97 0,96 0,94 0,92 0,84 DEC5 1,00 0,98 0,97 0,96 0,94 0,87 DEC6

5 In realiy, do we have only one facor? Correlaion beween Russia sock marke porfolios formed on size. Daa: MICEX capialisaion indices, based on daily price levels lef and daily reurns righ, BASE STANDARD HIGH BASE 1 0,97 0,92 STANDARD 1,00 0,96 HIGH 1,00 BASE STANDARD HIGH BASE 1,00 0,75 0,66 STANDARD 1,00 0,82 HIGH 1,00 Correlaion marix indicaes ha difference in price dynamics is increasing by he difference in marke equiy. Are here underlying facors ha influence differen porfolios in differen manner? 1-5

0,92 STANDARD 1,00 0,96 HIGH 1,00 BASE STANDARD HIGH BASE 1,00 0,75 0,66 STANDARD 1,00 0,82 HIGH 1,00 Correlaion marix indicaes

6 1-6 Idenifying principal componens* Le us find 3 facors ha explain he dynamics of he ME porfolios The analysis for more han 3 facors is a sraighforward exension Assume he following dynamics of porfolios reurns *The approach is adoped from modeling he shape of he yield curve, Fixed Income lecure noes a NES, professor Dmiri Makarov r r r n n n n n n meaning ha each facor impacs all of he reurns wih sensiiviies β alernaively he same regression for price levels can be used.

of he yield curve, Fixed Income lecure noes a NES, professor Dmiri Makarov 1 3 13 2 12 1 11 1 1 r 2 3 23 2 22 1 21 2 2 r 3 3 2 2 1

7 Principal componen analysis We look for facors ha are implici in he movemens over ime of he various reurns To deermine hem we are going apply Principal Componen Analysis Finding firs principal componen means ha we are maximizing a weighed average of he R-squares of he above regressions Afer compuing he firs componen, we can find he residuals, applying hem insead of reurns, and find he second facor, and so on 1-7

means ha we are maximizing a weighed average of he R-squares of he above regressions Afer compuing he

8 Principal componens in pracice USA sock marke Using monhly reurns for he US sock marke beween 1926 and 2000, we have he following sensiiviies of size-based porfolios reurns o he hree facors 1,2 1,0 0,8 0,6 0,4 0,2 0,0-0,2-0,4 Facor 1 Facor 2 Facor 3 LO20 QNT2 QNT3 QNT4 HI20 Correlaion marix f1 f2 f3 LO20 0,94-0,314 0,12 QNT2 0,98-0,151-0,04 QNT3 0,99-0,003-0,09 QNT4 0,98 0,124-0,09 HI20 0,93 0,350 0,12 Exracion Mehod: Principal Componen Analysis. 1-8

Facor 1 Facor 2 Facor 3 LO20 QNT2 QNT3 QNT4 HI20 Correlaion marix f1 f2 f3 LO20 0,94-0,314 0,12 QNT2

9 Principal componens in pracice Russia sock marke Using daily price daa for he Russia sock marke beween 2005 and 2011, we have he following sensiiviies of size-based porfolios price level 1 and reurns 2 o he hree facors 1,0 0,8 0,6 0,4 0,2 0,0-0,2-0,4 1,0 0,8 0,6 0,4 0,2 0,0-0,2-0,4-0,6 f1 f2 f3 BASE STANDARD HIGH f1 f2 f3 D_BASE D_STAND D_HIGH Correlaion marix, price levels f1 f2 f3 BASE 0,98-0,19 0,08 STANDARD 0,99-0,05-0,11 HIGH 0,97 0,24 0,04 Correlaion marix, reurns f1 f2 f3 D_BASE 0,90 0,41 0,12 D_STAND 0,96-0,05-0,29 D_HIGH 0,92-0,35 0,18 1-9

0,0-0,2-0,4-0,6 f1 f2 f3 BASE STANDARD HIGH f1 f2 f3 D_BASE D_STAND D_HIGH Correlaion marix, price levels f1 f2 f3 BASE 0,98-0,19 0,08

10 Empirical componens descripion Firs facor has roughly he same effec on all porfolio yields/prices. For his reason, i can be referred o as parallel shif/level. I explains mos of he variaion Second facor has negaive impac on small socks and posiive impac on big socks. This is a slope facor The hird facor is posiive a he small and big socks, and negaive a he medium socks. I can be referred o as curvaure 1-10

I explains mos of he variaion Second facor has negaive impac on small socks and posiive impac on big

11 Reurns or price levels? Applying facor analysis o reurns or price levels should lead o he same resuls Meaning reurn facors can be ransformed mahemaically ino level facors correspondingly and vice versa Empirically here are some difficulies for reurn daa Posiive correlaion in levels wih slighly lagged daa can lead o negaive correlaion in reurns Daily reurns are very noisy, for weekly or monhly reurns a long period of daa is needed For he above reasons price level componens have been exraced for Russia sock marke 1-11

ino level facors correspondingly and vice versa Empirically here are some difficulies for reurn daa Posiive correlaion in levels

12 Principal componens in dynamics Three facor componen in dynamic, Russia sock marke based on daily price levels 2 Facor dynamics ok.2011-feb.2012гг. f1 f2 f3 1 0 Октябрь Ноябрь Декабрь Январь Февраль

13 Consrucing marke curve Based on hree facor componens we can consruc sock marke curve which indicaes underpriced/overpriced segmens. 0,5 0,4 0,3 0,2 0,1 F1 parallel shif 0-0,1-0,2-0,3-0,4-0,5 f2, 5% + f2, 5% - f3, 5% + f3, 5% Marke curve, russian sock marke - micex F3 curvaure facor base sandard high Based on Russia sock marke daa daily price levels, Normal shape of he curve: all marke segmens around heir average F2 slope facor 1-13

0,5 0,4 0,3 0,2 0,1 F1 parallel shif 0-0,1-0,2-0,3-0,4-0,5 f2, 5% + f2, 5% - f3, 5% + f3, 5% - 11.11.2011 30.11.2011 30.12.

14 Applicaion mulifacor asse pricing model 1 Expeced reurn of any markeable securiy can be wrien as a funcion of he expeced reurn of he efficien porfolio eff E[ ri ] rf i * E[ reff ] rf I is exremely difficul o idenify porfolios ha are efficien bu we can use collecion of porfolios from which he efficien porfolio can be consruced o measure risk Each facor porfolio capures differen componens of he sysemaic risk Mulifacor models is significan improvemen over he CAPM, i is widely used in academic lieraure and in pracice o measure risk and o calculae cos of capial 1-14

which he efficien porfolio can be consruced o measure risk Each facor porfolio capures differen componens of he sysemaic risk Mulifacor

15 Applicaion mulifacor asse pricing model 2 The above described facors represen he following porfolios The level facor is a marke porfolio o he exen i can be derived from sock marke daa The slope facor is small-minus-big porfolio. I is similar bu no idenical o SMB facor from Fama-French model. The curvaure facor is small and big minus medium porfolio. Taking ino accoun he se of 3 facors he mulifacor models becomes f 1 f 2 E[ ri ] rf i * E[ rf 1] rf i * E[ rf 2] rf i * E[ rf 3] rf f

I is similar bu no idenical o SMB facor from Fama-French model. The curvaure facor is small and big minus medium porfolio.

16 Applicaion mulifacor asse pricing model 3 Mos popular mulifacor model specificaion by now is based on marke, small-minus-big and high-minus-low facors see hp://en.wikipedia.org/wiki/fama%e2%80 %93French_hree-facor_model : Mk SMB E[ ri ] rf i * E[ rmk] rf i * E[ rsmb] rf i * E[ rhml] rf Bu hese facors were seleced empirically whereas level, slope and curvaure facors are derived mahemaically using principal componens analysis HML 1-16

org/wiki/fama%e2%80 %93French_hree-facor_model : Mk SMB E[ ri ] rf i * E[ rmk] rf i * E[ rsmb] rf i * E[

17 Furher developmen Facor componens mehodology and componens values are published on hp://fmlab.hse.ru/ Ready for any discussion and cooperaion wih individuals/academics/financial insiuies of furher developmen and calculaion of he presened facor componens in form of sock indices or reurns. Conacs: Andrei Bouzaev,

ru/ Ready for any discussion and cooperaion wih individuals/academics/financial

A Note on Using the Svensson procedure to estimate the risk free rate in corporate valuation

A Note on Using the Svensson procedure to estimate the risk free rate in corporate valuation A Noe on Using he Svensson procedure o esimae he risk free rae in corporae valuaion By Sven Arnold, Alexander Lahmann and Bernhard Schwezler Ocober 2011 1. The risk free ineres rae in corporae valuaion