Topic 6 Introduction to Portfolio Theory

Transcription

1 Topc 6 Introducton to ortfolo Theory 1. racttoners fundamental ssues. ortfolo optmzaton usng Markowtz effcent fronter 3. ortfolo dversfcaton & beta coeffcent 4. Captal asset prcng model 04/03/015 r. Dder Folus 1

2 1. racttoners fundamental ssues Tradtonal assets classes: o Equtes o Bonds & rates Alternatve asset classes: o Currences o Commodtes o Real estate Does an optmal combnaton of asset classes exst? What does dversfcaton mean? What about the rsk-expected return relatonshp? 04/03/015 r. Dder Folus

3 . ortfolo optmzaton : Markowtz approach.1. Two rsky assets combnaton Securty X : E X = 8 % & s X = 1 % Securty Y : E Y = 10 % & s Y = 18 % ortfolo % 1 % E s A %.7 % A B C D D % 9.5 % 9.0 % 8.0 % 7.5 % 18.0 % 13.8 % 10.8 % 1.0 % 15.7 % /03/015 r. Dder Folus 3

4 Feasble portfolos hyperbola (case = 0) Expected return 10 % 9 % 8 % MV C D B X D A Y 5 % 10 % 15 % 0 % A Short n X Short n Y Standard devaton return E volatlty s E 1 X X E Y 1 s Y s XsY s 1 04/03/015 r. Dder Folus 4

5 Computng the Mnmum Varance ortfolo (case = 0) rogram : Frst order condton : Soluton : mns * s X d s 0 d s Y cov XY s cov Y XY MV : nvestng 69.3 % n X & % n Y expected return E MV = 8.7 % expec ted volatlty s MV = 10.1 % 04/03/015 r. Dder Folus 5

6 .. Optmal rsky assets combnaton wth r.f.a. Rsk free asset F : r = 5 % & s F = 0 % Combnatons between (X, Y) hyperbola and F Optmal rsky combnatons : (F, T, Y) curve Expected return T Y Rsk averson F 0.05 X Standard devaton 04/03/015 r. Dder Folus 6

7 .3. H. Markowtz (195, 1959) effcent fronter Lne [F,T) f rsk free asset short sales are allowed Segment [F, T] f not Expected return T E r ET r s s T 0.05 F Best rsk/reward rato Standard devaton 04/03/015 r. Dder Folus 7

8 3. ortfolo dversfcaton 3.1. Two rsky assets combnaton Securty X : E X = 8 % & s X = 1 % Securty Y : E Y = 10 % & s Y = 18 % ortfolo rsk s less than average assets rsk ortfolo 1 E s % % = 1 = 0 = 1 A % 5.5 %.7 % 19.5 % A % 18.0 % 18.0 % 18.0 % B % 10.0 % 13.8 % 16.5 % C % 3.0 % 10.8 % 15.0 % D % 1.0 % 1.0 % 1.0 % D % 19.5 % 15.7 % 10.5 % ortfolo return equals the average assets return 04/03/015 r. Dder Folus 8

9 3.. Several rsky assets combnaton ortfolo varance reducton 100 % 50 % 40 % 30 % 0 % 10 % Specfc rsk or dversfable rsk : - patent - productvty - strke Systematc rsk : - geopoltcal tensons - GD growth - IR fluctuatons Number of stocks Generally R # 0.40 common factor(s) 04/03/015 r. Dder Folus 9

10 3.3. Measurng a stock senstvty to market: beta Sharpe (1964) : Sngle securty rsk : Stock senstvty to market r t r Common factor var Lnear regresson qualty 0 < R < 1 04/03/015 Stock return eases ndex return, but r. Dder Systematc Folus rsk low b < 1 10 doesn t follow t closely Specfc rsk hgh R 0 mt t r cov var, r r r varr var Systematc rsk Data Rsk level Model Stock return follows closely and amplfes ndex return Stock return follows closely and eases ndex return Stock return amplfes ndex return, but doesn t follow t closely m Systematc rsk hgh Specfc rsk low Systematc rsk low Specfc rsk low Systematc rsk hgh Specfc rsk hgh m m Specfc rsk b > 1 R 1 b < 1 R 1 b > 1 R 0

11 Industral sectors beta coeffcents From Value Lne database frms 5 years weekly data aganst Nyse Composte By A. Damodaran Sector Ol/gas utltes Banks Food processng Medcal servces Botechnology Basc chemcal Auto. parts Number of companes Average market beta Jan 014 Jan 01 Jan /03/015 r. Dder Folus 11

12 3.4. Rsk reducton through dversfcaton Sharpe model r Sngle securty rsk.. var r ortfolo rsk... var r t varr m var varr var r mt m t Gven a portfolo, a -securty contrbutes to the portfolo rsk through cov r, r Dversfable rsk Systematc rsk Dversfed portfolo rsk : lm var n r varr m 04/03/015 r. Dder Folus 1

13 4. The Captal Asset rcng Model 4.1. Objectve & assumptons Implementng a rsk-expected return relaton requres observaton of the tangency portfolo (see Markowtz) CAM has been developped by Sharpe (1964), Lntner (1965), Mossn (1966) : A1. Investors care only about mean/varance of portfolo returns A. Markets are frctonless A3. There exsts one rsk-free asset A4. Investors have homogeneous belefs (they reach the same concluson about all feasble portfolos returns dstrbuton) 04/03/015 r. Dder Folus 13

14 4.. CAM major results Implcatons for optmal nvestment (A4) : the tangency portfolo s the market portfolo M (A3) : Markowtz effcent fronter reduces to CML : Expected return 0.05 F M E r r E r s m r m r s r Standard devaton 04/03/015 r. Dder Folus 14

15 Optmzaton example : Computng the weghts of the market portfolo AXA LVMH eugeot Shares,000 M 500 M 00 M Quote EUR 10 EUR 50 EUR 5 Cap. EUR 0 bn EUR 5 bn EUR 5 bn Weght 40 % 50 % 10 % Choosng an optmal combnaton nvestng EUR 100,000 Mster Hatersk Mss Loversk Rsk free 90 % 0 % Tblls EUR 90,000 EUR 0,000 Rsky assets 10 % 80 % AXA LVMH eugeot EUR 4,000 EUR 5,000 EUR 1,000 EUR 3,000 EUR 40,000 EUR 8,000 ortfolo EUR 100,000 EUR 100,000 04/03/015 r. Dder Folus 15

16 4... Implcatons for the rsk premum Only the securty systematc rsk s proftable Securty rsk premum : E r r Er m r Rsk free rate or safe rate Market rsk premum Average market rsk averson Market volatlty 04/03/015 r. Dder Folus 16

17 Source : Crédt Susse Global Investment Returns Yearbook /03/015 r. Dder Folus 17