CHAPTER 8. r E( r ) m e. Reduces the number of inputs for diversification. Easier for security analysts to specialize

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1 CHATE 8 Idex odels cgra-hll/ir Copyrght 0 by The cgra-hll Compaes, Ic. All rghts reserved. 8- Advatages of the Sgle Idex odel educes the umber of puts for dversfcato Easer for securty aalysts to specalze Sgle Factor odel 8-3 β respose of a dvdual securty s retur to the commo factor m. Beta measures systematc rsk. m a commo macroecoomc factor that affects all securty returs. The S& 500 s ofte used as a proxy for m. e r E( r ) m e frm-specfc surprses

2 Sgle-Idex odel egresso Equato: E t t e t The expectato of the resdual term e s zero, so the expected returbeta relatoshp s: E 8-4 Sgle-Idex odel 8-5 sk ad covarace: Varace - Systematc rsk ad Frmspecfc rsk, assume ose s ucorrelated: ( e ) Covarace - product of betas x market dex rsk: Cov( r, r ) j j Sgle-Idex odel - Correlato roduct of correlatos th the market dex: Corr Corr /, j r, rj Cov, j r, rj j, j r, rj j j r r j, Corrr r Corr, j j 8-6

3 Questos to test your tuto 8-7 What s the stock s E(r) f (r -r f )=0? What s the resposveess of the stock to market movemets relatve to r f? What s the stock-specfc compoet of retur (ot drve by the market)? What s the varace attrbutable to ucertaty of the market? Ad that attrbutable to frm-specfc evets? Idex odel ad Dversfcato 8-8 Cosder a Equally eghted portfolo ad take the expected retur as the average: e e e Idex odel ad Dversfcato 8-9 The portfolo varace by defto: σ = β σ + σ (e ) here the market compoet comes from the portfolo s sestvty to the market: ad the o-systemc compoet σ (e ) s the cotrbuto of all the stocks the portfolo.

4 Idex odel ad Dversfcato 8-0 Varace of the o-systemc compoet of a equally eghted portfolo s (e assume all the stock-specfc compoets are ucorrelated): e e e Whe gets large, σ (e ) becomes eglgble ad frm specfc rsk ca be dversfed aay. Fgure 8. The Varace of a Equally Weghted ortfolo th sk Coeffcet β p 8- Fgure 8. Excess eturs o H ad S&

5 Fgure 8.3 Scatter Dagram of H, the S& 500, ad H s Securty Characterstc Le (SCL) H t t e t H H S500 H 8-3 Table 8. Excel Output: egresso Statstcs for the SCL of Helett-ackard 8-4 correlato explaatory poer (mothly) Table 8. Iterpretato 8-5 Correlato of H th the S& 500 s The model explas about 5% of the varato H H s alpha s 0.86% per moth (0.3% p.a.), but t s ot statstcally sgfcat H s beta s.0348, but the 95% cofdece terval s +/- ~ stadard errors, hch s qute de

6 Fgure 8.4 Excess eturs o ortfolo Assets Study pars of securtes vs the market to estmate correlatos Compute stats to measure correlatos 8-6 Study portfolo stats 8-7 A closer look at correlatos 8-8

7 Study portfolo stats 8-9 j e Example: buld optmal portfolo 8-0 Alpha ad Securty Aalyss 8-. Use acroecoomc aalyss to estmate rsk premum ad rsk of the market dex (, σ ). Use statstcal aalyss to estmate the beta coeffcets of all securtes ad ther resdual varaces σ (e )

8 8- Alpha ad Securty Aalyss 3. Use umercal methods to establsh the expected retur of each securty depedetly of securty aalyss (β) 4. Use securty aalyss to develop your o forecast of the expected returs for each securty (α) Sgle-Idex odel cosderatos 8-3 Techques for estmatg β are ell ko Estmatg alpha requres a deep koledge of the compay behd the stock: ostve α meas overeght the portfolo What do you do f α s egatve? ecall the mum-varace Froter 7-4 Chapter 7 took the etre uverse of stocks ad used brute-force math to fd the effcet froter

9 Sgle-Idex odel Optmzato 8-5 Sgle-Idex model offers a smpler optmzato tha the model chapter 7 as the model s smplfed Iclude the market as asset + to mprove dversfcato. By defto: Beta of market dex = Alpha of market dex = 0 e market_dex = 0 Sgle-Idex odel Iput Lst 8-6 sk premum o the S&500 portfolo ( ) Estmate of the SD of the S&500 portfolo (σ ) sets of estmates for each stock of: Beta coeffcet Stock resdual varaces Alpha values Sgle-Idex odel steps 8-7 Use, alphas ad betas to costruct + expected returs Use betas ad σ to costruct the covarace matrx Set up the optmzato problem to mmze portfolo varace, gve a retur, subject to costrat that eghts add up to oe You could use excel solver to solve ths problem ad buld your effcet froter

10 8-8 Idex odel ecall α ad β Cosder a geerc portfolo ad take the excess retur as the average: e e e 8-9 Optmal sky ortfolo of the Sgle-Idex odel No take the portfolo expected excess retur: E E E 8-30 Optmal sky ortfolo of the Sgle-Idex odel Stadard Devato ad Sharpe ato: e e E S /

11 Optmal sky ortfolo of the Sgle-Idex odel 8-3 No eed to use Excel as there s a aalytcal soluto Soluto s a combato of: Actve portfolo (A), th eght A arket-dex passve portfolo () 8-3 Optmal sky ortfolo - A Assume for a momet beta= The the optmal eght A s proportoal to the rato σ A /σ (e A ) to balace excess retur ad resdual varace from Actve portfolo A: 0 A A ea E Optmal sky ortfolo of the Sgle-Idex odel 8-33 Next, modfy of actve portfolo eght A to optmze, as beta s ot ecessarly =: * A Notce that he A 0 A 0 A A * 0 the A A

12 The Iformato ato 8-34 The Sharpe rato of a optmally costructed rsky portfolo ll exceed that of the dex portfolo (the passve strategy): A s s ( e A ) Iformato rato The Iformato ato 8-35 The cotrbuto of the actve portfolo depeds o the rato of ts alpha to ts resdual stadard devato. The formato rato measures the extra retur e ca obta from securty aalyss. Fgure 8.5 Effcet Froters th the Idex odel ad Full-Covarace atrx 8-36

13 Table 8. ortfolos from the Sgle-Idex ad Full-Covarace odels 8-37 Is the Idex odel Iferor to the Full-Covarace odel? 8-38 Full arkotz model may be better prcple, but: Usg the full-covarace matrx vokes estmato rsk of thousads of terms Cumulatve errors may result a portfolo that s actually feror to that derved from the sgle-dex model The sgle-dex model s practcal ad decouples macro ad securty aalyss. Beta Book: Idustry Verso of the Idex odel 8-39 Use 60 most recet moths of prce data Use S& 500 as proxy for Compute total returs that gore dvdeds Estmate dex model thout excess returs: * r a brm e

14 Beta Book: Idustry Verso of the Idex odel 8-40 Adjust beta because: The average beta over all securtes s. Thus, our best forecast of the beta ould be that t s. Also, frms may become more typcal as they age, causg ther betas to approach. Table 8.4 Idustry Betas ad Adjustmet Factors 8-4