Chapter 11: Optmal Portolo Choce and the CAPM-1 Chapter 11: Optmal Portolo Choce and the Captal Asset Prcng Model Goal: determne the relatonshp between rsk and return => key to ths process: examne how nvestors buld ecent portolos Note: The chapter ncludes a lot o math and there are several places where the authors skp steps. For all o the places where I thought the skpped steps made ollowng the development dcult, I ve added the mssng steps. See Chapter 11 supplement or these addtonal steps. I. The o a Portolo Note: a portolo s dened by the percent o the portolo nvested n each asset MV x (11.1) MV j j R P xr (11.) R x ER E P (11.3) where: x = percent o portolo nvested n asset MV = market value o asset = number o shares o outstandng prce per share o MV = total value o all securtes n the portolo j j R P = realzed return on portolo R = realzed return on asset E[R P ] = expected return on portolo E[R ] = expected return on asset II. The Volatlty o a Two-Stock Portolo A. Basc dea 1) by combnng stocks, reduce rsk through dverscaton Q: What determnes the amount o rsk elmnated? => tend to move together, not much rsk cancels out. => don t tend to move together, more rsk cancels out
Chapter 11: Optmal Portolo Choce and the CAPM- ) amount o rsk that remans n a portolo depends on the amount o rsk that s common to the stocks => need to measure amount o common rsk n stocks n our portolo B. Covarance and Correlaton 1 1. Covarance: Cov R R j R, t R R j, t R j, (11.) T 1 t where: T = number o hstorcal returns Notes: 1) two stocks tend to move together, ther returns wll tend to be above or below the average at same tme => covarance wll be postve ) two stocks tend to move n opposte drectons, one stock s return wll tend to be above ts mean when the other s below ts mean => covarance wll be negatve => can go n opposte drecton as well = > covarance postve, tend to move together => covarance negatve, tend to move n opposte drectons 3) Covarance wll be larger : - stock s returns are more closely related - the stocks are more volatle Goal: solate the relatonshp part. Correlaton: CorrR R Notes: R, R j R SDR Cov, j (11.6) SD 1) Same sgn as covarance so same nterpretaton ) takes out the volatlty o ndvdual stocks => let wth common rsk j
Chapter 11: Optmal Portolo Choce and the CAPM-3 3) Correlaton s always between +1 and -1 => as correlaton changes rom to +1, movng more closely together => as correlaton changes rom to -1, movng more and more n opposte drectons Corr = +1: always move exactly together Corr = -1: always move n exactly opposte drectons 4) stocks wth hgh correlatons tend to be aected n the same way by the economy C. Portolo Varance and Volatlty R x VarR x VarR x x CovR R Var p (11.8) 1 1 1 1, R x SDR x SDR x x CorrR R SDR SDR Var p (11.9) 1 1 1 1, Ex. Use the ollowng returns on JPMorganChase (JPM) and General Dynamcs (GD) to estmate the covarance and correlaton between JPM and GD and the expected return and volatlty o returns on a portolo o $3, nvested n JPM and $, nvested n GD. Return on: Year JPM GD 1-1% 36% 7% -34% 3 14% 37% 4-3% 9% 3% 18% 6 19% 18% 1 RJPM, RGD RJPM,t RJPM R GD,t RGD Cov T 1 t 1 R JPM 6 1 R GD 6 36 34 37 9 18 18 Cov(R JPM, R GD ) 1 1 7 14 3 3 19 6.% 14.% 1 6.36 14 7 6. 34 14 14 6.37 14 3 6.9 14 3 6.18 14 19 6.18 14 = -8.6 1
Chapter 11: Optmal Portolo Choce and the CAPM-4 Corr(R JPM, R GD ) = SD Cov R JPM JPM,R SD GD GD 66.3 1 1 6. 7 6. 14 6. 3 6. 3 6. 19 6. Var R JPM SD 66.3 16.3% R JPM 674.8 1 36 14 34 14 37 14 9 14 18 14 18 14 Var R GD SD R GD 674.8.98% 8.6 Corr(R JPM, R GD ) = 16.3.98 E(R p ) =.7(6.) +.(14) = 8.37% x JPM x GD = -.138 3,.7 3,,,. 1.7 3,, R x1 VarR1 xvarr x1xcovr1, R R x SDR x SDR x x CorrR R SDR SDR Var p (11.8) Var p (11.9) 1 1 1 1, 11.8: Var(R P ) =.7 66.3. 674.8..7 8.6 = 169.99 11.9: Var(R P ) =.7 16.3..98..7.13816.3.98= 169.99 SD(R p ) = 169.99 = 13.4% Notes: 1) varance o portolo lower than ether stock by tsel 1
Chapter 11: Optmal Portolo Choce and the CAPM- ) can acheve wde range o rsk-return combnatons by varyng portolo weghts X(JPM) SD(Rp) E(Rp) 1. 16.3 6..9 14.6 7..8 13.37 8..7 1.91 8.7.6 13.6 9.. 14.3..4 16.4 11..3 18.17 11.7..9 1.. 3.1 13...98 14. Q: Why does expected return rse as X jpm alls? Q: Why does standard devaton ntally all then rse as X jpm alls? 3) the ollowng graph shows the volatlty and expected return o varous portolos Graph #1: Volatlty and or Portolos o JPM and GD 16 14 1 8 6 4 7% JPM % JPM % GD 1 3 Volatlty II. Rsk Verses Return: Choosng an Ecent Portolo Note: Can narrow down our choces a bt A. Ecent portolos wth two stocks Ecent portolo: no portolo has both hgher expected return and lower volatlty Q: Whch part o the graph s ecent?
Chapter 11: Optmal Portolo Choce and the CAPM-6 Graph #: Ecent Portolos o JPM and GD 16 14 1 8 6 4 Ecent Portolos % JPM % GD 1 3 Volatlty B. The Eect o Correlaton Key: Correlaton measures relatonshp between assets => How mpact portolos? => other thngs equal, the lower the correlaton the lower the volatlty o portolos (due to greater dverscaton). => more bend to curve o possble portolos Graph #3: The Eect o Correlaton 16 14 1 8 6 4 % GD % JPM 1 3 Volatlty Corr= -.8 Corr= -.14 Corr= +.6
Chapter 11: Optmal Portolo Choce and the CAPM-7 C. Short Sales I correlaton: +1: portolos le on a straght lne between ponts -1: portolos le on a straght lne that bounces o vertcal axs (rsk-ree) => add graphs wth same standard devaton 1. Short sale: sell stock don t own and buy t back later Notes: 1) borrow shares rom broker (who borrows them rom someone who owns the shares) ) sell shares n open market and receve cash rom sale 3) make up any dvdends pad on stock whle have short poston 4) can close out short poston at any tme by purchasng the shares and returnng them to broker ) broker can ask or shares at any tme to close out short poston => must buy at current market prce at that tme. 6) untl return stock to broker, have short poston (negatve nvestment) n stock 7) portolo weghts stll add up to % even when have short poston Ex. Assume short-sell $, o JPM and buy $, o GD. What s volatlty and expected return on portolo E(R JPM ) = 6.%, E(R GD ) = 14.%; SD(R JPM ) = 16.3%, SD(R GD ) =.98%; and Corr (R JPM, R GD ) =.138? Note: total nvestment = $4, x GD =,/4, = 1. x JPM = -,/4, = -. E(R P ) = -.(6.) + 1.(14) = 1.87% Q: What s allowng us to earn a hgher return than 14% (E(R) on GD)? Notes: 1) Expected dollar gan/loss on JPM = -,*.6 = $6 ) Expect dollar gan/loss on GD =,*.14 = 7, = 4,*.14 +,*.14 = 6, + 14, 3) Net expected gan = 7, 6 = 6, + (14, 6) = 63,
Chapter 11: Optmal Portolo Choce and the CAPM-8 63, 4) Expected return =.187 4, R x SDR x SDR x x CorrR R SDR SDR Var p (11.9) 1 1 1 1, Var(R P ) = -. 16.3 1..98 -.1..13816.3.98 SD(R P ) = 17.64 = 33.8% = 17.64 Q: Why s rsk hgher than smply nvestng $4, n GD (wth a standard devaton o returns o.98%)? 1) short-sellng JPM creates rsk ) gan/loss on a $, nvestment n GD s greater than the gan/loss on a $4, nvestment n GD 3) loss o dverscaton: Correlaton between a short and long poston n JPM s -1. Correlaton between short JPM and GD wll be +.138 => less dverscaton than between long poston n JPM and GD w/ correlaton o -.138. Impact on graphs => curve extends beyond endponts (o % n one stock or the other). 1 Graph #4: Portolos o JPM and GD wth Short Sellng 3 3 1 - - -1 % GD % JPM X(jpm) = -. SS GD, Buy JPM 4 6 8 Volatlty SS JPM, Buy GD
Chapter 11: Optmal Portolo Choce and the CAPM-9 Ecent ronter: portolos wth hghest expected return or gven volatlty Q: What part o the graph s ecent? Graph #: Ecent Fronter wth JPM and GD and Short Sellng 3 3 1 - - -1 4 6 8 Volatlty D. Rsk Versus Return: Many Stocks 1. Three stock portolos: long postons only Q: How does addng Sony mpact our portolo? E(R JPM ) = 6.%, SD(R JPM ) = 16.3%; E(R GD ) = 17%, SD(R GD ) = 6%; E(R Sony ) = 1%, SD(R Sony ) = 3%; Corr(R JPM,R GD ) = -.138; Corr(R Sony, R GD ) =.398; Corr(R Sony, R JPM ) =.4 Graph #6: Portolos o JPM, GD, and SNE 1 Long n all 3 JPM GD SNE 3 4 Note: Get area rather than curve when add 3 rd asset
Chapter 11: Optmal Portolo Choce and the CAPM- Q: How does graph change allow long and short stock postons?. Three Stock Portolos: long and short postons Q: What allow short postons n any o the three stocks? Graph #7: Porolos o 3 stocks (long and short) 3 3 1 Note: possble to acheve any pont nsde the curves w/ 3 or more All 3 JPM GD SNE JPMnGD - 3 4 6 - Volatlty (SD) Graph #8: Ecent ronter wth 3 stocks (long and short) 3 3 1 All 3 JPM GD SNE - 3 4 6 - Volatlty (SD) 3. More than 3 stocks (long and short): greater dverscaton so that ecent ronter curves urther to the let Note: addng necent stock (lower expected return and hgher volatlty) may mprove ecent ronter!
Chapter 11: Optmal Portolo Choce and the CAPM-11 III. Rsk-Free Securty A. Ways to change rsk 1. Ways to reduce rsk 1) move to let on ecent ronter ) sell some o rsky assets and nvest n rskless securtes. Ways to ncrease rsk 1) move to rght on ecent ronter ) short-sell rskless securtes and nvest n rsky assets Q: Whch approach s better? B. Portolo Rsk and Return Let: x = percent o portolo nvested n rsky portolo P 1-x = percent o portolo nvested n rsk-ree securty 1. ER 1 xr xer r x ER r (11.1) xp P P => expected return equals rsk-ree rate plus racton o rsk premum on P based on amount we nvest n P. SD R 1 x Varr x VarR 1 xxcovr, R xp (11.16a) Note: Var(r ) and Cov(r,R p ) both equal! P => SD(R xp ) = xsd(r P ) (11.16b) => volatlty equals racton o volatlty o rsky portolo 3. Note: ncrease x, ncrease rsk and return proportonally => combnatons o rsky portolo P and the rsk-ree securty le on a straght lne between the rsk-ree securty and P. P
Chapter 11: Optmal Portolo Choce and the CAPM-1 Ex. Assume that you nvest $8, n P (7% JPM and % n GD) and $3, n Treasures earnng a 4% return. What volatlty and return can you expect? Note: rom earler example: E(R p ) = 8.37%, and SD(R P ) = 13.4% x = 8,/4, =. $ nvested n JPM and GD: JPM =.7(8,) = $6,; GD =.(8,) = $, SD(R.P ) =.(13.4) =.61% E(R.P ) =.8(4) +.(8.37) = 4 +.(8.37 4) = 4.88% Ex. Assume you nvest $36, n P and $4, n Treasures x = 36,/4, =.9; $ nvested n JPM and GD: JPM =.7(36,) = 7,; GD =.(36,) = 9, SD(R.9P ) =.9(13.4) = 11.73% E(R.9P ) =.1(4) +.9(8.37) = 4 +.9(8.37 4) = 7.94% Graph #9: Combnng P wth rsk-ree securtes 18 16 14 1 8 6 4.9P.P P 3 4 Volatlty
Chapter 11: Optmal Portolo Choce and the CAPM-13 C. Short-sellng the Rsk-ree Securty Remnder: x = percent o portolo nvested n rsky portolo P 1-x = percent o portolo nvested n rsk-ree securty I x > 1 (x > %), 1-x < => short-sellng rsk-ree nvestment 11.16b: SD(R xp ) = xsd(r P ) 11:1: ER 1 xr xer r x ER xp P Ex. Assume that n addton to your $4,, you short-sell $, o Treasures that earn a rsk-ree rate o 4% and nvest $, n P. What volatlty and return can you expect? Note: E(R P ) = 8.37%, SD(R P ) = 13.4% x =,/4, = 1. $ nvested n JPM and GD: JPM =.7(,) = 37,; GD =.(,) = 1, SD(R 1.P ) = 1.(13.4) = 16.3% E(R 1.P ) = -.(4) + 1.(8.37) = 4 + 1.(8.37 4) = 9.47% P r
Chapter 11: Optmal Portolo Choce and the CAPM-14 Graph #: Combnng P wth rsk-ree securtes 18 16 14 1 8 6 4 1.P P Sharpe=.336 3 4 Volatlty Q: Can we do better than P? Goal => want hghest return or the rsk => want steepest possble lne D. Identyng the Optmal Rsky Portolo 1. E RP r Sharpe Rato (11.17) SD RP => slope o lne that create when combne rsk-ree nvestment wth rsky P Ex. Sharpe rato when nvest $3, n JPM and $, n GD. Sharpe Rato = => see graph 8.37 4 =.336 13.4 Q: What happens to the Sharpe Rato choose a pont just above P along curve? => ncreases Q: What s best pont on the curve?
Chapter 11: Optmal Portolo Choce and the CAPM-1. Optmal Rsky Portolo Key => tangent portolo gves hghest Sharpe rato o any portolo Ex. Hghest Sharpe rato when x JPM =.447, x GD = 1.447 =.78 Note: I solved or x w/ hghest Sharp rato usng Solver n Excel => nvest $4, total, then nvest $178,888 n JPM (.447x4,) and $1,11 n GD (.78x4,) Note: E(R JPM ) = 6.%, E(R GD ) = 14%; SD(R JPM ) = 16.3%, SD(R GD ) = 6%; and Corr (R JPM, R GD ) =.138 E(R T ) =.646% =.447(6.) +.78(14) SD.447 16.3.78 6.447.78.13816.36 1.18% R T.646 4 Sharpe Rato (Tangent) 1.18.4378.336 Sharpe Rato (P) Graph #11: Tangent Portolo Ecent Fronter w/ Rsky and Rsk-Free 1 Xjpm=.447 Ecent Fronter w/ Rsky Tangent Portolo = Ecent Portolo Sharpe=.4378 3 4 Volatlty
Chapter 11: Optmal Portolo Choce and the CAPM-16 Implcatons: 1) all nvestors wll buy or short-sell rsk-ree securty and nvest n the tangent portolo ) no other rsky portolo s ecent Graph #1: Tangent Portolo 1 Tangent Portolo Sharpe=.4378 3 4 Volatlty => show pont put. n Tangent Portolo (TP),.9 n TP, and 1. n TP IV. The Ecent Portolo and Requred Returns A. Basc Idea Q: Assume I own some portolo P. Can I ncrease my portolo s Sharpe rato by shortsellng rsk-ree securtes and nvestng the proceeds n asset? A: I can the extra return per unt o extra rsk exceeds the Sharpe rato o my current portolo Note: add graph to board that shows mprovng P by movng up and to rght 1. Addtonal return short-sell rsk-ree securtes and nvest proceeds n Use Eq. 11.3: E R x ER P =>
Chapter 11: Optmal Portolo Choce and the CAPM-17. Addtonal rsk short-sell rsk-ree securtes and nvest proceeds n Use Eq. 11.13 (rom text):, =>, 3. Addtonal return per rsk = 4. Improvng portolo,, => I mprove my portolo by short-sellng rsk-ree securtes and nvestng the proceeds n :, Or (equvalently): E R r SDR CorrR, R P E R SD P R r P (11.18) B. Impact o people mprovng ther portolos 1. As I (and lkely other people) start to buy asset, two thngs happen 1) E(R ) alls as the prce gets bd up ) Corr(R, Rp) rses as P becomes more lke. Opposte happens or any asset or whch 11.1 has < rather than > C. Equlbrum 1) people wll trade untl 11.18 becomes an equalty ) when 11.18 s an equalty, the portolo s ecent and can t be mproved by buyng or sellng any asset E R r SDR CorrR, R E E RE SD R r E (11.A)
Chapter 11: Optmal Portolo Choce and the CAPM-18 3) I rearrange 11.A and dene a new term, the ollowng must hold n equlbrum E R r r ER E E r (11.1) where: E SD R CorrR, RE SDRE (11.B) r = requred return on = expected return on necessary to compensate or the rsk the assets adds to the ecent portolo V. The Captal Asset Prcng Model A. Assumptons (and where 1 st made smlar assumptons) 1. Investors can buy and sell all securtes at compettve market prces (Ch 3). Investors pay no taxes on nvestments (Ch 3) 3. Investors pay no transacton costs (Ch 3) 4. Investors can borrow and lend at the rsk-ree nterest rate (Ch 3). Investors hold only ecent portolos o traded securtes (Ch 11) 6. Investors have homogenous (same) expectatons regardng the volatltes, correlatons, and expected returns o securtes (Ch 11) Q: Why even study a model based on such unrealstc assumptons? 1) helpul smplcaton o realty => gan understandng o the way the world works ) startng pont => examne mpact o gettng rd o the unrealstc assumptons 3) works despte assumptons B. The Captal Market Lne 1. Basc dea: the market portolo must be the ecent portolo (hghest Sharpe rato) held by all nvestors Ratonale: 1) By assumpton, all nvestors have the same expectatons ) all nvestors wll denty the same rsky portolo (n terms o x ) as ecent
Chapter 11: Optmal Portolo Choce and the CAPM-19 3) all nvestors wll hold the same portolo (n terms o x ) 4) the combned portolos o all nvestors must be ecent ) the combned portolo o all nvestors s the market portolo. Captal Market Lne: Optmal portolos or all nvestors: nvest x n the market and (1-x) n the rsk-ree nvestment Graph #13: CML 1 Tangent Portolo or all nvestors = market x > 1 x < 1 Ecent Fronter o rsky assets or all nvestors 3 4 Volatlty C. Market Rsk and Beta I the market portolo s ecent, then the expected and requred returns on any traded securty are equal as ollows: E R r r ER Mkt r (11.) where: R CorrR, RMkt SDR Mkt SD Cov, RMkt (11.3) Var R Mkt R Mkt Notes: Mkt 1) substtutng or ) wll use rather than E Mkt and E[R Mkt ] or E[R E ] nto 11.1
Chapter 11: Optmal Portolo Choce and the CAPM- 3) rather than usng equaton 11.3, can estmate beta by regressng excess returns (actual returns mnus rsk-ree rate) on securty aganst excess returns on the market => beta s slope o regresson lne Ex. Assume the ollowng returns on JPM and the market. What s the beta o JPM? What s the expected and requred return on JPM the rsk-ree rate s 4% and the expected return on the market s 9%? Return on: Year JPM Market 1-1% -19% 7% -% 3 14% 17% 4-3% 4% 3% 7% 6 19% 18% R JPM Var R JPM SD R JPM 6. 66. 3 16. 3 => see pages 3 and 4 or these calculatons Cov R βjmp Var JPM R,R Mkt Mkt 1 RJPM, RMkt RJPM,t RJPM R MKT,t RMkt Cov T 1 t 1 R Mkt 4. 19 17 4 7 18 6 => Cov(R JPM,R Mkt ) = Var 1 1 6. 19 4. 7 6. 4. 14 6.17 4. 3 6.4 4. 3 6.7 4. 19 6.18 4. = 19.3 187.7 1 19 4. 4. 17 4. 4 4. 7 4. 18 4. R Mkt 19.3 => β JMP 1.13 187.7 E(R JPM ) = r JPM = 4 + 1.13(9 4) =9.6%
Chapter 11: Optmal Portolo Choce and the CAPM-1 D. The Securty Market Lne (SML) 1. Denton: graph o equaton 11.: ER r r ER Mkt r => lnear relatonshp between beta and expected (and requred) return Graph #14: SML 1% % E[Rmkt] Market % %.. 1. 1.. Beta. All securtes must le on the SML => expected return equals the requred return or all securtes Reason: => an asset s not on the SML, then the market portolo s no longer ecent => tradng wll push the asset back on to the SML and the market back to ecency => JPM wll le on the SML just above and to the rght o the market 3. Betas o portolos P x (11.4) Note: see Equaton (11.) on separatng out x
Chapter 11: Optmal Portolo Choce and the CAPM- Ex. Assume beta or JPM s 1.13 and that beta or GD s.19. What s beta o portolo where nvest $3, n JPM and $, n GD? x JPM =.7, x GD =. => P =.7(1.13) +.(.19) =.799