Outline. CAPM: Introduction. The Capital Asset Pricing Model (CAPM) Professor Lasse H. Pedersen. Key questions: Answer: CAPM

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1 The Catal Asset Prcng odel (CAP) Proessor Lasse H. Pedersen Pro. Lasse H. Pedersen 1 Key questons: Outlne What s the equlbrum requred return, E(R), o a stock? What s the equlbrum rce o a stock? Whch ortolos should nvestors hold n equlbrum? Answer: CAP Assumtons Results: Identy the tangency ortolo n equlbrum Hence, denty nvestors ortolos Derve equlbrum returns (and hence rces) Pro. Lasse H. Pedersen CAP: Introducton Equlbrum model that redcts otmal ortolo choces redcts the relatonsh between rsk and exected return underles much o modern nance theory underles most o real-world nancal decson makng Derved usng arkowtz s rncles o ortolo theory, wth addtonal smlyng assumtons. Share, Lntner and ossn are researchers credted wth ts develoment. Wllam Share won the Nobel Prze n Pro. Lasse H. Pedersen 3

Pedersen CAP: Introducton Equlbrum model that redcts otmal ortolo choces redcts the relatonsh between rsk and exected return underles much o modern nance theory underles most o real-world nancal

2 CAP Assumtons Stylzed Assumtons: 1. The market s n a comettve equlbrum;. Sngle-erod nvestment horzon; 3. All assets are tradable; 4. No rctons; 5. Investors are ratonal mean-varance otmzers wth 6. homogeneous exectatons Some assumtons can be relaxed, and CAP stll holds. An mortant aroxmaton o realty n any case. I many assumtons are relaxed, generalzed versons o CAP ales. (Toc o ongong research.) Pro. Lasse H. Pedersen 4 1: The market s n a comettve equlbrum Equlbrum: Suly Demand Suly o securtes s xed (n the short-run). I Demand > Suly or a artcular securty, the excess demand drves u rce and reduces exected return. (Reverse Demand < Suly) Comettve market: Investors take rces as gven No nvestor can manulate the market. No monoolst Pro. Lasse H. Pedersen 5 : Sngle-erod horzon All nvestors agree on a horzon. Ensures that all nvestors are acng the same nvestment roblem. Pro. Lasse H. Pedersen 6

) Pro. Lasse H. Pedersen 4 1: The market s n a comettve equlbrum Equlbrum: Suly Demand Suly o securtes s xed (n the short-run).

3 3: All assets are tradable Ths ncludes n rncle: All nancal assets (ncludng nternatonal stocks) Real estate Human catal Ths ensures that every nvestor has the same assets to nvest n: all the assets n the world, the market ortolo Pro. Lasse H. Pedersen 7 No taxes 4: No rctons No transacton costs (no bd-ask sread) Same nterest rate or lendng and borrowng All nvestors Investors can borrow or lend unlmted amounts. (No margn requrements.) Pro. Lasse H. Pedersen 8 5-6: Investors are ratonal meanvarance otmzers wth homogeneous exectatons Investors choose ecent ortolos consstent wth ther rsk-return reerences Investors have the same vews about exected returns, varances, and covarances (and hence correlatons). Pro. Lasse H. Pedersen 9

Pedersen 7 No taxes 4: No rctons No transacton costs (no bd-ask sread) Same nterest rate or lendng and borrowng All nvestors Investors can borrow or lend unlmted amounts.

4 What s the Equlbrum Tangency Portolo? Recall rom ortolo theory: All nvestors should have a (ostve or negatve) racton o ther wealth nvested n the rsk-ree securty, and The rest o ther wealth s nvested n the tangency ortolo. The tangency ortolo s the same or all nvestors (homogeneous exectatons). In equlbrum, sulydemand so: the tangency ortolo must be the ortolo o all exstng rsky assets, the market ortolo!! Pro. Lasse H. Pedersen 10 The arket Portolo rce o one share o rsky securty n number o shares outstandng or rsky securty arket Portolo. The ortolo n whch each rsky securty has the ollowng weght: n ω n market catalzaton o securty total market catalzaton In words, the market ortolo s the ortolo consstng o all assets (everythng!). However, you also nvest n the market ortolo you buy a ew shares o every securty, wegthed by ther market ca. Pro. Lasse H. Pedersen 11 The Catal arket Lne (CL) Recall: The CAL wth the hghest Share rato s the CAL wth resect to the tangency ortolo. In equlbrum, the market ortolo s the tangency ortolo. The market ortolo s CAL s called the Catal arket Lne (CL). The CL gves the rsk-return combnatons acheved by ormng ortolos rom the rsk-ree securty and the market ortolo: E( R ) R [ E( R ) R + Pro. Lasse H. Pedersen 1

The tangency ortolo s the same or all nvestors (homogeneous exectatons). In equlbrum, sulydemand so: the tangency ortolo must be the ortolo o all exstng rsky assets, the market ortolo!! Pro. Lasse H.

5 The E(R)-SD Fronter and The Catal arket Lne E[R R Pro. Lasse H. Pedersen 13 The Requred Return on Indvdual Stocks CAP s most amous or ts redcton concernng the relatonsh between rsk and return or ndvdual securtes: E[ R R + β [ E[ R R cov[ R, R where β The model redcts that exected return o an asset s lnear ts beta. Ths lnear relaton s called the Securty arket Lne (SL). The beta measures the securty s senstvty to market movements. Pro. Lasse H. Pedersen 14 Dervng CAP Equaton usng FOC The market ortolo s the tangency ortolo and thereore t solves: E( R ) R max SR w1,... wn R where E( R ), j w E( R ) + (1 The rst-order condton (FOC) s: 0 SR that s, w w w 0 (E( R ) R ) j w w cov( R, R ) Pro. Lasse H. Pedersen 15 j w ) R cov( R, R ) ( E( R ) R )

The model redcts that exected return o an asset s lnear ts beta. Ths lnear relaton s called the Securty arket Lne (SL). The beta measures the securty s senstvty to market movements. Pro.

6 Securty arket Lne (SL) Securtes wth β>1 E(R) Securtes wth 0<β<1 R5% E(R) SL Beta coecent β 1 β Pro. Lasse H. Pedersen 16 E(R)-SD Fronter and the Betas The grah relates the locaton o the ndvdual securtes wth resect to the -SD ronter to ther betas CL E(R)-SD Fronter 0. arket Portolo Exected Return 0.15 β>1 β R 0<β<1 β<0 β0 0 Pro. Lasse H. Pedersen Standard Devaton The Catal arket Lne and the Securty arket Lne E[r E[r Q E[rK r E[r E[r Q E[rK r 1 E[r r E[r r Pro. Lasse H. Pedersen 18

Pedersen 16 E(R)-SD Fronter and the Betas The grah relates the locaton o the ndvdual securtes wth resect to the -SD ronter to ther betas. 0.3 0.

7 Systematc and Non-Systematc Rsk The CAP equaton can be wrtten as: R R + β ( R R ) + error cov[ R, R where β E( error ) 0 cov[ error, R 0 Ths mles the total rsk o a securty can be arttoned nto two comonents: { β + { 13 var( R ) market var( error ) total rsk dosyncratc rsk rsk Pro. Lasse H. Pedersen 19 Systematc and Non-Systematc Rsk: Examle ABC Internet stock has a volatlty o 90% and a beta o 3. The market ortolo has an exected return o 14% and a volatlty o 15%. The rskree rate s 7%. What s the equlbrum exected return on ABC stock? What s the roorton o ABC Internet s varance whch s dversed away n the market ortolo? β + (0.9) ( ) Hence 75% o varances dversed away (0.9) Pro. Lasse H. Pedersen 0 Systematc and Non-Systematc Rsk β measures securty s contrbuton o to the total rsk o a well-dversed ortolo, namely the market ortolo. Hence, β measures the non-dversable rsk o the stock Investors must be comensated or holdng non-dversable rsk. Ths exlans the CAP equaton: E(R ) R + β [ E(R ) - R, 1,,N Pro. Lasse H. Pedersen 1

Pedersen 19 Systematc and Non-Systematc Rsk: Examle ABC Internet stock has a volatlty o 90% and a beta o 3. The market ortolo has an exected return o 14% and a volatlty o 15%. The rskree rate s 7%.

8 Rsk Premum Recall the SL: E(R ) R + β [ E(R ) - R Stock s systematc or market rsk s: β Investors are comensated or holdng systematc rsk n orm o hgher returns. The sze o the comensaton deends on the equlbrum rsk remum, [ E(R ) - R. The equlbrum rsk remum s ncreasng n: 1. the varance o the market ortolo. the degree o rsk averson o average nvestor Pro. Lasse H. Pedersen Estmatng Beta An Examle: any nsttutons estmate beta s, e.g.: Bloomberg errll Lynch Value Lne Yahoo Pro. Lasse H. Pedersen 3 Estmatng Beta by Lnear Regressons (OLS) [ CAP equaton: E[R R β E[R m R Get data on excess returns : R e (t) R (t) R R e (t) R (t) R where R s the rsk-ree rate rom tme t-1 to tme t. Estmate β by runnng the regresson: e e R ( t) α + β R ( t) error ( t) m + Tycally, 60 months o data are used. Pro. Lasse H. Pedersen 4

Lasse H. Pedersen Estmatng Beta An Examle: any nsttutons estmate beta s, e.g.: Bloomberg errll Lynch Value Lne Yahoo Pro. Lasse H.

9 Securty Characterstc Lne (SCL) The SCL s the regresson lne : R ( t) R α + β ( R ( t) R ) error ( t) + Note: CAP mles α 0 r r rm r Pro. Lasse H. Pedersen 5 Estmatng Beta: Real Le Examle, AT&T Take 5 years ( ) o monthly data on AT&T returns, S&P500 returns and 1 month US T-blls. Construct excess returns Run the regresson, or nstance usng Excel: aly Tools, Add-ns, Analyss ToolPak use Tools, Data Analyss, Regresson The result s n the sreadsheet betareg.xls Excel Regresson outut: Coecents SE t Stat P-value Intercet X Varable Pro. Lasse H. Pedersen 6 Estmatng Beta: Real-Le SCL or AT&T AT&T vs arket Return AT&T excess return Actual AT&T Excess Returns Predcted AT&T Excess Returns Excess market return Pro. Lasse H. Pedersen 7

Construct excess returns Run the regresson, or nstance usng Excel: aly Tools, Add-ns, Analyss ToolPak use Tools, Data Analyss, Regresson The result s n the sreadsheet betareg.

10 Alcatons o the CAP Portolo choce Shows what a ar securty return s Gves benchmark or securty analyss Requred return used n catal budgetng to comute NPV o rsky roject or hurdle rate or IRR Evaluaton o und manager erormance. Pro. Lasse H. Pedersen 8 Stock Selecton and Actve anagement One ossble benchmark or stock selecton s to nd assets that are chea relatve to CAP (or more advanced models). A securty s alha s dened as: α E[ R R cov[ R, R where β β [ E[ R R Some und managers try to buy ostve-alha stocks and sell negatve-alha stocks. CAP redcts that all alha s are zero. Pro. Lasse H. Pedersen 9 Stock Selecton 1.40% 1.0% 1.00% 0.80% 0.60% } Overrced α B B ˆ >0 α A ˆ <0 A SL Underrced { 0.40% R 0.0% 0.00% Beta Pro. Lasse H. Pedersen 30

Pedersen 8 Stock Selecton and Actve anagement One ossble benchmark or stock selecton s to nd assets that are chea relatve to CAP (or more advanced models).

11 Actve and Passve Strateges An actve strategy tres to beat the market buy stock ckng, by tmng, or other methods But, CAP mles that securty analyss s not necessary every nvestor should just buy a mx o the rsk-ree securty and the market ortolo, a assve strategy. Pro. Lasse H. Pedersen 31 Summary The CAP ollows rom equlbrum condtons n a rctonless mean-varance economy wth ratonal nvestors. Predcton 1: Everyone should hold a mx o the market ortolo and the rsk-ree asset. (That s, everyone should hold a ortolo on the CL.) Predcton : The exected return on a stock s a lnear uncton o ts beta. (That s, stocks should be on SL.) The beta s gven by: cov[ R, R β A stock s beta can be estmated usng hstorcal data by lnear regresson. (That s, by estmatng the SCL.) Pro. Lasse H. Pedersen 3

Predcton 1: Everyone should hold a mx o the market ortolo and the rsk-ree asset. (That s, everyone should hold a ortolo on the CL.

Outline. Investment Opportunity Set with Many Assets. Portfolio Selection with Multiple Risky Securities. Professor Lasse H.

Outline. Investment Opportunity Set with Many Assets. Portfolio Selection with Multiple Risky Securities. Professor Lasse H. Portfolo Selecton wth Multple Rsky Securtes. Professor Lasse H. Pedersen Prof. Lasse H. Pedersen Outlne Investment opportunty set wth many rsky assets wth many rsky assets and a rsk-free securty Optmal