σ σ σ = = im i im is constant and then βi If is constant and ρim R i -rf σ ρ If i = M-V efficient portfolio p then Rp = = =

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1 F8000 Valuaton of Fnancal Assets Srng Seester 2010 Dr. Isabel Tkatch Assstant Professor of Fnance Today The Catal Asset Prcng Model Catal allocaton n rsky assets one rsk free asset CAPM equlbru β - A new easure of rsk Market Effcency next te The Catal Asset Prcng Model Shar (1968), Black (1969) and Lntner (1970) A odel that t tells us the far (rsk- adjusted) exected return for every ndvdual asset A arket equlbru odel The Catal Asset Prcng Model (CAPM): Outlne The assutons of the odel The arket equlbru: SML equaton The two coonents of rsk: Systeatc (non-dversfable) Non-systeatc (dversfable) Beta as a easure of systeatc rsk The returns and the rces of rsky assets The Catal Asset Prcng Model (CAPM): Assutons There are any nvestors each nvestor s a rce taker All nvestors lan for one dentcal holdng erod All rsky assets are ublcly traded All nvestors are rsk-averse and Mean-Varance otzers Hoogeneous exectatons - all nvestors have the sae nforaton and nterret t the sae way The CAPM: Assutons The eect arket assuton There are no taxes or transacton costs or nforaton costs There are no frctons Stocks can be bought and sold n any quantty (even fractons) There s one rsk-free asset and all nvestors can borrow or lend at that rate 1

2 The Market Portfolo n the - Plane The Catal Market Lne: = +[( -) / ] CAPM: Market Equlbru All the nvestors wll nvest n the sae ortfolo of rsky assets: - the arket ortfolo. The rsk references of the nvestors wll result n ther catal allocaton between the arket ortfolo and the rsk-free asset.e. the locaton of ther ortfolo on the Catal Market Lne (CML). (The searaton / utual fund theore) Passve Investent Strateges n the - Plane The CML: = + [( -) / ] q CAPM: Market Equlbru The rsk reu of each rsky assets wll be roortonal to the rsk reu of the arket ortfolo and to the beta coeffcent of the rsky asset: [ ] = β Catal Asset Prcng Model: The Securty Market Lne (SML) The SML: = + [ -] β q What s Beta? Beta s a easure of rsk Beta easures how senstve are the returns of asset to the returns of the arket ortfolo Beta s the sloe (coeffcent) n the regresson of asset s return (rsk reu) on the arket s return (arket rsk reu) Beta s a relatve easure of rsk Beta < 1 : defensve asset Beta = 1: neutral asset Beta > 1: aggressve asset 2

3 Beta Calculatng Beta R - β ρ β = = 2 β What Affects Beta Rsk? ρ ρ = = = 2 2 If s constant and ρ then β If ρ s constant and then β The CAPM Equlbru: Outlne of the Proof The rsk-free asset s on the SML Calculate the beta of the rsk-free asset The arket ortfolo s on the SML Calculate the beta of the arket ortfolo Any M-V effcent ortfolo s on the SML Calculate the beta of an effcent ortfolo Any rsky asset s on the SML Portfolos on the SML If = then ρ = ρ =0 and β = β = 0 :, = + ( ) β = + ( ) 0= If = then β = β = 1: = + ( ) β = + ( ) 1= If = M-V effcent ortfolo then R = (1 α) + αr, ρ ( α ) 1 ρ = ρ = 1 and β = β = = = α. Therefore, f the CML holds we can show that the SML also holds: = + = + ( ) α = + ( ) β Project Valuaton Exale Fr XYZ usually nvests n rojects wth a rsk level of β=0.8. It s consderng an nvestent n a new roject whch s exected to roduce a CF of $12.6M a year fro now, and ths CF s exected to grow at a constant rate of 2% er year forever. Ths CF s only an exectaton and the fr s econost estates t s Std to be $3M. What s the resent value of the CFs of ths roject, f the exected annual return of the arket ortfolo s 12%, the annual return of oney arket nstruents s 4% and the arket s n equlbru (CAPM)? (k = 10.4%; PV = $150M) 3

4 Benefts of Dversfcaton Dversfable Rsk Systeatc Rsk Rsk Coonents The rsk of any rsky asset has two coonents D - The dversfable (non-systeatc, dosyncratc, fr-secfc) rsk can be elnated by addng assets to the ortfolo ND - The systeatc t (non-dversfable, f arket) rsk can not be elnated through dversfcaton Accordng to the CAPM, nvestors are coensated only for the systeatc coonent of the total asset rsk ( ND ). Rsk Coonents n the - Plane CAPM Equlbru: Rsk and Return n the -β Plane The CML The SML: = + [ -] β = =, ND D β =β β =1 Project Valuaton Exale Joseh s lookng for a treasure sh n the Medterranean sea. He lans to kee lookng for a year, and at the end of that year the value of hs fr wll be deterned by the outcoe of hs quest. The robablty of fndng the $25M treasure s only 10% but he s ore lkely to end u wth a saller catch of only $5M. Obvously, the outcoe of Joseh s quest s ndeendent of any acroeconoc rsks, but we know that the exected annual return of the arket ortfolo s 14%, t s Std s 22% and the annual return of oney arket nstruents s 6%. What s the value of Joseh s fr f the arket s n equlbru (CPAM)? (PV = $6.604M) CAPM Equlbru: Overrcng and Underrcng Asset j s underrced return s too hgh SML: = + [ -] β Asset k s overrced return s too low 4

5 Overrcng Relatve to CAPM Equlbru Prce Underrcng Relatve to CAPM Equlbru Prce SML: = + [ -] β SML: = + [ -] β CAPM Market j Market k CAPM β k β j The Return and the Current Prce: Inversely Related [ ] [ P + Dv ] 1 1 = = = E R k 0 A and B are two rsky stocks. An analyst found that they have the followng araeters: A =15% and β A =0.5; B =22% and β B =2. The rsk-free rate s =10% and the exected return of the arket ortfolo s =18%. Relatve to CAPM equlbru rces, whch stock s underrced and whch s overrced? (A s underrced; B s overrced) E P 1 Practce Probles BKM 7th Ed. Ch. 9: 1-2, 4-17, 21-28; 28; BKM 8th Ed. Ch. 9: 1-2, 4-17, CFA: