EQUILIBRIUM ASSET PRICING MODELS

Transcription

1 EQUILIBRIUM ASSET PRICING MODELS 2 Asse pricing derived rom heory o consumpion and invesmen behavior 2 Pricing equaions oen ake he orm o PV models: 4 Asse value equals expeced sum o discouned uure CFs generaed by owning he asse 4 Alernaive heories imply dieren discoun raes 2 Two classes o pricing models 4 Preerence-based heories 4 Arbirage-based heories

2 2 Preerence-Based Models 4 Consider a represenaive invesor whose iniial wealh is w &1. Represens aggregae o many separae invesors operaing in he marke 6 allows preerence and wealh o characerize marke equilibrium. Invesor allocaes wealh over N risky asses and a risk-ree asse (1) w ' j n i'1 r i x i &1 % r 0 (w &1 & j n i'1 x i &1 ) r i = "gross" reurn on asse i rom -1 o x i &1 = invesmen in i a -1 r 0 = "gross" reurn on risk-ree asse Objecive: Max E U(w &1 ) s.. (1) E -1 = expecaion operaor condiional on inormaion a -1

3 u ) ' Mu(w) > 0, u )) ' M 2 u(w ) Mw Mw 2 <0 FOC ME &1 u(w ) Mu(w ) Mw Mx &1 ' E &1 Mw Mx &1 ' E &1 [u ) (w )(r & r 0 )] ' 0 Le y ' (r & r 0 ) = "excess reurn" E (y ) &1 = risk premium FOC: E &1 [u ) (w )y ] ' 0 y Assume join densiy o w and condiional on inormaion a -1 can be approximaed by a bivariae normal densiy.

4 Useul propery cov(x,y) ' E(xy) & E(x)E(y) Y E(xy) ' E(x)E(y) % cov(x,y) Le x ' u ) (w )andy ' y FOC: E &1 [u ) (w )y ] ' E &1 [u ) (w )] E &1 [y ] % cov &1 (u ) (w ),y ) Useul propery : Sein s Lemma I x and y are bivariae normal and g(y) is a leas once diereniable in erms o y, hen cov(x,g(y)) ' E[g ) (y)]cov(x,y) Le x ' y and g(y) ' u ) (w ) FOC: E &1 [u ) (w )] E &1 [y ] % E &1 (u )) (w ))cov &1 (w,y ) ' 0

5 Rearranging, E &1 [y ] ' &E &1 [u )) (w )] E &1 [u ) (w )] Cov &1 (w,y ) This is he "undamenal equaion o asse valuaion" Leads o alernaive pricing heories depending on srucure placed on R.H.S. R.H.S Y risk premium

6 Implicaions or asse pricing Le = &E &1 [u )) (w )] E &1 [u ) (w )] Cov &1 (w,y ) = ime-varying risk premium FEAV: E &1 (y ) ' Remember r ' p % D P &1 p D = price o asse a. = cash low o a. Subsiuing y ' r & r 0 E &1 p % D P &1 & r 0 '

7 Rearranging E &1 (p ) % E &1 (D ) ' p &1 (r 0 % ) p &1 ' 1 [E (D (r 0 &1 ) % % ) E &1 (p )] moving orward one period p ' 1 [E r 0 %1 % (D %1 ) % E (P %1 )] %1 Subsiuing p &1 ' 1 E (D (r 0 % ) &1 ) % E &1 1 E (r 0 %1 % ) (D %1 ) % E (p %1 %1 ) Repeaedly subsiuing orward or expecaions p %i and using he law o ieraed L.I.E. : E &1 (E %j (D %j%1 )) ' E &1 (D %j%1 ) p &1 ' E &1 D r 0 % % E &1 D %1 (r 0 % )(r 0 %1 % %1 ) % ä

8 So asse value in any period is p %1 ' E D (r 0 %1 % %1 ) % D %2 (r 0 %1 % %1 )(r0 %2 % %2 ) % ä Remember r 0 ' 1 % k 0 p %1 ' E D (1 % k 0 %1 % %1 ) % D %2 (1 % k 0 %1 % %1 )(1 % k 0 %2 % %2 ) % ä Noe in he general model: k 0 %j is ime-varying %j is ime-varying In mos applicaions and ess: = %j k 0 %j ' k 0 (someimes relaxed) D %1 ' E (1 % k 0 % ) % D%2 (1 % k 0 % ) 2 % ä 8 8 (1 % k) (1 % k) 2 p This is he PV model we have been using in class.

9 2 Implicaions o class model hus ar: 4 Risk-ree reurn is consan over ime 4 Risk premium is consan over ime We have derived an equilibrium pricing model bu have no been explici abou wha he risk premium will look like in equilibrium. 2 Capial Asse Pricing Model (CAPM) Suppose invesor holds marke porolio 4 Marke porolio risk is sysemaic 4 Reurn on marke is only source o risk FEAV m : where y m E (y m &1 &1 ) ' &E [u )) &1 (w )] E [u cov &1 ) &1 (w )] (w,y m ) ' r m & r 0

10 Remember FEAV ' E (y &1 ) ' &E [u )) (w &1 )] E [u cov &1 ) &1 (w )] (w,y ) Subsiue FEAV M ino FEAV ) E (y &1 ) ' E (y m &1 ) cov (w,y &1 cov (w,y m &1 ) ) ' E (y m &1 ) cov (w,y &1 var (y m &1 ) Using y ' r & r 0 and y m ' r m & r 0 we ge E (r ) ' r 0 &1 % E &1 (r m & r 0 ) CAPM where ' cov (y m &1,y var (y m &1 ) )

11 Alernaively, using our earlier noaion E (k &1 ) ' k 0 % E &1 (k m & k 0 ) ' cov (k m &1,k var (k m &1 ) ) This is he CAPM We can rewrie he CAPM in he orm o a PV model using earlier approach. Le %j ' E [k m &1 %j & k o %j ] %j ). P %1 ' E D (1 % k 0 %1 % %1 ) % D %2 (1 % k 0 %1 % %1 )(1 % k 0 %2 % %2 ) % ä Oen assume k 0 %j ' k 0 and/or %j ' D %1 ' E (1 % k 0 % ) % D%2 (1 % k 0 % ) % ä % 2 8 k'k 0 %E (k m &k 0 ) P

12 This, again, is in he orm o he PV model we ve used in class (assumes consan discoun rae) P ' E D %1 (1 % k) % D %2 (1 % k) 2 % ä k j ' k 0 % E(k M & k 0 ) j 2 Discoun rae is deermined by CAPM (could allow or ime-varying reurns) 2 Expeced reurn rom holding an asse in equilibrium equals risk-ree reurn plus risk premium Risk premium equals expeced excess reurn on he marke imes he asses conribuion o porolio risk as measured by.

13 2 measures residual risk ha can be diversiied away by holding a porolio o asses. (measures relaive risk o asse o he porolio raher han he variance o he asse) Properies o he CAPM 2 Securiy Marke Line (graph o CAPM) E(k ) E(k m ) Ä A k 0 C Ä Ä B = 1 E(k ) ' k 0 % E(k m & k 0 )

14 RELATIONSHIP TO CML 2 All invesors hold marke porolio, hus w ' Marke value o individual asse Marke value o all asses 2 Porolio o a% in risky asse and (1 - a%) in he marke porolio E( r p ) ' ae( r ) % (1 & a)e( r m ) 1 ' a 2 2 p % (1 & a) 2 2 m % 2a(1 & a) 2,m 2 Change in E( r p ) and p ME( r p ) Ma ' E( r ) & E( r m ) M p Ma ' 1 2 a 2 2 % (1 & a) 2 2 m % 2a(1 & a),m & 1 2 x 2a 2 & 2 2 m % 2a 2 m % 2 m & 4a m

15 2 Sharpe and Liner insigh In equilibrium marke porolio has w percen inves in asse. Thus a is he "excess" demand or, which mus be zero in equilibrium. 2 Evaluaing ME(r p )/Ma and M p / Ma a a = 0 (no access demand) deermines equilibrium pricing relaionship a poin m. Ä m r 0 ME(r p ) Ma /0 a ' 0 ' E(r ) & E(r m ) M p Ma / 0 a ' 0 ' r,m & 2 m m

16 2 This gives he slope o he risk-reurn rade o a poin M. ME(r p )/Ma M p / Ma / 0a ' 0 ' E(r ) & E(r m ) ( m & 2 m )/ m 2 This mus equal slope o CML: E(r m ) & r 0 m E(r m ) & r 0 m ' E(r ) & E(r m ) ( m & 2 m )/ m Rearrange E(r ) ' r 0 % (E(r m ) & r 0 ) m / 2 m ' r 0 % (E(r m ) & r 0 ) (SML)

17 Noe he dierence rom CML E(k j ) CML MV Ä Ä M Ä Ä Ä Ä Ä Ä k 0 r p 2 Each poin on eicien se is a dieren porolio 2 Asses and porolios inside eicien se will lie on SML (no on CML) Reason no all variances o he asses reurn is o concern o he invesor only covariance risk is imporan remainder is diversiied away

18 Key poin: 2 Invesors will only pay o avoid covariance risk Toal risk = sysemaic risk + unsysemaic risk (marke) (asse speciic) k j ' a j % b j k m + j k m and j are independen 2 j ' b 2 j 2 m % 2 j 2 j can be diversiied away. 2 b j in linear regression is exacly j in CAPM OLS min 2 E ' E[k j & a j & b j k m & E(k j & a j & b j k m )] 2 ' E[(k j & E(k j )] & b j (k m & E(k m ))] 2 ' 2 j % b 2 j 2 m & 2b j cov(k j,k m )

19 M 2 Mb j ' 2b j 2 m & 2cov(k j,k m ) ' 0 b ( j ' cov(k j,k m ) 2 M / j So we can esimae j by simple OLS regression k j ' a j % b j k m % e j ˆb j is he esimae o j Example: Year Asse A Marke

20 E(k m ) '.154 k 0 '.04 (&bills) k a '&.1513 % k m % e a (1.66) (3.31) ˆ j ' E(k a ) '.04 % (.154 &.04)(1.906) (CAPM) =.257 or 25.7% 8 discoun rae

21 2 Measure o risk or individual asses is linearly addiive when he asses are added o a porolio p ' w 1 1 % w 2 2 % ä w n n (Prove using deiniion o porolio bea) 2 Sysemaic risk o porolio can be measured by beas o individual asses. 2 We could derive equilibrium pricing models or cases where invesors hold only a ew asses.

22 2 Exensions o CAPM 4 no riskless asse (can borrow and level a risk ree rae) E(k j ) ' E(k z ) % E(k m & k z ) j E(k z ) = expeced reurn on min-variance zero bea porolio CAPM doesn require risk-ree securiy (in pracice, requires use o shor sales o creae zero-bea porolio).

23 OTHER EXTENSION TO THE CAPM 2 Symmeric (bu non-normal) disribuions 2 Exisence o non-markeable asses 2 Coninuous ime rading (wih log normal reurns) 2 Sochasic risk-ree rae 2 Heerogeneous expecaions 2 Taxes 2 Consumpion beas

24 EMPIRICAL TESTS OF CAPM 2 Numerous aemps o es model (diicul) 2 Form o es: y j ' 0 % 1 j % j Some predicions: 4 0 = 0 4 only should explain risk 4 relaionship should be linear 4 ' E(k m 1 & k 0 ) 4 E( j ) ' 0 &n

25 Typical resuls 4 ˆ 0 ú 0 4 ˆ 1 <E(k m & k 0 ) (bu ˆ 1 >0) E(k) SML Ä Ä Ä Ä Ä Ä Ä Ä Ä Ä Ä Ä Ä Ä Ä Ä Ä ÄÄ Ä Ä Ä Ä Ä Ä Ä Ä Ä Ä Ä Ä Ä Ä 4 ( ) 2 adds lile o explain reurns 4 Facors oher han useul in explaining reurns 3 P/E 3 Firm size 3 Dividend yield 4 Land Values: E(, &n ) ú 0

26 2 Theoreical CAPM useul in undersanding price behavior doesn' always agree wih realiy 2 Empirical Marke Line k j ' ˆ 0 % ˆ 1b j % e j 4 does good job o modeling reurns in many cases 4 diicul o obain unbiased esimaes o 0 and 1 2 Roll's Criique 4 Convincing argumen ha cross-secional daa ess o CAPM can' be done wih expos daa 4 ime-series ess sill possible

27 ARBITRAGE PRICING THEORY (APT) CAPM suggess asse reurn is linear uncion o a single "acor," he rae o reurn on he marke porolio Implicaion is ha all invesors hold he asses in same proporions in heir porolios No rue, bu many invesors do Y hold diversiied porolios broad-based marke index represen many invesors porolios Empirical evidence sugges reurn on mos asses is sysemaically relaed o oher "marke acors" besides he reurn on he marke porolio Accouning or hese acors improve predicion o reurns relaive o CAPM

28 2 APT makes "linear acor" assumpion r ' E ( r &1 ) % F 1 1 % F 2 2 % ä % F n n % F k = zero mean sochasic reurn rom acor k 3 common o all asses under consideraion k = sensiiviy o reurn o change in F k = zero mean reurn speciic o 4 Facors no explicily ideniied by APT 3 Characerisic economic evens in macro economy or indusry which end o have sysemaic impac on asse reurns in an economy or indusry 4 Remaining "non-sysemaic" deerminans o reurn capured by

29 STRUCTURE OF LINEAR FACTOR MODEL Example: Facor 1 = inlaion surprise Facor 2 = GNP surprise Facor 3 = ineres rae surprise Suppose E(inlaion) = 5% E(GNP) = 2% E( ineres rae) = 0% I ' 2 G ' 1 r '&1.8 e.g. 1% increase inlaion Y 2% increase asse s reurn Suppose during he year inlaion is 7%, GNP rises by only 1%, and ineres raes all by 2%. Eecs o sysemaic risks on he asse s reurn F I = =.02 F G = = -.01 F r = -.02

30 M = I F I + G F G + r F r M = [2 (.02)] + [1 (-.01)] + [-1.8 (-.02)] M =.066 or 6.6% Also, suppose here was "good" news abou which boosed reurns 5% above he expeced 4% reurn or he year. So he acual reurn or he year was k ' E(k ) % [ I F I % G F G % r F r ] % ' 4% % 6.6% % 5% ' 15.6%

31 DIVERSIFYING AWAY NON-SYSTEMATIC COMPONENTS Consider a 1 aser model where k j ' k j % j F % j Le k p ' w 1 k 1 % ä % w n k n ' w 1 ( k 1 % 1 F % 1 ) % ä w n ( k n % n F % n ) Suppose w j ' 1 n k p ' k % ä k n n % B 1 % ä B n n F % 1 n % 2 n % ä n n As n ges big k p ' k p % B p F

32 2 Asse speciic acors ( ) can be diversiied away 4 invesors don need o hold marke porolio or even he same porolio Y hold enough asses o reduce non-sysemaic risk o insigniican level 2 Aer eliminaing non sysemaic risk we are le wih risk associaed wih he random sysemaic acors. Turns ou we can eliminae sysemaic risk associaed wih each acor by correcly choosing amoun o each asse o hold in he porolio. Leaves a porolio ree o risk which mus earn risk-ree reurn. Leads o APT: E (r ) ' r 0 % Ø 1 1 % &1 ä Ø n n Ø k = risk premium or k h acor

33 2 APT based sricly on arbirage argumens 4 no risk preerence assumpions 4 no disribuional assumpions Example: 2 asses, 1 acor. Suppose asse speciic risk or each acor has been diversiied away. Single acor model o reurns r ' E &1 (r ) % 1 F 1 r g ' E &1 (r g ) % g 1 F 1 Reurns sill risky. Suppose we hold porolio r p ' qr % (1 & q)r g q = proporion invesed in

34 r p ' [E &1 (r g ) % q(e &1 (r ) & E &1 (r g ))] %[ g 1 F 1 % q( 1 & g 1 )F 1 ] Now choose q so ha sysemaic risk associaed wih F 1 is eliminaed se q '& g 1 ( 1 & g 1 ) r p ' E &1 (r g ) % & g 1 ( 1 & g 1 ) E &1 (r ) & E &1 (r g ) Now r p is non-sochasic and mus earn risk-ree reurn. Seing r p ' r 0 and rearranging [ 1 & g 1 ][E &1 (r g ) & r 0 ] ' g 1 [E &1 (r ) & E &1 (r g )] Add and subrac r 0 o R.H.S. and rearrange (E &1 (r g ) & r 0 ) g &1 ' (E(r ) & r 0 ) &1 &1

35 L.H.S. = sandardized risk premium or g R.H.S. = sandardized risk premium or APT implies sandardized risk premiums are equal across all asses or acor 1. Le his sandardized risk prem. or acor 1 equal Ø 1, hen E (r &1 ) & r 0 &1 ' Ø 1 E (r g &1 ) & r 0 g 1 ' Ø 1 So E &1 (r ) ' r 0 % Ø 1 1 E &1 (r g ) ' r 0 % Ø 1 g 1 We can generalize o n acors.

36 2 Noe in single acor case, i 1 ' cov (y m &1,y ) var &1 (y m ) Ø 1 ' E &1 (r m & r 0 ) hen E (r &1 ) ' r 0 % E &1 (r m & r 0 ) (CAPM) CAPM is special case o APT wih one acor where he acor is he marke porolio.

37 APT leads o asse pricing model as beore. Le ' Ø 1 1 % ä Ø n n be risk premium rom APT. E &1 (r ) ' r 0 % Subsiuing prices and cash lows in or r and solving as beore gives. P %1 ' E D (r 0 %1 % %1 ) % D %2 (r %1 % %1 )(r0 %2 % %2 ) % ä Seing r 0 %j ' 1 % k 0 and %j ' D %1 ' E (1 % k 0 % ) % D%2 (1 % k 0 % ) % ä 2 P Same PV model as we have used. Discoun rae depends on risk-ree reurn plus risk premium associaed wih non-diversiiable common acors.

38 BASIC APPROACH TO ESTIMATING APT MODELS Sep 1: Ideniy Facors Sep 2: Esimae risk premium or each acor (Ø) k p & k 0 ' a % bg i % p where G i = acor i (no zero mean) ˆØ ' ˆbḠ i Sep 3: Esimae Facor sensiiviies k ' c % 1 F 1 % ä n F n % 4 Some economeric concerns 4 Facor ideniicaion issues 4 Srong explanaory power