Chapter 9: Factor pricing models. Asset Pricing Zheng Zhenlong

Transcription

1 Chaper 9: Facor prcng models Asse Prcng

2 Conens Asse Prcng Inroducon CAPM ICAPM Commens on he CAPM and ICAPM APT APT vs. ICAPM

3 Bref nroducon Asse Prcng u β u ( c + 1 ) a + b f + 1 ( c )

4 Bref nroducon Asse Prcng More drecly, he essence of asse prcng s ha here are specal saes of he world n whch nvesors are especally concerned ha her porfolos no do badly. The facors are varables ha ndcae ha hese bad saes have occurred. Any varable ha forecass asse reurns ( changes n he nvesmen opporuny se ) or macroeconomc varables s a canddae facor. Such as :erm premum, dvdend/prce rao, sock reurns

5 Should facors be unpredcable over me? Asse Prcng Facors ha proxy for margnal uly growh, hough hey don have o be oally unpredcable, should no be hghly predcable. If one chooses hghly predcable facors, he model wll counerfacually predc large neres rae varaon. ( c ) u 1 u ( c ) = R E [ u ( c )] = + f + 1 β + 1 ε f + 1 u ( c ) β R In pracce, hs consderaon means ha one should choose he rgh uns: Use GNP growh raher han level, porfolo reurns raher han prces or prce/dvdend raos, ec.

6 The dervaons of facor prcng model Asse Prcng Deermne one parcular ls of facors ha can proxy for margnal uly growh Prove ha he relaon should be lnear. Remark: all facor models are derved as specalzaons of he consumpon-based model.

7 guard agans fshng Asse Prcng One should call for beer heores or dervaons, more carefully amed a lmng he ls of poenal facors and descrbng he fundamenal macroeconomc sources of rsk, and hus provdng more dscplne for emprcal work.

8 Capal Asse Prcng Model (CAPM) Asse Prcng W R 1 m = a br W wealh porfolo reurn. In expeced reurn / bea language, E R = γ + β, R W + ( ) ( ) [ γ ] W E R CAPM can be derved from consumpon-based model by dfferen assumpon.

9 Dfferen assumpon Asse Prcng 1) wo-perod quadrac uly 2) exponenal uly and normal reurns, 3) Infne horzon, quadrac uly and..d. reurns 4) Log uly. Same assumpon: no labor ncome

10 Two-perod quadrac uly,no labor ncome Asse Prcng Invesors have quadrac preferences and only lve wo perods, U ( ) ( ) 2, ( c c = c c βe c c ) margnal rae of subsuon s hus m [ ] ( c + 1 ) ( ) + 1 ( c c ) + 1 ( ) u = β = β u c c c

11 Asse Prcng he budge consran s c + 1 = W +1 W W + 1 = R + 1 ( W c ) R N N W + = w R = 1 = 1 w = 1

12 Asse Prcng m ( ) ( c c ) ( ) c R W c β W c W + 1 β c W + 1 = β = R + 1 c c c c Jus as m = a b R W

13 Exponenal uly, normal dsrbuons, no labor ncome Asse Prcng If consumpon only n he las perod and s normally dsrbued, we have E [ ()] [ ] ac u c = E e a s he coeffcen of absolue rsk averson. E éu( c) ù ê ú =-e ë û 2 2 ( /2) s ( ) - ae écù êë úû + a c

14 Asse Prcng he budge consran s c = y R + y f f R W = y + y f 1

15 Asse Prcng E éu () c ù ê ú = -e ë û ù ( ) ú ( 2 /2) é f f ê ë û å - a y R + y E R + a y y y = 1 E ( R ) a R f f ( ) ( W R R = a y = a cov R R ) E, ( W ) f = σ 2 ( W ) E R R a R

16 Quadrac value funcon, dynamc programmng U ( c ) + β E V ( W ) = u +1 Asse Prcng frs order condon So, p u ( c ) = βe [ V ( W ) x ] m + 1 = β V u ( W + 1 ) ( c )

17 Asse Prcng suppose he value funcon were quadrac, Then, m V η 2 ( W ) = ( W W ) η W ( W c ) ( ) W + 1 = β + R + 1 u ( c) u c Some addon assumpons: The value funcon only depends on wealh. The value funcon s quadrac. I needs he followng assumpons: he neres rae s consan, reurns are d, no labor ncome. βη

18 he exsence of value funcon (Proof ) Asse Prcng Suppose nvesors las forever, and have he sandard sor of uly funcon ( c ) U = j E β u + j = 0 Defne he value funcon as he maxmzed value of he uly funcon n hs envronmen. V ( W ) j max E ( ) { } β u c + j c c..., w w..., + 1, + 1 j j= 0

19 Value funcons allow you o express an nfne perod problem as a wo perod problem Asse Prcng

20 Why s he value funcon quadrac? Asse Prcng Remark: quadrac uly funcon leads o a quadrac value funcon n hs envronmen Specfy: Guess: V u ( ) ( c = 0.5 c c ) 2 ( ) ( W = 0. η W W ) Thus, V ( W) = max 0.5 c c 0.5 E W W 2 2 ( ) ( ηβ ) + 1 { c } s.. W W + 1 = R + 1 ( W c )

21 Asse Prcng cˆ c = βηe R ( W c ) W R { W W } cˆ = c βη E W W 2 ( R + 1 ) W + βη E [ R + 1 ] W βη E [ R ] + 1 W

22 Asse Prcng V ( ) ( ) [ ( ) ] 2 W W = 0.5 cˆ c 0.5ηβE R W cˆ W 2 + 1

23 Log uly, no labor ncome Asse Prcng p W ( c ) u c β = 1 j + j j = E β c = E β c j u + = 1 j= 1 + j + j j ( c ) c β c R ( β ( β) ) /1 ( ) ( c ) ( ) p + c /1 + 1c c u 1 W W = = = = = W p β β c βc βu c+ 1 m+ 1

24 Asse Prcng Log uly has a specal propery ha ncome effecs offse subsuon effecs, or n an asse prcng conex ha dscoun rae effecs offse cash flow effecs.

25 How o lnearze he model? Asse Prcng The wn goals of a lnear facor model dervaon are o derve wha varables derve he dscoun facor, and o derve a lnear relaon beween he dscoun facor and hese varables. Ths secon covers hree rcks ha are used o oban a lnear funconal form. Taylor approxmaon he connuous me lm normal dsrbuon

26 Taylor approxmaon Asse Prcng The mos obvous way o lnearze he model s by a Taylor approxmaon m + 1 = g ( f + 1 ) g ( E ( f )) + g ( E ( f ))( f E ( f ))

27 Connuous me lm Asse Prcng If he dscree me s shor enough, we can apply he connuous me resul as an approxmaon d Λ E = g For a shor dscree me nerval, = dp p g 1 ( f ) Λ = g, + d ( f, ) D p + g g f d ( f, ) f 2 ( f, ) g 2 r f E d df = dp p + E df 0.5 f 1 g( f, ) E ( R ) R cov ( R, f )( ) β g( f, ) f dp p 2 df d Λ Λ f f , f; λ

28 Normal dsrbuon n dscree me Asse Prcng Sen s lemma : If f and R are bvarae normal, g(f) s dfferenable and E g f <,hen ( ) [ g ( f ), R] = E[ g ( f )] cov( f, R) cov

29 Asse Prcng Remark: If m=g(f), f f and a se of he payoffs prced by m are normally dsrbued reurns, and f E g ( f ) <, hen here s a lnear model m=a+bf ha prces he normally dsrbued reurns.

30 Asse Prcng ( ) ( ( ) ) é ( ) ù ( ) cov é ( ), ù êë úû êë úû é ( ) ù ( ) é ( ) ù ê cov (, ë úû êë úû ) ({ é ( ) ù é ( ) ù( ( ))} ) ê ë úû êë úû ( ) ( ) ( ) ( ) ( ) ( ) p = E mx = E g f x = E g f E x + g f x = E g f E x + E g f f x = E E g f + E g f f - E f x ({ é ù é ù é ù } ) = E E êg f - E g f E f + E g f f x ë úû êë úû êë úû = E m x = E a + b f x ( )

31 Asse Prcng Smlar, allows us o derve an expeced reurn-bea model usng he facors ( ) f ( ) + 1 = cov + 1, + 1 E R R R m = R E g f R f f f, f ; ( ) ( ) + 1 cov + 1, + 1 f = R + β λ

32 Two perod CAPM Asse Prcng Sen s lemma allows us o subsue a normal dsrbuon assumpon for he quadrac assumpon n he wo perod CAPM. m + 1 W u ( c 1) u + ( R+ 1( W c)) = β = β u ( c ) u ( c ) Assumng R W and R are normally dsrbued, we have: ( W c ) u [ R ( W c )] cov (, ) [ ]cov (, ) W + 1 W R+ 1 m+ 1 = E β R+ 1 R+ 1 u ( c )

33 Log uly CAPM Sen s lemma canno be appled o he log uly CAPM because he marke reurn canno be normally dsrbued. For log uly CAPM, g(f)=1/r W, so f 1 W E( R+ 1) = R + E( 2 )cov ( R 1, 1) W + R+ R If R W + 1 s normally dsrbued, E(1/R W2 ) does no exs. The Sen s lemma condon s volaed. Asse Prcng

34 Ineremporal Capal Asse Prcng Model (ICAPM) Asse Prcng The ICAPM generaes lnear dscoun facor models m + 1 = a + b f+1 n whch he facors are sae varables for he nvesor s consumpon-porfolo decson.

35 Asse Prcng he value funcon depends on he sae varables V W z so we can wre m ( ), VW + 1 = β V ( W + 1, z + 1 ) ( W, z ) W

36 Asse Prcng Sar from Λ = ( W z ) δ e V W, We have d Λ Λ + V V Wz W = δ d + ( W, z ) ( W, z ) dz W V V W +... ( W, z ) ( W, z ) WW dw W

37 Asse Prcng Defne he coeffcen of relave rsk averson, WV WW ( W, z ) rra = V W, z Then we oban he ICAPM, W ( ) E dp p + D p d r f d = rra E dp p dw W V V Wz, W, E dp p dz

38 Asse Prcng Thus, n dscree me ( ) ( ) , cov, cov Δ + Δ z f z R W W R rra R R E λ

39 9.3 Commens on he CAPM and ICAPM Asse Prcng

40 Is he CAPM condonal or uncondonal? Asse Prcng

41 Asse Prcng

42 Asse Prcng The log uly CAPM expressed wh he nverse marke reurn s a beauful model, snce holds boh condonally and uncondonally. There are no free parameers ha can change wh condonng nformaon. 1 1 = E = R E R W W + R + 1 R Fnally requres no specfcaon of he nvesmen opporuny se, or no specfcaon of echnology. However, he expecaons n he lnearzed log uly CAPM are condonal.

43 Should he CAPM prce opons? Asse Prcng he quadrac uly CAPM and he nonlnear log uly CAPM should apply o all payoffs: socks, bonds, opons, conngen clams, ec. However, f we assume normal reurn dsrbuons o oban a lnear CAPM, we can no longer hope o prce opons, snce opon reurns are non-normally dsrbued

44 Why boher lnearzng a model? Asse Prcng

45 Wha abou he wealh porfolo? Asse Prcng To own a (share of) he consumpon sream, you have o own no only all socks,bu all bonds, real esae, prvaely held capal, publcly held capal (roads, parks, ec.), and human capal. Clearly, he CAPM s a poor defense of common proxes such as he value-weghed NYSE porfolo.

46 Implc consumpon-based models Asse Prcng ( c ) u ( c ) m + 1 = βu + 1 /

47 Asse Prcng The log uly model also allows us for he frs me o look a wha moves reurns ex-pos as well as ex-ane. R W + 1 = Aggregae consumpon and asse reurns are lkely o be de-lnked a hgh frequences, bu how hgh (quarerly?) and by wha mechansm are mporan quesons o be answered. In sum, he poor performance of he consumpon-based model s an mporan nu o chew on, no jus a blnd alley or faled aemp ha we can safely dsregard and go on abou our busness. c βc + 1

48 Ideny of sae varables Asse Prcng The ICAPM does no ell us he deny of he sae varables z, leadng Fama (1991) o characerze he ICAPM as a fshng lcense. The ICAPM.

49 Porfolo Inuon and Recesson Sae Varables Asse Prcng The covarance (or bea) of wh measures how much a margnal ncrease n affecs he porfolo varance. Modern asse prcng sars when we realze ha nvesors care abou porfolo reurns, no abou he behavor of specfc asses. Tha s he cenral nsgh n CAPM. The ICAPM adds long nvesmen horzons and me-varyng nvesmen opporunes o hs pcure. People are unhappy when news comes ha fuure reurns are lower, hey wll hus prefer socks ha do well on such news. Mos curren heorzng and emprcal work, whle cng he ICAPM, really consders anoher source of addonal rsk facors: Invesors have jobs. Or hey own houses and shares of small busnesses. People wh jobs wll prefer socks ha don fall n recessons.

50 Arbrage Prcng Theory (APT) Asse Prcng The nuon behnd he APT s ha he compleely dosyncrac movemens n asse reurns should no carry any rsk prces, snce nvesors can dversfy hem away by holdng porfolos. Therefore, rsk prces or expeced reurns on a secury should be relaed o he secury s covarance wh he common componens or facors only.

51 Asse Prcng The APT models he endency of asse payoffs (reurns) o move ogeher va a sascal facor decomposon Defne So, x x M = a + j=1 ~ f β f + ε = a + β f + ε f j j E M ( f ) ( x ) + β f j + ε = E j = 1 j ~

52 Asse Prcng E ~ ( ) ε = 0 ; E ( ε f j ) = 0 E ( ε ε ) = 0 j j ( x x ) = E ( β )( ) f + ε β j f + ε j cov, β β σ ( f ) 2 = j + 2 σ ε f = 0 f j j

53 Asse Prcng Thus, wh N= number of secures, he N(N-1)/2 elemens of a varance-covarance marx are descrbed by N beas, and N+1 varances. Wh mulple (orhogonalzed) facors, we oban ( ) ( ) + = , cov σ σ σ β β f x x ( ) ( ) ( ) ( ) rx dagonalma f f x x = , cov σ β β β σ β

54 Asse Prcng If we know he facors we wan o use ahead of me, we can esmae a facor srucure by runnng regressons. If we don, we use facor analyss o esmae he facor model.

55 Exac facor prcng Asse Prcng x = E ( x )1 + b f p ( x ) E ( x ) p () p ( f ) ~ = 1 + β ~ E ( ) [ ( )] f f f R = R + β R p f = R + β λ

56 Approxmae APT usng he law of one prce Asse Prcng There s some dosyncrac or resdual rsk; we canno exacly replcae he reurn of a gven sock wh a porfolo of a few large facor porfolos. However, he dosyncrac rsks are ofen small. There s reason o hope ha he APT holds approxmaely, especally for reasonably large porfolos.

57 Asse Prcng Suppose Agan ake prces of boh sdes, ( ) f x E x ε β + + = ~ 1 ( ) ( ) () ( ) ( ) m E f p p x E x p ε β + + = ~ 1

58 Asse Prcng

59 Lmng argumens Asse Prcng ( x ) = var ( β f ) var ( ε ) var + var var ( ε ) ( x ) = 1 R 2

60 These wo heorems can be nerpreed o say ha he APT holds approxmaely (n he usual lmng sense) for eher porfolos ha naurally have hgh R 2, or well-dversfed porfolos n large enough markes. Asse Prcng

61 Law of one prce argumens fal Asse Prcng

62 Asse Prcng Remark: he effor o exend prces from an orgnal se of secures (f n hs case) o new payoffs ha are no exacly spanned by he orgnal se of secures, usng only he law of one prce, s fundamenally doomed. To exend a prcng funcon, you need o add some resrcons beyond he law of one prce.

63 he law of one prce: arbrage and Sharpe raos Asse Prcng The approxmae APT based on he law of one prce fell apar because we could always choose a dscoun facor suffcenly far ou o generae an arbrarly large prce for an arbrarly small resdual. Bu hose dscoun facors are surely unreasonable. Surely, we can rule hem ou.

64 Asse Prcng ( m ) = σ ( m ) + E ( m ) = ( m ) / R m = E σ + 1 f

65 Theorem Asse Prcng

66 Asse Prcng APT vs. ICAPM In he ICAPM here s no presumpon ha facors f n a prcng model = descrbe he covarance marx of reurns. The facors do no have o be orhogonal or..d. eher. Hgh n me-seres regressons of he reurns on he facors may mply facor prcng (APT), bu agan are no necessary (ICAPM). Facors such as ndusry may descrbe large pars of reurns varances bu no conrbue o he explanaon of average reurns. The bgges dfference beween APT and ICAPM for emprcal work s n he nspraon for facors. The APT suggess ha one sar wh a sascal analyss of he covarance marx of reurns and fnd porfolos ha characerze common movemen. The ICAPM suggess ha one sar by hnkng abou sae varables ha descrbe he condonal dsrbuon of fuure asse reurns.

67 Asse Prcng