On the Derivation of the Black-Litterman Equation for Expected Excess Returns 1. Harald Bogner

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1 O th Drvato of th Black-Lttrma Equato for Epctd Ecss Rturs Harald Bogr Frst vrso: May 7 th 05 hs vrso: Octobr 8 th 05 h Black-Lttrma approach s a mthod to comb subjctv vws about th dstrbutos of pctd css rturs of portfolos cosstg of substs of avalabl assts, wth dstrbutos of rsk prmums mpld currt markt prcs for all avalabl assts udr a assumpto about pctd css rturs qulbrum. h mthod has tsvly b plad ad dscussd th ltratur. h followg prsts th approach thrfor oly a vry comprssd cutv summary styl. h focus of ths tt s stad o th drvato of a ma rsult of th Black-Lttrma modl, thr formula for pctd css rturs. h drvato of ths rsult has ot b gv th orgal paprs by Black ad Lttrma. Black ad Lttrma suggstd thr usg th hl md stmato mthod to drv th formula, or th Black-Lttrma approach 3 whch s oly vry brfly plad th appd of o of thr orgal paprs o th modl. 4 A dot aothr o of thr paprs rfrrd to a Baysa approach ad also a latr publcato by H ad Lttrma gav a ht, statg that th modl uss th Baysa approach to fr th assts pctd rturs, but thr dd provd a comprhsv drvato. 5 Svral proofs usg th Bays thorm hav b prstd by othr authors. 6 h drvato gv th followg dffrs from ths by rlyg plctly o gral rsults rgardg th proprts of th product of two dffrt dsts for th sam multvarat ormal radom vctor. For covc, a h quato rfrrd to hr s gv stp 8. of th appd o pag 4 to Global Portfolo Optmzato, by Fschr Black ad Robrt Lttrma, publshd th Facal Aalysts Joural, 48, 5, Sp/Oct 99, p ths papr ( th followg rfrrd to as Black/Lttrma 99) s also th ma sourc of th modl dscrpto gv ad otato usd hr. For a dtald ad comprhsv ovrvw s th blog by Jay Waltrs. 3 Black/Lttrma 99, p Ibd, appd, p.4 5 S dot Asst Allocato, h Joural of Fd Icom, Sptmbr 99, Fschr Black ad Rbrt B Lttrma ad pag of h Ituto Bhd Black-Lttrma Modl Portfolos by Guaglag H ad Robrt Lttrma, Avalabl at SSRN: or - ad th followg rfrrd to as H/Lttrma. h abov ad all followg ol rfrcs wr rtrvd o May 7 th Aga, for a comprhsv ovrvw s Black/Lttrma 99. corrspodg author harald.bogr@lv.d

2 proof of ths rsults that follows gv rfrcs, ad rls o th formato form 7 for ormal dsts, s show th appd. 8 Markt-mpld rsk prmums h Black-Lttrma mthod s basd o th assumpto, that from th vstor prspctv, ot oly assts css rturs,.. rturs aftr dductg th rsk-fr rat, but also rsk prmums,.. pctd css rturs, ar radom mor prcsly: th rsk prmums, th followg wrtt as radom vctor M, wth μ bg th ralzato of M,.. th tru rsk prmums vctor, hav a multvarat ormal dstrbuto. hs radomss of rsk prmums ca b cosdrd as th rsult of markt prcs fluctuatg aroud qulbrum prcs. By assumg th markt was currtly a CAPM qulbrum, ad wth (a stmat for) th markt prc of rsk (aka th markt rsk avrso paramtr) ad a kow covarac matr of css rturs Σ, qulbrum rsk prmums ca b drvd,.. rsk prmums that would b th tru rsk prmums f th markt was qulbrum at currt prc lvls hs vctor of qulbrum rsk prmums s th followg wrtt as π Black ad Lttrma suggst to us ths vctor as th ma vctor of a markt-mpld dstrbuto of rsk prmums, wth a covarac matr that s proportoal to th covarac matr of css rturs,.. vry covarac btw two rsk prmums (cludg th varacs) s th covarac of th corrspodg css rturs tms a costat. - For th markt-mpld dstrbuto, th covarac matr of rsk prmums s hc wrtt as: τ I summary, whl th tru rsk prmums ar ukow, as o possbl modl o ca stmat a multvarat ormal dstrbuto for th vctor of rsk prmums wth a ma corrspodg to rsk prmums mpld currt markt prcs udr a qulbrum assumpto (th qulbrum rsk prmums), ad a covarac matr proportoal to th (stmatd) covarac matr of css rturs: I. M ~ N, 7 O could crtaly argu, that th gral drvato of th formato form from th momt form s part of th proof, ad dpdg how dtald o formulats ths stps, t may actually ot b much shortr tha th proofs gv othr sourcs. Howvr, v vwd lk ths, t may stll provd a altratv structur that may cotrbut to a comprhsv udrstadg of th Black-Lttrma rsult. Aga for covc ad rlyg o hts a rfrc gv thr, scto A.I of th appd shows how a dsty fucto wrtt formato form ca b drvd from a dsty fucto wrtt momt form. 8 Not that th last part of th appd, th rwrtg of th scalg factor, s gv oly for th sak of compltss, t s ot rlvat th cott of th Black-Lttrma quato. 9 h qulbrum Black/Lttrma rfr to s a tso of th Sharp-Ltr-Moss-ryor-CAPM, wth th currcy markt also bg qulbrum,.. thr s a qulbrum lvl of currcy hdgg, as dscrbd a 989 papr ttld Uvrsal Hdgg: Optmzg Currcy Rsk ad Rward Itratoal Equty Portfolos by Fschr Black ad drvd Equlbrum Echag Rat Hdgg, a 989 workg papr by th sam author. 0 hs rqurs th stmato of a rsk avrso paramtr for th markt (aka markt prc of rsk s th corrspodg scto th post o Froot/St ad ryor/black ths blog ad th rfrcs gv thr), s H/Lttrma pag 3.

3 hs wll th followg b rfrrd to as markt-mpld or followg H ad Lttrma pror dstrbuto for rsk prmums. Rsk prmums mpld th vstor s subjctv vws Bsds th markt-mpld dstrbuto, a vstor may also hav dvdual subjctv vws o o or mor lar combatos of th assts rsk prmums.. o rsk prmums of portfolos (cosstg of log ad short postos). Black ad Lttrma prss ths vws as follows: - P s a k matr of wghts thos portfolos spcfd by th vstor, whr k s th total umbr of vws, s th umbr of assts th markt - h vws hold for th tru vctor of asst rsk prmums,.. th ralzato of M, ad μ ar hc th valus pctd by th vstor for th css rturs of th portfolos o whch th vstor has a vw o. Whl th vstor dos ot kow th tru rsk prmum for ay of th portfolos, thy kow a modl of thr dstrbuto, codtoal o : II. μ q ε whr q s a -dmsoal (colum) vctor of costats, kow by th vstor. h lmts of th vctor ε ar radom ad ormally dstrbutd wth ma zro ad a dagoal covarac matr,..: ~ N 0, III. μ~n q,, such that μ q ε ca also b (appromatly) solvd for th pctd rtur vctor: IV. μ q ε A vw o a sgl asst could b udrstood as a spcal cas of a lar combato, whr all th wghts of othr assts ar zro. h trprtato of vws as portfolos was troducd H/Lttrma. 3

4 whr P + s th psudovrs of P. hs vctor dscrbs th vstor s formato o rsk prmums mpld thr vws. h vctor wll oly cota rsk prmums for assts whch ar affctd by th vstor s vws, ad zros for othr assts. From IV t follows that ths pctd rtur vctor s radom wth a multvarat ormal dstrbuto: 3 v q, ' V. ~N M whr M v s th radom pctd rtur vctor, ad th ralzato of M v,.. th tru pctd rtur vctor mpld th vws, s lablld v. Hr th d v dcats that ths ar th vw-mpld paramtrs for th dstrbuto of rsk prmums. Not that th covarac matr wll clud zros for assts ot affctd by th vws. As th vws ar codtoal o th ukow tru rsk prmums vctor, th rsk prmums vctor mpld th vws s also codtoal o. h dsty fucto for th pctd css rtur vctor v mpld th vws ad codtoal o th ukow tru rsk prmums vctor s hc th followg wrtt as: f M v v M I summary, th vstor has vws th form of a dstrbuto for th rsk prmums of o or mor portfolos formd wth o or mor of th assts avalabl th markt. hs dstrbuto s codtoal o th tru vctor of rsk prmums. It mpls a codtoal probablty dsty fucto for th radom vctor of asst rsk prmums. h two modls for rsk prmums dscussd abov, th markt-mpld ad th subjctv vwsmpld rsk prmums ca b combd wth th Bays rul for probablty dsty fuctos, as dscrbd th followg. Combg th markt-mpld ad vws-mpld modls for pctd css rturs Accordg to th Bays formula for dsts 4, th dsty of th tru vctor of rsk prmums codtoal o th rsk prmums mpld th vws,.. th postror dsty fucto for th rsk prmums, s th product of th markt-mpld pror dsty ad th codtoal dsty mpld th As P has full row rak (o vw s a lar combato of o or mor othr vws), th rght-vrs P (PP ) - could b usd as psudovrs. Solvg th quato as abov wth th psudovrs rsults a appromato of, such that th Euclda orm of th dstac btw th stmat of basd o th multplcato wth P + ad th tru caot b mad smallr by chagg th stmat of. S pag 45, h Oprator hory of th Psudo-Ivrs, I. Boudd Oprators, by Frdrck J. Butlr, Joural of Mathmatcal Aalyss ad Applcatos, 0, 965, p Hr howvr, du to a d vrso at a latr stag, whch rvrss th vrso hr, t s ot cssary to actually fd th psudovrs. 3 As s a dagoal, th covarac matr of th lmts of th vctor v Ρ ε s a dagoal as wll, wth th lmts of th dagoal bg p matr otato, ths ca b wrtt as: '. 4 S for ampl 4

5 vws for th asst rsk prmums, dvdd by th ucodtoal dsty of th rsk prmums mpld th vws: f M M v v f Mv v M f M M f M Mv v v As show scto A.II of th appd (ad th rfrcs gv thr), th umrator of th fracto o th rght had sd of ths quato s a Gaussa. For a gv μ v, th domator of th fracto o th rght had sd, f M μ, s a costat. Furthr, as th Bays rul for dsts mpls that th Mv v v lft had sd s a dsty fucto, th valu of f must sur that th rght had sd s a M M v propr dsty fucto as wll, partcular that t tgrats to. I.. v v f M v M v μv taks th rol of a ormalzg factor, ad th rght had sd s ot oly a Gaussa, but a dsty fucto of a multvarat ormal dstrbuto. hs holds for vry μ v, so that (ow slghtly chagg th otato to mphasz that μv s varabl), f M μ μv f M v f M v μv μ f M μ μ v s th probablty dsty fucto of a varabl wth a multvarat ormal dstrbuto. h paramtrs of ths dstrbuto,.. ts ma vctor ad covarac matr, ar gv by th gral quatos for paramtrs of products of multvarat ormal dsts, whch ar drvd scto A.II of th appd followg th rfrcs gv thr. h quatos ar (A.9.d scto A.II of th appd): o apply ths rsults hr, us th covarac matrcs gv quatos I. ad V. to spcfy th covarac matr of th postror dstrbuto: 5 τ τ Ad wth that th ma vctor of th postror dstrbuto bcoms: μ τ τ π 5 Not for th followg that th vrs of a psudovrs of a matr A s th orgal matr A. 5

6 whch s th Black-Lttrma quato for pctd css rturs. Appd 6 For th multplcato of dsts th t scto, th followg dscrbs th formato form for dsts. A.I Dsty of a multvarat ormal dstrbuto formato form h dsty fucto of a multvarat-ormally dstrbutd -dmsoal vctor,,,, s: 7 j A. f wth a ma vctor ad a covarac matr. Dfg th varabls A. ad A.3 th dsty ca b wrtt th so-calld formato form (aka atural form or caocal form) 8 6 As all fuctos apparg th whol followg tt ar dsts o th sam radom vctor, th otato s smplfd.. stad of f X (X=), ad usg.g. g X (X=) for aothr dsty fucto for th sam radom vctor, f () s wrtt, whr dffrt valus for th d ar usd to dstgush dffrt dsts o th sam radom vctor. 7 Whl ths frst scto of th appd oly o dsty s cosdrd, th d would ot b cssary ad may b rmovd futur vrsos of ths documt to th b rtroducd th t scto o products of dsts. 6

7 7 f Proof: Prform multplcatos th pot of A.: A.4 wth ad as s a scalar:, A.4 bcoms: ad hc th pot A. s: so that A bcoms: 8 S.g. p. 4 Som Proprts Of th Gaussa Dstrbuto Ja Wu, GVU Ctr ad Collg of Computg, Gorga Isttut of chology, Aprl, 004, ad p. Mapulatg th Multvarat Gaussa Dsty, homas B. Scho ad Frdrk Ldst, Dvso of Automatc Cotrol, Lkopg Uvrsty, Jauary, 0,

8 8 A.5 f Bcaus th fracto o th lft ca b wrtt as follows: A , ad wth th prssos troducd A.: th pot o th rght had sd of A6 ca b wrtt as A3: ad th dsty A5 ca b prssd as: 9 A.7 f whch complts th proof. A.II Product of two multvarat ormal dsts: Usg A.7, a product of two multvarat ormal dsts for th sam radom vctor ca b wrtt as: A.8 f f 9 S for ampl th rfrcs gv footot 5 ad also pag 5 Products ad Covolutos of Gaussa Probablty Dsty Fuctos, P.A. Bromly, Imagg Sccs Rsarch Group, Isttut of Populato Halth, School of Mdc, Uvrsty of Machstr, a Mmo No , Itral Rport, last updatd 4 / 8 / th followg rfrrd to as Bromly.

9 wth A.8a ad dfd aalogously to A.3:, A.8 ca b prssd as: 0 A.9 whr A.9.a s a Gaussa pdf as dfd by A.7 ad th prssos troducd A. ad A.3. h product of th two dsts A.8 s hc a scald Gaussa pdf ad ca b wrtt as: A.9.b S Whr A.9.c S s th scalg factor. I othr words, th product of a multvarat ormal dsty multpld wth th rcprocal of that scalg factor, s a multvarat ormal dsty. h rcprocal of that scalg factor s also rfrrd to as ormalzg costat. A.III Paramtrs of a scald product of two multvarat Gaussa dsts: From A.9.b t follows that a product of two multvarat-ormal dsts dvdd by a scalg factor s a multvarat ormal dsty: f f f ~ N, S 0 hs s a spcal cas wth = of th prsso for th product of multvarat ormal dsts gv Bromly. 9

10 0 (wth th scalg factor S dfd as A.9.c) From A. ad A.8.a, o ca s th paramtrs of th multvarat ormal vctor wth dsty f: A.9.d A.IV Rwrtg th scalg factor Wth th pot of th scalg factor ca b wrtt as: Rarragg ad smplfyg ths prsso gvs: a Whr a s dfd as follows: A.0 a. Wth

11 ad 0.5 th scalg factor S ca th b wrtt as: A. a 0.5 Wth ad a bcoms: a hs prsso s ow smplfd ough to rplac th prssos troducd for th formato form: a Corrspods to quato 6d Gaussa Idtts, Sam Rows, (rvsd July 999),

12 A. h thr-brackts product o th rght ca b wrtt as: A.3 A spcal cas of th matr vrso lmma s: A.4 So that th frst summad A.3 ca b wrtt as: Ad th scod summad: So that th sum A.3 bcoms: S p. 0 of Gaussa Procsss for Mach Larg, by C. E. Rasmuss & C. K. I. Wllams, th MI Prss, 006, - th followg rfrrd to as Rasmuss.

13 3 Ad wth that o gts for A.: A.5 o procd, us th followg rsult: 3 Proof: Ad: Hc: 3 For ampl gv at

14 4 Ad: Apply th rsult just prov o A.5: Apply ow th spcal cas of th matr vrso lmma gv arlr aga ad smplfy: Rarrag: Us ths rsult for a th quato for th scalg factor A.: a S 0.5 Substtut for a:

15 A.6 So th scalg factor s th vctor of valus of th dsty fucto of a multvarat Gaussa wth ma ad covarac matr valuatd at. 4 4 hs s th form for th ormalzg factor as gv Rasmuss, quato A.8, pag 00. Not that, as wrtt thr, ad ca b chagd. 5