Internet Appendix to: Idea Sharing and the Performance of Mutual Funds

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1 Coes Iere Appedx o: Idea harg ad he Perforace of Muual Fuds Jule Cujea IA. Proof of Lea A IA. Proof of Lea A IA.3 Proof of Lea B IA.4 Populao dyacs he geeral seg of eco IA.5 Average rajecores of uber of deas IA.6 Proof of Proposo 9 (dsrbuo shf relave o luck) IA.7 Deals o copuaos eco 6 (fud flows ad fees) IA.. Proof of Lea A. Deoe by p( F ) he codoal desy of wh respec o F.Fxgae = k ad applyg Bayes rule, he codoal desy p( F ) sasfes he recursve relao p( F )= p( F )f( ) R R p(x F )f( x)dx (IA.) where f( ) deoes he desy of a vecor of sgals codoal o ad where p( F ) sasfes p( F )= o ( exp b! ) (IA.) sce, fro Theore A., p( F ) s codoally Gaussa for ay ( k, k ) (Lpser ad h hryaev (), Theore.6). Frs, le =... > be he vecor of sgals he sequece ( :applej apple j ). Codoal o, hese sgals are depede ad hus f( ) = ( ) Y j= j A A. (IA.3) Afer subsug (IA.3) (IA.) ad egrag, he codoal desy p( F )sexplcly gve by s p( F )= o B o b o P j= j C A o (IA.4) C A C A.

2 ecod, le = Y be he aggregae sgal, whch case f(y ) = exp Y. (IA.5) ubsug (IA.5) (IA.) ad egrag, he codoal desy (IA.4) becoes s p( F )= o o b o Y / o Clearly, he expressos (IA.4) ad (IA.6) are equvale f ad oly f Y =( ad, by duco, hs resul us be rue for all = k ad k {,,...,N T }. IA.. Proof of Lea A.3 (IA.6) ) P j= j! A. Observe ha he Browa oo B bc s adaped o F c ad, herefore, o F cobe (57) ad (74) ad oba F c. ecod, d b B =d b B c (,, (, a, )( b b c ))d. (IA.7) ce F c F, follows ha b b c =,,, whch, subsued (IA.7) ad usg (6) gves (86). Thrd, for EP bc [Z] = o hold,.e., for P b o be absoluely couous wh respec o bp c uder F, he Rado-Nkody dervave (85) us be a argale. A su ce codo uder whch Z s a argale s he Novkov codo (Karazas ad hreve (988), Proposo 5.): P E appleexp bc Z T (k ) d <, apple T apple (IA.8) where he process uder b P c sasfes d = o ( )k d k (o ( ) o c )d b B c. (IA.9) Observg ha he process (IA.9) s Gaussa, Exaple 3 (a) (Lpser ad hryaev (), p. 33) shows ha he Novkov codo (IA.8) bols dow o sup applet k E b P c <, I hs seup, he codos (IA.) splfy o supapplet k V b P c <. (IA.) sup applet k R k sds <, sup applet k R k sds <. (IA.) Usg (6) ad acpag o he equlbru resul of Lea C., he wo codos (IA.) are equvale o requrg ha he fuco ( ) be couous, whch s by assupo. Theore A.3 he follows fro Grsaov heore (Karazas ad hreve (988), Theore 5.).

3 IA.3. Proof of Lea B. Cosder frs he rgh-had sde of (38). Fro (34), I oba (, ) A> Q, A Q, BQ, (, ) =! >! k Furherore, fro (35), I ca wre o ()o ( ) = k o ()o ( ). (IA.) B,( )A Q, B,()A Q, A, (, ) = A, (, ) k B Q, B o ()o ( ), Q, (IA.3) whch, subsued (IA.) yelds B >,()A Q, (, ) = k o ()o ( ) (, ) A, B Q, = k o ()o ( ) k o () o ()o ( ) B,()A Q, A, (, ) B Q, (IA.4) (IA.5) where he secod equaly follows fro splfcaos based o (6). ecod, cosder he lef-had sde of (38) ad drecly d ereae (7) usg (69) o oba (, ) = o ()o ( ) k o ()o ( )(o ()o ( )). (IA.6) Fally, regroupg ers (IA.4) ad usg Eq. 68 shows ha (IA.4) ad (IA.6) cocde ad he d ereal equao (38) us herefore hold. IA.4. Populao dyacs he geeral seg of eco. I hs appedx, I derve he dyacs of forao sharg for he geeral seg of eco. (Appedx IA.4). I he geeral seg of eco., he dyacs of a aager s uber of deas sasfy d = dn,, ( ; ) (IA.7) where (N ) deoes a Posso process wh esy ( ). These dyacs ply a cera cross-secoal dsrbuo, µ, of uber of deas,.e., he dsrbuo µ us sasfy a cera Kologorov Forward Equao (KFE), whch I derve usg he resul forulaed Lea IA.4.. Lea IA.4.. Defe he expecao g E[f( )] = kn f(k)µ (k) (IA.8) 3

4 for a arbrary fuco f( ). The, he fuco g us sasfy he d ereal equao d d g = ()µ () f( ) (; ) N N ()µ ()f() N (IA.9) wh al codo g = P kn f(k) (k). Proof. Observe ha he geeraor of he process (IA.7) sasfes Af() = () N (; )(f( ) f()) (IA.) ad rewre he expecao (IA.8) as g = E[f( )] Z Z E[Af( s )]ds = E[f( )] Af()µ s ()ds. N (IA.) D ereag Eq. (IA.) wh respec o e ad rearragg yelds (IA.9). To oba he KFE (3), I he chage he suao order (IA.9). pecfcally, roduce he chage of varable k ad rewre he frs er (IA.9) as N N f( ) () (; )µ () = kn k = Pluggg (IA.) o (IA.9) ad rearragg, I oba N f() d d µ () = N f() = f(k) (k ) (; k )µ (k ). (IA.) ( ) (; )µ ( ) ()µ () Observg ha he fuco f( ) s arbrary, he KFE (3) follows.!. (IA.3) IA.5. Average rajecores of uber of deas I hs appedx, I derve a aager s average rajecory of uber of deas, whch I use for he a aalyss eco 5 (Appedx IA.5). I parcular, I oba he average rajecory of a aager s uber of deas codoal o aager holdg T = k deas a he horzo dae,.e., E T = k. Applyg Bayes rule, frs observe ha P = T = k = P[ = ]P[ T = k = ] P[ T = k] = µ () T (k; ) µ T (k) (IA.4) where s he probably ha aager ges k deas by he horzo dae codoal o holdg deas a e. To copue hs probably, apply he resul of Lea IA.4. o g s P N f() T s(; ) for s>wh al codo, g =, whchyelds: T (k; ) = {k } e (k )T e T k e (T ) {k } (IA.5) 4

5 uder dea sharg, whereas uder dea orgao: d d s(k; ) = s (k; ) s (k; ), (IA.6) wh al codo, (; ) = =,where = s a Drac ass a =. I he follows ha he average rajecory of aager s uber of deas s gve by E T = k = µ T (k) k µ () T (k; ), (IA.7) = whch has a closed-for soluo uder dea sharg: E T = k = et e k(t ) e k e T k e (T ). (IA.8) IA.6. Proof of Proposo 9 (dsrbuo shf relave o luck) I hs appedx I show ha he cross-secoal dsrbuo of sascs s shfed o he lef whe he dsrbuo of uber of deas s syerc or sasfes Corollary. I sar by copug he probably ha a aager s sasc s egave: P[, apple ] = = Z l (x) N µ ()erf xp R ()! dx (IA.9) p! µ ()a E[R p ]. (IA.3) k N Provg ha Eq. (43) holds s hus equvale o provg: µ ()a p ' <, (IA.3) N where ' > s posve a all fe es. Assug he fucoal for for µ Corollary, I sar by boudg a ( ) Eq. (IA.3) by a pecewse lear fuco: a ' p apple appleb c a (' ( )) >b c p, o g(), N, (IA.3) where he frs er explos ha he suppor s bouded a ad he secod er explos boh ha he fuco s cocave for all ad hus bouded above by s frs-order Taylor expaso ad ha a ( ) s bouded above a /. Alhough hs boud ca be gheed, s su ce o prove Eq. (IA.3) ad sple eough o copue explc expressos: rewrg µ Eq. (7) as µ () = ( ), (IA.33) 5

6 ad akg expecaos over he fuco g( ) Eq. (IA.3) yelds he explc boud: < b c ' b c p p ' l p p a (' ( ' p ' )) ' ( ) p (b c) C A. (IA.34) By assupo, >, whch ples ha > ad hus he er sde he bracke s oposve, fro whch he equaly Eq. (IA.3) follows. Noe ha he resul also obas whe µ s srcly syerc a all es. ce he fuco a ( ) Eq. (IA.3) s odd ad egave for all appleb c, I defe a ew fuco f :[b c, b c]! R by reflecg a ( ) Eq. (IA.3) frs horzoally abou zero ad he vercally abou o oba: f() = a (b ' c ), = b c,...,b c. (IA.35) p b c The cocavy of he skll-o-luck rao he ples ha f() > a ' p, = b c,...,b c. (IA.36) Usg hs resul I oba a src boud for Eq. (IA.3): < [,b c][(b c,...,) µ ()a p ' a all fe es. yery furher ples ha [,b c] µ ()a p ' ad ha µ has zero ass beyod wce s ea: (b c,b c] (b c,b c] µ ()f(), (IA.37) µ ()f() = (IA.38) µ () =, b c, (IA.39) sce s syerc ad s suppor s bouded fro he lef a. Hece, he syerc case Eq. (IA.37) drecly leads o Eq. (IA.3). IA.7. Deals o copuaos eco 6 (fud flows ad fees) I hs appedx I dscuss copuaoal deals relaed o he equlbru soluo he presece of fud flows ad fees. For he sake of brevy, I sply ppo resuls ha d er fro he basele odel. Noe frs ha flows ad fees do o a ec he resuls of Proposo. However, hey odfy porfolo sraeges. To see how, I go over he a seps of Appedx B he presece of fees ad 6

7 flows. Maxzg aager s expeced uly over copesao Eq. (48) ples: ax E exp f(( )W T B T ) F. (IA.4) The soluo o hs axzao proble s T (,,T )= f( ) (o T ()) > T,T,T T b T. (IA.4) whch subsued he value fuco yelds he boudary codo J(W, B,,,T )= exp f(( )W B) (o T ()) > T. (IA.4) The proble has a addoal sae varable, B, whch applyg Io s lea o Eq. (5) sasfes: db = b c, dp = dp! > dp. (IA.43), The assocaed HJB equao ow sasfes = ax J W A Q J WWBQ( ) B Q B ( ) > J W J WB BQ! > J J > A r(j B ( )B ( ) > )J B ( ( )E L( b Y, ) h J(W,B, ) >! > A Q J BBB Q( ) >! >! (, ) b Y,,) J(W,B, The frs-order codo he yelds he followg porfolo polcy: B Q! > (IA.44) B ( ) > J B (IA.45),,). (IA.46) (, )= J W A Q B Q B ( ) > J W J WB B Q!> J WW B Q. (IA.47) ubsug back he HJB equao, edous dervaos show he asaz of Theore B. becoes: J(W, B,,,)= exp f(( )W B) u () > R ()R () > > M (), (IA.48) where R ad M sasfy he syse of equaos Theore B.. The equao for u d ers, bu s rreleva for porfolo sraeges ad s hus oed. As a resul, he soluo for R ad M s decal o Lea B.3. ubsug hese expressos he opal polcy above yelds Eq. (5). To oba prce coe ces I ow go over he a seps Appedx C. Aggregag frs porfolos a he horzo dae: Z T d = N µ T () f (o T ()) > T T () T! T T =( )? T, (IA.49) 7

8 whch yelds he boudary codos,t = o c T T ( )( T o c T ) ad,t = f o c T ( T o c T ). (IA.5) larly, aggregag porfolos a dae, Eq. (84) becoes: N µ () A Q, B Q, (o ()) B,() > ()! =,, f( )B Q,, (IA.5) sce! >! =, whch yelds,, =, ad hus, f( ) o c = f( ) Z f( )! d ds s ds. (IA.5) Fally, spellg ou he secod equao of he syse Eq. (9) becoes d d, = k, B Q, BQ,, o c he soluo of whch s Eq. (5). Furherore, gog hrough he seps of Appedx D ad splfyg yelds b ( ) o c = f ( )( oc f o c ) ( ) k (f ( ) k o c ) ( ( )(f k ) k o c ) f ( ) o c f, (IA.53) p f ( ) (IA.54) (IA.55) a, ( ) a, ( ) a, ( ), (IA.56) where he secod le correspods o he fuco H( ) Eq. (53). To copue foraoal alphas, I subsue he equlbru soluo Eq. (8) o oba ad A Q, = k o c (f 4 ( )k o c ( oc )) ( )( oc )! fo c 4 k o c (k ( ) f) ( oc ) (IA.57) b, ( ) b, ( ), (IA.58) B Q, = oc ( )(k f ) k o c ( )( o c ). (IA.59) Wh hese expressos ca he copue sascs by sulaos usg Eq. (). Refereces Karazas, I., hreve,., 988. Browa Moo ad ochasc Calculus. prger, New York, Graduae Texs Maheacs. 8

9 Lpser, R.., hryaev, A. N.,. ascs of Rado Processes I. prger Verlag, New York. Lpser, R.., hryaev, A. N.,. ascs of Rado Processes II. prger Verlag, New York. 9