Workin aper Series Investors Direct Stock Holdins and erormance Evaluation or Mutual Funds by Wolan Breuer and Marc Gürtler No.: FW06V4/04 First Drat: 004-06-18 Tis Version: 005-0-01 (erscienen in: Kredit und Kapital, 38. J., 005) Tecnisce Universität Braunscwei Institut ür Wirtscatswissenscaten Lerstul BWL, insbes. Finanzwirtscat Abt-Jerusalem-Str. 7 38106 Braunscwei
Investors Direct Stock Holdins and erormance Evaluation or Mutual Funds von Wolan Breuer und Marc Gürtler Abstract. Investors need perormance measures particularly as a means or unds selection in te process o exante portolio optimization. Unortunately, tere are various perormance measures recommended or dierent decision situations. Since an investor may be uncertain wic kind o decision problem is best apt to describe is personal situation te question arises up to wic extent unds rankins react sensitive wit respect to canes in perormance measurement. To be more precise, an investor wit mean-variance preerences is considered wo is tryin to identiy te best und out o a set consistin o F unds and to combine tis one optimally wit te direct oldin o a broadly diversiied (reerence) portolio o stocks as well as riskless lendin or borrowin. For an investor just startin to acquire risky securities all tree ractions o te various assets in question as part o is overall portolio can be considered variable, wile tere also mit be investors wit already iven direct oldins o stocks amountin to a certain raction o teir total wealt wic cannot or sall not be altered. For bot situations dierent adequate perormance measures ave been suested by Breuer/Gürtler (1999, 000) and Scolz/Wilkens (003). We analyze teoretically as well as empirically possible deviations in resultin unds rankins or te two decision situations described previously. Wile tere are indeed only loose teoretical relationsips between te perormance measures under consideration, empirical evidence suests almost identical unds rankins. As a consequence, potential investors need not boter muc about weter teir situation is best described by an already ixed or a still variable amount o direct stock oldins. Moreover, traditional perormance measures like te Sarpe ratio or te Treynor ratio will in eneral lead to reasonable unds selection in bot situations. Zusammenassun. erormancemaße werden von Investoren insbesondere als Mittel zur Selektion von Investmentonds im Ramen von Ex-ante-Optimierunen verwandt. Unlücklicerweise existieren versciedene erormancemaße ür untersciedlice Entsceidunsprobleme. Da ein Anleer im Unklaren darüber sein ma, welces Entsceidunsproblem am besten seine persönlice Situation bescreibt, dränt sic die Frae au, in welcem Ausmaß Fondsrankins au einen Wecsel des erormancemaßes reaieren. räziser ormuliert, wird ein Investor mit µ-σ-räerenzen betractet, der versuct, den besten Fonds aus einer Mene von F zur Auswal steenden zu bestimmen und diesen in optimaler Weise mit dem direkten Halten eines breit diversiizierten (Reerenz-) ortolios aus Aktien sowie risikoloser Anlae bzw. Versculdun zu kombinieren. Aus Sict eines Investors, der erade beinnt, seine Mittel in riskanten Aktiva anzuleen, können alle Anteile der versciedenen Aktivaklassen an seinem Gesamtvermöen als variabel aueasst werden, wärend auc Anleer mit bereits vor Fondsselektion eebenem positiven Aktienenaement existieren möen, das nict one weiteres eändert werden kann oder soll. Für beide Situationen wurden eeinete erormancemaße voresclaen, und zwar von Breuer/Gürtler (1999, 000) und Scolz/Wilkens (003). Mölice Untersciede in den jeweilien Fondsreiunen ür die beiden enannten Entsceidunssituationen werden teoretisc wie empirisc untersuct. Wärend sic nur lockere teoretisce Zusammenäne beleen lassen, weist der empirisce Beund au tatsäclic ast identisce Reiunen in. Als Konsequenz ieraus müssen sic potentielle Anleer nict allzu viele Gedanken darüber macen, ob ire Situation besser durc ein ixes oder ein variables Aktienenaement bescrieben wird. Ferner üren traditionelle erormancemaße wie die Sarpe Ratio oder die Treynor Ratio in beiden Entsceidunssituationen zu akzeptablen Reiunen. Sticworte: erormancemessun, Sarpe-Maß, Treynor-Maß, optimale ortolioselektion Keywords: erormance measurement, Sarpe ratio, Treynor ratio, optimal portaolio selection JEL classiication: G11 roessor Dr. Wolan Breuer roessor Dr. Marc Gürtler Aacen University o Tecnoloy Braunscwei University o Tecnoloy Department o Finance Department o Finance Templerraben 64, 5056 Aacen, Germany Abt-Jerusalem-Str. 7, 38106 Braunscwei, Germany one: 49 41 8093539 - Fax: 809163 one: 49 531 391895 - Fax: 391899 email: wolan.breuer@rwt-aacen.de email: marc.uertler@tu-bs.de
I. Introduction Issues o perormance measurement or investment unds lie at te root o modern portolio manaement researc. Investors need perormance measures as a means or te ex-post assessment o unds perormance wic in turn under te assumption o stable return caracteristics over time can be utilized or adequate unds selection in te process o ex-ante portolio optimization as well. Unortunately, tere are various perormance measures recommended or dierent decision situations. Since an investor may be uncertain wic kind o decision problem is best apt to describe is personal situation te question arises up to wic extent unds rankins react sensitive wit respect to canes in perormance measurement. It is tis issue to wic we want to contribute. To be more precise, an investor wit mean-variance preerences is considered wo is tryin to identiy te best und out o a set consistin o F unds = 1,, F and to combine tis one optimally wit te direct oldin o a broadly diversiied (reerence) portolio o stocks as well as riskless lendin or borrowin at a rate r 0. For an investor just startin to acquire risky securities all tree ractions o te various assets in question as part o is overall portolio can be considered variable, wile tere also mit be investors wit already iven direct oldins o stocks amountin to a raction o teir total wealt. I tose investors are not able or not willin to cane tis part o teir investment, we call suc a situation te exoenous case since te allocation o a certain part o te investor s initial wealt is ixed. Oterwise, we speak o te endoenous case because tere are no restrictions (besides possible sort-sales constraints) wit respect to te allocation o te investor s monetary wealt. Fiure 1 visualizes te two dierent decision problems and introduces x0, x, and x as symbols or ractions o initial wealt invested in riskless assets, equity portolio, and und eventually cosen by an individual. x >>> Insert Fiure 1 about ere <<< In bot te exoenous case and te endoenous one, we are able to identiy central subportolios o te investor s respective overall portolio wic tereore sould be explicitly caracterized by adequate symbols. In te exoenous case, te investor only aims at optimizin is 1
subportolio o direct stock investments as represented by a combination o reerence portolio and riskless lendin or borrowin and tus we explicitly denote tis subportolio or cosen und as Q(). In te endoenous case, as an application o te well-known two-unds separation teorem irstly identiied by Tobin (1958) and later eneralized by Cass/Stilitz (1970), te structure o te risky subportolio is independent o te investor s deree o risk aversion and tus tis subportolio or cosen und sall be caracterized by R(). For te endoenous case, te application o so-called optimized Sarpe measures as been suested by Breuer/Gürtler (1999, 000), wile or te exoenous case Scolz/Wilkens (003) derived te investor speciic perormance measure as te adequate one. Bot kinds o perormance measures assure te selection o suc a und and correspondin overall portolio structure tat an investor s mean-variance preerence unction is maximized. Tereby, in te endoenous case, te optimized unrestricted (restricted) Sarpe measure recommends tat und tat leads to te maximum Sarpe ratio o te correspondin best overall portolio consistin o riskless assets, te reerence portolio and just one und witout (wit) sortsales constraints. Te investor speciic perormance measure ulills te same task or te exoenous case. Contrary to te situation in te endoenous case, te two-unds separation teorem does not old in te exoenous case so tat te investor s deree o risk aversion as to be speciied in order to identiy te best und tus explainin te denomination o te perormance measure as investor speciic. Funds rankins accordin to te optimized Sarpe measures are independent o an investor s deree o risk aversion. rima acie, we deem te endoenous case as well as te exoenous one as o equal practical importance. Hence, it seems to be interestin to analyze somewat more in detail teoretical as well as empirical relationsips between unds rankins in tese two alternative settins in order to better assess te necessity o explicitly dierentiatin between tese two decision problems. Tereby, te question sould be answered weter te correspondin somewat new perormance measures could be replaced by a simple application o te well-establised and widespread used traditional perormance measures by Treynor (1965), Sarpe (1966), Jensen (1968), and Treynor/Black (1973). Findins o tis kind would be o immediate practical importance as tey mit support practitioners in te process o adequate unds selection or portolio optimization by distinuisin more critical issues rom less critical ones. Unortunately, neiter Breuer/Gürtler (1999,
000) nor Scolz/Wilkens (003) examine torouly teoretical and empirical connections between teir proposed ways o perormance measurement. In wat ollows we want to close tis ap. To do so, in section II we briely ive an overview o te main indins in Breuer/Gürtler (1999, 000) and in Scolz/Wilkens (003). Section III is devoted to te teoretical analysis o te relationsips amon te optimized Sarpe measures o Breuer/Gürtler (1999, 000), and te investor speciic perormance measure o Scolz/Wilkens (003) as well as te traditional perormance measures mentioned above. Section IV presents a possible empirical application o te indins o section III and tereby examines te practical relevance o te teoretically justiied clear dierentiation between te optimized Sarpe measures and te investor speciic perormance measure and tus te distinction between te endoenous case and te exoenous one. Section V concludes. II. erormance Evaluation in te Endoenous Case and te Exoenous One Reconsidered As a uidance, Table 1 ives an overview o all relevant matematical symbols o te investor s portolio selection problem. Moreover, wile stands or just one arbitrary und out o all F accessible ones, we use te symbols and to distinuis between two dierent speciic unds simultaneously considered wen discussin evolvin unds rankins as a consequence o te application o te various perormance measures. >>> Insert Table 1 about ere <<< In wat ollows, we only look at unds wit expected excess return u > 0, because oterwise even simple riskless lendin will in eneral be preerred to an investment in und. Wit te backround o Table 1, we are able to (re-) introduce ormally te ollowin classical or traditional measures or perormance evaluation wit respect to a und or reerence portolio : u (1) ϕ =, te Sarpe measure 1 o, σ u () ϕ =, te Treynor measure o, β 3
(J) (3) ϕ = u β u, te Jensen measure 3 o, (4) ϕ ϕ = te Treynor/Black measure 4 o. σ (J) (TB), ε Moreover, we need tree additional perormance measures, i.e. (invj) (5) ϕ = u β u, te inverse Jensen measure o, (6) u R () ϕ =, te optimized Sarpe measure o witout sort sales restrictions ( unre- σ R () stricted optimized Sarpe measure ), and (7) u R restr. () ϕ,restr. =, te optimized Sarpe measure o wit sort sales restrictions ( re- σ R restr. () stricted optimized Sarpe measure ). 5 Te inverse Jensen measure o a und corresponds to te Jensen measure o reerence portolio in te case o a linear reression o te excess return u o portolio wit respect to te excess return u o und and tus reverses te oriinal roles o und and reerence portolio. Te restricted optimized Sarpe measure o und reers to te Sarpe measure o te optimal risky subportolio R () o und and reerence portolio wen sort sales restrictions reardin und and reerence portolio are nelected. 6 Correspondinly, te restricted optimized Sarpe measure allows or te requirements x 0 as well as x 0 and tus is based on te optimal risky portolio R () restr. in te case o sort sales restrictions. In wat ollows, optimal solutions are enerally caracterized by an asterisk ( ). Wit tese deinitions in mind, Breuer/Gürtler (1999, 000) were able to derive te ollowin results or te endoenous case: 1 As already mentioned in section I, also known as te Sarpe ratio. See Sarpe (1966). Also known as te Treynor ratio. See Treynor (1965). 3 Also known as Jensen s Alpa. See Jensen (1968). 4 Also known as te Treynor/Black appraisal ratio. See Treynor/Black (1973). 5 For te unrestricted as well as te restricted optimized Sarpe measure see in particular Breuer/Gürtler (000). 6 It sould be mentioned tat te maximization o te optimized unrestricted (restricted) Sarpe measure as deined in (6) (in (7)) is equivalent to te maximization o te Sarpe measure o an investor s overall portolio in te case witout (wit) sort sales constraints. Tis equivalence was used in section I or te inormal description o te optimized Sarpe measure. 4
(BG1) Funds, wic sould best be sold sort, i.e. x < 0, or lead to x = 0 wen combined wit reerence portolio, are caracterized by a neative Jensen measure or a Jensen measure o 0, respectively, and mit be called inerior unds. (BG) A rankin by te Jensen measure or te (neative inverse o te) Treynor measure can be justiied (only) to rank inerior unds wit te latter measure in contrast to te ormer one not bein prone to manipulation by te variation o te amount o riskless lendin or borrowin o te manaer o a und. In te case o all beta coeicients bein positive, a rankin accordin to te neative inverse o te Treynor measure is equivalent to a rankin accordin to te Treynor measure itsel. (BG3) Amon several unds, tat one sould be cosen wic oers te iest optimized (restricted or unrestricted) Sarpe measure. (BG4) Funds wic sould best be combined wit sort sales o equity portolio ( x < 0) coincide wit a neative inverse Jensen measure o. 7 In suc a case sort sales restrictions imply an evaluation o te respective und by its simple Sarpe measure. (BG5) Te optimized unrestricted Sarpe measure implies te same rankin as te square o te Treynor/Black measure. 8 (BG6) All unds or wic R () = R (), i.e. wic do not lead to violations o sort restr. sales restrictions, can be ranked amon eac oter accordin to teir Treynor/Black measure. Tus, in tis case te latter rankin is equivalent to tat based on te (restricted or unrestricted) optimized Sarpe measure. For te exoenous case, Scolz/Wilkens (003) accomplised to derive a perormance measure wic diers in some respect rom te ones introduced above. 9 Tey call it te investor speciic perormance measure and deine it oriinally as (8) u Q( )(x ) 1 1 1 ISM,ori. (x ) : = x u Q() (x ) ( ). σ ϕ ϕ 1 x Tereby, u Q( )(x) stands or te contribution o subportolio Q() to an investor s overall acievable expected excess return, i.e., u Q( )(x) is te product o raction 1 x o subportolio Q() and its correspondin expected excess return u Q( ). As pointed out earlier, contrary to 7 To be precise, tis result is not mentioned in Breuer/Gürtler (1999, 000), but it immediately ollows rom (BG1), i one reverses te roles o und and reerence portolio. 8 A riorous portolio-teoretical oundation or te application o te square o te Treynor/Black measure was irst derived by Jobson/Korkie (1984). 9 See also Breuer/Gürtler (1998) or a similar, but earlier approac. 5
te endoenous case te two-unds separation teorem does not apply in te exoenous case. Tereore te investor s deree o risk aversion must be at least implicitly taken into account. Tis can be done by ixin te desired expected return o te investor s overall portolio at a certain value u > 0. Wit iven raction x o reerence portolio and iven overall expected excess return u, te contribution u Q( )(x) o te complementary subportolio Q() is necessarily iven, too. In act, we ave u = x u (1 x ) uq() = x u u Q() (x) and tus Q( ) Q u (x ) = u x u = :u (x ) as a unction o x, but bein independent o und under consideration. Wit tis ormal backround, and under te additional assumptions o all reression coeicients β : =σ / σ bein positive and sort sales o unds bein impossible te ollowin results ave been stated by Scolz/Wilkens (003) or te exoenous case: (SW1) For iven raction x wit 0 x < 1 and desired overall expected return u, unds sould be ranked accordin to (8). (SW) I te Sarpe measure as well as te Treynor measure o a und is not smaller tan tat o a und, we ave ISM,ori. (x ) ISM,ori. (x ). (SW3) For te case x = 0 unds rankins accordin to te investor speciic perormance measure (8) reduce to rankins on te basis o te conventional Sarpe measure. (SW4) For desired overall expected return u =u δ, witδ >0andδ 0, unds rankins accordin to te investor speciic perormance measure (8) reduce to rankins on te basis o te conventional Treynor measure. Unortunately, as sown in te Appendix, result (SW1) can only be eneralized to te case 1 x > 0. Oterwise, i.e. or 1 x < 0, a unds rankin accordin to (8) will lead to te best score or tat und wic minimizes te investor s mean-variance preerence unction. Situations wit 1 x < 0 occur, or example, wen direct equity oldins are inanced by riskless borrowin and tus cannot a priori be excluded. For te eneral case, i.e. or arbitrary sins o 1 x te ollowin modiied investor speciic perormance measure ISM (x ) as to be applied: (9) Q,ori. Q ( ) u (x ) 1 1 ISM (x ) : = ISM (x ) (1 x ) = x u (x ) σ ϕ ϕ In wat ollows we simply speak o te investor speciic perormance measure, altou we mean ISM (x ) instead o te oriinal one (8) as introduced by Scolz/Wilkens (003).. 6
We consider bot te endoenous as well as te exoenous case to be o practical importance. Apparently, tere must be some teoretical relationsips between unds rankins in tese bot cases since te investor speciic perormance measure o a und is determined by its Sarpe measure and its Treynor measure. Tese relationsips are examined in te next section. III. Teoretical Relationsips amon Traditional Measures, Optimized Ones, and te Investor Speciic erormance Measure We start wit te consideration o possible connections between te investor speciic per- ormance measure and te traditional ones. Tereby, contrary to Scolz/Wilkens (003) we allow or neative reressions coeicients β and sort sales possibilities in order to ormulate our results as enerally as possible. Result 1: (S (T ) (T ) (R1.1) ϕ ϕ 1/ ϕ 1/ ϕ implies ISM ( x ) ISM ( x ) or all 0 x ) u /u. (R1.) ϕ (T ) (T ) ϕ 1/ ϕ 1/ ϕ implies ISM ( x ) ISM ( x ) or all x < 0 x > u /u. ( S) (R1.3) ϕ ϕ ϕ (T ) ϕ (T ) > implies ISM ( x ) ISM ( x ) in te absence o sort 0 (R1.4) sales possibilities. For x = 0 we ave ϕ ϕ ISM ( x ) ISM ( x ). (R1.5) For desired overall expected return u =u δ, wit δ > 0 and δ 0, we ave (T ) (T ) 1/ ϕ 1/ ϕ ISM ( x ) ISM ( x ) wic, in te case o positive Treynor (T ) (T ) (T ) measures ϕ and ϕ d to (T ), can be simpliie ϕ ϕ ISM ( x ) ISM ( x ). (J) (J) (R1.6) ϕ ϕ does not imply 0 0 ISM ( x ) ISM ( x ) or any x \{u / u }. (R1.7) ISM ( x ) ISM ( x ) or all x X \{u / u } (wit X bein an arbitrary sub- (J) set o ) does not imply ϕ. 0 roo. See te Appendix. 7
Statements (R1.1) to (R1.3) indicate tat it is possible to draw some eneral conclusions wit respect to te relationsip between traditional perormance measures and te investor speciic one. To be precise, under certain conditions te knowlede o Sarpe and (te neative inverse o) Treynor measures may be suicient to deduce te resultin rankin o two unds accordin to te investor speciic perormance measure. Neverteless, te preerability o simultaneously ier values o bot te Sarpe and te Treynor measure cannot enerally be concluded, as is indicated by te eneral relevance o te neative inverse o te Treynor measure as well as by (R1.). Fortunately, (R1.3) describes one important special case, wen indeed simultaneously ier Sarpe and Treynor measure imply a better investor speciic perormance measure. Tereby, (R1.3) obviously is a direct extension o te result (SW) by Scolz/ Wilkens (003), as te ormer one is based on te perormance measure ISM (x ) instead o te oriinally by Scolz/Wilkens (003) suested one IFM,ori. (x ) and tus olds true even or te case 1 x < 0. In act, results analoous to (SW3) as well as (SW4) or te oriinal investor speciic perormance measure (8) can be derived on te basis o te more eneral investor speciic perormance (9) as well. Tis is stated by (R1.4) and (R1.5) wic bot are valid even i sort sales o risky assets are not proibited. Moreover, (R1.4) and te eneral ormulation o (R1.5) old true reardless o te sins o te Treynor measures o unds and under consideration. As as already been sketced in section II we mit deine an inerior und in te endoenous case as suc a one wic leads to x 0. In te exoenous case we mit call a und inerior compared to a und or iven possible exoenous ractions x X, i we ave ISM (x ) ISM (x ) or all x X. Obviously, accordin to (R1.1) to (R1.3) inerior unds in te exoenous case cannot as enerally be determined as in te endoenous case. In particular, it enerally is not possible to identiy an inerior und in te exoenous case by simply lookin at its Sarpe and Treynor measure because te relevance o te Treynor measure is ambiuous. However, or most practical purposes we may expect te assumptions underlyin (R1.3) to old and ten an inerior und is in act (suiciently) caracterized by a simultaneously smaller Sarpe and Treynor measure tan a superior und. Finally, as statements (R1.6) and (R1.7) reveal, inerior unds in te endoenous case need not be inerior in te exoenous one et vice versa. Bot cases tereore must be considered separately. 8
Wit Result 1 in mind, we are now able to take a closer look at te possible connections between te investor speciic perormance measure and te (restricted or unrestricted) optimized Sarpe measure. Result : (R.1) Let x, describe te optimal investment in reerence portolio i we combine tis portolio wit portolio Q() (consistin o a und and riskless lendin or borrowin). Ten ISM ( x ) ISM ( x ) implies ϕ ϕ.,, (R.) Let x,,restr. describe te optimal investment in reerence portolio i we combine tis portolio wit Q() and ave to consider sort sales restrictions. Ten ISM ( x ) ISM ( x ) implies ϕ ϕ,,restr.,,restr.,restr.,restr.. (R.3) ISM ( x ) ISM ( x ) eiter or all 0 x u /u or or all x < 0 x > u /u does not imply (TB) (TB) ( ϕ ) ( ϕ ) nor ϕ ϕ. (R.4) (R.5) ϕ ϕ does not imply ISM ( x ) ISM ( x ) or any x \{u / u }. ϕ,restr. ϕ,restr. does not imply ISM ( x ) ISM ( x ) or any x \{u / u }. roo. See te Appendix. Summarizin, tere are only rater loose eneral connections between te investor speciic perormance measure or te exoenous case and its counterparts or te endoenous one. In particular, it may be possible to draw some conclusions rom te rankin accordin to te investor speciic perormance measure to te rankin accordin to te restricted or unrestricted optimized Sarpe measure i one knows optimal restricted or unrestricted und investments. However, in suc a situation optimized Sarpe measures can directly be calculated. In Breuer/Gürtler (1999, 000) several relations between te classical perormance measures and te optimized ones ave been derived as as already been described in section I. Rater interestinly, te introduction o te investor speciic perormance measure by Scolz/Wilkens (003) enables us to add some more indins to te ones stated above. 9
Result 3: (R3.1) (J) (invj) (J) (invj) ϕ ϕ sn( ϕ ) sn( ϕ ) ( 1 / ϕ ) sn( ϕ ) sn( ϕ ) ( 1 / ϕ ) implies ϕ ϕ. (R3.) (J) (invj) (J) (invj) ϕ ϕ sn( ϕ ) sn( ϕ ) ( 1 / ϕ ) sn( ϕ ) sn( ϕ ) ( 1 / ϕ ) (TB) implies ϕ (TB) ϕ. (R3.3) (T ) (T ) ϕ ϕ 1/ ϕ 1/ ϕ implies ϕ ϕ,restr.,restr. roo. See te Appendix. In particular, under certain conditions it now becomes possible to reconize te superiority o a und in relation to a und accordin to te optimized Sarpe measure by simply lookin at its oriinal Sarpe, Treynor, and Jensen measure. Tereby, rom Breuer/Gürtler (000) it is already known tat conditions ϕ (J) 0 (J) ϕ 0 imply ϕ, since (only) und,restr. ϕ,restr. is inerior in te endoenous case. 10 From (R3.3) we learn tat a und wit a better Sarpe measure and a better (neative inverse o te) Treynor measure is once aain also caracterized by a ier restricted optimized Sarpe measure. Tins become more complicated i we allow or two unds and wit exibitin a ier Sarpe measure but bein caracterized by a ier Treynor measure. In suc a situation conclusions are only possible reardin te unrestricted optimized Sarpe measure and in addition we need some more inormation wit respect to te optimality o sort sales o unds or te reerence portolio. As tis inormation is iven by te sins o te Jensen measure and te inverse Jensen measure tey bot are necessary in order to derive results wit respect to te relation between te unrestricted optimized Sarpe measures o unds and. In act, te dense ormulations accordin to (R3.1) and (R3.) are valid or ϕ ϕ as well 1/ 1/ as 1/ ϕ < 1/ ϕ. Once aain, it is not possible to reverse te conclusions (R3.1) to (R3.3), i.e. or example ϕ >ϕ,restr.,restr. does not imply ϕ >ϕ 1/ ϕ 1/ ϕ tou tis seems to be o only minor importance. 10 See also (BG1) o section II. 10
Ater all, rom a teoretical point o view te introduction o te exoenous case by Scolz/Wilkens (003) seems to be an interestin extension o te endoenous one analyzed by Breuer/Gürtler (1999, 000). Quite remarkably, by derivin te investor speciic perormance measure it also becomes possible to clariy somewat more in dept relationsips between classical perormance measures and te optimized ones o te endoenous case. Te empirical analysis o te ollowin section aims at identiyin te useulness o te relationsips teoretically ound in Results 1 to 3. Moreover, te practical relevance o te caveats o Results 1 to 3 wit respect to implications wic are not enerally valid are examined. IV. Empirical Analysis Similarly to Breuer/Gürtler (003) we consider (post tax) return data or 45 mutual unds investin in German equity sares 11 over a period rom July 1996 to Auust 1999 wic are calculated on te basis o te development o te respective montly repurcase prices per sare. 1 We assume tat all earnins paid out to te investors by a und are reinvested in tis und. Te riskless interest rate r 0 can be approximated by te expected return o German time deposit runnin or one mont and coverin te respective period o time to be observed. We use te DAX 100 as a broadly diversiied reerence portolio. 13 For all 45 unds and te DAX 100 unbiased estimators or te relevant moments o one-montly returns are calculated and listed in Table. 14 Te expected excess returns as well as te beta coeicients o all unds under consideration are positive so tat te neative inverse o a Treynor measure can be enerally replaced by te (positive) Treynor measure itsel. >>> Insert Table about ere <<< 11 In wat ollows we briely speak o German unds, tou we do not mean teir country o oriin but te eorapical ocus o teir investments. 1 Tis means tat possible sellin markups are not taken into account. In tis respect, te perormance o unds enerally tends to be overestimated wen compared to te perormance o any reerence index. However, te determination ere (in accordance wit many oter approaces) o ross perormance measures allows at least some conclusions to be made wit reard to te sensitivity o rankins wen dierent types o perormance measures are observed. Exactly tis aspect orms te central issue o tis paper as pointed out in section I. 13 Te DAX 100 was an index (listed until 03/1/003) tat consisted o 100 continuously traded sares o German companies includin te 30 blue cips o te DAX 30 and te (ormer) 70 midcap-stocks o te MDAX. For urter inormation see e.. Deutsce Boerse Group (003), p. 6. 14 See Roati (1976) or te unbiased estimators o te expectation value and te second central moment. 11
On tis basis, we are mainly interested in te question weter te teoretical distinction between te endoenous and te exoenous case carries over to siniicantly dierent unds rankins in bot cases in practical applications. Tereby, we ocus on a situation wit sort sales restrictions because at least sort sales o mutual unds are not realizable by private investors. Most importantly, we know rom (R3.3) and (R1.3), respectively, tat a und wit a ier Sarpe measure and a ier Treynor measure tan a und simultaneously exibits a reater optimized restricted Sarpe measure and a reater investor speciic perormance measure ISM in te case o sort sales restrictions. Wit tis result, it is possible to identiy 8 o our 45 unds or wic te rankin accordin to teir Sarpe measure and teir Treynor measure, respectively, is identical so tat or tem rankins in te exoenous case (wit sort sales restrictions) will always coincide wit te correspondin rankin or te endoenous case. For tese unds numbered rom # 1 to # 8 in Table and separately listed in Table 3, investors may not boter weter te endoenous or te exoenous case is o more practical importance. We tereore restrict our remainin analyses to te 17 unds or wic rankins accordin to teir Sarpe and teir Treynor measure dier. For suc unds te application o te investor speciic perormance measure will lead to rankins wic are not necessarily identical to tat o te endoenous one and even may vary or dierent exoenous ractions x o te reerence portolio and desired overall expected excess returns u. >>> Insert Table 3 about ere <<< In order to better assess resultin dierences in rankins we calculate Spearman rankin correlation coeicients between rankins accordin to ISM (in wat ollows: ISM-rankins ) or iven identical desired overall expected excess returns (1) () u u u = = wit u {1.7719 %, 1.9 %,.0 %,,.7 %, 10 % 15 (1) () (1) } and dierent values and wit, {0, 5 %,., 100 %}. We ind tat correlation coeicients between two ISM-rankins are x x x () x very similar or iven dierence x : = x x. For illustrative purposes, Table 4 presents (1) () all resultin dierent correlation coeicients or te special case o a desired expected return u =.3 % and varyin x (1) and x (). For example, wit u (1) =.3 % rankins o r x = 10 15 We add u = 10 % as an extreme value in order to better assess te stability o our results. 1
() % and x = 30 % exibit a correlation coeicient o 99.0196 % wile ISM-rankins o r u =.3 %, x (1) () = 30 % and x = 50 % lead to a correlation coeicient o 99.647 %. >>> Insert Table 4 about ere <<< Table 4 is based on 1 dierent unds rankins as tis is te number o exoenous values (1) x and () x taken into account. 10 more tables o tis kind based on 10 additional unds rankins could be presented or all oter overall desired expected excess returns u under consideration. Certainly, because o space constraints all tese data sould be presented in a somewat more condensed way. In order to so, we summarize our indins in Table 5 by presentin averae correlation coeicients between ISM-rankins or dierent identical values o desired expected returns u and varyin dierences x between exoenously iven oldins o te reerence portolio. For example, accordin to te saded cell in Table 5 te averae rankin correlation coeicient or te pair (u, x ) = (.3 %,0 %) amounts to about 99.36563 % and is computed as te averae value o all rankin correlation coeicients in Table 4 wic are saded as well. As can easily be learnt rom Table 5, correlations are rater i even i we restrict ourselves to unds wic cannot be unambiuously ranked accordin to te Sarpe and te Treynor measure. Moreover, averae rankin correlation coeicients are decreasin wit allin value or u. Neverteless, since we rerain rom considerin situations wit sort sales o stocks or unds, te minimum accessible value or u amounts to 1.7719 % because u Q (x ) = u x u >0 (and tus x > 0) is only ulilled or all 0 x 1 i u > u 1.77189 %. >>> Insert Table 5 about ere <<< Finally, rankin correlation coeicients in Table 5 are smallest or i dierences x wic is intuitively appealin. Since (R1.4) and (R1.5), or x = 1 implies eiter (1) x 1 () = or x = 1, accordin to x = 1 and u = u δ (δ positive and small) te correspondin (averae) rankin correlation coeicient is identical to te correlation coeicient between te rankins accordin to te Sarpe and te (positive) Treynor measure. For correlations bein 13
increasin in u and decreasin in x, a i correlation between rankins accordin to Sarpe measure and Treynor measure tus implies only minor importance o ISM. 16 Te limited independent relevance o te exoenous case is also underpinned by rankin correlation coeicients between unds rankins accordin to te optimized restricted Sarpe measure and ISM or dierent values u and x as Table 6 points out. Moreover, Table 6 indicates tat two ISM-rankins wit identical equity oldins as described by x, but dierent values and () u or desired overall expected excess return will enerally be very similar since rankin correlation coeicients between ISM-rankins and te (iven) restricted optimized Sarpe measure do not cane muc or varyin expected excess returns u. In act, Table 6 suests tat variations o u aect te ISM-rankin even less tan canes in x. u (1) >>> Insert Table 6 about ere <<< Table 7 explicitly presents te rankin o te 17 unds wic underlie Tables 5 and 6 or x = 0 in te exoenous case (just leadin to a unds rankin accordin to te Sarpe measure and tus bein independent o u 17 ) and or te endoenous case. As expected, dierences in rankins seem to be almost neliible wic is underlined by a i rankin correlation coeicient o approximately 93.14 %. >>> Insert Table 7 about ere <<< Summarizin, at least or our empirical example tere seems to be no need to explicitly distinuis between te exoenous case and te endoenous one. 18 In act, we repeated all calculations underlyin Tables to 7 or te period rom June 1993 to July 1996 or all but our 19 16 In act, i correlations between Sarpe and Treynor measure seem to be typical or practical decision problems as Scolz/Wilkens (003), p. 4, point out. See also, or example Mölmann (1993), pp. 178-179, or Reilly/Brown (1997), p. 1010. 17 For tis last result see also Breuer/Gürtler (1999), pp. 75-76. 18 It sould be mentioned tat results would be quite dierent i one allows or sort sales o risky assets as even inerior unds may become very attractive wen sold sort tus possibly leadin to an almost perectly neative correlation between te unds rankin based on te unrestricted optimized Sarpe measure and te restricted optimized one. Neverteless, as mentioned earlier, we do not deem suc sort sales possibilities to be o practical importance. 19 Tree unds (# 30, # 3, # 34) were opened at a later date and one und (# 45) realized a neative averae excess return tus violatin our basic assumptions. For te latter reason it was not possible to analyze as a tird subperiod durin te nineties te time interval rom May 1991 to June 1993 as or tese years actually none o te unds under consideration realized a positive averae excess return. Since tis paper is not primarily empirically oriented we rerain rom discussin issues reardin te adequate estimation o a priori unknown return moments. See e.. Breuer/Gürtler/Scumacer (004), pp. 40-93. 14
unds under consideration. Essentially, our empirical indins are te same as or te period rom July 1996 to Auust 1999. 0 Bot te endoenous case and te exoenous one seem to be o practical importance and rom a teoretical point o view tere mit be siniicantly varyin unds rankins dependin on te situation under consideration. Yet, tere is empirical evidence tat resultin overall unds rankins are in eneral almost identical. In act, investors may restrict temselves to unds rankins accordin to te Sarpe measure (or even te Treynor measure) and will probably arrive at outcomes very similar to tose by application o te optimized (restricted) Sarpe measure or ISM. Tus, rom an empirical point o view, te use o te optimized (restricted) Sarpe measure or te ISM seems to make no dierence. Neverteless, we recommend te use o te optimized Sarpe measure because it represents te correct solution or te endoenous case and tere is no more inormation required tan or te calculation o te traditional perormance measures so tat tere is no need to apply just an approximation o te correct unds rankin in te endoenous case. On te contrary, ISM can only be computed or iven desired expected overall excess return u tou te inluence o tis variable on unds rankins seems to be only limited. Moreover, as a by-product o considerin te endoenous case one can determine optimal investments in und and reerence portolio, i.e te (preerence-independent) optimal structure o risky subportolio R(). V. Conclusion Tis paper examined teoretically relationsips between unds rankins or iven exoenous investor s oldins o a certain reerence portolio o stocks ( exoenous case ) and or decision situations were purcases o unds, stocks and riskless assets can simultaneously be optimized by investors ( endoenous case ). For te exoenous case Scolz/Wilkens (003) recommend a so-called investor speciic perormance measure wile or te endoenous case Breuer/Gürtler (1999, 000) derived te adequacy o te application o an optimized Sarpe measure. From a teoretical point o view te concept o te investor speciic perormance measure in particular enabled us to draw new conclusions reardin te relationsip between 0 Details are available rom te autors upon request. 15
classical perormance measures and te optimized Sarpe measure. Most importantly, in a situation wit sort sales restrictions, a und wit bot a ier Sarpe and a ier (positive) Treynor measure tan a und will be better tan und in te endoenous as well as te exoenous case reardless o te speciic parameters o te investor s portolio selection problem (i.e. or any desired overall expected excess return and any exoenously iven oldin o reerence portolio o direct equity oldin). Tis teoretical indin contributes particularly to our understandin o te relevance o te traditional perormance measures by Sarpe (1966) and Treynor (1965). Neverteless, teoretically, rankins in te exoenous case and in te endoenous one may dier considerably since tere are only a ew (loose) connections between tem. For tis reason, we analyzed empirically dierences in rankins or bot cases. Ater all, we did not ind suicient evidence tat a distinction between te endoenous and te exoenous case is o real practical importance. Certainly, tis result is practically important, since it indicates possibilities or te simpliication o unds selection problems. In particular, investors need not care muc about te question weter teir iven oldin o a portolio o stocks can be altered or not wen searcin or a ood und investment. Optimal und selections tus seem to be quite robust or mean-variance preerences and in eneral may be approximated rater well by a simple application o te traditional Sarpe or Treynor measure. Tis result is anoter indicator or te useulness o tese classical perormance measures even in complicated settins. Despite tis we recommend te application o te restricted optimized Sarpe measure as developed in Breuer/Gürtler (1999, 000) since it is based on te same inormation as te traditional perormance measures and ives te correct solution or te endoenous case so tat tere is no need to use an approximation o tat unds rankin as supplied by te simple Sarpe or Treynor measure. Appendix roo o te statement result (SW1) only olds true in te case 1 x > 0 : Denote wit σ Q() te covariance between te excess returns o subportolio Q() and o reerence portolio. In te case o a risk-averse investor wit mean-variance preerences, or iven overall expected return u, und is better tan und i te overall variance resultin rom te coice o und is lower tan te overall variance wen selectin und, i.e., 16
(A1) Var(x u (1 x ) u ) < Var(x u (1 x ) u ) Q() Q() (1 x ) σ x (1 x ) σ x σ Q() Q() < (1 x ) σ x (1 x ) σ x σ Q() Q() Q Q Q() x Q() uq() uq() u (x ) u (x ) σ σ Q(x ) u Q(x ) σ Q() x σq() uq() uq() u < Q 1 1 Q 1 x u(x) Q x u(x) Q σ u Q() uq() σ u Q() uq() σq() βq() σq() βq() u (x ) u (x ) 1 > ISM (x ) (1 x ) > ISM,ori. (x ) (1 x ). (8),ori. For te last equivalence we use te act tat te Sarpe measure and te Treynor measure cannot be inluenced by riskless lendin or borrowin and tus ϕ =ϕ Q() and ϕ =ϕ Q (). (A1) immediately implies te postulated statement. roo o Result 1: (R1.1) and (R1.). Results (R1.1) and (R1.) are obvious since te product x u (x ) = x (u x u ) is positive or all 0 < x < u / u and neative or all x Q \[0, u / u ], respectively. ϕ (R1.3). Since and are bot positive we ave ϕ 1/ ϕ 1/ ϕ ϕ ϕ. u /u = (x u u (x ))/u = x (u (x )/u ) > x i we rerain rom sort Moreover, Q Q sales possibilities. Tis directly yields 0 x rom part (R1.1). u /u and (R1.3) ten immediately ollows (R1.4). Te acts u Q(x = 0) = u > 0 and x = 0 directly imply te postulated statement under consideration o te deinition o ISM (x ). (R1.5). An overall expected return u = u δ = x u (1 x ) u Q() wit δ > 0 is only acievable in te case o u Q( ) u. Tus, u =u x = 1. Tis in turn implies ISM (x = 1) = 0 = ISM (x = 1) or arbitrary unds and. Because o 17
(A) ISM (x ) ISM (x ) u u x = 1, u = u x = 1, u = u 1 1 1 1 u(1) Q 1 u(1) Q 1 () σ () () σ ( ) ( ϕs ) ϕt ( ϕs ) ϕt = 0 1 1 ϕ ϕ () T () T result (R1.5) is obvious. (R1.6). First o all we sow tat a situation wit ϕ ( J ) 0 and ϕ ( J ) 0 may coincide simulta- 0 0 neously wit ϕ as well as ϕ >. To tis end, or eac und we use a linear reression accordin to te ordinary-least-squares metod to determine parameters α and β wit u = α β u ε, unds and may exibit te properties E( ε ) = 0, and Cov(u, ε ) = 0. In particular, te parameters or 0 u <β < and u u β as well as u Var( ε ) (J) and Var( ε ) and consequentially ϕ = u β u > 0 and ϕ (J) = u β u 0. In addition, we ave (A3) Var(u ) =β Var(u ) Var( ε ) and Var(u ) = β Var(u ) Var( ε ). p p Tus, te correspondin Sarpe measures are (A4) u ϕ = σ = βp Var(u ) Var( ε ) u 0, u ϕ = σ = βp Var(u ) Var( ε ) > u 0. Since te Treynor measures o bot unds are inite, we et ISM (x ) and ISM (x ) or all x u / u wic immediately implies (R1.6). (R1.7). Result (R1.7) ollows directly rom te situation presented in te proo o (R1.6). I we reverse te roles o unds and, it results ϕ > and ISM (x ) ISM (x ) or all x (J) 0 u / u. roo o Result : (R.1). From te derivation o te investor speciic perormance measure ISM at te beinnin o te Appendix and te assumption ISM (x ) ISM (x ),, underlyin statement 18
(R.1), we know tat an investment in und wit x = x, implies lower overall portolio x, risk tan an investment in und (wit x = ) since we ave a iven overall expected excess return u reardless o te und actually cosen by te investor. Tereore, better unds are caracterized by a lower variance o overall portolio return. As a consequence, or x = x, te Sarpe measure o te overall portolio includin und is ier tan te Sarpe measure o te overall portolio includin und, i.e. (A5) ϕ (x ) ϕ (x ).,, Since an optimal investment x, in und does not lead to a lower Sarpe measure in comparison to te Sarpe measure o an investment x, in und, (R.1) is obvious: (A6) ϕ =ϕ (x ) ϕ (x ) ϕ (x ) =ϕ.,,, (A5) (R.). Te proo o tis part is analoue to te proo o (R.1), i we replace x, wit x,,restr. and wit. ϕ ϕ,restr. (R.3). It is suicient to ive just one numerical example in wic ISM (x ) ISM (x ) or all x < 0 x > u /u (TB) (TB) coincides wit ( ϕ ) > ( ϕ ) and ϕ >ϕ. Under consideration o te linear reressions presented in te proo o (R1.6) we tereore look at two unds and wit α = 0.04, β = 0.8, and Var( ε ) = 0.004 as well as α = 0.0, β = 1.7, and Var( ε ) = 0.0005. In addition, we assume u 1.77 % = as well as σ = 0.004 and tus ave 1/ ϕ 168.48 < 14.77 1/ ϕ and ϕ 0.09 < 0.67 ϕ. From (R1.1) we know tat ISM (x ) ISM (x ) or all 0 x u /u. Since ( ϕ ) = 0.8> 0.4 = ( ϕ TB) ) (TB) ( and tus ϕ >ϕ te irst part o (R.3) is proven. I we cane te parameters to α = 0.0 and β = 0.8 we et 1/ ϕ 136.99 > 14.77 1/ ϕ and ϕ 0.11 < 0.67 ϕ. From (R1.), it ollows ISM (x ) ISM (x ) or all x < 0 x > u /u. In (TB) (TB) tis situation aain we ave ( ϕ ) = 0.8> 0.4 = ( ϕ ) and ϕ >ϕ. Tus, te second part o (R.3) is veriied, too. 19
(J) (R.4). Consider two unds and wit bot ϕ > and ϕ =0. Accordin to (BG1) o (J) 0 section II we ave x > 0 as well as x = 0 and tus ϕ >ϕ =ϕ. Te proo o (R1.6). points out tat te conditions ϕ > ( J ) 0 and ϕ ( J ) 0 do not necessarily lead to ISM (x ) ISM (x ) or any x u / u, so tat (R.4) is proven. (R.5). Aain, we look at te two unds and o te proo o (R.4). On te one and we ave ϕ =ϕ =ϕ. On te oter and we know rom (BG4) tat ϕ { ϕ, ϕ }.,restr.,restr. articularly, tis implies ϕ,restr. ϕ and (R.5) is proven by te same aruments as (R.4). roo o Result 3: (R3.1). From (R.1) we know we can restrict ourselves to sow ISM (x ) ISM (x ). Tis statement is equivalent to,, (A7) 1 1 u Q(x,) () x, σ u Q(x,) () ( ϕs ) ϕt 1 1 u Q(x,) x (), σ u Q(x,). () ( ϕs ) ϕt Under consideration o te assumptions o tis part as well as (A7) it is suicient to prove tat (A8) sn(x u (x )) = sn( ϕ ) sn( ϕ ). (invj) (J), Q, From x u x u = u and x u u (x ) = u we et x = u (x )/u. Tus, and Q, Q, Q, u (x ) ave te same sin. Toeter wit (BG1) and (BG4) o section II tis implies x (A9) sn( ϕ ) = sn(u (x )), (J) Q, (BG1) sn( ϕ ) = sn(x ). (invj), (BG4) (A9) immediately leads to te asserted result (A8). (R3.). Result (R3.) is a direct consequence o (R3.1) and (BG5). (R3.3). In accordance wit (R3.1) and under consideration o (R.), we only ave to sow: 0
(A10) Q,,restr. x,,restr. u(x Q,,restr. ) σ ( ϕ ) ϕ u (x ) 1 1 Q,,restr.,,restr. Q,,restr. σ ( ϕ ) u (x ) 1 1 x u(x ). ϕ Since tis result is obvious or x,,restr. = 0 (because o ϕ ϕ ), we only ave to treat te case and x > 0. Even in suc a situation we immediately et (A10), since,,restr. sn(x u (x )) = 1.,,restr. Q,,restr. 1/ ϕ 1/ ϕ Reerences Breuer, W./Gürtler, M. (1998): erormance Evaluation wit reard to Investor ortolio Structures and Skewness reerences. An Empirical Analysis or German Equity Funds, Bonn Workin apers FW 1/98U1. - Breuer, W./Gürtler, M. (1999): erormancemessun mittels Sarpe-, Jensen- und Treynor-Maß: Eine Anmerkun, Zeitscrit ür Bankrect und Bankwirtscat, Vol. 11, pp. 73-86. - Breuer, W./Gürtler, M. (000): erormancemessun mittels Sarpe-, Jensen- und Treynor-Maß: Eine Eränzun, Zeitscrit ür Bankrect und Bankwirtscat, Vol. 1, pp. 168-176. - Breuer, W./Gürtler, M. (003): erormance Evaluation and reerences beyond Mean-Variance, Financial Markets and ortolio Manaement, Vol. 17, pp. 13-33. - Breuer, W./Gürtler, M./Scumacer, F. (004): ortoliomanaement I. Grundlaen, nd ed., Wiesbaden: Gabler. - Cass, D./Stilitz, J. (1970): Te Structure o Investor reerences and Asset Returns, and Separability in ortolio Allocation: A Contribution to te ure Teory o Mutual Funds, Journal o Economic Teory, Vol., pp. 1-160. - Deutsce Boerse Group (003): Leitaden zu den Aktienindizes der Deutscen Börse Version 5.1, Gruppe Deutsce Börse Inormation Services. - Jensen, M.C. (1968): Te erormance o Mutual Funds in te eriod 1956-1964, Journal o Finance, Vol. 3, pp. 389-416. - Jobson, J. D./Korkie, B. (1984): On te Jensen Measure and Marinal Improvements in ortolio erormance a Note, Journal o Finance, Vol. 39, pp. 45-51. - Mölmann, J. (1993): Teoretisce Grundlaen und Metoden zweidimensionaler erormancemessun von Investmentonds: Eine metodisce und empirisce Untersucun zur Messun der Manaementleistunen deutscer Aktienanlaeonds, Stuttart: Scäer- oescel. - Reilly, F. K./Brown, K. C. (1997): Investment Analysis and ortolio Manaement, 5 t ed., Fort Wort: Dryden ress. - Roati, V. K. (1976): An Introduction to robability Teory and Matematical Statistics, New York: Wiley. - Scolz, H./Wilkens, M. (003): Zur Relevanz von Sarpe Ratio und Treynor Ratio, Zeitscrit ür Bankrect und Bankwirtscat, Vol. 15, pp. 1-8. - Sarpe, W. F. (1966): Mutual Fund erormance, Journal o Business, Vol. 39, pp. 119-138. - Tobin, J. (1958): Liquidity reerence as Beaviour towards Risk, Review o Economic Studies, Vol. 5, pp. 65-86. - Treynor, J. L. (1965): How to Rate Manaement o Investment Funds, Harvard Business Review, Vol. 43, January/February, pp. 63-75. - Treynor, J. L./ Black, F. (1973): How to Use Security Analysis to Improve ortolio Selection, Journal o Business, Vol. 46, pp. 66-86. 1
Fiure 1 Structure o te Investor s Optimal Overall ortolio Structure (x 0, x, x ) o te investor s optimal overall portolio or eac und Relative investment x in equity portolio Relative investment 1 x in portolio Q() Investment in und Investment in riskless assets risky subportolio R() In te endoenous case, or any und under consideration te investor simultaneously optimizes relative sares x 0 (o riskless assets), x (o te equity portolio ), and x (o te unds ). In te exoenous case, only x and x 0 (subportolio Q()) can be optimized, since x = x = const. (i.e. te saded component in Fiure 1 is iven).
Table 1 Synopsis o Relevant Symbols Assets:,,: investment unds, F: total number o unds, : best und out o all unds = 1,, F, : portolio o direct stock oldins (servin as te reerence portolio ). Investor s subportolios (bein part o te investor s total asset oldins): R(): risky subportolio, i.e. (only) investment in und and in reerence portolio, Q(): subportolio wic in te exoenous case is not already ixed, i.e. (only) investment in und and riskless lendin or borrowin. Return caracteristics: r 0 : riskless interest rate, r: % return o und, r % : return o reerence portolio, u % : excess return r% r 0 o wit expectation value u and standard deviation σ, u % : excess return r% r 0 o wit expectation value u and standard deviation σ, u % Q( ) : excess return r% Q( ) r 0 o Q() wit expectation value u Q( ) and standard deviation σ Q( ), u % R() : excess return r% R() r 0 o R() wit expectation value u R() and standard deviation σ R(), σ : covariance between u% and u, % β : =σ / σ (reression coeicient o a linear reression o u% wit respect to u % ), β : =σ / σ (reression coeicient o a linear reression o u% wit respect to u), % ε% : error term o a linear reression o u% wit respect to u, % σ standard deviation o error term ε%. ε : Decision variables: x 0 : raction o monetary wealt risklessly invested (x 0 < 0: borrowin o money), x : raction o monetary wealt invested in reerence portolio, x : raction o monetary wealt invested in sares o und. Speciic parameters or te exoenous case: u : overall expected excess return desired by te investor, x : percentae o initial wealt already ixed by an investment in te reerence portolio, u Q( )(x ) = u Q(x ): contribution o subportolio Q() to an investor s overall acievable expected excess return (independent o ). erormance measures: ϕ Sarpe measure o, : ϕ : Treynor measure o, (J) ϕ : Jensen measure o, (TB) ϕ : Treynor/Black measure o, (invj) ϕ : inverse Jensen measure o, ϕ : unrestricted optimized Sarpe measure o, ϕ,restr. : restricted optimized Sarpe measure o, ISM (x ) : oriinal investor speciic perormance measure o,,ori. ISM (x ) : modiied investor speciic perormance measure o. Optimal values are enerally caracterized by an asterisk ( ) and in te endoenous case wit sort sales restrictions additionally by an index restr.. Tildes ( ~ ) denote random variables.
Table Unbiased Estimators or Expectation Values u, Standard Deviations Covariances σ, and σ o Excess Returns o German Funds and Reerence ortolio No. name o und u 1 Aberdeen Global German Eq 0.46351 % 5.77708 % 0.33096 % ABN AMRO Germany Equity.4189 % 7.09676 % 0.409 % 3 ADIFONDS.1643 % 7.614 % 0.44304 % 4 Barin German Growt.85000 % 7.05836 % 0.33608 % 5 CB Lux ortolio Euro Aktien 1.79676 % 6.77890 % 0.4088 % 6 Concentra 1.85919 % 6.71783 % 0.41575 % 7 CS EF (Lux) Germany 1.5897 % 6.66003 % 0.40816 % 8 DekaFonds 1.91459 % 6.81638 % 0.4138 % 9 DELBRÜCK Aktien UNION-Fonds 1.4919 % 6.5 % 0.38175 % 10 Dexia Eq L Allemane C 1.67865 % 6.3957 % 0.38700 % 11 DIT Wacstumsonds 1.88919 % 6.8905 % 0.37674 % 1 DVG Fonds SELECT INVEST.0743 % 6.6111 % 0.4079 % 13 EMIF Germany Index plus B 1.57108 % 6.45667 % 0.40139 % 14 Flex Fonds 1.39730 % 5.98888 % 0.3654 % 15 Frankurter Sparinvest Deka 1.8134 % 6.41583 % 0.39600 % 16 FT Deutscland Dynamik Fonds 1.79459 % 6.5969 % 0.40786 % 17 Hauck Main I Universal Fonds 1.45865 % 6.5848 % 0.4051 % 18 Incoonds.13865 % 6.04074 % 0.3491 % 19 Interselex Equity Germany B 1.7514 % 6.60614 % 0.40989 % 0 Lux Linea 1.71378 % 7.60317 % 0.46976 % 1 Metallbank Aktienonds DWS.0734 % 5.14655 % 0.6836 % MK Alakapital 1.9843 % 7.41669 % 0.45851 % 3 MMWI ROGRESS Fonds 1.76081 % 6.71760 % 0.41379 % 4 arvest Germany C 1.60108 % 6.31697 % 0.39 % 5 lusonds.4034 % 6.83304 % 0.40050 % 6 ortolio artner Universal G 1.09946 % 6.08717 % 0.340 % 7 SMH Special UBS Fonds 1 1.90811 % 6.60503 % 0.40739 % 8 Tesaurus 1.7811 % 6.36330 % 0.39459 % 9 AC Deutscland 1.86378 % 7.0976 % 0.41137 % 30 Baer Multistock German Stk A 1.7770 % 5.4860 % 0.387 % 31 BBV Invest Union 1.90946 % 6.3097 % 0.38537 % 3 Berlinwerte Weberbank O 1.57595 % 5.68085 % 0.33807 % 33 DIT Fonds ür Vermöensbildun 1.3405 % 5.79650 % 0.34777 % 34 DWS Deutscland 1.60784 % 6.08441 % 0.36909 % 35 Fidelity Fds Germany 1.789 % 6.4931 % 0.37989 % 36 Gerlin Deutscland Fonds 1.41054 % 5.19347 % 0.3136 % 37 HANSAeekt 1.73973 % 6.49867 % 0.40096 % 38 INVESCO GT German Growt C 1.71649 % 5.67770 % 0.4657 % 39 Investa.11541 % 6.9485 % 0.4699 % 40 Köln Aktienonds DEKA 1.83865 % 6.5477 % 0.40355 % 41 Oppeneim Select 1.69757 % 6.47148 % 0.39475 % 4 Rin Aktienonds DWS 1.86784 % 6.15453 % 0.37430 % 43 Trinkaus Capital Fonds INKA 1.71541 % 6.49609 % 0.40013 % 44 UniFonds 1.74784 % 6.4735 % 0.39665 % 45 Universal Eect Fonds 1.74568 % 6.741 % 0.38306 % DAX 100 1.77189 % 6.4936 % 0.39055 % Only te rankin o unds # 1 to # 8 is te same or te Sarpe measure as well as te Treynor measure (see Table 3) and tus tey are separated by a orizontal line rom unds # 9 to # 45. σ σ
Sarpe ( ϕ ) and Treynor ( ϕ Table 3 ) Measures o German Funds wit Identical Resultin Rankins No. name o und ϕ ϕ 4 Barin German Growt 40.37767 % 3.31185 % 1 Metallbank Aktienonds DWS 40.8415 % 3.01718 % 18 Incoonds 35.40378 %.3943 % 5 lusonds 35.17095 %.34349 % ABN AMRO Germany Equity 34.1671 %.409 % 1 DVG Fonds SELECT INVEST 31.34768 % 1.98418 % 11 DIT Wacstumsonds 30.0393 % 1.95841 % 3 ADIFONDS 9.9514 % 1.906 % 7 SMH Special UBS Fonds 1 8.88873 % 1.891 % 15 Frankurter Sparinvest Deka 8.60 % 1.7887 % 8 DekaFonds 8.08814 % 1.77449 % 6 Concentra 7.67543 % 1.74649 % 16 FT Deutscland Dynamik Fonds 7.096 % 1.71840 % 8 Tesaurus 7.15740 % 1.71039 % 10 Dexia Eq L Allemane C 6.9036 % 1.69401 % MK Alakapital 6.7935 % 1.68859 % 5 CB Lux ortolio Euro Aktien 6.50514 % 1.6676 % 3 MMWI ROGRESS Fonds 6.1190 % 1.66191 % 19 Interselex Equity Germany B 6.11410 % 1.64370 % 4 arvest Germany C 5.34571 % 1.5946 % 13 EMIF Germany Index plus B 4.3368 % 1.5863 % 7 CS EF (Lux) Germany 3.7684 % 1.51466 % 14 Flex Fonds 3.33154 % 1.49411 % 9 DELBRÜCK Aktien UNION-Fonds.85890 % 1.4614 % 0 Lux Linea.54039 % 1.4479 % 17 Hauck Main I Universal Fonds.15170 % 1.40585 % 6 ortolio artner Universal G 18.06190 % 1.3444 % 1 Aberdeen Global German Eq 8.033 % 0.54696 %
Table 4 Correlation Coeicients between ISM-Rankins or Desired Expected Excess Return u =.3 % and dierent values (1) x and () x (in %) (1) x () x 0 % 5 % 10 % 15 % 0 % 5 % 30 % 35 % 40 % 45 % 50 % 55 % 60 % 65 % 70 % 75 % 80 % 85 % 90 % 95 % 100 % 0 % 100.0000 98.594 98.843 98.039 97.3039 97.3039 97.3039 96.335 96.335 95.588 95.0980 95.0980 94.859 94.859 94.859 93.675 93.675 93.675 93.675 93.1373 93.1373 5 % 98.594 100.0000 99.7549 99.5098 98.7745 98.7745 98.7745 98.594 98.594 97.7941 97.3039 97.3039 97.0588 97.0588 97.0588 95.8333 95.8333 95.8333 95.8333 95.3431 95.3431 10 % 98.843 99.7549 100.0000 99.7549 99.0196 99.0196 99.0196 98.7745 98.7745 98.039 97.5490 97.5490 97.3039 97.3039 97.3039 96.335 96.335 96.335 96.335 95.8333 95.8333 15 % 98.039 99.5098 99.7549 100.0000 99.5098 99.5098 99.5098 99.647 99.647 98.7745 98.843 98.843 98.039 98.039 98.039 97.3039 97.3039 97.3039 97.3039 97.0588 97.0588 0 % 97.3039 98.7745 99.0196 99.5098 100.0000 100.0000 100.0000 99.7549 99.7549 99.5098 99.647 99.647 99.0196 99.0196 99.0196 98.594 98.594 98.594 98.594 98.843 98.843 5 % 97.3039 98.7745 99.0196 99.5098 100.0000 100.0000 100.0000 99.7549 99.7549 99.5098 99.647 99.647 99.0196 99.0196 99.0196 98.594 98.594 98.594 98.594 98.843 98.843 30 % 97.3039 98.7745 99.0196 99.5098 100.0000 100.0000 100.0000 99.7549 99.7549 99.5098 99.647 99.647 99.0196 99.0196 99.0196 98.594 98.594 98.594 98.594 98.843 98.843 35 % 96.335 98.594 98.7745 99.647 99.7549 99.7549 99.7549 100.0000 100.0000 99.7549 99.5098 99.5098 99.647 99.647 99.647 98.7745 98.7745 98.7745 98.7745 98.594 98.594 40 % 96.335 98.594 98.7745 99.647 99.7549 99.7549 99.7549 100.0000 100.0000 99.7549 99.5098 99.5098 99.647 99.647 99.647 98.7745 98.7745 98.7745 98.7745 98.594 98.594 45 % 95.588 97.7941 98.039 98.7745 99.5098 99.5098 99.5098 99.7549 99.7549 100.0000 99.7549 99.7549 99.5098 99.5098 99.5098 99.647 99.647 99.647 99.647 99.0196 99.0196 50 % 95.0980 97.3039 97.5490 98.843 99.647 99.647 99.647 99.5098 99.5098 99.7549 100.0000 100.0000 99.7549 99.7549 99.7549 99.5098 99.5098 99.5098 99.5098 99.647 99.647 55 % 95.0980 97.3039 97.5490 98.843 99.647 99.647 99.647 99.5098 99.5098 99.7549 100.0000 100.0000 99.7549 99.7549 99.7549 99.5098 99.5098 99.5098 99.5098 99.647 99.647 60 % 94.859 97.0588 97.3039 98.039 99.0196 99.0196 99.0196 99.647 99.647 99.5098 99.7549 99.7549 100.0000 100.0000 100.0000 99.7549 99.7549 99.7549 99.7549 99.5098 99.5098 65 % 94.859 97.0588 97.3039 98.039 99.0196 99.0196 99.0196 99.647 99.647 99.5098 99.7549 99.7549 100.0000 100.0000 100.0000 99.7549 99.7549 99.7549 99.7549 99.5098 99.5098 70 % 94.859 97.0588 97.3039 98.039 99.0196 99.0196 99.0196 99.647 99.647 99.5098 99.7549 99.7549 100.0000 100.0000 100.0000 99.7549 99.7549 99.7549 99.7549 99.5098 99.5098 75 % 93.675 95.8333 96.335 97.3039 98.594 98.594 98.594 98.7745 98.7745 99.647 99.5098 99.5098 99.7549 99.7549 99.7549 100.0000 100.0000 100.0000 100.0000 99.7549 99.7549 80 % 93.675 95.8333 96.335 97.3039 98.594 98.594 98.594 98.7745 98.7745 99.647 99.5098 99.5098 99.7549 99.7549 99.7549 100.0000 100.0000 100.0000 100.0000 99.7549 99.7549 85 % 93.675 95.8333 96.335 97.3039 98.594 98.594 98.594 98.7745 98.7745 99.647 99.5098 99.5098 99.7549 99.7549 99.7549 100.0000 100.0000 100.0000 100.0000 99.7549 99.7549 90 % 93.675 95.8333 96.335 97.3039 98.594 98.594 98.594 98.7745 98.7745 99.647 99.5098 99.5098 99.7549 99.7549 99.7549 100.0000 100.0000 100.0000 100.0000 99.7549 99.7549 95 % 93.1373 95.3431 95.8333 97.0588 98.843 98.843 98.843 98.594 98.594 99.0196 99.647 99.647 99.5098 99.5098 99.5098 99.7549 99.7549 99.7549 99.7549 100.0000 100.0000 100 % 93.1373 95.3431 95.8333 97.0588 98.843 98.843 98.843 98.594 98.594 99.0196 99.647 99.647 99.5098 99.5098 99.5098 99.7549 99.7549 99.7549 99.7549 100.0000 100.0000
Table 5 Averae Correlation Coeicients between ISM-Rankins or Varyin Identical Values o Desired Expected Excess Return u and Identical Dierences x = x x between Exoenous Investments in Reerence ortolio (1) () x u 1.7719 % 1.90 %.00 %.10 %.0 %.30 %.40 %.50 %.60 %.70 % 10.00 % 0 % 100.00000 % 100.00000 % 100.00000 % 100.00000 % 100.00000 % 100.00000 % 100.00000 % 100.00000 % 100.00000 % 100.00000 % 100.00000 % 5 % 99.75490 % 99.77941 % 99.79167 % 99.79167 % 99.79167 % 99.8039 % 99.8039 % 99.8039 % 99.81618 % 99.84069 % 99.914 % 10 % 99.57430 % 99.61300 % 99.6590 % 99.6590 % 99.63880 % 99.65170 % 99.66460 % 99.66460 % 99.69040 % 99.67750 % 99.81940 % 15 % 99.41449 % 99.45534 % 99.45534 % 99.46895 % 99.4857 % 99.50980 % 99.534 % 99.534 % 99.55065 % 99.53704 % 99.7767 % 0 % 99.791 % 99.338 % 99.30796 % 99.338 % 99.3511 % 99.36563 % 99.38005 % 99.38005 % 99.40888 % 99.40888 % 99.61073 % 5 % 99.06556 % 99.1779 % 99.15748 % 99.18811 % 99.3407 % 99.4939 % 99.6471 % 99.800 % 99.800 % 99.800 % 99.551 % 30 % 98.8561 % 98.93791 % 98.93791 % 98.97059 % 99.03595 % 99.10131 % 99.13399 % 99.16667 % 99.15033 % 99.15033 % 99.4810 % 35 % 98.63445 % 98.7199 % 98.7199 % 98.75700 % 98.84454 % 98.84454 % 98.89706 % 98.96709 % 99.0010 % 99.0010 % 99.33473 % 40 % 98.4169 % 98.51056 % 98.49170 % 98.5941 % 98.6368 % 98.6368 % 98.6804 % 98.77451 % 98.73680 % 98.75566 % 99.4585 % 45 % 98.061 % 98.34559 % 98.8431 % 98.3516 % 98.36601 % 98.36601 % 98.44771 % 98.54984 % 98.50899 % 98.5941 % 99.1416 % 50 % 97.95009 % 98.1506 % 98.08378 % 98.1834 % 98.19519 % 98.1506 % 98.3975 % 98.8431 % 98.3975 % 98.603 % 99.01961 % 55 % 97.59804 % 97.8916 % 97.81863 % 97.8916 % 97.9900 % 97.94118 % 98.039 % 98.06373 % 98.01471 % 98.039 % 98.9157 % 60 % 97. % 97.5490 % 97.49455 % 97.5765 % 97.68519 % 97.68519 % 97.8135 % 97.90305 % 97.84858 % 97.8758 % 98.8898 % 65 % 96.84436 % 97.1010 % 97.08946 % 97.18137 % 97.4647 % 97.4647 % 97.57966 % 97.701 % 97.6109 % 97.67157 % 98.7134 % 70 % 96.4857 % 96.74370 % 96.63866 % 96.77871 % 97.0588 % 97.0588 % 97.6891 % 97.40896 % 97.33894 % 97.37395 % 98.56443 % 75 % 96.03758 % 96.4053 % 96.868 % 96.4053 % 96.56863 % 96.60948 % 96.85458 % 97.0588 % 96.9771 % 97.01797 % 98.40686 % 80 % 95.5884 % 95.98039 % 95.83333 % 95.98039 % 96.1745 % 96.549 % 96.7451 % 96.51961 % 96.4157 % 96.51961 % 98.359 % 85 % 94.97549 % 95.46569 % 95.8186 % 95.34314 % 95.5696 % 95.5884 % 95.71078 % 95.71078 % 95.7706 % 95.89461 % 98.10049 % 90 % 94.03595 % 94.60784 % 94.5614 % 94.60784 % 94.68954 % 94.93464 % 95.09804 % 95.09804 % 95.6144 % 95.01634 % 97.9575 % 95 % 9.64706 % 93.50490 % 93.99510 % 93.99510 % 94.11765 % 94.400 % 94.4859 % 94.4859 % 94.73039 % 94.3675 % 97.67157 % 100 % 91.17647 % 91.91176 % 9.8916 % 9.8916 % 9.8916 % 93.1375 % 93.1375 % 93.1375 % 93.6745 % 93.6745 % 97.3039 %