47:
containing all securities in proportion to their market value and an 'active portfolio' containing the securities for which the investor has made a prediction about alpha. In the active portfolio the weight of each stock is proportional to the alpha value divided by the variance of the residual risk.
884:≤ 1 i.e short positions in the market portfolio or active portfolio could be initiated to leverage a position in the other portfolio. This is often regarded as the major flaw of the model, as it often yields an unrealistic weight in the active portfolio. Imposing lower and upper bounds for
404:
726:
210:
596:
816:
473:
527:
290:
865:
240:
267:
95:
614:
409:(Note that if an alpha is negative the corresponding portfolio weight will also be negative, i.e. the active portfolio is in general a long–short portfolio).
107:
915:
Treynor, J. L. and F. Black, 1973, How to Use
Security Analysis to Improve Portfolio Selection, Journal of Business, January, pages 66–88.
926:
533:
737:
608:
invested in the active portfolio and the remainder invested in the market portfolio. This active fraction is found as follows:
425:
412:
The alpha, beta and residual risk of the constructed active portfolio are found using the previously computed weights
479:
945:
33:
925:
Kane, Kim and White: Active
Portfolio Management - The power of the Treynor–Black model, December 2003,
399:{\displaystyle w_{i}={\frac {\alpha _{i}/\sigma _{i}^{2}}{\sum _{j=1}^{N}\alpha _{j}/\sigma _{j}^{2}}}}
940:
101:
securities that have been analyzed and are thought to be mispriced, with expected returns given by:
824:
218:
245:
73:
36:, but who believes they have information that can be used to predict the abnormal performance (
269:, and are mutually uncorrelated. (This is the so-called Diagonal Model of Stock Returns, or
721:{\displaystyle w_{0}={\frac {\alpha _{A}/\sigma _{A}^{2}}{(R_{M}-R_{F})/\sigma _{M}^{2}}}}
8:
40:) of a few of them; the model finds the optimum portfolio to hold under such conditions.
916:
270:
37:
274:
32:
in 1973. The model assumes an investor who considers that most securities are
934:
205:{\displaystyle r_{i}=R_{F}+\beta _{i}(R_{M}-R_{F})+\alpha _{i}+\epsilon _{i}}
43:
In essence the optimal portfolio consists of two parts: a passively invested
25:
29:
920:
44:
601:
The overall risky portfolio for the investor consists of a fraction
280:
Then it was shown by
Treynor and Black that the active portfolio
17:
731:
And corrected for the beta exposure of the active portfolio:
591:{\displaystyle \sigma _{A}^{2}=\sum w_{i}^{2}\sigma _{i}^{2}}
811:{\displaystyle w_{A}={\frac {w_{0}}{1+(1-\beta _{A})w_{0}}}}
24:
is a mathematical model for security selection published by
242:
are normally distributed with mean 0, standard deviation
827:
740:
617:
536:
482:
428:
293:
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221:
110:
76:
859:
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720:
590:
521:
467:
398:
261:
234:
204:
89:
468:{\displaystyle \alpha _{A}=\sum w_{i}\alpha _{i}}
932:
522:{\displaystyle \beta _{A}=\sum w_{i}\beta _{i}}
933:
13:
63:and the expected market return is
56:Assume that the risk-free rate is
14:
957:
284:is constructed using the weights
901:
891:is a measure to counter this.
792:
773:
692:
666:
173:
147:
1:
894:
870:The model is not bounded 0 ≤
860:{\displaystyle w_{M}=1-w_{A}}
235:{\displaystyle \epsilon _{i}}
7:
262:{\displaystyle \sigma _{i}}
90:{\displaystyle \sigma _{M}}
10:
962:
70:with standard deviation
51:
215:where the random terms
861:
812:
722:
592:
523:
469:
400:
362:
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236:
206:
91:
862:
813:
723:
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92:
825:
738:
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534:
480:
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291:
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108:
74:
714:
663:
587:
572:
551:
392:
339:
22:Treynor–Black model
946:Portfolio theories
857:
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649:
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558:
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519:
465:
396:
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325:
271:Single-index model
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87:
34:priced efficiently
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394:
275:William F. Sharpe
953:
941:Financial models
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27:
26:Fischer Black
23:
19:
903:
885:
878:
877:≤ 1 and 0 ≤
871:
869:
730:
602:
600:
413:
411:
408:
281:
279:
214:
98:
97:. There are
64:
57:
55:
42:
30:Jack Treynor
21:
15:
907:Kane et al.
935:Categories
895:References
45:index fund
845:−
784:β
780:−
702:σ
680:−
651:σ
636:α
575:σ
556:∑
539:σ
511:β
497:∑
485:β
457:α
443:∑
431:α
380:σ
365:α
344:∑
327:σ
312:α
251:σ
224:ϵ
194:ϵ
181:α
161:−
139:β
79:σ
921:2351280
273:due to
18:finance
919:
917:JSTOR
52:Model
38:Alpha
28:and
20:the
419::
277:).
16:In
937::
888:A
886:w
881:M
879:w
874:A
872:w
853:A
849:w
842:1
839:=
834:M
830:w
801:0
797:w
793:)
788:A
777:1
774:(
771:+
768:1
762:0
758:w
752:=
747:A
743:w
711:2
706:M
697:/
693:)
688:F
684:R
675:M
671:R
667:(
660:2
655:A
646:/
640:A
629:=
624:0
620:w
605:A
603:w
584:2
579:i
569:2
564:i
560:w
553:=
548:2
543:A
515:i
505:i
501:w
494:=
489:A
461:i
451:i
447:w
440:=
435:A
416:i
414:w
389:2
384:j
375:/
369:j
359:N
354:1
351:=
348:j
336:2
331:i
322:/
316:i
305:=
300:i
296:w
282:A
255:i
228:i
198:i
190:+
185:i
177:+
174:)
169:F
165:R
156:M
152:R
148:(
143:i
135:+
130:F
126:R
122:=
117:i
113:r
99:N
83:M
67:M
65:R
60:F
58:R
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