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  • 8/9/2019 Machine Learning and Finance III

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    M a c h i n e l e a r n i n g a n d p o r t f o l i o s e l e c t i o n s .

    L s z l G y r

    D e p a r t m e n t o f C o m p u t e r S c i e n c e a n d I n f o r m a t i o n T h e o r y

    B u d a p e s t U n i v e r s i t y o f T e c h n o l o g y a n d E c o n o m i c s

    g y o r f i @ s z i t . b m e . h u

    A b s t r a c t

    I n t h i s p a p e r t h e s t a t i c p o r t f o l i o a n d t h e d y n a m i c l o g - o p t i m a l p o r t -

    f o l i o f o r m e m o r y l e s s a n d f o r s t a t i o n a r y m a r k e t p r o c e s s e s i s s t u d i e d .

    W e i l l u s t r a t e t h e i r p e r f o r m a n c e ( a v e r a g e a n n u a l y i e l d ) f o r N Y S E

    d a t a .

    1 N o t a t i o n s .

    g o n d m k o n n o d F o l o n o m k n

    m n d n o o sI

    ; sP

    ; : : : PR

    d

    C

    D

    sn

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    n

    ; : : : ; s

    @ d A

    n

    A

    j E o m o n n s

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    o sn

    d n o o j E

    o n n E d n o d F s n o d o n o m l z D s

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    a I F f sn

    o n n l n d X

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    W

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    n d m o o

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  • 8/9/2019 Machine Learning and Finance III

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    s n o d o l l d o n n o m n l

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    n o m o o l

    o n n o n o f xn

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    d n o m o n

    o n d n n n l n j E o n n E d n

    o d F

    2 S t a t i c p o r t f o l i o .

    o o l o n l o d n m n F e o o l o

    o d n o d b a @ b @ I A ; : : : b @ d A A F j E o m o n n b @ j A o b

    d n o o o o n o n o 9 l n d n j F

    m o o l o o b n o n n o m o n n m o

    I F o o o l o o d n o d

    d

    a

    8

  • 8/9/2019 Machine Learning and Finance III

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    s b

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    > H o l l j a I ; : : : ; d n o n d m l

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    n

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    D n o o l o m m l o m x

    j

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    3 D y n a m i c p o r t f o l i o s e l e c t i o n : s p e c i a l c a s e

    y n n n o o l o n n n m n F

    o n n l l n d o o l o @ g A l d n m o o l o l E

    o n F d n m o o l o l o n m l E o d n m n E

    D n n n o d n o d n l

    m o n F o n o g n o d d n d d d

    u l l P R D v n P S D f m n U D w k o z P V D p n k l n n d E

    l I Q D w Q H D w n d z k l Q Q n d f o n n d g o R F l o

    g T n d I S n g o n d o m I P D n d g I S n v n

    P T F

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    n

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  • 8/9/2019 Machine Learning and Finance III

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    m o o o o o l o l o n

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    a I F

    4 L o g - o p t i m a l p o r t f o l i o f o r m e m o r y l e s s m a r k e t

    p r o c e s s .

    s m k o fX

    i

    m m o l D F F D n o n d E

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    o o n n l l n d o o l o @ f g A l o E

    o m l o o l o X

    b X a r g m x b

    E

    f l n h b ; XI

    i :

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    n

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    n d l o E o m m o o l o b D n o n o o l o

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    i A

    P

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    I

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    i :

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    n

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    n n d n o n F

    g o n n n o n d o n @ A D f l m n d u l S D s n n d g o P Q D

    s n P I D P P D f o k n d n T D n d Q T o n d d

    o l m o n o n o F p o m m o l o w k o n m k o D

    o m l n n o d d d o n o m k

    S

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    f o

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    i

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    ;

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    r g m x

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    o o l o o n I o o n D E

    f XI

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    ~b a r g m x bX Var@ h b ; X

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    p o o o o D o m n @ o A o

    ~b n l o o o m n o o m l

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    l l o n o

    ~b n o n l n o m m n @ x v A o l m D

    l n n o n m m z d n d d o n n F

    s n o d o l l l o E o m l o o l o b D o n o k n o

    d o n o XI

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    o m l o o l o n d ( n d

    bn

    a r g m x

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    n

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    i

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    o o m l o o l o bn

    n d m o d ( o n d d

    w Q I D Q P n d w o Q R D Q S F

    U

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    l l o n o bn

    x v o l m D o o F g o I H n o d d n

    l o m o l l n b

    n

    F e n l n o l o o E

    n d o n l p 2 o l l Q V F o n d o n n l d

    o m m n m o d D o m n l l l o l o l o n o x v

    o l n d o m m n o l m n d m l o l

    m m m o d n o l o l m m m F

    5 E x a m p l e s f o r c o n s t a n t l y r e b a l a n c e d p o r t f o -

    l i o .

    E x a m p l e 1 . g o n d m l o d a P n d X a @ X @ I A ; X @ P A A

    ( o m o n n X

    @ I A

    o n o X n ( l o k X

    X

    @ I A

    a

    (

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    I = P o l I = P D

    @ I A

    n d o n d o m o n n X

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    X

    X

    @ P A

    a I :

    y o l D z o o F n o n o (

    o m o n n

    E

    f X

    @ I A

    a I = P @ P C I = P A a S = R > I ;

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    E

    f S

    @ I A

    n

    aE

    (

    n

    Y

    i a I

    X

    @ I A

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    )

    a @ S = R A

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    ; @ P A

    o E

    f S

    @ I A

    n

    o o n n l l F r o D d o n o m l

    n d o m l S

    @ I A

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    o o n n l l D o o F v 9 l l o

    W

    @ I A

    X

    W

    @ I A

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    n

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    n

    n

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    i a I

    l n X

    @ I A

    i

    aE

    f l n X

    @ I A

    a I = P l n P C I = P l n @ I = P A a H ;

    V

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    m n ( o m o n n z o o D o o F v 9

    l l l o E o m l o o l o o n o D o o m E

    o n n z o o F o o l o o o m

    b a @ b ; I b A :

    n

    W @ bA a E f l n h b ; Xi

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    a I = P l n @ I C b A @ I b = P A :

    y n n k W @ bA m m m o b a I = P D o l o E o m l

    o o l o

    b a @ I = P ; I = P A ;

    n d m o o

    W

    aE

    f l n h b ; Xi a I = P l n @ W = V A a H : H S W ;

    o o F

    E x a m p l e 2 . g o n d m l o d a Q n d X a @ X @ I A ; X @ P A ; X @ Q A A

    ( n d o n d o m o n n o n o X

    ( l o k o o m @ I A D l d o m o n n F y n

    n o l o E o m l o o l o

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    n d m m l m o o

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    ; : : : ; X

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    ( d I o m o n n o n o X

    ( l

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    l o E o m l o o l o

    b a @ I = @ d I A ; : : : ; I = @ d I A ; H A ;

    W

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    m n D o d > Q D o d n o l o E o m l o o l o

    z o F v N d n o n m o o m o n n o X l

    o P D n N n o m l l d d m @ d I ; I = P A D n d

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    !

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    n

    ;

    o m n o k

    S

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    f S

    n

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    e m o j n o l p

    j

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    j

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    m n n n I 6 o n o j l n o

    j

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    H 6 F n n o o o m

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    o

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    6 S e m i - l o g - o p t i m a l p o r t f o l i o .

    j d R H d n o m o n o b n d bn

    n

    h @ z A X a z I

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    @ z I A

    P

    ;

    o n d o d l o n o n o n o n l n z z a I F

    n D m E l o E o m l o o l o l o n

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    E

    f h @ h b ; xI

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    "bn

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    I P

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    o m m n D d o l D F F D n o n Q u a d P r o g + + o h

    q o I R F o m q o l d E s d n n d l m o d o o l n

    d o m m n o l m I S F s o m C

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    l C o d ( n F d ' n n o d n o l m

    n m n D n o l m l o m m C

    l l d o m d D n d o C n o l l E n k m D m l

    C o n l o m E d ( n F

    7 N u m e r i c a l c o m p a r i s o n o f l o g - o p t i m a l a n d s e m i -

    l o g - o p t i m a l p o r t f o l i o s .

    l I m m z n m l l o N Y S E d F e

    w w w F s z i t F m e F h u G o t i G p o r t f o l i o o n m k d o m

    N Y S E X

    ( d o n o d l d o Q T o k l n P P

    @ S T S I d n d n d n n I W V S A F d n d

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    v o D d n I U D n q ( D d n D l k I W n d n q ( D n D

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