machine learning and finance ii

8/9/2019 Machine Learning and Finance II

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Machine learning and portfolio selections. II.

Laszlo (Laci) Gyorfi1

1Department of Computer Science and Information TheoryBudapest University of Technology and Economics

Budapest, Hungary

August 8, 2007

e-mail: [email protected]/gyorfiwww.szit.bme.hu/oti/portfolio

Gyorfi Machine learning and portfolio selections. II.
http://find/http://goback/


2/103

Dynamic portfolio selection: general case

xi = (x(1)i , . . . x

(d)i ) the return vector on day i

b = b1 is the portfolio vector for the first day

initial capital S0

S1 = S0 b1 , x1



3/103


xi = (x(1)i , . . . x



initial capital S0

S1 = S0 b1 , x1

for the second day, S1 new initial capital, the portfolio vectorb2 = b(x1)

S2 = S0 b1 , x1 b(x1) , x2 .



4/103


xi = (x(1)i , . . . x



initial capital S0

S1 = S0 b1 , x1

for the second day, S1 new initial capital, the portfolio vectorb2 = b(x1)

S2 = S0 b1 , x1 b(x1) , x2 .

nth day a portfolio strategy bn = b(x1, . . . , xn1) = b(xn11 )

Sn = S0

n

i=1b(xi11 ) , xi = S0enWn(B)

with the average growth rate

Wn(B) =1

n

n

i=1ln

b(xi11 ) , xi

.



5/103

log-optimum portfolio

X1, X2, . . . drawn from the vector valued stationary and ergodicprocesslog-optimum portfolio B = {b()}

E{ln

b(Xn11 ) , Xn

| Xn11 } = maxb()

E{ln

b(Xn11 ) , Xn

| Xn11 }

Xn11 = X1, . . . , Xn1



6/103

Optimality

Algoet and Cover (1988): If Sn = Sn(B) denotes the capital after

day n achieved by a log-optimum portfolio strategy B, then for

any portfolio strategy B with capital Sn = Sn(B) and for anyprocess {Xn}

,

lim supn

1

nln Sn

1

nln Sn 0 almost surely



7/103

Optimality

Algoet and Cover (1988): If Sn = Sn(B) denotes the capital after

day n achieved by a log-optimum portfolio strategy B, then for

any portfolio strategy B with capital Sn = Sn(B) and for anyprocess {Xn}

,

lim supn

1

nln Sn

1

nln Sn 0 almost surely

for stationary ergodic process {Xn},

limn

1

nln Sn = W

almost surely,

where

W = E

maxb()

E{ln

b(X1) , X0

| X1}

is the maximal growth rate of any portfolio.



8/103

Martingale difference sequences

for the proof of optimality we use the concept of martingale

differences:

Definition

there are two sequences of random variables:

{Zn} {Xn}

Zn is a function of X1, . . . , Xn,

E{Zn | X1, . . . , Xn1} = 0 almost surely.

Then {Zn} is called martingale difference sequence with respect to{Xn}.



9/103

A strong law of large numbers

Chow Theorem: If {Zn} is a martingale difference sequence withrespect to {Xn} and

n=1

E{Z2n }

n2

<

then

limn

1

n

ni=1

Zi = 0 a.s.


A k l f l b


10/103

A weak law of large numbers

Lemma: If {Zn} is a martingale difference sequence with respectto {Xn} then {Zn} are uncorrelated.


A k l f l b


11/103



Proof. Put i < j.E{ZiZj}


A k l f l b


12/103



Proof. Put i < j.E{ZiZj} = E{E{ZiZj | X1, . . . , Xj1}}


A k l f l b


13/103




= E{ZiE{Zj | X1, . . . , Xj1}}


A k l f l b s


14/103




= E{ZiE{Zj | X1, . . . , Xj1}}

= E{Zi 0}




15/103




= E{ZiE{Zj | X1, . . . , Xj1}}

= E{Zi 0} = 0




16/103




= E{ZiE{Zj | X1, . . . , Xj1}}

= E{Zi 0} = 0

Corollary

E

1

n

ni=1

Zi

2




17/103




= E{ZiE{Zj | X1, . . . , Xj1}}

= E{Zi 0} = 0

Corollary

E

1

n

ni=1

Zi

2

=1

n2

ni=1

nj=1

E{ZiZj}




18/103




= E{ZiE{Zj | X1, . . . , Xj1}}

= E{Zi 0} = 0

Corollary

E

1

n

ni=1

Zi

2

=1

n2

ni=1

nj=1

E{ZiZj}

=1

n2

ni=1

E{Z2i }




19/103




= E{ZiE{Zj | X1, . . . , Xj1}}

= E{Zi 0} = 0

Corollary

E

1

n

ni=1

Zi

2

=1

n2

ni=1

nj=1

E{ZiZj}

=1

n2

ni=1

E{Z2i }

0

if, for example,E

{Z2i } is a bounded sequence.


Constructing martingale difference sequence


20/103


{Yn} is an arbitrary sequence such that Yn is a function of

X1, . . . , Xn




21/103



X1, . . . , XnPut

Zn = Yn E{Yn | X1, . . . , Xn1}

Then {Zn} is a martingale difference sequence:




22/103



X1, . . . , XnPut

Zn = Yn E{Yn | X1, . . . , Xn1}






23/103



X1, . . . , XnPut

Zn = Yn E{Yn | X1, . . . , Xn1}



E{Zn | X1, . . . , Xn1}




24/103

C g g q


X1, . . . , XnPut

Zn = Yn E{Yn | X1, . . . , Xn1}



E{Zn | X1, . . . , Xn1}

= E{Yn E{Yn | X1, . . . , Xn1} | X1, . . . , Xn1}

= 0

almost surely.


Optimality


25/103

p y

log-optimum portfolio B = {b()}

E{ln

b(Xn11 ) , Xn

| Xn11 } = maxb()

E{ln

b(Xn11 ) , Xn

| Xn11 }


Optimality


26/103

p y

log-optimum portfolio B = {b()}

E{ln

b(Xn11 ) , Xn

| Xn11 } = maxb()

E{ln

b(Xn11 ) , Xn

| Xn11 }

If Sn = Sn(B) denotes the capital after day n achieved by a

log-optimum portfolio strategy B, then for any portfolio strategyB with capital Sn = Sn(B) and for any process {Xn}

,

lim supn

1n ln S

n 1n ln Sn

0 almost surely


Proof of optimality


27/103

p y

1

n

ln Sn =1

n

n

i=1

lnb(Xi11

) , Xi


Proof of optimality


28/103

1

n

ln Sn =1

n

n

i=1

lnb(Xi11 ) , Xi=

1

n

ni=1

E{ln

b(Xi11 ) , Xi

| Xi11 }

+1

n

ni=1

ln

b(Xi11 ) , Xi

E{ln

b(Xi11 ) , Xi

| Xi11 }


Proof of optimality


29/103

1

n

ln Sn =1

n

n

i=1

lnb(Xi11 ) , Xi=

1

n

ni=1

E{ln

b(Xi11 ) , Xi

| Xi11 }

+1

n

ni=1

ln

b(Xi11 ) , Xi

E{ln

b(Xi11 ) , Xi

| Xi11 }

and

1

n ln Sn =

1

n

ni=1

E{ln

b

(Xi11 ) , Xi

| X

i11 }

+1

n

ni=1

ln

b(Xi11 ) , Xi

E{ln

b(Xi11 ) , Xi

| Xi11 }


Universally consistent portfolio


30/103

These limit relations give rise to the following definition:

Definition

An empirical (data driven) portfolio strategy B is called

universally consistent with respect to a class C of stationaryand ergodic processes {Xn}

, if for each process in the class,

limn

1

nln Sn(B) = W

almost surely.


Empirical portfolio selection


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E{ln b(Xn11 ) , Xn | X

n11 } = max

b()E{ln b(X

n11 ) , Xn | X

n11 }




32/103


n11 } = max

b()E{ln b(X

n11 ) , Xn | X

n11 }

fixed integer k > 0

E{ln

b(Xn11 ) , Xn

| Xn11 } E{ln

b(Xn1nk) , Xn

| Xn1nk}




33/103


n11 } = max

b()E{ln b(X

n11 ) , Xn | X

n11 }

fixed integer k > 0

E{ln

b(Xn11 ) , Xn

| Xn11 } E{ln

b(Xn1nk) , Xn

| Xn1nk}

and

b(Xn11 ) bk(Xn1nk) = argmaxb()

E{ln

b(Xn1nk) , Xn

| Xn1nk}




34/103


n11 } = max

b()E{ln b(X

n11 ) , Xn | X

n11 }

fixed integer k > 0

E{ln

b(Xn11 ) , Xn

| Xn11 } E{ln

b(Xn1nk) , Xn

| Xn1nk}

and


E{ln

b(Xn1nk) , Xn

| Xn1nk}

because of stationarity

bk(xk1 ) = argmax

b()E{lnb(x

k1 ) , Xk+1 | X

k1 = x

k1 }

= argmaxb

E{ln b , Xk+1 | Xk1 = x

k1 },




35/103


n11 } = max

b()E{ln b(X

n11 ) , Xn | X

n11 }

fixed integer k > 0

E{ln

b(Xn11 ) , Xn

| Xn11 } E{ln

b(Xn1nk) , Xn

| Xn1nk}

and


E{ln

b(Xn1nk) , Xn

| Xn1nk}

because of stationarity

bk(xk1 ) = argmax

b()E{lnb(x

k1 ) , Xk+1 | X

k1 = x

k1 }

= argmaxb

E{ln b , Xk+1 | Xk1 = x

k1 },

which is the maximization of the regression function

mb(xk1 ) = E{ln b , Xk+1 | X

k1 = x

k1 }


Regression function


36/103

Y real valuedX observation vector


Regression function


37/103

Y real valuedX observation vectorRegression function

m(x) = E{Y | X = x}

i.i.d. data: Dn = {(X1, Y1), . . . , (Xn, Yn)}


Regression function


38/103


m(x) = E{Y | X = x}

i.i.d. data: Dn = {(X1, Y1), . . . , (Xn, Yn)}Regression function estimate

mn(x) = mn(x, Dn)


Regression function


39/103


m(x) = E{Y | X = x}


mn(x) = mn(x, Dn)

local averaging estimates

mn(x) =

ni=1

Wni(x; X1, . . . , Xn)Yi


Regression function


40/103


m(x) = E{Y | X = x}


mn(x) = mn(x, Dn)

local averaging estimates

mn(x) =

ni=1

Wni(x; X1, . . . , Xn)Yi

L. Gyorfi, M. Kohler, A. Krzyzak, H. Walk (2002) ADistribution-Free Theory of Nonparametric Regression,Springer-Verlag, New York.


Correspondence


41/103

X Xk1


Correspondence


42/103

X Xk1

Y ln b , Xk+1


Correspondence


43/103

X Xk1

Y ln b , Xk+1m(x) = E{Y | X = x} mb(x

k1 ) = E{ln b , Xk+1 | X

k1 = x

k1 }


Partitioning regression estimate


44/103

Partition Pn = {An,1, An,2 . . . }




45/103

Partition Pn = {An,1, An,2 . . . }An(x) is the cell of the partition Pn into which x falls




46/103


mn(x) =

ni=1 YiI[XiAn(x)]ni=1 I[XiAn(x)]




47/103


mn(x) =


Let Gn be the quantizer corresponding to the partition Pn:Gn(x) = j if x An,j.




48/103


mn(x) =


Let Gn be the quantizer corresponding to the partition Pn:Gn(x) = j if x An,j.the set of matches

In = {i n : Gn(x) = Gn(Xi)}

Then

mn(x) =

iIn

Yi

|In|.


Partitioning-based portfolio selection


49/103

fix k, = 1, 2, . . .P = {A,j,j = 1, 2, . . . , m} finite partitions ofRd,




50/103

fix k, = 1, 2, . . .P = {A,j,j = 1, 2, . . . , m} finite partitions ofRd,G be the corresponding quantizer: G(x) = j, if x A,j.




51/103

fix k, = 1, 2, . . .

P = {A,j,j = 1, 2, . . . , m} finite partitions ofRd,G be the corresponding quantizer: G(x) = j, if x A,j.G(x

n1) = G(x1), . . . , G(xn),




52/103

fix k, = 1, 2, . . .


n1) = G(x1), . . . , G(xn),

the set of matches:

Jn = {k < i < n : G(xi1ik) = G(xn1nk)}




53/103

fix k, = 1, 2, . . .


n1) = G(x1), . . . , G(xn),

the set of matches:

Jn = {k < i < n : G(xi1ik) = G(xn1nk)}

b(k,)(xn11 ) = argmaxb iJn

ln b , xi

if the set In is non-void, and b0 = (1/d, . . . , 1/d) otherwise.


Elementary portfolios


54/103

for fixed k, = 1, 2, . . .,B(k,) = {b(k,)()}, are called elementary portfolios


Elementary portfolios


55/103

for fixed k, = 1, 2, . . .,B(k,) = {b(k,)()}, are called elementary portfolios

That is, b(k,)n quantizes the sequence x

n11 according to the

partition P, and browses through all past appearances of the lastseen quantized string G(x

n1nk) of length k.

Then it designs a fixed portfolio vector according to the returns onthe days following the occurence of the string.


Combining elementary portfolios


56/103

How to choose k,

small k or small : large bias

large k and large : few matching, large variance


Combining elementary portfolios


57/103

How to choose k,

small k or small : large bias

large k and large : few matching, large variance

Machine learning: combination of experts

N. Cesa-Bianchi and G. Lugosi, Prediction, Learning, and Games.Cambridge University Press, 2006.


Exponential weighing

combine the elementary portfolio strategies B(k,) = {b(k,)n }


58/103

y p g { n }





59/103

y p g { n }let {qk,} be a probability distribution on the set of all pairs (k, )such that for all k, , qk, > 0.





60/103

y p g { }let {qk,} be a probability distribution on the set of all pairs (k, )such that for all k, , qk, > 0.

for > 0 putwn,k, = qk,e

lnSn1(B(k,))





61/103

g { }let {qk,} be a probability distribution on the set of all pairs (k, )such that for all k, , qk, > 0.


lnSn1(B(k,))

for = 1,

wn,k, = qk,elnSn

1(B(k,)

) = qk,Sn1(B(k,

))


Exponential weighingcombine the elementary portfolio strategies B(k,) = {b

(k,)n }


62/103

let {qk,} be a probability distribution on the set of all pairs (k, )such that for all k, , qk, > 0.


lnSn1(B(k,))

for = 1,

wn,k, = qk,elnSn

1(B(k,)

) = qk,Sn1(B(k,

))

andvn,k, =

wn,k,

i,jwn,i,j

.


Exponential weighingcombine the elementary portfolio strategies B(k,) = {b

(k,)n }


63/103

let {qk,} be a probability distribution on the set of all pairs (k, )such that for all k, , qk, > 0.


lnSn1(B(k,))

for = 1,

wn,k, = qk,elnS

n

1(B(k,))

= qk,Sn1(B(k,)

)

andvn,k, =

wn,k,

i,jwn,i,j

.

the combined portfolio b:

bn(xn11 ) =

k,

vn,k,b(k,)n (x

n11 ).


n


64/103

Sn(B) =n

i=1bi(x

i11 ) , xi


n


65/103

Sn(B) =n

i=1bi(x

i11 ) , xi

=n

i=1

k, wi,k,

b

(k,)i (x

i11 ) , xi

k, wi,k,


n


66/103

Sn(B) =n

i=1bi(x

i11 ) , xi

=n

i=1

k, wi,k,

b

(k,)i (x

i11 ) , xi

k, wi,k,

=

ni=1

k, qk,Si1(B(k,))b(k,)i (xi11 ) , xik, qk,Si1(B

(k,))


n


67/103

Sn(B) =n

i=1bi(x

i11 ) , xi

=n

i=1

k, wi,k,

b

(k,)i (x

i11 ) , xi

k, wi,k,

=

ni=1


(k,))

=n

i=1

k, qk,Si(B(k,))

k, qk,Si1(B(k,))


n


68/103

Sn(B) =n

i=1bi(x

i11 ) , xi

=n

i=1

k, wi,k,

b

(k,)i (x

i11 ) , xi

k, wi,k,

=

ni=1


(k,))

=n

i=1

k, qk,Si(B(k,))

k, qk,Si1(B(k,))

=k,

qk,Sn(B(k,)),



69/103

The strategy B = BH then arises from weighing the elementary

portfolio strategies B(k,) = {b(k,)n } such that the investors capital

becomesSn(B) =

k,

qk,Sn(B(k,)).


Theorem


70/103

Assume that

(a) the sequence of partitions is nested, that is, any cell of P+1 isa subset of a cell of P, = 1, 2, . . .;

(b) ifdiam(A) = supx,yA x y denotes the diameter of a set,then for any sphere S centered at the origin

lim

maxj:A,jS=

diam(A,j) = 0 .

Then the portfolio scheme BH defined above is universallyconsistent with respect to the class of all ergodic processes suchthat E{| ln X(j)| < , for j = 1, 2, . . . , d.



71/103

L. Gyorfi, D. Schafer (2003) Nonparametric prediction, inAdvances in Learning Theory: Methods, Models and Applications,J. A. K. Suykens, G. Horvath, S. Basu, C. Micchelli, J. Vandevalle(Eds.), IOS Press, NATO Science Series, pp. 341-356.

www.szit.bme.hu/gyorfi/histog.ps


ProofWe have to prove that

1


72/103

lim infn

Wn(B) = lim infn

1

nln Sn(B) W

a.s.

W.l.o.g. we may assume S0 = 1, so that

Wn(B) =1

nln Sn(B)



1


73/103

lim infn

Wn(B) = lim infn

1

nln Sn(B) W

a.s.


Wn(B) =1

nln Sn(B)

=

1

n ln

k,qk,Sn(B

(k,)

)



1


74/103

lim infn

Wn(B) = lim infn

1

nln Sn(B) W

a.s.


Wn(B) =1

nln Sn(B)

=

1

n ln

k,qk,Sn(B

(k,)

)

1

nln

supk,

qk,Sn(B(k,))



1


75/103

lim infn

Wn(B) = lim infn

1

nln Sn(B) W

a.s.


Wn(B) =1

nln Sn(B)

=

1

n ln

k,qk,Sn(B

(k,)

)

1

nln

supk,

qk,Sn(B(k,))

= 1n

supk,

ln qk, + ln Sn(B(k,))



1


76/103

lim infn

Wn(B) = lim infn

1

nln Sn(B) W

a.s.


Wn(B) =1

nln Sn(B)

=

1

n ln

k,qk,Sn(B

(k,)

)

1

nln

supk,

qk,Sn(B(k,))

= 1n

supk,

ln qk, + ln Sn(B(k,))

= sup

k,

Wn(B

(k,)) +ln qk,

n

.


Thus

l


77/103

lim infn

Wn(B) lim infn

sup

k,Wn(B(k,)) +

ln qk,n


Thus

l


78/103

lim infn

Wn(B) lim infn

sup

k,Wn(B(k,)) +

ln qk,n

supk,

lim infn

Wn(B

(k,)) +ln qk,

n


Thus

ln q


79/103

lim infn

Wn(B) lim infn

sup

k,Wn(B(k,)) +

ln qk,n

supk,

lim infn

Wn(B

(k,)) +ln qk,

n

= supk,

lim infn

Wn(B(k,))


Thus

ln q


80/103

lim infn

Wn(B) lim infn

supk,Wn(B(k,)) +

ln qk,n

supk,

lim infn

Wn(B

(k,)) +ln qk,

n

= supk,

lim infn

Wn(B(k,))

= supk,

k,

Since the partitions P are nested, we have that

supk,

k,

= limk

liml

k,

= W.


Kernel regression estimate


81/103

Kernel function K(x) 0Bandwidth h > 0

mn(x) =

ni=1 YiK

xXih

ni

=1

K xXih

Naive (window) kernel function K(x) = I{x1}

mn(x) = ni=1 YiI{xXih}

ni=1 I{xXih}


Kernel-based portfolio selection


82/103

choose the radius rk, > 0 such that for any fixed k,

lim

rk, = 0.


Kernel-based portfolio selection


83/103

choose the radius rk, > 0 such that for any fixed k,

lim

rk, = 0.

for n > k + 1, define the expert b(k,) by

b(k,)(xn11 ) = argmaxb

{k


84/103

The kernel-based portfolio scheme is universally consistent withrespect to the class of all ergodic processes such thatE{| ln X(j)| < , for j = 1, 2, . . . , d.

L. Gyorfi, G. Lugosi, F. Udina (2006) Nonparametric kernel-basedsequential investment strategies, Mathematical Finance, 16, pp.337-357www.szit.bme.hu/gyorfi/kernel.pdf


k-nearest neighbor (NN) regression estimate


85/103

mn(x) =n

i=1Wni(x; X1, . . . , Xn)Yi.

Wni is 1/k if Xi is one of the k nearest neighbors of x amongX1, . . . , Xn, and Wni is 0 otherwise.


Nearest-neighbor-based portfolio selection

choose p (0 1) such that lim p = 0


86/103

choose p (0, 1) such that lim p 0for fixed positive integers k, (n > k + + 1) introduce the set ofthe = pn nearest neighbor matches:

J(k,)n = {i; k + 1 i n such that x

i1ik is among the NNs of x

n1nk

in xk1 , . . . , xn1nk}.


Nearest-neighbor-based portfolio selection

choose p (0, 1) such that lim p = 0


87/103

choose p (0, 1) such that lim p 0for fixed positive integers k, (n > k + + 1) introduce the set ofthe = pn nearest neighbor matches:

J(k,)n = {i; k + 1 i n such that x

i1ik is among the NNs of x

n1nk

in xk1 , . . . , xn1nk}.

Define the portfolio vector by

b(k,)(xn11 ) = argmaxb

iJ

(k,)n

ln b , xi

if the sum is non-void, and b0 = (1/d, . . . , 1/d) otherwise.

Gyorfi Machine learning and portfolio selections II

Theorem

If for any vector s = sk1 the random variable


88/103

y 1

Xk1 s

has continuous distribution function, then the nearest-neighborportfolio scheme is universally consistent with respect to the classof all ergodic processes such that E{| ln X(j)|} < , for

j = 1, 2, . . . d.

NN is robust, there is no scaling problem


Theorem

If for any vector s = sk1 the random variable


89/103

y 1

Xk1 s

has continuous distribution function, then the nearest-neighborportfolio scheme is universally consistent with respect to the classof all ergodic processes such that E{| ln X(j)|} < , for

j = 1, 2, . . . d.

NN is robust, there is no scaling problem

L. Gyorfi, F. Udina, H. Walk (2006) Nonparametric nearest

neighbor based empirical portfolio selection strategies,(submitted), www.szit.bme.hu/gyorfi/NN.pdf


Semi-log-optimal portfolio

empirical log-optimal:

(k ) 1


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h(k,)(xn11 ) = argmax

biJn

ln b , xi




(k ) 1


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biJn

ln b , xi

Taylor expansion: ln z h(z) = z 1 12 (z 1)2




(k )( n 1)


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biJn

ln b , xi

Taylor expansion: ln z h(z) = z 1 12 (z 1)2 empirical

semi-log-optimal:

h(k,)(xn11 ) = argmaxb

iJn

h(b , xi) = argmaxb

{b , mb , Cb}




h(k )( n 1) l b


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biJn

ln b , xi

Taylor expansion: ln z h(z) = z 1 12 (z 1)2 empirical

semi-log-optimal:

h(k,)(xn11 ) = argmaxb

iJn

h(b , xi) = argmaxb

{b , mb , Cb}

smaller computational complexity: quadratic programming



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Conditions of the model:


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Assume that

the assets are arbitrarily divisible,

the assets are available in unbounded quantities at the currentprice at any given trading period,

there are no transaction costs,

the behavior of the market is not affected by the actions ofthe investor using the strategy under investigation.


NYSE data sets


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At www.szit.bme.hu/~oti/portfolio there are two benchmarkdata set from NYSE:

The first data set consists of daily data of 36 stocks withlength 22 years.

The second data set contains 23 stocks and has length 44years.


NYSEdata sets


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At www.szit.bme.hu/~oti/portfolio there are two benchmarkdata set from NYSE:

The first data set consists of daily data of 36 stocks withlength 22 years.

The second data set contains 23 stocks and has length 44years.

Our experiment is on the second data set.


Experiments on average annual yields (AAY)


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Kernel based semi-log-optimal portfolio selection withk = 1, . . . , 5 and l = 1, . . . , 10

r2k,l = 0.0001 d k ,




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r2k,l = 0.0001 d k ,

AAY of kernel based semi-log-optimal portfolio is 222.7%




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r2k,l = 0.0001 d k ,

AAY of kernel based semi-log-optimal portfolio is 222.7%double the capital




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r2k,l = 0.0001 d k ,

AAY of kernel based semi-log-optimal portfolio is 222.7%double the capitalMORRIS had the best AAY, 30.1%




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r2k,l = 0.0001 d k ,

AAY of kernel based semi-log-optimal portfolio is 222.7%double the capitalMORRIS had the best AAY, 30.1%the BCRP had average AAY 35.2%


The average annual yields of the individual experts.

k 1 2 3 4 5


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1 29.4% 27.8% 22.9% 23.9% 23.7%

2 201.0% 124.6% 98.3% 36.1% 90.8%

3 114.2% 62.9% 38.8% 91.4% 31.6%

4 172.5% 155.0% 100.6% 162.3% 50.8%

5 233.5% 170.2% 166.6% 171.1% 107.7%6 245.4% 216.2% 176.9% 182.0% 143.0%

7 261.8% 211.0% 181.2% 165.6% 158.7%

8 229.4% 189.2% 171.0% 138.8% 131.3%

9 219.3% 172.8% 162.4% 118.7% 116.0%

10 210.6% 151.5% 131.8% 103.4% 110.9%


machine learning and finance ii

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