Page 1
Growth Optimal Portfolio Selection Strategies withTransaction Cost
Laszlo Gyorfi1
Gyorgy OttucsakIstvan Vajda
1Department of Computer Science and Information TheoryBudapest University of Technology and Economics
Budapest, Hungary
September 24, 2007
e-mail: [email protected] /∼gyorfiwww.szit.bme.hu/∼oti/portfolio
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 2
Notation
investment in the stock marketd assetss(j)n price of asset j at the end of trading period (day) n
initial price s(j)0 = 1, j = 1, . . . , d
x(j)n =
s(j)n
s(j)n−1
xn = (x(1)n , . . . , x
(d)n ) the return vector on trading period n
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 3
Notation
investment in the stock marketd assetss(j)n price of asset j at the end of trading period (day) n
initial price s(j)0 = 1, j = 1, . . . , d
x(j)n =
s(j)n
s(j)n−1
xn = (x(1)n , . . . , x
(d)n ) the return vector on trading period n
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 4
Portfolio selection without transaction cost
nth trading period a portfolio strategy
bn = (b(1)n , . . . , b
(d)n ) = b(x1, . . . , xn−1) = b(xn−1
1 )
b(j)n ≥ 0 gives the proportion of the investor’s capital invested in
stock j for trading period n (∑d
j=1 b(j)n = 1)
for the nth trading period, Sn−1 is the initial capital (it is invested).
Sn = Sn−1
d∑j=1
b(j)n x
(j)1 = Sn−1 〈bn , xn〉 = S0
n∏i=1
〈bi , xi 〉 = S0enWn(b)
with the average growth rate
Wn(B) =1
n
n∑i=1
log 〈bi , xi 〉 .
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 5
Portfolio selection without transaction cost
nth trading period a portfolio strategy
bn = (b(1)n , . . . , b
(d)n ) = b(x1, . . . , xn−1) = b(xn−1
1 )
b(j)n ≥ 0 gives the proportion of the investor’s capital invested in
stock j for trading period n (∑d
j=1 b(j)n = 1)
for the nth trading period, Sn−1 is the initial capital (it is invested).
Sn = Sn−1
d∑j=1
b(j)n x
(j)1
= Sn−1 〈bn , xn〉 = S0
n∏i=1
〈bi , xi 〉 = S0enWn(b)
with the average growth rate
Wn(B) =1
n
n∑i=1
log 〈bi , xi 〉 .
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 6
Portfolio selection without transaction cost
nth trading period a portfolio strategy
bn = (b(1)n , . . . , b
(d)n ) = b(x1, . . . , xn−1) = b(xn−1
1 )
b(j)n ≥ 0 gives the proportion of the investor’s capital invested in
stock j for trading period n (∑d
j=1 b(j)n = 1)
for the nth trading period, Sn−1 is the initial capital (it is invested).
Sn = Sn−1
d∑j=1
b(j)n x
(j)1 = Sn−1 〈bn , xn〉
= S0
n∏i=1
〈bi , xi 〉 = S0enWn(b)
with the average growth rate
Wn(B) =1
n
n∑i=1
log 〈bi , xi 〉 .
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 7
Portfolio selection without transaction cost
nth trading period a portfolio strategy
bn = (b(1)n , . . . , b
(d)n ) = b(x1, . . . , xn−1) = b(xn−1
1 )
b(j)n ≥ 0 gives the proportion of the investor’s capital invested in
stock j for trading period n (∑d
j=1 b(j)n = 1)
for the nth trading period, Sn−1 is the initial capital (it is invested).
Sn = Sn−1
d∑j=1
b(j)n x
(j)1 = Sn−1 〈bn , xn〉 = S0
n∏i=1
〈bi , xi 〉
= S0enWn(b)
with the average growth rate
Wn(B) =1
n
n∑i=1
log 〈bi , xi 〉 .
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 8
Portfolio selection without transaction cost
nth trading period a portfolio strategy
bn = (b(1)n , . . . , b
(d)n ) = b(x1, . . . , xn−1) = b(xn−1
1 )
b(j)n ≥ 0 gives the proportion of the investor’s capital invested in
stock j for trading period n (∑d
j=1 b(j)n = 1)
for the nth trading period, Sn−1 is the initial capital (it is invested).
Sn = Sn−1
d∑j=1
b(j)n x
(j)1 = Sn−1 〈bn , xn〉 = S0
n∏i=1
〈bi , xi 〉 = S0enWn(b)
with the average growth rate
Wn(B) =1
n
n∑i=1
log 〈bi , xi 〉 .
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 9
1
nlog Sn ≈ 1
n
n∑i=1
E{log⟨b(Xi−1
1 ) , Xi
⟩| Xi−1
1 }
and
1
nlog S∗n ≈ 1
n
n∑i=1
E{log⟨b∗(Xi−1
1 ) , Xi
⟩| Xi−1
1 }
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 10
1
nlog Sn ≈ 1
n
n∑i=1
E{log⟨b(Xi−1
1 ) , Xi
⟩| Xi−1
1 }
and
1
nlog S∗n ≈ 1
n
n∑i=1
E{log⟨b∗(Xi−1
1 ) , Xi
⟩| Xi−1
1 }
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 11
Portfolio selection with transaction cost
S0 = 1, gross wealth Sn, net wealth Nn
for the nth trading period, Nn−1 is the initial capital
Sn = Nn−1 〈bn , xn〉
Calculate the transaction cost for selecting the portfolio bn+1.
Before rearranging, at the j-th asset there is b(j)n x
(j)n Nn−1 dollars.
After rearranging, we need b(j)n+1Nn dollars.
If b(j)n x
(j)n Nn−1 ≥ b
(j)n+1Nn then we have to sell and the transaction
cost at the j-th asset is
c(b
(j)n x
(j)n Nn−1 − b
(j)n+1Nn
),
otherwise we have to buy and the transaction cost at the j-th assetis
c(b
(j)n+1Nn − b
(j)n x
(j)n Nn−1
).
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 12
Portfolio selection with transaction cost
S0 = 1, gross wealth Sn, net wealth Nn
for the nth trading period, Nn−1 is the initial capital
Sn = Nn−1 〈bn , xn〉
Calculate the transaction cost for selecting the portfolio bn+1.
Before rearranging, at the j-th asset there is b(j)n x
(j)n Nn−1 dollars.
After rearranging, we need b(j)n+1Nn dollars.
If b(j)n x
(j)n Nn−1 ≥ b
(j)n+1Nn then we have to sell and the transaction
cost at the j-th asset is
c(b
(j)n x
(j)n Nn−1 − b
(j)n+1Nn
),
otherwise we have to buy and the transaction cost at the j-th assetis
c(b
(j)n+1Nn − b
(j)n x
(j)n Nn−1
).
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 13
Portfolio selection with transaction cost
S0 = 1, gross wealth Sn, net wealth Nn
for the nth trading period, Nn−1 is the initial capital
Sn = Nn−1 〈bn , xn〉
Calculate the transaction cost for selecting the portfolio bn+1.
Before rearranging, at the j-th asset there is b(j)n x
(j)n Nn−1 dollars.
After rearranging, we need b(j)n+1Nn dollars.
If b(j)n x
(j)n Nn−1 ≥ b
(j)n+1Nn then we have to sell and the transaction
cost at the j-th asset is
c(b
(j)n x
(j)n Nn−1 − b
(j)n+1Nn
),
otherwise we have to buy and the transaction cost at the j-th assetis
c(b
(j)n+1Nn − b
(j)n x
(j)n Nn−1
).
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 14
Portfolio selection with transaction cost
S0 = 1, gross wealth Sn, net wealth Nn
for the nth trading period, Nn−1 is the initial capital
Sn = Nn−1 〈bn , xn〉
Calculate the transaction cost for selecting the portfolio bn+1.
Before rearranging, at the j-th asset there is b(j)n x
(j)n Nn−1 dollars.
After rearranging, we need b(j)n+1Nn dollars.
If b(j)n x
(j)n Nn−1 ≥ b
(j)n+1Nn then we have to sell and the transaction
cost at the j-th asset is
c(b
(j)n x
(j)n Nn−1 − b
(j)n+1Nn
),
otherwise we have to buy and the transaction cost at the j-th assetis
c(b
(j)n+1Nn − b
(j)n x
(j)n Nn−1
).
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 15
Portfolio selection with transaction cost
S0 = 1, gross wealth Sn, net wealth Nn
for the nth trading period, Nn−1 is the initial capital
Sn = Nn−1 〈bn , xn〉
Calculate the transaction cost for selecting the portfolio bn+1.
Before rearranging, at the j-th asset there is b(j)n x
(j)n Nn−1 dollars.
After rearranging, we need b(j)n+1Nn dollars.
If b(j)n x
(j)n Nn−1 ≥ b
(j)n+1Nn then we have to sell and the transaction
cost at the j-th asset is
c(b
(j)n x
(j)n Nn−1 − b
(j)n+1Nn
),
otherwise we have to buy and the transaction cost at the j-th assetis
c(b
(j)n+1Nn − b
(j)n x
(j)n Nn−1
).
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 16
Portfolio selection with transaction cost
S0 = 1, gross wealth Sn, net wealth Nn
for the nth trading period, Nn−1 is the initial capital
Sn = Nn−1 〈bn , xn〉
Calculate the transaction cost for selecting the portfolio bn+1.
Before rearranging, at the j-th asset there is b(j)n x
(j)n Nn−1 dollars.
After rearranging, we need b(j)n+1Nn dollars.
If b(j)n x
(j)n Nn−1 ≥ b
(j)n+1Nn then we have to sell and the transaction
cost at the j-th asset is
c(b
(j)n x
(j)n Nn−1 − b
(j)n+1Nn
),
otherwise we have to buy and the transaction cost at the j-th assetis
c(b
(j)n+1Nn − b
(j)n x
(j)n Nn−1
).
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 17
Let x+ denote the positive part of x .
Thus,
Nn = Sn −d∑
j=1
c(b
(j)n x
(j)n Nn−1 − b
(j)n+1Nn
)+
−d∑
j=1
c(b
(j)n+1Nn − b
(j)n x
(j)n Nn−1
)+,
or equivalently
Sn = Nn + cd∑
j=1
∣∣∣b(j)n x
(j)n Nn−1 − b
(j)n+1Nn
∣∣∣ .
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 18
Let x+ denote the positive part of x .Thus,
Nn = Sn −d∑
j=1
c(b
(j)n x
(j)n Nn−1 − b
(j)n+1Nn
)+
−d∑
j=1
c(b
(j)n+1Nn − b
(j)n x
(j)n Nn−1
)+,
or equivalently
Sn = Nn + cd∑
j=1
∣∣∣b(j)n x
(j)n Nn−1 − b
(j)n+1Nn
∣∣∣ .
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 19
Let x+ denote the positive part of x .Thus,
Nn = Sn −d∑
j=1
c(b
(j)n x
(j)n Nn−1 − b
(j)n+1Nn
)+
−d∑
j=1
c(b
(j)n+1Nn − b
(j)n x
(j)n Nn−1
)+,
or equivalently
Sn = Nn + cd∑
j=1
∣∣∣b(j)n x
(j)n Nn−1 − b
(j)n+1Nn
∣∣∣ .
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 20
Dividing both sides by Sn and introducing ratio
wn =Nn
Sn,
0 < wn < 1,
we get
1 = wn + cd∑
j=1
∣∣∣∣∣ b(j)n x
(j)n
〈bn , xn〉− b
(j)n+1wn
∣∣∣∣∣ .
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 21
Dividing both sides by Sn and introducing ratio
wn =Nn
Sn,
0 < wn < 1,we get
1 = wn + cd∑
j=1
∣∣∣∣∣ b(j)n x
(j)n
〈bn , xn〉− b
(j)n+1wn
∣∣∣∣∣ .
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 22
Sn = Nn−1〈bn , xn〉 = Sn−1wn−1〈bn , xn〉 =n∏
i=1
[w(bi−1,bi , xi−1) 〈bi , xi 〉].
Introduce the notation
g(bi−1,bi , xi−1, xi ) = log(w(bi−1,bi , xi−1) 〈bi , xi 〉),
then the average growth rate becomes
1
nlog Sn =
1
n
n∑i=1
log(w(bi−1,bi , xi−1) 〈bi , xi 〉)
=1
n
n∑i=1
g(bi−1,bi , xi−1, xi ).
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 23
Sn = Nn−1〈bn , xn〉 = Sn−1wn−1〈bn , xn〉 =n∏
i=1
[w(bi−1,bi , xi−1) 〈bi , xi 〉].
Introduce the notation
g(bi−1,bi , xi−1, xi ) = log(w(bi−1,bi , xi−1) 〈bi , xi 〉),
then the average growth rate becomes
1
nlog Sn =
1
n
n∑i=1
log(w(bi−1,bi , xi−1) 〈bi , xi 〉)
=1
n
n∑i=1
g(bi−1,bi , xi−1, xi ).
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 24
Sn = Nn−1〈bn , xn〉 = Sn−1wn−1〈bn , xn〉 =n∏
i=1
[w(bi−1,bi , xi−1) 〈bi , xi 〉].
Introduce the notation
g(bi−1,bi , xi−1, xi ) = log(w(bi−1,bi , xi−1) 〈bi , xi 〉),
then the average growth rate becomes
1
nlog Sn =
1
n
n∑i=1
log(w(bi−1,bi , xi−1) 〈bi , xi 〉)
=1
n
n∑i=1
g(bi−1,bi , xi−1, xi ).
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 25
In the sequel xi will be random variable and is denoted by Xi .Let’s use the decomposition
1
nlog Sn
=1
n
n∑i=1
E{g(bi−1,bi ,Xi−1,Xi )|Xi−11 }
+1
n
n∑i=1
(g(bi−1,bi ,Xi−1,Xi )− E{g(bi−1,bi ,Xi−1,Xi )|Xi−11 }),
therefore
1
nlog Sn ≈
1
n
n∑i=1
E{g(bi−1,bi ,Xi−1,Xi )|Xi−11 }
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 26
In the sequel xi will be random variable and is denoted by Xi .Let’s use the decomposition
1
nlog Sn
=1
n
n∑i=1
E{g(bi−1,bi ,Xi−1,Xi )|Xi−11 }
+1
n
n∑i=1
(g(bi−1,bi ,Xi−1,Xi )− E{g(bi−1,bi ,Xi−1,Xi )|Xi−11 }),
therefore
1
nlog Sn ≈
1
n
n∑i=1
E{g(bi−1,bi ,Xi−1,Xi )|Xi−11 }
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 27
If the market process {Xi} is a homogeneous and first orderMarkov process then
E{g(bi−1,bi ,Xi−1,Xi )|Xi−11 }
= E{log(w(bi−1,bi ,Xi−1) 〈bi , Xi 〉)|Xi−11 }
= log w(bi−1,bi ,Xi−1) + E{log 〈bi , Xi 〉 |Xi−11 }
= log w(bi−1,bi ,Xi−1) + E{log 〈bi , Xi 〉 |bi ,Xi−1}def= v(bi−1,bi ,Xi−1),
therefore the maximization of the average growth rate
1
nlog Sn
is asymptotically equivalent to the maximization of
1
n
n∑i=1
v(bi−1,bi ,Xi−1).
dynamic programming problem
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 28
If the market process {Xi} is a homogeneous and first orderMarkov process then
E{g(bi−1,bi ,Xi−1,Xi )|Xi−11 }
= E{log(w(bi−1,bi ,Xi−1) 〈bi , Xi 〉)|Xi−11 }
= log w(bi−1,bi ,Xi−1) + E{log 〈bi , Xi 〉 |Xi−11 }
= log w(bi−1,bi ,Xi−1) + E{log 〈bi , Xi 〉 |bi ,Xi−1}def= v(bi−1,bi ,Xi−1),
therefore the maximization of the average growth rate
1
nlog Sn
is asymptotically equivalent to the maximization of
1
n
n∑i=1
v(bi−1,bi ,Xi−1).
dynamic programming problem
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 29
If the market process {Xi} is a homogeneous and first orderMarkov process then
E{g(bi−1,bi ,Xi−1,Xi )|Xi−11 }
= E{log(w(bi−1,bi ,Xi−1) 〈bi , Xi 〉)|Xi−11 }
= log w(bi−1,bi ,Xi−1) + E{log 〈bi , Xi 〉 |Xi−11 }
= log w(bi−1,bi ,Xi−1) + E{log 〈bi , Xi 〉 |bi ,Xi−1}def= v(bi−1,bi ,Xi−1),
therefore the maximization of the average growth rate
1
nlog Sn
is asymptotically equivalent to the maximization of
1
n
n∑i=1
v(bi−1,bi ,Xi−1).
dynamic programming problem
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 30
If the market process {Xi} is a homogeneous and first orderMarkov process then
E{g(bi−1,bi ,Xi−1,Xi )|Xi−11 }
= E{log(w(bi−1,bi ,Xi−1) 〈bi , Xi 〉)|Xi−11 }
= log w(bi−1,bi ,Xi−1) + E{log 〈bi , Xi 〉 |Xi−11 }
= log w(bi−1,bi ,Xi−1) + E{log 〈bi , Xi 〉 |bi ,Xi−1}def= v(bi−1,bi ,Xi−1),
therefore the maximization of the average growth rate
1
nlog Sn
is asymptotically equivalent to the maximization of
1
n
n∑i=1
v(bi−1,bi ,Xi−1).
dynamic programming problemGyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 31
Algorithm 1
empirical portfolio selection
Naive approachFor the optimization, neglect the transaction costkernel based log-optimal portfolio selectionDefine an infinite array of experts B(`) = {b(`)(·)}, where ` is apositive integer.For fixed positive integer `, choose the radius r` > 0 such that
lim`→∞
r` = 0.
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 32
Algorithm 1
empirical portfolio selectionNaive approach
For the optimization, neglect the transaction costkernel based log-optimal portfolio selectionDefine an infinite array of experts B(`) = {b(`)(·)}, where ` is apositive integer.For fixed positive integer `, choose the radius r` > 0 such that
lim`→∞
r` = 0.
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 33
Algorithm 1
empirical portfolio selectionNaive approachFor the optimization, neglect the transaction cost
kernel based log-optimal portfolio selectionDefine an infinite array of experts B(`) = {b(`)(·)}, where ` is apositive integer.For fixed positive integer `, choose the radius r` > 0 such that
lim`→∞
r` = 0.
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 34
Algorithm 1
empirical portfolio selectionNaive approachFor the optimization, neglect the transaction costkernel based log-optimal portfolio selection
Define an infinite array of experts B(`) = {b(`)(·)}, where ` is apositive integer.For fixed positive integer `, choose the radius r` > 0 such that
lim`→∞
r` = 0.
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 35
Algorithm 1
empirical portfolio selectionNaive approachFor the optimization, neglect the transaction costkernel based log-optimal portfolio selectionDefine an infinite array of experts B(`) = {b(`)(·)}, where ` is apositive integer.
For fixed positive integer `, choose the radius r` > 0 such that
lim`→∞
r` = 0.
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 36
Algorithm 1
empirical portfolio selectionNaive approachFor the optimization, neglect the transaction costkernel based log-optimal portfolio selectionDefine an infinite array of experts B(`) = {b(`)(·)}, where ` is apositive integer.For fixed positive integer `, choose the radius r` > 0 such that
lim`→∞
r` = 0.
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 37
putb1 = {1/d , . . . , 1/d}
for n > 1, define the expert b(`) by
b(`)n = arg max
b∈∆d
∑{i<n:‖xi−1−xn−1‖≤r`}
ln 〈b , xi 〉 ,
if the sum is non-void,and b1 = (1/d , . . . , 1/d) otherwise, where ‖ · ‖ denotes theEuclidean norm.
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 38
putb1 = {1/d , . . . , 1/d}
for n > 1, define the expert b(`) by
b(`)n = arg max
b∈∆d
∑{i<n:‖xi−1−xn−1‖≤r`}
ln 〈b , xi 〉 ,
if the sum is non-void,
and b1 = (1/d , . . . , 1/d) otherwise, where ‖ · ‖ denotes theEuclidean norm.
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 39
putb1 = {1/d , . . . , 1/d}
for n > 1, define the expert b(`) by
b(`)n = arg max
b∈∆d
∑{i<n:‖xi−1−xn−1‖≤r`}
ln 〈b , xi 〉 ,
if the sum is non-void,and b1 = (1/d , . . . , 1/d) otherwise, where ‖ · ‖ denotes theEuclidean norm.
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 40
Aggregations: mixtures of experts
let {q`} be a probability distribution over the set of all positiveintegers `
Sn(B(`)) is the capital accumulated by the elementary strategyB(`) after n periods with an initial capital S0 = 1
after period n, aggregations with the wealths:
Sn =∑
`
q`Sn(B(`)). (1)
after period n, aggregations with the portfolios:
bn =
∑` q`Sn−1(B(`))b
(`)n∑
` q`Sn−1(B(`)). (2)
the investor’s capital is
Sn = Sn−1〈bn , xn〉w(bn−1,bn, xn−1).
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 41
Aggregations: mixtures of experts
let {q`} be a probability distribution over the set of all positiveintegers `Sn(B(`)) is the capital accumulated by the elementary strategyB(`) after n periods with an initial capital S0 = 1
after period n, aggregations with the wealths:
Sn =∑
`
q`Sn(B(`)). (1)
after period n, aggregations with the portfolios:
bn =
∑` q`Sn−1(B(`))b
(`)n∑
` q`Sn−1(B(`)). (2)
the investor’s capital is
Sn = Sn−1〈bn , xn〉w(bn−1,bn, xn−1).
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 42
Aggregations: mixtures of experts
let {q`} be a probability distribution over the set of all positiveintegers `Sn(B(`)) is the capital accumulated by the elementary strategyB(`) after n periods with an initial capital S0 = 1
after period n, aggregations with the wealths:
Sn =∑
`
q`Sn(B(`)). (1)
after period n, aggregations with the portfolios:
bn =
∑` q`Sn−1(B(`))b
(`)n∑
` q`Sn−1(B(`)). (2)
the investor’s capital is
Sn = Sn−1〈bn , xn〉w(bn−1,bn, xn−1).
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 43
Aggregations: mixtures of experts
let {q`} be a probability distribution over the set of all positiveintegers `Sn(B(`)) is the capital accumulated by the elementary strategyB(`) after n periods with an initial capital S0 = 1
after period n, aggregations with the wealths:
Sn =∑
`
q`Sn(B(`)). (1)
after period n, aggregations with the portfolios:
bn =
∑` q`Sn−1(B(`))b
(`)n∑
` q`Sn−1(B(`)). (2)
the investor’s capital is
Sn = Sn−1〈bn , xn〉w(bn−1,bn, xn−1).
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 44
Aggregations: mixtures of experts
let {q`} be a probability distribution over the set of all positiveintegers `Sn(B(`)) is the capital accumulated by the elementary strategyB(`) after n periods with an initial capital S0 = 1
after period n, aggregations with the wealths:
Sn =∑
`
q`Sn(B(`)). (1)
after period n, aggregations with the portfolios:
bn =
∑` q`Sn−1(B(`))b
(`)n∑
` q`Sn−1(B(`)). (2)
the investor’s capital is
Sn = Sn−1〈bn , xn〉w(bn−1,bn, xn−1).
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 45
Algorithm 2
empirical portfolio selection
a one-step optimization as follows:
b1 = {1/d , . . . , 1/d}
for n ≥ 1,
b(`)n = arg max
b∈∆d
∑{i<n:‖xi−1−xn−1‖≤r`}
(ln 〈b , xi 〉+ lnw(b
(`)n−1,b, xn−1)
),
if the sum is non-void,and b1 = (1/d , . . . , 1/d) otherwise.These elementary portfolios are mixed as before (1) or (2).
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 46
Algorithm 2
empirical portfolio selectiona one-step optimization as follows:
b1 = {1/d , . . . , 1/d}
for n ≥ 1,
b(`)n = arg max
b∈∆d
∑{i<n:‖xi−1−xn−1‖≤r`}
(ln 〈b , xi 〉+ lnw(b
(`)n−1,b, xn−1)
),
if the sum is non-void,and b1 = (1/d , . . . , 1/d) otherwise.These elementary portfolios are mixed as before (1) or (2).
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 47
Algorithm 2
empirical portfolio selectiona one-step optimization as follows:
b1 = {1/d , . . . , 1/d}
for n ≥ 1,
b(`)n = arg max
b∈∆d
∑{i<n:‖xi−1−xn−1‖≤r`}
(ln 〈b , xi 〉+ lnw(b
(`)n−1,b, xn−1)
),
if the sum is non-void,
and b1 = (1/d , . . . , 1/d) otherwise.These elementary portfolios are mixed as before (1) or (2).
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 48
Algorithm 2
empirical portfolio selectiona one-step optimization as follows:
b1 = {1/d , . . . , 1/d}
for n ≥ 1,
b(`)n = arg max
b∈∆d
∑{i<n:‖xi−1−xn−1‖≤r`}
(ln 〈b , xi 〉+ lnw(b
(`)n−1,b, xn−1)
),
if the sum is non-void,and b1 = (1/d , . . . , 1/d) otherwise.
These elementary portfolios are mixed as before (1) or (2).
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 49
Algorithm 2
empirical portfolio selectiona one-step optimization as follows:
b1 = {1/d , . . . , 1/d}
for n ≥ 1,
b(`)n = arg max
b∈∆d
∑{i<n:‖xi−1−xn−1‖≤r`}
(ln 〈b , xi 〉+ lnw(b
(`)n−1,b, xn−1)
),
if the sum is non-void,and b1 = (1/d , . . . , 1/d) otherwise.These elementary portfolios are mixed as before (1) or (2).
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 50
NYSE data sets
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 (5651 trading days ending in 1985).
The second data set contains 23 stocks and has length 44years (11178 trading days ending in 2006).
Our experiment is on the second data set.
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 51
NYSE data sets
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 (5651 trading days ending in 1985).
The second data set contains 23 stocks and has length 44years (11178 trading days ending in 2006).
Our experiment is on the second data set.
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 52
Experiments on average annual yields (AAY)
Kernel based log-optimal portfolio selection with` = 1, . . . , 10
r2` = 0.0001 · d · `,
MORRIS had the best AAY, 20%
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 53
Experiments on average annual yields (AAY)
Kernel based log-optimal portfolio selection with` = 1, . . . , 10
r2` = 0.0001 · d · `,
MORRIS had the best AAY, 20%
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 54
The average annual yields of the individual experts and ofthe aggregations with c = 0.0015.
` c = 0 Algorithm 1 Algorithm 2
1 20% -18% -14%
2 118% -2% 25%
3 71% 14% 55%
4 103% 28% 73%
5 134% 33% 77%
6 140% 43% 92%
7 148% 37% 83%
8 132% 38% 74%
9 127% 42% 66%
10 123% 44% 62%
Aggregation with wealth (1) 137% 40% 83%Aggregation with portfolio (2) 137% 49% 89%
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 55
Strategy 1
non-empirical strategy
0 < δ < 1 denotes a discount factordiscounted Bellman equation:
Fδ(b, x) = maxb′
{v(b,b′, x) + (1− δ)E{Fδ(b
′,X2) | X1 = x}}
.
b∗1 = {1/d , . . . , 1/d}
and
b∗i+1 = arg maxb′
{v(b∗i ,b
′,Xi ) + (1− δi )E{Fδi(b
′,Xi+1)|Xi}},
for 1 ≤ i , where 0 < δi < 1 is a discount factor such that δi ↓ 0.non-stationary policy
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 56
Strategy 1
non-empirical strategy0 < δ < 1 denotes a discount factor
discounted Bellman equation:
Fδ(b, x) = maxb′
{v(b,b′, x) + (1− δ)E{Fδ(b
′,X2) | X1 = x}}
.
b∗1 = {1/d , . . . , 1/d}
and
b∗i+1 = arg maxb′
{v(b∗i ,b
′,Xi ) + (1− δi )E{Fδi(b
′,Xi+1)|Xi}},
for 1 ≤ i , where 0 < δi < 1 is a discount factor such that δi ↓ 0.non-stationary policy
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 57
Strategy 1
non-empirical strategy0 < δ < 1 denotes a discount factordiscounted Bellman equation:
Fδ(b, x) = maxb′
{v(b,b′, x) + (1− δ)E{Fδ(b
′,X2) | X1 = x}}
.
b∗1 = {1/d , . . . , 1/d}
and
b∗i+1 = arg maxb′
{v(b∗i ,b
′,Xi ) + (1− δi )E{Fδi(b
′,Xi+1)|Xi}},
for 1 ≤ i , where 0 < δi < 1 is a discount factor such that δi ↓ 0.non-stationary policy
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 58
Strategy 1
non-empirical strategy0 < δ < 1 denotes a discount factordiscounted Bellman equation:
Fδ(b, x) = maxb′
{v(b,b′, x) + (1− δ)E{Fδ(b
′,X2) | X1 = x}}
.
b∗1 = {1/d , . . . , 1/d}
and
b∗i+1 = arg maxb′
{v(b∗i ,b
′,Xi ) + (1− δi )E{Fδi(b
′,Xi+1)|Xi}},
for 1 ≤ i ,
where 0 < δi < 1 is a discount factor such that δi ↓ 0.non-stationary policy
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 59
Strategy 1
non-empirical strategy0 < δ < 1 denotes a discount factordiscounted Bellman equation:
Fδ(b, x) = maxb′
{v(b,b′, x) + (1− δ)E{Fδ(b
′,X2) | X1 = x}}
.
b∗1 = {1/d , . . . , 1/d}
and
b∗i+1 = arg maxb′
{v(b∗i ,b
′,Xi ) + (1− δi )E{Fδi(b
′,Xi+1)|Xi}},
for 1 ≤ i , where 0 < δi < 1 is a discount factor such that δi ↓ 0.
non-stationary policy
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 60
Strategy 1
non-empirical strategy0 < δ < 1 denotes a discount factordiscounted Bellman equation:
Fδ(b, x) = maxb′
{v(b,b′, x) + (1− δ)E{Fδ(b
′,X2) | X1 = x}}
.
b∗1 = {1/d , . . . , 1/d}
and
b∗i+1 = arg maxb′
{v(b∗i ,b
′,Xi ) + (1− δi )E{Fδi(b
′,Xi+1)|Xi}},
for 1 ≤ i , where 0 < δi < 1 is a discount factor such that δi ↓ 0.non-stationary policy
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 61
Theorem 1
Assume
(i) that {Xi} is a homogeneous and first order Markov process,
(ii) and there exist 0 < a1 < 1 < a2 < ∞ such thata1 ≤ X (j) ≤ a2 for all j = 1, . . . , d .
Choose the discount factor δi ↓ 0 such that
(δi − δi+1)/δ2i+1 → 0
as i →∞, and∞∑
n=1
1
n2δ2n
< ∞.
Then, for Strategy 1, the portfolio {b∗i } with capital S∗n is optimalin the sense that for any portfolio strategy {bi} with capital Sn,
lim infn→∞
(1
nlog S∗n −
1
nlog Sn
)≥ 0
a.s.
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 62
Theorem 1
Assume
(i) that {Xi} is a homogeneous and first order Markov process,
(ii) and there exist 0 < a1 < 1 < a2 < ∞ such thata1 ≤ X (j) ≤ a2 for all j = 1, . . . , d .
Choose the discount factor δi ↓ 0 such that
(δi − δi+1)/δ2i+1 → 0
as i →∞, and∞∑
n=1
1
n2δ2n
< ∞.
Then, for Strategy 1, the portfolio {b∗i } with capital S∗n is optimalin the sense that for any portfolio strategy {bi} with capital Sn,
lim infn→∞
(1
nlog S∗n −
1
nlog Sn
)≥ 0
a.s.
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 63
Theorem 1
Assume
(i) that {Xi} is a homogeneous and first order Markov process,
(ii) and there exist 0 < a1 < 1 < a2 < ∞ such thata1 ≤ X (j) ≤ a2 for all j = 1, . . . , d .
Choose the discount factor δi ↓ 0 such that
(δi − δi+1)/δ2i+1 → 0
as i →∞, and∞∑
n=1
1
n2δ2n
< ∞.
Then, for Strategy 1, the portfolio {b∗i } with capital S∗n is optimalin the sense that for any portfolio strategy {bi} with capital Sn,
lim infn→∞
(1
nlog S∗n −
1
nlog Sn
)≥ 0
a.s.
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 64
Theorem 1
Assume
(i) that {Xi} is a homogeneous and first order Markov process,
(ii) and there exist 0 < a1 < 1 < a2 < ∞ such thata1 ≤ X (j) ≤ a2 for all j = 1, . . . , d .
Choose the discount factor δi ↓ 0 such that
(δi − δi+1)/δ2i+1 → 0
as i →∞, and∞∑
n=1
1
n2δ2n
< ∞.
Then, for Strategy 1, the portfolio {b∗i } with capital S∗n is optimalin the sense that for any portfolio strategy {bi} with capital Sn,
lim infn→∞
(1
nlog S∗n −
1
nlog Sn
)≥ 0
a.s.Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 65
Strategy 2
non-empirical strategy
For any integer 1 ≤ k, put
b(k)1 = {1/d , . . . , 1/d}
and
b(k)i+1 = arg max
b′
{v(b
(k)i ,b′,Xi ) + (1− δk)E{Fδk
(b′,Xi+1)|Xi}},
for 1 ≤ i .The portfolio B(k) = {b(k)
i } is called the portfolio of expert k withcapital Sn(B(k)).Choose an arbitrary probability distribution qk > 0, and introducethe combined portfolio with its capital
Sn =∞∑
k=1
qkSn(B(k)).
stationary policy
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 66
Strategy 2
non-empirical strategyFor any integer 1 ≤ k, put
b(k)1 = {1/d , . . . , 1/d}
and
b(k)i+1 = arg max
b′
{v(b
(k)i ,b′,Xi ) + (1− δk)E{Fδk
(b′,Xi+1)|Xi}},
for 1 ≤ i .
The portfolio B(k) = {b(k)i } is called the portfolio of expert k with
capital Sn(B(k)).Choose an arbitrary probability distribution qk > 0, and introducethe combined portfolio with its capital
Sn =∞∑
k=1
qkSn(B(k)).
stationary policy
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 67
Strategy 2
non-empirical strategyFor any integer 1 ≤ k, put
b(k)1 = {1/d , . . . , 1/d}
and
b(k)i+1 = arg max
b′
{v(b
(k)i ,b′,Xi ) + (1− δk)E{Fδk
(b′,Xi+1)|Xi}},
for 1 ≤ i .The portfolio B(k) = {b(k)
i } is called the portfolio of expert k withcapital Sn(B(k)).
Choose an arbitrary probability distribution qk > 0, and introducethe combined portfolio with its capital
Sn =∞∑
k=1
qkSn(B(k)).
stationary policy
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 68
Strategy 2
non-empirical strategyFor any integer 1 ≤ k, put
b(k)1 = {1/d , . . . , 1/d}
and
b(k)i+1 = arg max
b′
{v(b
(k)i ,b′,Xi ) + (1− δk)E{Fδk
(b′,Xi+1)|Xi}},
for 1 ≤ i .The portfolio B(k) = {b(k)
i } is called the portfolio of expert k withcapital Sn(B(k)).Choose an arbitrary probability distribution qk > 0, and introducethe combined portfolio with its capital
Sn =∞∑
k=1
qkSn(B(k)).
stationary policy
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 69
Strategy 2
non-empirical strategyFor any integer 1 ≤ k, put
b(k)1 = {1/d , . . . , 1/d}
and
b(k)i+1 = arg max
b′
{v(b
(k)i ,b′,Xi ) + (1− δk)E{Fδk
(b′,Xi+1)|Xi}},
for 1 ≤ i .The portfolio B(k) = {b(k)
i } is called the portfolio of expert k withcapital Sn(B(k)).Choose an arbitrary probability distribution qk > 0, and introducethe combined portfolio with its capital
Sn =∞∑
k=1
qkSn(B(k)).
stationary policyGyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 70
Theorem 2
Assume (i) and (ii) of Theorem 1.
Choose the discount factor δi ↓ 0 as i →∞.Then, for Strategy 2,
limn→∞
(1
nlog S∗n −
1
nlog Sn
)= 0
a.s.
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 71
Theorem 2
Assume (i) and (ii) of Theorem 1.Choose the discount factor δi ↓ 0 as i →∞.
Then, for Strategy 2,
limn→∞
(1
nlog S∗n −
1
nlog Sn
)= 0
a.s.
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 72
Theorem 2
Assume (i) and (ii) of Theorem 1.Choose the discount factor δi ↓ 0 as i →∞.Then, for Strategy 2,
limn→∞
(1
nlog S∗n −
1
nlog Sn
)= 0
a.s.
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost
Page 73
Open problem
How to construct empirical (data-driven) optimal portfolioselection strategy?
Gyorfi, Ottucsak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost