Growth Optimal Portfolio Selection Strategies with Transaction Cost

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Growth Optimal Portfolio Selection Strategies with Transaction Cost

László Györfi¹ György Ottucsák

Istv´an Vajda

1Department of Computer Science and Information Theory Budapest University of Technology and Economics

Budapest, Hungary

September 24, 2007 e-mail: gyorfi@szit.bme.hu

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Notation

investment in the stock market d assets

s_n^(j) price of assetj at the end of trading period (day)n initial prices₀^(j) = 1,j = 1, . . . ,d

xn^(j⁾= sn^(j⁾

s_n−1^(j)

x_n= (x_n⁽¹⁾, . . . ,x_n^(d)) the return vector on trading period n

Gy¨orfi, Ottucs´ak, Vajda Growth Optimal Port. Sel. Strategies with Transaction Cost

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Notation

investment in the stock market d assets

s_n^(j) price of assetj at the end of trading period (day)n initial prices₀^(j) = 1,j = 1, . . . ,d

xn^(j⁾= sn^(j⁾

s_n−1^(j)

x_n= (x_n⁽¹⁾, . . . ,x_n^(d)) the return vector on trading period n

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Portfolio selection without transaction cost

nth trading period a portfolio strategy

b_n= (b⁽¹⁾_n , . . . ,b^(d)_n ) =b(x₁, . . . ,xn−1) =b(xⁿ⁻¹₁ ) b^(jn⁾≥0 gives the proportion of the investor’s capital invested in stockj for trading period n (Pd

j=1b^(jn⁾= 1)

for thenth trading period,Sn−1 is the initial capital (it is invested). Sn=Sn−1

d

X

j=1

bn^(j)x₁^(j) =Sn−1hb_n,xni =S0 n

Y

i=1

hb_i,xii =S0e^nWⁿ^(b)

with the average growth rate W_n(B) = 1

n

X

i=1

loghb_i,x_ii.

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Portfolio selection without transaction cost

j=1b^(jn⁾= 1)

for thenth trading period,Sn−1 is the initial capital (it is invested).

Sn=Sn−1 d

X

j=1

bn^(j)x₁^(j)

=Sn−1hb_n,xni =S0 n

Y

i=1

n

X

i=1

loghb_i,x_ii.

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Portfolio selection without transaction cost

j=1b^(jn⁾= 1)

Sn=Sn−1 d

X

j=1

bn^(j)x₁^(j) =Sn−1hb_n,xni

=S0 n

Y

i=1

n

X

i=1

loghb_i,x_ii.

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Portfolio selection without transaction cost

j=1b^(jn⁾= 1)

Sn=Sn−1 d

X

j=1

Y

i=1

hb_i,xii

=S0e^nWⁿ^(b)

n

X

i=1

loghb_i,x_ii.

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Portfolio selection without transaction cost

j=1b^(jn⁾= 1)

Sn=Sn−1 d

X

j=1

Y

i=1

n

X

i=1

loghb_i,x_ii.

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1

nlogS_n ≈ 1 n

n

X

i=1

E{logD

b(Xⁱ⁻¹₁ ),X_iE

|Xⁱ⁻¹₁ }

and 1

nlogS_n^∗ ≈ 1 n

n

X

i=1

E{logD

b^∗(Xⁱ₁⁻¹),X_iE

|Xⁱ⁻¹₁ }

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1

nlogS_n ≈ 1 n

n

X

i=1

E{logD

b(Xⁱ⁻¹₁ ),X_iE

|Xⁱ⁻¹₁ }

and 1

nlogS_n^∗ ≈ 1 n

n

X

i=1

E{logD

b^∗(Xⁱ₁⁻¹),X_iE

|Xⁱ⁻¹₁ }

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Portfolio selection with transaction cost

S₀= 1, gross wealthS_n, net wealth N_n

for thenth trading period, Nn−1 is the initial capital S_n=Nn−1hb_n,x_ni

Calculate the transaction cost for selecting the portfoliobn+1. Before rearranging, at thej-th asset there is b^(jn⁾xn^(j⁾Nn−1 dollars. After rearranging, we needb^(j_n+1⁾ N_n dollars.

Ifb^(j_n⁾x_n^(j)Nn−1 ≥b_n+1^(j) N_n then we have to sell and the transaction cost at thej-th asset is

c

bn^(j)xn^(j)Nn−1−b^(j_n+1⁾ Nn

,

otherwise we have to buy and the transaction cost at thej-th asset is

c

b_n+1^(j) N_n−b_n^(j)x_n^(j⁾Nn−1

.

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Portfolio selection with transaction cost

Calculate the transaction cost for selecting the portfoliobn+1.

Before rearranging, at thej-th asset there is b^(jn⁾xn^(j⁾Nn−1 dollars. After rearranging, we needb^(j_n+1⁾ N_n dollars.

c

,

c

.

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Portfolio selection with transaction cost

Calculate the transaction cost for selecting the portfoliobn+1. Before rearranging, at thej-th asset there is b^(jn⁾xn^(j⁾Nn−1 dollars.

After rearranging, we needb^(j_n+1⁾ N_n dollars.

c

,

c

.

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Portfolio selection with transaction cost

c

,

c

.

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Portfolio selection with transaction cost

c

,

c

.

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Portfolio selection with transaction cost

c

,

c

.

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Letx⁺ denote the positive part of x.

Thus,

N_n=S_n −

d

X

j=1

c

bn^(j)xn^(j)Nn−1−b^(j_n+1⁾ N_n+

−

d

X

j=1

c

+

,

or equivalently

S_n=N_n + c

d

X

j=1

bn^(j)xn^(j⁾Nn−1−b^(j_n+1⁾ N_n .

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Thus,

N_n=S_n −

d

X

j=1

c

−

d

X

j=1

c

+

,

or equivalently

S_n=N_n + c

d

X

j=1

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Thus,

N_n=S_n −

d

X

j=1

c

−

d

X

j=1

c

+

,

or equivalently

S_n=N_n + c

d

X

j=1

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Dividing both sides byS_n and introducing ratio wn= Nn

S_n, 0<w_n<1,

we get

1 =wn+c

d

X

j=1

b^(jn⁾xn^(j⁾

hb_n,xni −b_n+1^(j) wn

.

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Dividing both sides byS_n and introducing ratio wn= Nn

S_n, 0<w_n<1,

we get

1 =wn+c

d

X

j=1

b^(jn⁾xn^(j⁾

hb_n,xni −b_n+1^(j) wn

.

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S_n=Nn−1hb_n,x_ni=Sn−1wn−1hb_n,x_ni=

n

Y

i=1

[w(b_i−1,b_i,x_i−1)hb_i,x_ii].

Introduce the notation

g(bi−1,bi,xi−1,xi) = log(w(bi−1,bi,xi−1)hb_i,xii), then the average growth rate becomes

1

nlogS_n = 1 n

n

X

i=1

log(w(bi−1,b_i,xi−1)hb_i,x_ii)

= 1

n

X

i=1

g(bi−1,b_i,xi−1,x_i).

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n

Y

i=1

g(bi−1,bi,xi−1,xi) = log(w(bi−1,bi,xi−1)hb_i,xii),

then the average growth rate becomes 1

nlogS_n = 1 n

n

X

i=1

= 1

n

X

i=1

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n

Y

i=1

g(bi−1,bi,xi−1,xi) = log(w(bi−1,bi,xi−1)hb_i,xii), then the average growth rate becomes

1

nlogS_n = 1 n

n

X

i=1

= 1

n

X

i=1

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In the sequelx_i will be random variable and is denoted byX_i. Let’s use the decomposition

1 nlogS_n

= 1

n

X

i=1

E{g(bi−1,bi,Xi−1,Xi)|Xⁱ⁻¹₁ }

+ 1

n

X

i=1

(g(bi−1,bi,Xi−1,Xi)−E{g(bi−1,bi,Xi−1,Xi)|Xⁱ⁻¹₁ }),

therefore 1

nlogSn≈ 1 n

n

X

i=1

E{g(b_i−1,b_i,X_i−1,X_i)|Xⁱ⁻¹₁ }

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In the sequelx_i will be random variable and is denoted byX_i. Let’s use the decomposition

1 nlogS_n

= 1

n

X

i=1

E{g(bi−1,bi,Xi−1,Xi)|Xⁱ⁻¹₁ }

+ 1

n

X

i=1

(g(bi−1,bi,Xi−1,Xi)−E{g(bi−1,bi,Xi−1,Xi)|Xⁱ⁻¹₁ }),

therefore 1

nlogSn≈ 1 n

n

X

i=1

E{g(b_i−1,b_i,X_i−1,X_i)|Xⁱ⁻¹₁ }

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If the market process{X_i}is ahomogeneous and first order Markov processthen

E{g(bi−1,bi,Xi−1,Xi)|Xⁱ₁⁻¹}

= E{log(w(b_i−1,b_i,Xi−1)hb_i,X_ii)|Xⁱ⁻¹₁ }

= logw(bi−1,b_i,Xi−1) +E{loghb_i,X_ii |Xⁱ⁻¹₁ }

= logw(bi−1,b_i,Xi−1) +E{loghb_i,X_ii |b_i,X_i−1}

def= v(bi−1,b_i,Xi−1),

therefore the maximization of the average growth rate 1

n logSn

is asymptotically equivalent to the maximization of 1

n

X

i=1

v(bi−1,b_i,Xi−1). dynamic programming problem

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n logSn

n

X

i=1

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n logSn

n

X

i=1

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n logSn

n

X

i=1

v(bi−1,b_i,Xi−1).

dynamic programming problem

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Algorithm 1

empirical portfolio selection

Naive approach

For the optimization, neglect the transaction cost kernel based log-optimal portfolio selection

Define an infinite array of expertsB^(`) ={b^(`)(·)}, where`is a positive integer.

For fixed positive integer`, choose the radius r_` >0 such that

`→∞lim r` = 0.

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Algorithm 1

empirical portfolio selection Naive approach

`→∞lim r` = 0.

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Algorithm 1

For the optimization, neglect the transaction cost

kernel based log-optimal portfolio selection

`→∞lim r` = 0.

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Algorithm 1

`→∞lim r` = 0.

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Algorithm 1

`→∞lim r` = 0.

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Algorithm 1

`→∞lim r` = 0.

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put

b1={1/d, . . . ,1/d}

forn >1, define the expert b^(`) by b^(`)n =arg max

b∈∆_d

X

{i<n:kxi−1−xn−1k≤r_`}

lnhb,xii ,

if the sum is non-void,

andb₁ = (1/d, . . . ,1/d) otherwise, wherek · k denotes the Euclidean norm.

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put

b1={1/d, . . . ,1/d} forn >1, define the expert b^(`) by

b^(`)n =arg max

b∈∆_d

X

lnhb,xii ,

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put

b1={1/d, . . . ,1/d} forn >1, define the expert b^(`) by

b^(`)n =arg max

b∈∆_d

X

lnhb,xii ,

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Aggregations: mixtures of experts

let{q_`}be a probability distribution over the set of all positive integers`

Sn(B^(`)) is the capital accumulated by the elementary strategy B^(`) aftern periods with an initial capitalS₀ = 1

after period n, aggregations with the wealths: Sn=X

`

q`Sn(B^(`)). (1)

after period n, aggregations with the portfolios: bn=

P

`q_`Sn−1(B^(`))b^(`)_n P

`q_`Sn−1(B^(`)) . (2)

the investor’s capital is

Sn=Sn−1hb_n,xniw(bn−1,bn,xn−1).

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Aggregations: mixtures of experts

after period n, aggregations with the wealths: Sn=X

`

q`Sn(B^(`)). (1)

P

`q_`Sn−1(B^(`))b^(`)_n P

`q_`Sn−1(B^(`)) . (2)

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Aggregations: mixtures of experts

after period n, aggregations with the wealths:

Sn=X

`

q`Sn(B^(`)). (1)

P

`q_`Sn−1(B^(`))b^(`)_n P

`q_`Sn−1(B^(`)) . (2)

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Aggregations: mixtures of experts

Sn=X

`

q`Sn(B^(`)). (1)

after period n, aggregations with the portfolios:

bn= P

`q_`Sn−1(B^(`))b^(`)_n P

`q_`Sn−1(B^(`)) . (2)

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Aggregations: mixtures of experts

Sn=X

`

q`Sn(B^(`)). (1)

after period n, aggregations with the portfolios:

bn= P

`q_`Sn−1(B^(`))b^(`)_n P

`q_`Sn−1(B^(`)) . (2)

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Algorithm 2

empirical portfolio selection

a one-step optimization as follows:

b₁={1/d, . . . ,1/d}

forn ≥1, b^(`)_n =arg max

b∈∆_d

X

{i<n:kx_i−1−xn−1k≤r_`}

lnhb,x_ii+ lnw(b^(`)_n−1,b,xn−1) ,

andb₁ = (1/d, . . . ,1/d) otherwise.

These elementary portfolios are mixed as before (1) or (2).

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Algorithm 2

empirical portfolio selection a one-step optimization as follows:

b₁={1/d, . . . ,1/d}

b∈∆_d

X

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Algorithm 2

b₁={1/d, . . . ,1/d}

b∈∆_d

X

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Algorithm 2

b₁={1/d, . . . ,1/d}

b∈∆_d

X

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Algorithm 2

b₁={1/d, . . . ,1/d}

b∈∆_d

X

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NYSE data sets

Atwww.szit.bme.hu/~oti/portfolio there are two benchmark data set fromNYSE:

The first data set consists of daily data of 36 stocks with length 22 years (5651 trading days ending in 1985).

The second data set contains 23 stocks and has length 44 years (11178 trading days ending in 2006).

Our experiment is on the second data set.

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NYSE data sets

Atwww.szit.bme.hu/~oti/portfolio there are two benchmark data set fromNYSE:

The first data set consists of daily data of 36 stocks with length 22 years (5651 trading days ending in 1985).

The second data set contains 23 stocks and has length 44 years (11178 trading days ending in 2006).

Our experiment is on the second data set.

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Experiments on average annual yields (AAY)

Kernel based log-optimal portfolio selection with

`= 1, . . . ,10

r_`² = 0.0001·d ·`,

MORRIS had the best AAY, 20%

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Experiments on average annual yields (AAY)

Kernel based log-optimal portfolio selection with

`= 1, . . . ,10

r_`² = 0.0001·d ·`, MORRIS had the best AAY, 20%

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The average annual yields of the individual experts and of the 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%

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Strategy 1

non-empirical strategy

0< δ <1 denotes a discount factor discounted Bellman equation:

Fδ(b,x) = max

b⁰

v(b,b⁰,x) + (1−δ)E{F_δ(b⁰,X2)|X1=x} .

b^∗₁={1/d, . . . ,1/d} and

b^∗_i+1 =arg max

b⁰

v(b^∗_i,b⁰,X_i) + (1−δ_i)E{F_δ_i(b⁰,X_i+1)|X_i}}, for 1≤i, where 0< δi <1 is a discount factor such thatδi ↓0. non-stationary policy

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Strategy 1

0< δ <1 denotes a discount factor

discounted Bellman equation: Fδ(b,x) = max

b⁰

v(b,b⁰,x) + (1−δ)E{F_δ(b⁰,X2)|X1=x} .

b^∗₁={1/d, . . . ,1/d} and

b^∗_i+1 =arg max

b⁰

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Strategy 1

Fδ(b,x) = max

b⁰

v(b,b⁰,x) + (1−δ)E{F_δ(b⁰,X2)|X1=x} .

b^∗₁={1/d, . . . ,1/d} and

b^∗_i+1 =arg max

b⁰

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Strategy 1

Fδ(b,x) = max

b⁰

v(b,b⁰,x) + (1−δ)E{F_δ(b⁰,X2)|X1=x} .

b^∗₁={1/d, . . . ,1/d} and

b^∗_i+1 =arg max

b⁰

v(b^∗_i,b⁰,X_i) + (1−δ_i)E{F_δ_i(b⁰,X_i+1)|X_i}}, for 1≤i,

where 0< δi <1 is a discount factor such thatδi ↓0. non-stationary policy

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Strategy 1

Fδ(b,x) = max

b⁰

v(b,b⁰,x) + (1−δ)E{F_δ(b⁰,X2)|X1=x} .

b^∗₁={1/d, . . . ,1/d} and

b^∗_i+1 =arg max

b⁰

v(b^∗_i,b⁰,X_i) + (1−δ_i)E{F_δ_i(b⁰,X_i+1)|X_i}}, for 1≤i, where 0< δi <1 is a discount factor such thatδi ↓0.

non-stationary policy

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Strategy 1

Fδ(b,x) = max

b⁰

v(b,b⁰,x) + (1−δ)E{F_δ(b⁰,X2)|X1=x} .

b^∗₁={1/d, . . . ,1/d} and

b^∗_i+1 =arg max

b⁰

v(b^∗_i,b⁰,X_i) + (1−δ_i)E{F_δ_i(b⁰,X_i+1)|X_i}}, for 1≤i, where 0< δi <1 is a discount factor such thatδi ↓0.

non-stationary policy

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Theorem 1

Assume

(i) that {X_i} is a homogeneous and first order Markov process,

(ii) and there exist 0<a1<1<a2 <∞ such that a1 ≤X^(j⁾≤a2 for all j = 1, . . . ,d.

Choose the discount factorδ_i ↓0 such that (δ_i −δ_i+1)/δ_i+1² →0 asi → ∞, and

∞

X

n=1

1

n²δ²_n <∞.

Then, for Strategy 1, the portfolio{b^∗_i} with capitalS_n^∗ is optimal in the sense that for any portfolio strategy{b_i} with capitalSn,

lim inf

n→∞

1

nlogS_n^∗−1 nlogSn

≥0 a.s.

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Theorem 1

Assume

(i) that {X_i} is a homogeneous and first order Markov process, (ii) and there exist 0<a1<1<a2 <∞ such that

a1 ≤X^(j⁾≤a2 for all j = 1, . . . ,d.

Choose the discount factorδ_i ↓0 such that (δ_i −δ_i+1)/δ_i+1² →0 asi → ∞, and

∞

X

n=1

1

n²δ²_n <∞.

lim inf

n→∞

1

≥0 a.s.

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Theorem 1

Assume

a1 ≤X^(j⁾≤a2 for all j = 1, . . . ,d. Choose the discount factorδ_i ↓0 such that

(δ_i −δ_i+1)/δ_i+1² →0 asi → ∞, and

∞

X

n=1

1

n²δ²_n <∞.

lim inf

n→∞

1

≥0 a.s.

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Theorem 1

Assume

a1 ≤X^(j⁾≤a2 for all j = 1, . . . ,d. Choose the discount factorδ_i ↓0 such that

(δ_i −δ_i+1)/δ_i+1² →0 asi → ∞, and

∞

X

n=1

1

n²δ²_n <∞.

lim inf

n→∞

1

≥0 a.s.

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Strategy 2

For any integer 1≤k, put

b^(k)₁ ={1/d, . . . ,1/d} and

b^(k)_i+1 =arg max

b⁰

v(b^(k)_i ,b⁰,Xi) + (1−δ_k)E{F_δ_k(b⁰,Xi+1)|X_i}}, for 1≤i.

The portfolioB^(k)={b^(k)_i } is called the portfolio of expertk with capitalS_n(B^(k)).

Choose an arbitrary probability distributionqk >0, and introduce the combined portfolio with its capital

S˜_n=

∞

X

k=1

q_kS_n(B^(k)). stationary policy

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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

b⁰

v(b^(k)_i ,b⁰,Xi) + (1−δ_k)E{F_δ_k(b⁰,Xi+1)|X_i}}, for 1≤i.

The portfolioB^(k)={b^(k)_i } is called the portfolio of expertk with capitalS_n(B^(k)).

Choose an arbitrary probability distributionqk >0, and introduce the combined portfolio with its capital

S˜_n=

∞

X

k=1

q_kS_n(B^(k)). stationary policy