Principal component and constantly re-balanced portfolio

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Principal component and constantly re-balanced portfolio

György Ottucsák¹ László Györfi

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

Budapest, Hungary

September 20, 2007 e-mail: oti@szit.bme.hu

www.szit.bme.hu/˜oti

www.szit.bme.hu/˜oti/portfolio

Ottucs´ak, Gy¨orfi Principal component and constantly re-balanced portfolio

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Investment in the stock market: Growth rate

The model:

d assets

S_n^(j) price of asset j at the end of trading period (day) n initial price S₀^(j⁾= 1,

S_n^(j)=e^nWⁿ^(j) ≈e^nW^(j) j = 1, . . . ,d average growth rate

W_n^(j)= 1 nlnS_n^(j⁾ asymptotic average growth rate

W^(j⁾= lim

n→∞

1 nlnS_n^(j)

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Investment in the stock market: Growth rate

The model:

d assets

W^(j⁾= lim

n→∞

1 nlnS_n^(j)

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Investment in the stock market: Growth rate

The model:

d assets

S_n^(j)=e^nWⁿ^(j)

≈e^nW^(j)

j = 1, . . . ,d average growth rate

W_n^(j)= 1 nlnS_n^(j⁾

asymptotic average growth rate W^(j⁾= lim

n→∞

1 nlnS_n^(j)

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Investment in the stock market: Growth rate

The model:

d assets

W^(j⁾= lim

n→∞

1 nlnS_n^(j)

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Static portfolio selection: single period investment

The aim is to achieve maxjW^(j).

Static portfolio selection:

Fix a portfolio vectorb= (b⁽¹⁾, . . .b^(d)).

S0b^(j⁾ denotes the proportion of the investor’s capital invested in asset j. Assumptions:

no short-salesb^(j) ≥0 self-financingP

jb^(j)= 1

Aftern day

S_n =S₀X

j

b^(j)Sn^(j)

Use the following simple bound S₀max

j b^(j)S_n^(j)≤S_n≤dS₀max

j b^(j⁾S_n^(j⁾

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Static portfolio selection: single period investment

The aim is to achieve maxjW^(j). Static portfolio selection:

jb^(j)= 1

Aftern day

S_n =S₀X

j

b^(j)Sn^(j)

j b^(j⁾S_n^(j⁾

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Static portfolio selection: single period investment

jb^(j)= 1

Aftern day

S_n =S₀X

j

b^(j)Sn^(j)

j b^(j⁾S_n^(j⁾

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Static portfolio selection: single period investment

jb^(j)= 1

Aftern day

S_n =S₀X

j

b^(j)Sn^(j)

j b^(j⁾S_n^(j⁾

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assume thatb^(j)>0 1

nln max

j

S₀b^(j⁾S_n^(j⁾

≤ 1

nlnS_n≤ 1 n ln

dS₀max

j b^(j⁾S_n^(j)

maxj

1

nlnS₀b^(j)+ 1 nlnSn^(j⁾

≤ 1 n lnS_n

≤max

j

1

nln(dS₀b^(j⁾) +1 nlnS_n^(j)

n→∞lim 1

nlnS_n= lim

n→∞max

j

1

nlnS_n^(j)= max

j W^(j)

Conclusion: any static portfolio achieves the maximal growth rate max_jW^(j). We can do much better!

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nln max

j

≤ 1

nlnS_n≤ 1 n ln

dS₀max

j b^(j⁾S_n^(j)

maxj

1

≤ 1 n lnS_n

≤max

j

1

n→∞lim 1

nlnS_n= lim

n→∞max

j

1

nlnS_n^(j)= max

j W^(j)

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nln max

j

≤ 1

nlnS_n≤ 1 n ln

dS₀max

j b^(j⁾S_n^(j)

maxj

1

≤ 1 n lnS_n

≤max

j

1

n→∞lim 1

nlnS_n= lim

n→∞max

j

1

nlnS_n^(j)= max

j W^(j)

Conclusion: any static portfolio achieves the maximal growth rate max_jW^(j).

We can do much better!

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nln max

j

≤ 1

nlnS_n≤ 1 n ln

dS₀max

j b^(j⁾S_n^(j)

maxj

1

≤ 1 n lnS_n

≤max

j

1

n→∞lim 1

nlnS_n= lim

n→∞max

j

1

nlnS_n^(j)= max

j W^(j)

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Dynamic portfolio selection: multi-period investment

The model:

Let xi = (x_i⁽¹⁾, . . .x_i^(d)) the return vector on trading periodi, where

x_i^(j⁾= S_i^(j) S_i−1^(j) .

is the price relatives of two consecutive days.

x_i^(j) is the factor by which capital invested in stock j grows during the market periodi

One of the simplest dynamic portfolio strategy is the Constantly Re-balanced Portfolio (CRP):

Fix a portfolio vectorb= (b⁽¹⁾, . . .b^(d)), where b^(j) gives the proportion of the investor’s capital invested in stockj. Thisbis the portfolio vector for each trading day.

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Dynamic portfolio selection: multi-period investment

The model:

x_i^(j⁾= S_i^(j) S_i−1^(j) .

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Dynamic portfolio selection: multi-period investment

The model:

x_i^(j⁾= S_i^(j) S_i−1^(j) .

Fix a portfolio vectorb= (b⁽¹⁾, . . .b^(d)), where b^(j) gives the proportion of the investor’s capital invested in stockj.

Thisbis the portfolio vector for each trading day.

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Dynamic portfolio selection: multi-period investment

The model:

x_i^(j⁾= S_i^(j) S_i−1^(j) .

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Dynamic portfolio selection: multi-period investment:2

Repeatedly investment:

for the first trading period S₀ denotes the initial capital S1=S0

d

X

j=1

b^(j)x₁^(j)=S0hb,x1i

for the second trading period, S₁ new initial capital S2 =S1· hb,x2i=S0· hb,x1i · hb,x2i. for the nth trading period:

S_n=Sn−1hb,x_ni=S₀

n

Y

i=1

hb,x_ii

=S₀e^nWⁿ^(b)

with the average growth rate Wn(b) = 1

n

X

i=1

lnhb,xii.

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Dynamic portfolio selection: multi-period investment:2

d

X

j=1

b^(j)x₁^(j)=S0hb,x1i for the second trading period, S₁ new initial capital

S2 =S1· hb,x2i=S0· hb,x1i · hb,x2i.

for the nth trading period: S_n=Sn−1hb,x_ni=S₀

n

Y

i=1

hb,x_ii

=S₀e^nWⁿ^(b)

with the average growth rate Wn(b) = 1

n

X

i=1

lnhb,xii.

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Dynamic portfolio selection: multi-period investment:2

d

X

j=1

S2 =S1· hb,x2i=S0· hb,x1i · hb,x2i. for the nth trading period:

n

Y

i=1

hb,x_ii

=S₀e^nWⁿ^(b) with the average growth rate

Wn(b) = 1 n

n

X

i=1

lnhb,xii.

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Dynamic portfolio selection: multi-period investment:2

d

X

j=1

S2 =S1· hb,x2i=S0· hb,x1i · hb,x2i. for the nth trading period:

n

Y

i=1

hb,x_ii =S₀e^nWⁿ^(b) with the average growth rate

Wn(b) = 1 n

n

X

i=1

lnhb,xii.

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log-optimum portfolio

The CRP is the optimal portfolio for special market process, where X₁,X₂, . . . is independent and identically distributed (i.i.d.)

Log-optimum portfoliob^∗

E{lnhb^∗,X₁i}= max

b E{lnhb,X₁i}

Best Constantly Re-balanced Portfolio (BCRP) Properties:

needed full-knowledge on the distribution

in experiments: not a causal strategy. We can calculate it only in hindsight.

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log-optimum portfolio

b E{lnhb,X₁i}

Best Constantly Re-balanced Portfolio (BCRP)

Properties:

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log-optimum portfolio

b E{lnhb,X₁i}

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log-optimum portfolio

b E{lnhb,X₁i}

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Optimality

IfS_n^∗ =S_n(b^∗) denotes the capital after trading periodn achieved by a log-optimum portfolio strategyb^∗, then for any portfolio strategybwith capital Sn=Sn(b) and for any i.i.d. process {X_n}^∞_−∞,

n→∞lim 1

nlnS_n≤ lim

n→∞

1

nlnS_n^∗ almost surely

and

n→∞lim 1

n lnS_n^∗ =W^∗ almost surely, where

W^∗ =E{lnhb^∗,X₁i} is the maximal growth rate of any portfolio.

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Optimality

IfS_n^∗ =S_n(b^∗) denotes the capital after trading periodn achieved by a log-optimum portfolio strategyb^∗, then for any portfolio strategybwith capital Sn=Sn(b) and for any i.i.d. process {X_n}^∞_−∞,

n→∞lim 1

nlnS_n≤ lim

n→∞

1

nlnS_n^∗ almost surely and

n→∞lim 1

n lnS_n^∗ =W^∗ almost surely, where

W^∗ =E{lnhb^∗,X₁i}

is the maximal growth rate of any portfolio.

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Semi-log-optimal portfolio

log-optimal:

arg max

b

E{lnhb,X1i}

It is a non-linear (convex) optimization problem with linear constraints. Calculation: not cheap.

Idea: use the Taylor expansion:

lnz ≈h(z) =z−1−1

2(z−1)²

Only the two biggest principal components, others are drop. semi-log-optimal:

arg max

b

E{h(hb,X1i)}=arg max

b

{hb,mi − hb,Cbi}

Cheaper: Quadratic Programming (QP) Connection to the Markowitz theory.

Gy. Ottucs´ak and I. Vajda, ”An Asymptotic Analysis of the

Mean-Variance portfolio selection”,Statistics&Decisions, 25, pp. 63-88, 2007. http://www.szit.bme.hu/˜oti/portfolio/articles/marko.pdf

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Semi-log-optimal portfolio

log-optimal:

arg max

b

E{lnhb,X1i}

lnz ≈h(z) =z−1−1

2(z−1)²

Only the two biggest principal components, others are drop.

semi-log-optimal: arg max

b

{hb,mi − hb,Cbi}

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Semi-log-optimal portfolio

log-optimal:

arg max

b

E{lnhb,X1i}

lnz ≈h(z) =z−1−1

2(z−1)²

semi-log-optimal:

arg max

b

E{h(hb,X1i)}

=arg max

b

{hb,mi − hb,Cbi}

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Semi-log-optimal portfolio

log-optimal:

arg max

b

E{lnhb,X1i}

lnz ≈h(z) =z−1−1

2(z−1)²

semi-log-optimal:

arg max

b

{hb,mi − hb,Cbi}

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Semi-log-optimal portfolio

log-optimal:

arg max

b

E{lnhb,X1i}

lnz ≈h(z) =z−1−1

2(z−1)²

semi-log-optimal:

arg max

b

{hb,mi − hb,Cbi}

Cheaper: Quadratic Programming (QP)

Connection to the Markowitz theory.

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Semi-log-optimal portfolio

log-optimal:

arg max

b

E{lnhb,X1i}

lnz ≈h(z) =z−1−1

2(z−1)²

semi-log-optimal:

arg max

b

{hb,mi − hb,Cbi}

(34)

Semi-log-optimal portfolio

log-optimal:

arg max

b

E{lnhb,X1i}

lnz ≈h(z) =z−1−1

2(z−1)²

semi-log-optimal:

arg max

b

{hb,mi − hb,Cbi}

Mean-Variance portfolio selection”,Statistics&Decisions, 25, pp. 63-88,

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Principal component

We may write

E{hb,X1i −1} −1

2E{(hb,X1i −1)²}

= 2E{hb,X1i} −1

2E{hb,X1i²} −3 2

= −1

2E{(hb,X1i −2)²}+1 2 then

arg max

b

−1

2E{(hb,X₁i −2)²}+1 2 =

arg min

b

E{(hb,X1i −2)²}, that is, we are looking for the portfolio which minimize the expected squared error.

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Principal component

We may write

E{hb,X1i −1} −1

2E{(hb,X1i −1)²}

= 2E{hb,X1i} −1

2E{hb,X1i²} −3 2

= −1

2E{(hb,X1i −2)²}+1 2 then

arg max

b

−1

2E{(hb,X₁i −2)²}+1 2 =

arg min

b

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Principal component

We may write

E{hb,X1i −1} −1

2E{(hb,X1i −1)²}

= 2E{hb,X1i} −1

2E{hb,X1i²} −3 2

= −1

2E{(hb,X1i −2)²}+1 2

then

arg max

b

−1

2E{(hb,X₁i −2)²}+1 2 =

arg min

b

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Principal component

We may write

E{hb,X1i −1} −1

2E{(hb,X1i −1)²}

= 2E{hb,X1i} −1

2E{hb,X1i²} −3 2

= −1

2E{(hb,X1i −2)²}+1 2 then

arg max

b

−1

2E{(hb,X₁i −2)²}+1 2 =

arg min

b

E{(hb,X1i −2)²}, that is, we are looking for the portfolio which minimize the

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Conditions of the model:

Assume that

the assets are arbitrarily divisible,

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

there are no transaction costs, (go to Session 1 today at 17.30)

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

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Conditions of the model:

Assume that

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Conditions of the model:

Assume that

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Conditions of the model:

Assume that

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NYSE data sets

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

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

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

Both sets are corrected with the dividends.

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 sets fromNYSE:

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

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

Both sets are corrected with the dividends.

Our experiment is on the second data set.

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Experimental results on CRP

Stock’s name AAY BCRP

log-NLP weights semi-log-QP weights

COMME 18% 0.3028 0.2962

HP 15% 0.0100 0.0317

KINAR 4% 0.2175 0.2130

MORRIS 20% 0.4696 0.4590

AAY 24% 24%

running time (sec) 9002 3

Table: Comparison of the two algorithms for CRPs.

The other 19 assets have 0 weight KINAR had the smallest AAY

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Experimental results on CRP

COMME 18% 0.3028 0.2962

HP 15% 0.0100 0.0317

KINAR 4% 0.2175 0.2130

MORRIS 20% 0.4696 0.4590

AAY 24% 24%

The other 19 assets have 0 weight

KINAR had the smallest AAY

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Experimental results on CRP

COMME 18% 0.3028 0.2962

HP 15% 0.0100 0.0317

KINAR 4% 0.2175 0.2130

MORRIS 20% 0.4696 0.4590

AAY 24% 24%

The other 19 assets have 0 weight KINAR had the smallest AAY

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Dynamic portfolio selection: Causal-CRP

BCRP is not a causal strategy. A simple causal version could be, that we use the CRP that was optimal up ton−1 for the next (nth) day.

Stock’s name AAY BCRP CCRP Static

log-NLP w. semi-log-QP w.

COMME 18% 0.3028 0.2962

HP 15% 0.0100 0.0317

KINAR 4% 0.2175 0.2130

MORRIS 20% 0.4696 0.4590

AAY 24% 24% 14% 16%

running t. (sec) 9002 3 111 3

we can even do much better!!

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Dynamic portfolio selection: Causal-CRP

BCRP is not a causal strategy. A simple causal version could be, that we use the CRP that was optimal up ton−1 for the next (nth) day.

Stock’s name AAY BCRP CCRP Static

log-NLP w. semi-log-QP w.

COMME 18% 0.3028 0.2962

HP 15% 0.0100 0.0317

KINAR 4% 0.2175 0.2130

MORRIS 20% 0.4696 0.4590

AAY 24% 24% 14% 16%

running t. (sec) 9002 3 111 3

we can even do much better!!

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Dynamic portfolio selection: general case

x_i = (x_i⁽¹⁾, . . .x_i^(d)) the return vector on day i b=b1 is the portfolio vector for the first day initial capitalS0

S1=S0· hb₁,x1i

for the second day,S₁ new initial capital, the portfolio vector b2=b(x1)

S₂ =S₀· hb₁,x₁i · hb(x₁),x₂i.

nth day a portfolio strategy b_n=b(x₁, . . . ,xn−1) =b(xⁿ⁻¹₁ ) Sn=S0

n

Y

i=1

D

b(xⁱ⁻¹₁ ),x_i E

=

S0e^nWⁿ^(B)

with the average growth rate Wn(B) = 1

n

X

i=1

ln D

b(xⁱ⁻¹₁ ),xi

E .

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Dynamic portfolio selection: general case

S1=S0· hb₁,x1i

S₂ =S₀· hb₁,x₁i · hb(x₁),x₂i.

n

Y

i=1

D

=

S0e^nWⁿ^(B)

n

X

i=1

ln D

b(xⁱ⁻¹₁ ),xi

E .

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Dynamic portfolio selection: general case

S1=S0· hb₁,x1i

S₂ =S₀· hb₁,x₁i · hb(x₁),x₂i.

n

Y

i=1

D

=

S0e^nWⁿ^(B)

n

X

i=1

ln D

b(xⁱ⁻¹₁ ),xi

E .

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Dynamic portfolio selection: general case

S1=S0· hb₁,x1i

S₂ =S₀· hb₁,x₁i · hb(x₁),x₂i.

n

Y

i=1

D

=S0e^nWⁿ^(B) with the average growth rate

Wn(B) = 1 n

n

X

i=1

ln D

b(xⁱ⁻¹₁ ),xi

E .

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log-optimum portfolio

X1,X2, . . . drawn from the vector valued stationary and ergodic process

log-optimum portfolioB^∗ ={b^∗(·)}

E{ln

b^∗(Xⁿ⁻¹₁ ),X_n

|Xⁿ⁻¹₁ }= max

b(·) E{ln

b(Xⁿ⁻¹₁ ),X_n

|Xⁿ⁻¹₁ }

Xⁿ⁻¹₁ =X₁, . . . ,Xn−1

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Optimality

Algoet and Cover (1988): IfS_n^∗ =Sn(B^∗) denotes the capital after dayn achieved by a log-optimum portfolio strategyB^∗, then for any portfolio strategyBwith capital S_n=S_n(B) and for any process{X_n}^∞_−∞,

lim sup

n→∞

1

n lnS_n−1 nlnS_n^∗

≤0 almost surely for stationary ergodic process{X_n}^∞_−∞.

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Kernel-based portfolio selection

fix integersk, `= 1,2, . . . elementary portfolios

choose the radiusr_k,`>0 such that for any fixed k,

`→∞lim r_k,`= 0.

forn >k+ 1, define the expertb^(k,`) by b^(k,`)(xⁿ⁻¹₁ ) =arg max

b

X

{k<i<n:kxⁱ⁻¹_i−k−xⁿ⁻¹_n−kk≤rk,`}

lnhb,x_ii,

if the sum is non-void, andb0= (1/d, . . . ,1/d) otherwise.

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Kernel-based portfolio selection

fix integersk, `= 1,2, . . . elementary portfolios

choose the radiusr_k,`>0 such that for any fixed k,

`→∞lim r_k,`= 0.

forn >k+ 1, define the expertb^(k,`) by b^(k,`)(xⁿ⁻¹₁ ) =arg max

b

X

{k<i<n:kxⁱ⁻¹_i−k−xⁿ⁻¹_n−kk≤rk,`}

lnhb,x_ii,

if the sum is non-void, andb0= (1/d, . . . ,1/d) otherwise.

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Combining elementary portfolios

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

The strategyBis the combination of the elementary portfolio strategiesB^(k,`)={b^(k,`)_n }such that the investor’s capital becomes

Sn(B) =X

k,`

qk,`Sn(B^(k,`)).

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Combining elementary portfolios

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

The strategyBis the combination of the elementary portfolio strategiesB^(k,`)={b^(k,`)_n }such that the investor’s capital becomes

Sn(B) =X

k,`

qk,`Sn(B^(k,`)).

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Experiments on average annual yields (AAY)

Kernel based log-optimal portfolio selection with k= 1, . . . ,5 and `= 1, . . . ,10

r_k,`² = 0.0001·d·k·`,

AAY of kernel based semi-log-optimal portfolio is 128% double the capital

MORRIS had the best AAY, 20% the BCRP had average AAY 24%

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Experiments on average annual yields (AAY)

r_k,`² = 0.0001·d·k·`,

AAY of kernel based semi-log-optimal portfolio is 128%

double the capital

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Experiments on average annual yields (AAY)

r_k,`² = 0.0001·d·k·`,

double the capital

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Experiments on average annual yields (AAY)

r_k,`² = 0.0001·d·k·`,

double the capital

MORRIS had the best AAY, 20%

the BCRP had average AAY 24%

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Experiments on average annual yields (AAY)

r_k,`² = 0.0001·d·k·`,

double the capital

MORRIS had the best AAY, 20%

the BCRP had average AAY 24%

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The average annual yields of the individual experts.

k 1 2 3 4 5

`

1 20% 19% 16% 16% 16%

2 118% 77% 62% 24% 58%

3 71% 41% 26% 58% 21%

4 103% 94% 63% 97% 34%

5 134% 102% 100% 102% 67%

6 140% 125% 105% 108% 87%

7 148% 123% 107% 99% 96%

8 132% 112% 102% 85% 81%

9 127% 103% 98% 74% 72%

10 123% 92% 81% 65% 69%