[PPT] - Online Learning and Online Investing Jia Mao February 20, 2006 Jia PowerPoint Presentation

SLIDE 1

Online Learning and Online Investing

Jia Mao February 20, 2006

Jia Mao () Online Learning and Online Investing February 20, 2006 1 / 20

SLIDE 2

Outline

1

Online Investing

2

Constant Rebalanced Portfolios

3

Algorithms competing against best CRP Algorithms competing against best CRP “Universal” algorithm

4

Implementation

5

Semi-Constant-Rebalanced Portfolios (SCRP)

Jia Mao () Online Learning and Online Investing February 20, 2006 2 / 20

SLIDE 3

Portfolio Selection

Consider n stocks

Jia Mao () Online Learning and Online Investing February 20, 2006 3 / 20

SLIDE 4

Portfolio Selection

Consider n stocks Our distribution of wealth is some vector b e.g. (1/3, 1/3, 1/3)

Jia Mao () Online Learning and Online Investing February 20, 2006 3 / 20

SLIDE 5

Portfolio Selection

Consider n stocks Our distribution of wealth is some vector b e.g. (1/3, 1/3, 1/3) At end of one period, we get a vector of “price relatives” x e.g. (0.98, 1.02, 1.00)

Jia Mao () Online Learning and Online Investing February 20, 2006 3 / 20

SLIDE 6

Portfolio Selection

Consider n stocks Our distribution of wealth is some vector b e.g. (1/3, 1/3, 1/3) At end of one period, we get a vector of “price relatives” x e.g. (0.98, 1.02, 1.00) Our wealth becomes b · x

Jia Mao () Online Learning and Online Investing February 20, 2006 3 / 20

SLIDE 7

Log Wealth

In each period, algorithm A performs 85% as well as algorithm B

Jia Mao () Online Learning and Online Investing February 20, 2006 4 / 20

SLIDE 8

Log Wealth

In each period, algorithm A performs 85% as well as algorithm B After t steps, we have A’s wealth = (0.85)t (B’s wealth)

Jia Mao () Online Learning and Online Investing February 20, 2006 4 / 20

SLIDE 9

Log Wealth

In each period, algorithm A performs 85% as well as algorithm B After t steps, we have A’s wealth = (0.85)t (B’s wealth) Nicer if we take logs ln(A) = ln(B) − 0.16t

Jia Mao () Online Learning and Online Investing February 20, 2006 4 / 20

SLIDE 10

Log Wealth

In each period, algorithm A performs 85% as well as algorithm B After t steps, we have A’s wealth = (0.85)t (B’s wealth) Nicer if we take logs ln(A) = ln(B) − 0.16t Problematic when stock price goes to zero.

Jia Mao () Online Learning and Online Investing February 20, 2006 4 / 20

SLIDE 11

References

Tom M. Cover, Universal Portfolios, 1991 Tom M. Cover, Erik Ordentlich, Universal Portfolios with Side Information, 1996 Avrim Blum and Adam Kalai, Universal Portfolios With and Without Transaction Costs, 1997 David P. Helmbold, Robert E. Schapire, Yoram Singer, Manfred K. Warmuth, Online Portfolio Selection Using Multiplicative Updates, 1998 Adam Kalai, slides, 1997 Avrim Blum, slides, 2000

Jia Mao () Online Learning and Online Investing February 20, 2006 5 / 20

SLIDE 12

Online Learning vs. Online Investing

Jia Mao () Online Learning and Online Investing February 20, 2006 6 / 20

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Online Learning vs. Online Investing

Similarities

Jia Mao () Online Learning and Online Investing February 20, 2006 6 / 20

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Online Learning vs. Online Investing

Similarities

◮ Stocks ↔ Experts Jia Mao () Online Learning and Online Investing February 20, 2006 6 / 20

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Online Learning vs. Online Investing

Similarities

◮ Stocks ↔ Experts ◮ Wealth allocation ↔ probability distribution (i.e. weights) Jia Mao () Online Learning and Online Investing February 20, 2006 6 / 20

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Online Learning vs. Online Investing

Similarities

◮ Stocks ↔ Experts ◮ Wealth allocation ↔ probability distribution (i.e. weights) ◮ Stock i drops by li% ↔ Expert i has loss li Jia Mao () Online Learning and Online Investing February 20, 2006 6 / 20

SLIDE 17

Online Learning vs. Online Investing

Similarities

◮ Stocks ↔ Experts ◮ Wealth allocation ↔ probability distribution (i.e. weights) ◮ Stock i drops by li% ↔ Expert i has loss li

Differences

Jia Mao () Online Learning and Online Investing February 20, 2006 6 / 20

SLIDE 18

Online Learning vs. Online Investing

Similarities

◮ Stocks ↔ Experts ◮ Wealth allocation ↔ probability distribution (i.e. weights) ◮ Stock i drops by li% ↔ Expert i has loss li

Differences

◮ Initial wealth allocation (dot) Price relatives vector

vs. Sum of losses

Jia Mao () Online Learning and Online Investing February 20, 2006 6 / 20

SLIDE 19

Online Learning vs. Online Investing

Similarities

◮ Stocks ↔ Experts ◮ Wealth allocation ↔ probability distribution (i.e. weights) ◮ Stock i drops by li% ↔ Expert i has loss li

Differences

◮ Initial wealth allocation (dot) Price relatives vector

vs. Sum of losses

◮ Stock price change automatically changes fraction of wealth

vs. Explicit update of weights

Jia Mao () Online Learning and Online Investing February 20, 2006 6 / 20

SLIDE 20

Outline

1

Online Investing

2

Constant Rebalanced Portfolios

3

Algorithms competing against best CRP Algorithms competing against best CRP “Universal” algorithm

4

Implementation

5

Semi-Constant-Rebalanced Portfolios (SCRP)

Jia Mao () Online Learning and Online Investing February 20, 2006 7 / 20

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Compete against the best stock

Jia Mao () Online Learning and Online Investing February 20, 2006 8 / 20

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Compete against the best stock

It’s hard, in a sense, because in worst case, we can’t hope to do better than

1 n ×(performance of best stock in hindsight)

Jia Mao () Online Learning and Online Investing February 20, 2006 8 / 20

SLIDE 23

Compete against the best stock

It’s hard, in a sense, because in worst case, we can’t hope to do better than

1 n ×(performance of best stock in hindsight)

However, there is a simple strategy Split to perform at least this well

Jia Mao () Online Learning and Online Investing February 20, 2006 8 / 20

SLIDE 24

Compete against the best stock

It’s hard, in a sense, because in worst case, we can’t hope to do better than

1 n ×(performance of best stock in hindsight)

However, there is a simple strategy Split to perform at least this well

◮ Initially invest an equal amount in each stock Jia Mao () Online Learning and Online Investing February 20, 2006 8 / 20

SLIDE 25

Compete against the best stock

It’s hard, in a sense, because in worst case, we can’t hope to do better than

1 n ×(performance of best stock in hindsight)

However, there is a simple strategy Split to perform at least this well

◮ Initially invest an equal amount in each stock ◮ Let it sit. (no trades) Jia Mao () Online Learning and Online Investing February 20, 2006 8 / 20

SLIDE 26

Compete against the best stock

It’s hard, in a sense, because in worst case, we can’t hope to do better than

1 n ×(performance of best stock in hindsight)

However, there is a simple strategy Split to perform at least this well

◮ Initially invest an equal amount in each stock ◮ Let it sit. (no trades) ◮ wealth of Split = avg. of stocks Jia Mao () Online Learning and Online Investing February 20, 2006 8 / 20

SLIDE 27

Compete against the best stock

It’s hard, in a sense, because in worst case, we can’t hope to do better than

1 n ×(performance of best stock in hindsight)

However, there is a simple strategy Split to perform at least this well

◮ Initially invest an equal amount in each stock ◮ Let it sit. (no trades) ◮ wealth of Split = avg. of stocks

wealth of Split wealth of best stock ≥ 1 n

Jia Mao () Online Learning and Online Investing February 20, 2006 8 / 20

SLIDE 28

Compete against the best stock

It’s hard, in a sense, because in worst case, we can’t hope to do better than

1 n ×(performance of best stock in hindsight)

However, there is a simple strategy Split to perform at least this well

◮ Initially invest an equal amount in each stock ◮ Let it sit. (no trades) ◮ wealth of Split = avg. of stocks

wealth of Split wealth of best stock ≥ 1 n

◮ i.e. ln(Split) ≥ ln(best stock) - ln(n) Jia Mao () Online Learning and Online Investing February 20, 2006 8 / 20

SLIDE 29

Compete against the best stock

It’s hard, in a sense, because in worst case, we can’t hope to do better than

1 n ×(performance of best stock in hindsight)

However, there is a simple strategy Split to perform at least this well

◮ Initially invest an equal amount in each stock ◮ Let it sit. (no trades) ◮ wealth of Split = avg. of stocks

wealth of Split wealth of best stock ≥ 1 n

◮ i.e. ln(Split) ≥ ln(best stock) - ln(n) ◮ avg. per-day ratio ≥ ( 1

n)1/t → 1 as t → ∞

Jia Mao () Online Learning and Online Investing February 20, 2006 8 / 20

SLIDE 30

Constant Rebalanced Portfolios (CRPs)

Definition

CRP(b): at end of each period, rebalance back to same distribution of wealth b.

Jia Mao () Online Learning and Online Investing February 20, 2006 9 / 20

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Constant Rebalanced Portfolios (CRPs)

Definition

CRP(b): at end of each period, rebalance back to same distribution of wealth b. Why CRP?

Jia Mao () Online Learning and Online Investing February 20, 2006 9 / 20

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Constant Rebalanced Portfolios (CRPs)

Definition

CRP(b): at end of each period, rebalance back to same distribution of wealth b. Why CRP? Intuition: Take advantage of market volatility – “Buy low, Sell high”

Jia Mao () Online Learning and Online Investing February 20, 2006 9 / 20

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CRP Example

Two stocks: first one stays constant (cash), second one alternately doubles and halves

Jia Mao () Online Learning and Online Investing February 20, 2006 10 / 20

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CRP Example

Two stocks: first one stays constant (cash), second one alternately doubles and halves Investing in a single stock will not increase the wealth by more than a factor of 2

Jia Mao () Online Learning and Online Investing February 20, 2006 10 / 20

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CRP Example

Two stocks: first one stays constant (cash), second one alternately doubles and halves Investing in a single stock will not increase the wealth by more than a factor of 2 Consider CRP(1/2, 1/2):

Jia Mao () Online Learning and Online Investing February 20, 2006 10 / 20

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CRP Example

Two stocks: first one stays constant (cash), second one alternately doubles and halves Investing in a single stock will not increase the wealth by more than a factor of 2 Consider CRP(1/2, 1/2): Stock #1 Stock #2 Our wealth

Jia Mao () Online Learning and Online Investing February 20, 2006 10 / 20

SLIDE 37

CRP Example

Two stocks: first one stays constant (cash), second one alternately doubles and halves Investing in a single stock will not increase the wealth by more than a factor of 2 Consider CRP(1/2, 1/2): Stock #1 Stock #2 Our wealth 1 1 1 = 1/2 + 1/2

Jia Mao () Online Learning and Online Investing February 20, 2006 10 / 20

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CRP Example

Two stocks: first one stays constant (cash), second one alternately doubles and halves Investing in a single stock will not increase the wealth by more than a factor of 2 Consider CRP(1/2, 1/2): Stock #1 Stock #2 Our wealth 1 1 1 = 1/2 + 1/2 1 2 3/2 = 1/2 + 1 → 3/4 + 3/4

Jia Mao () Online Learning and Online Investing February 20, 2006 10 / 20

SLIDE 39

CRP Example

Two stocks: first one stays constant (cash), second one alternately doubles and halves Investing in a single stock will not increase the wealth by more than a factor of 2 Consider CRP(1/2, 1/2): Stock #1 Stock #2 Our wealth 1 1 1 = 1/2 + 1/2 1 2 3/2 = 1/2 + 1 → 3/4 + 3/4 1 1 9/8 = 3/4 + 3/8 → 9/16 + 9/16

Jia Mao () Online Learning and Online Investing February 20, 2006 10 / 20

SLIDE 40

CRP Example

Two stocks: first one stays constant (cash), second one alternately doubles and halves Investing in a single stock will not increase the wealth by more than a factor of 2 Consider CRP(1/2, 1/2): Stock #1 Stock #2 Our wealth 1 1 1 = 1/2 + 1/2 1 2 3/2 = 1/2 + 1 → 3/4 + 3/4 1 1 9/8 = 3/4 + 3/8 → 9/16 + 9/16 1 2 (3/2)(9/8) = 9/16 + 9/8 → 27/32 + 27/32

Jia Mao () Online Learning and Online Investing February 20, 2006 10 / 20

SLIDE 41

CRP Example

Two stocks: first one stays constant (cash), second one alternately doubles and halves Investing in a single stock will not increase the wealth by more than a factor of 2 Consider CRP(1/2, 1/2): Stock #1 Stock #2 Our wealth 1 1 1 = 1/2 + 1/2 1 2 3/2 = 1/2 + 1 → 3/4 + 3/4 1 1 9/8 = 3/4 + 3/8 → 9/16 + 9/16 1 2 (3/2)(9/8) = 9/16 + 9/8 → 27/32 + 27/32 1 1 (9/8)2 = 27/32 + 27/64 . . .

Jia Mao () Online Learning and Online Investing February 20, 2006 10 / 20

SLIDE 42

CRP Example

Two stocks: first one stays constant (cash), second one alternately doubles and halves Investing in a single stock will not increase the wealth by more than a factor of 2 Consider CRP(1/2, 1/2): Stock #1 Stock #2 Our wealth 1 1 1 = 1/2 + 1/2 1 2 3/2 = 1/2 + 1 → 3/4 + 3/4 1 1 9/8 = 3/4 + 3/8 → 9/16 + 9/16 1 2 (3/2)(9/8) = 9/16 + 9/8 → 27/32 + 27/32 1 1 (9/8)2 = 27/32 + 27/64 . . . 1 1 1.12t/2

Jia Mao () Online Learning and Online Investing February 20, 2006 10 / 20

SLIDE 43

CRP Example

Two stocks: first one stays constant (cash), second one alternately doubles and halves Investing in a single stock will not increase the wealth by more than a factor of 2 Consider CRP(1/2, 1/2): Stock #1 Stock #2 Our wealth 1 1 1 = 1/2 + 1/2 1 2 3/2 = 1/2 + 1 → 3/4 + 3/4 1 1 9/8 = 3/4 + 3/8 → 9/16 + 9/16 1 2 (3/2)(9/8) = 9/16 + 9/8 → 27/32 + 27/32 1 1 (9/8)2 = 27/32 + 27/64 . . . 1 1 1.12t/2

Remark

In hindsight, we see that a (1

2, 1 2)-CRP is optimal among all CRPs.

Jia Mao () Online Learning and Online Investing February 20, 2006 10 / 20

SLIDE 44

Outline

1

Online Investing

2

Constant Rebalanced Portfolios

3

Algorithms competing against best CRP Algorithms competing against best CRP “Universal” algorithm

4

Implementation

5

Semi-Constant-Rebalanced Portfolios (SCRP)

Jia Mao () Online Learning and Online Investing February 20, 2006 11 / 20

SLIDE 45

Compete against the best CRP

Jia Mao () Online Learning and Online Investing February 20, 2006 12 / 20

SLIDE 46

Compete against the best CRP

What if we split up our wealth evenly among all CRPs and let it sit?

Jia Mao () Online Learning and Online Investing February 20, 2006 12 / 20

SLIDE 47

Compete against the best CRP

What if we split up our wealth evenly among all CRPs and let it sit? [Cover’91], [Cover & Ordentlich’96]

◮ algorithm Universal ◮ wealth ≥ (best CRP)/(t + 1)n−1 ◮ better split → wealth ≥ (best CRP)/

(t + 1)n−1

◮ per-day ratio → 1 Jia Mao () Online Learning and Online Investing February 20, 2006 12 / 20

SLIDE 48

Compete against the best CRP

What if we split up our wealth evenly among all CRPs and let it sit? [Cover’91], [Cover & Ordentlich’96]

◮ algorithm Universal ◮ wealth ≥ (best CRP)/(t + 1)n−1 ◮ better split → wealth ≥ (best CRP)/

(t + 1)n−1

◮ per-day ratio → 1

[Blum & Kalai’97]

◮ simpler proof of previous result ◮ extension to include transaction costs Jia Mao () Online Learning and Online Investing February 20, 2006 12 / 20

SLIDE 49

Compete against the best CRP

What if we split up our wealth evenly among all CRPs and let it sit? [Cover’91], [Cover & Ordentlich’96]

◮ algorithm Universal ◮ wealth ≥ (best CRP)/(t + 1)n−1 ◮ better split → wealth ≥ (best CRP)/

(t + 1)n−1

◮ per-day ratio → 1

[Blum & Kalai’97]

◮ simpler proof of previous result ◮ extension to include transaction costs

[Helmbold, Schapire, Singer, Warmuth’98]

◮ “experts”-based algorithm EG(η) ◮ multiplicative update rule, as used in online regression [Kivinen &

Warmuth]: find new wealth distribution vector bt+1 that maximizes η log(wt+1 · xt) − D(wt+1, wt)

◮ worse guarantees, but better performance on historical data Jia Mao () Online Learning and Online Investing February 20, 2006 12 / 20

SLIDE 50

Universal algorithm

Split money evenly among all CRPs Let it sit (i.e. Do not transfer between CRPs)

Jia Mao () Online Learning and Online Investing February 20, 2006 13 / 20

SLIDE 51

Universal algorithm

Split money evenly among all CRPs Let it sit (i.e. Do not transfer between CRPs) 4 CRPs CRP(1

3, 1 3, 1 3)

CRP(0,0,1) CRP(0,1,0) CRP(1,0,0)

Jia Mao () Online Learning and Online Investing February 20, 2006 13 / 20

SLIDE 52

Universal algorithm

Split money evenly among all CRPs Let it sit (i.e. Do not transfer between CRPs) 4 CRPs CRP(1

3, 1 3, 1 3)

CRP(0,0,1) CRP(0,1,0) CRP(1,0,0) 100 CRPs CRP(1

3, 1 3, 1 3)

. . . CRP(1

7, 2 7, 4 7)

. . . CRP(0,0,1)

Jia Mao () Online Learning and Online Investing February 20, 2006 13 / 20

SLIDE 53

Universal algorithm

Split money evenly among all CRPs Let it sit (i.e. Do not transfer between CRPs) 4 CRPs CRP(1

3, 1 3, 1 3)

CRP(0,0,1) CRP(0,1,0) CRP(1,0,0) 100 CRPs CRP(1

3, 1 3, 1 3)

. . . CRP(1

7, 2 7, 4 7)

. . . CRP(0,0,1) . . . . . . . . .

Jia Mao () Online Learning and Online Investing February 20, 2006 13 / 20

SLIDE 54

Universal algorithm

Split money evenly among all CRPs Let it sit (i.e. Do not transfer between CRPs) 4 CRPs CRP(1

3, 1 3, 1 3)

CRP(0,0,1) CRP(0,1,0) CRP(1,0,0) 100 CRPs CRP(1

3, 1 3, 1 3)

. . . CRP(1

7, 2 7, 4 7)

. . . CRP(0,0,1) . . . . . . . . . Limit is Universal algorithm.

Jia Mao () Online Learning and Online Investing February 20, 2006 13 / 20

SLIDE 55

Universal algorithm

Split money evenly among all CRPs Let it sit (i.e. Do not transfer between CRPs) 4 CRPs CRP(1

3, 1 3, 1 3)

CRP(0,0,1) CRP(0,1,0) CRP(1,0,0) 100 CRPs CRP(1

3, 1 3, 1 3)

. . . CRP(1

7, 2 7, 4 7)

. . . CRP(0,0,1) . . . . . . . . . Limit is Universal algorithm. Guarantee: wealth of Universal ≥ (best CRP)/(t + 1)n−1

Jia Mao () Online Learning and Online Investing February 20, 2006 13 / 20

SLIDE 56

Universal algorithm

Split money evenly among all CRPs Let it sit (i.e. Do not transfer between CRPs) 4 CRPs CRP(1

3, 1 3, 1 3)

CRP(0,0,1) CRP(0,1,0) CRP(1,0,0) 100 CRPs CRP(1

3, 1 3, 1 3)

. . . CRP(1

7, 2 7, 4 7)

. . . CRP(0,0,1) . . . . . . . . . Limit is Universal algorithm. Guarantee: wealth of Universal ≥ (best CRP)/(t + 1)n−1 More applications: data compression, language modelling, etc.

Jia Mao () Online Learning and Online Investing February 20, 2006 13 / 20

SLIDE 57

Simple analysis (without commission)

Jia Mao () Online Learning and Online Investing February 20, 2006 14 / 20

SLIDE 58

Simple analysis (without commission)

Universal achieves avg. wealth of all CRP’s.

Jia Mao () Online Learning and Online Investing February 20, 2006 14 / 20

SLIDE 59

Simple analysis (without commission)

Universal achieves avg. wealth of all CRP’s. “Near” CRP’s do nearly as well.

Jia Mao () Online Learning and Online Investing February 20, 2006 14 / 20

SLIDE 60

Simple analysis (without commission)

Universal achieves avg. wealth of all CRP’s. “Near” CRP’s do nearly as well. Lots of CRP’s are “near” the optimal CRP.

Jia Mao () Online Learning and Online Investing February 20, 2006 14 / 20

SLIDE 61

Simple analysis (without commission)

Universal achieves avg. wealth of all CRP’s. “Near” CRP’s do nearly as well. Lots of CRP’s are “near” the optimal CRP.

Jia Mao () Online Learning and Online Investing February 20, 2006 14 / 20

SLIDE 62

Proof.

Jia Mao () Online Learning and Online Investing February 20, 2006 15 / 20

SLIDE 63

Proof.

1 b is “near” b∗ if b = (1 − α)b∗ + αz Jia Mao () Online Learning and Online Investing February 20, 2006 15 / 20

SLIDE 64

Proof.

1 b is “near” b∗ if b = (1 − α)b∗ + αz 2

Wealth of CRPb Wealth of CRPb∗ ≥ (1 − α)t

Jia Mao () Online Learning and Online Investing February 20, 2006 15 / 20

SLIDE 65

Proof.

1 b is “near” b∗ if b = (1 − α)b∗ + αz 2

Wealth of CRPb Wealth of CRPb∗ ≥ (1 − α)t

3 Prob{a random b is “near” b∗} is

Vol{(1 − α)b∗ + αz|z ∈ β} Volβ = Vol{αz|z ∈ β} Volβ = αt−1

Jia Mao () Online Learning and Online Investing February 20, 2006 15 / 20

SLIDE 66

Proof.

1 b is “near” b∗ if b = (1 − α)b∗ + αz 2

Wealth of CRPb Wealth of CRPb∗ ≥ (1 − α)t

3 Prob{a random b is “near” b∗} is

Vol{(1 − α)b∗ + αz|z ∈ β} Volβ = Vol{αz|z ∈ β} Volβ = αt−1

4 if we choose α =

1 t+1, we will get

Wealth of CRPx Wealth of CRPy ≥ ( t t + 1)t ≥ 1/e

Jia Mao () Online Learning and Online Investing February 20, 2006 15 / 20

SLIDE 67

Proof.

1 b is “near” b∗ if b = (1 − α)b∗ + αz 2

Wealth of CRPb Wealth of CRPb∗ ≥ (1 − α)t

3 Prob{a random b is “near” b∗} is

Vol{(1 − α)b∗ + αz|z ∈ β} Volβ = Vol{αz|z ∈ β} Volβ = αt−1

4 if we choose α =

1 t+1, we will get

Wealth of CRPx Wealth of CRPy ≥ ( t t + 1)t ≥ 1/e Wealth of Universal Wealth of best CRP ≥ ( 1 t + 1)n−1 1 e

Jia Mao () Online Learning and Online Investing February 20, 2006 15 / 20

SLIDE 68

Proof.

1 b is “near” b∗ if b = (1 − α)b∗ + αz 2

Wealth of CRPb Wealth of CRPb∗ ≥ (1 − α)t

3 Prob{a random b is “near” b∗} is

Vol{(1 − α)b∗ + αz|z ∈ β} Volβ = Vol{αz|z ∈ β} Volβ = αt−1

4 if we choose α =

1 t+1, we will get

Wealth of CRPx Wealth of CRPy ≥ ( t t + 1)t ≥ 1/e Wealth of Universal Wealth of best CRP ≥ ( 1 t + 1)n−1 1 e

5 A more refined analysis can get rid of the 1

e

Jia Mao () Online Learning and Online Investing February 20, 2006 15 / 20

SLIDE 69

Transaction Costs

Jia Mao () Online Learning and Online Investing February 20, 2006 16 / 20

SLIDE 70

Transaction Costs

Fixed % commission charged on purchases, paid for by sales

Jia Mao () Online Learning and Online Investing February 20, 2006 16 / 20

SLIDE 71

Transaction Costs

Fixed % commission charged on purchases, paid for by sales CRPs pay commission (c < 1) as well

Jia Mao () Online Learning and Online Investing February 20, 2006 16 / 20

SLIDE 72

Transaction Costs

Fixed % commission charged on purchases, paid for by sales CRPs pay commission (c < 1) as well x = (1 − α)y + αz

Jia Mao () Online Learning and Online Investing February 20, 2006 16 / 20

SLIDE 73

Transaction Costs

Fixed % commission charged on purchases, paid for by sales CRPs pay commission (c < 1) as well x = (1 − α)y + αz CRPx day’s gain CRPy day’s gain ≥ (1 − α)(1 − αc) ≥ (1 − α)(1 − α)c ≥ (1 − α)1+c

Jia Mao () Online Learning and Online Investing February 20, 2006 16 / 20

SLIDE 74

Transaction Costs

Fixed % commission charged on purchases, paid for by sales CRPs pay commission (c < 1) as well x = (1 − α)y + αz CRPx day’s gain CRPy day’s gain ≥ (1 − α)(1 − αc) ≥ (1 − α)(1 − α)c ≥ (1 − α)1+c Wealth of CRPx Wealth of CRPy ≥ (1 − α)(1+c)t

Jia Mao () Online Learning and Online Investing February 20, 2006 16 / 20

SLIDE 75

Transaction Costs

Fixed % commission charged on purchases, paid for by sales CRPs pay commission (c < 1) as well x = (1 − α)y + αz CRPx day’s gain CRPy day’s gain ≥ (1 − α)(1 − αc) ≥ (1 − α)(1 − α)c ≥ (1 − α)1+c Wealth of CRPx Wealth of CRPy ≥ (1 − α)(1+c)t So with commission, t → t(1 + c)

Jia Mao () Online Learning and Online Investing February 20, 2006 16 / 20

SLIDE 76

Transaction Costs

Fixed % commission charged on purchases, paid for by sales CRPs pay commission (c < 1) as well x = (1 − α)y + αz CRPx day’s gain CRPy day’s gain ≥ (1 − α)(1 − αc) ≥ (1 − α)(1 − α)c ≥ (1 − α)1+c Wealth of CRPx Wealth of CRPy ≥ (1 − α)(1+c)t So with commission, t → t(1 + c) Wealth of Universal Wealth of best CRP ≥

1

(1 + c)t + 1 n−1

Jia Mao () Online Learning and Online Investing February 20, 2006 16 / 20

SLIDE 77

Outline

1

Online Investing

2

Constant Rebalanced Portfolios

3

Algorithms competing against best CRP Algorithms competing against best CRP “Universal” algorithm

4

Implementation

5

Semi-Constant-Rebalanced Portfolios (SCRP)

Jia Mao () Online Learning and Online Investing February 20, 2006 17 / 20

SLIDE 78

Implementing “Universal”

Uniform randomized approximation [Blum & Kalai]

Jia Mao () Online Learning and Online Investing February 20, 2006 18 / 20

SLIDE 79

Implementing “Universal”

Uniform randomized approximation [Blum & Kalai]

◮ if the best CRP achieves wealth

R· Universal

Jia Mao () Online Learning and Online Investing February 20, 2006 18 / 20

SLIDE 80

Implementing “Universal”

Uniform randomized approximation [Blum & Kalai]

◮ if the best CRP achieves wealth

R· Universal

◮ Chebyshev’s inequality ensures that using N ≥ (R−1)

ǫδ

random CRP’s, with probability at least 1 − δ, wealth of approximation ≥ (1 − ǫ)· (wealth of Universal)

Jia Mao () Online Learning and Online Investing February 20, 2006 18 / 20

SLIDE 81

Implementing “Universal”

Uniform randomized approximation [Blum & Kalai]

◮ if the best CRP achieves wealth

R· Universal

◮ Chebyshev’s inequality ensures that using N ≥ (R−1)

ǫδ

random CRP’s, with probability at least 1 − δ, wealth of approximation ≥ (1 − ǫ)· (wealth of Universal)

◮ for a given market, can determine in hindsight the optimal CRP

[Helmbold] and estimate R

Jia Mao () Online Learning and Online Investing February 20, 2006 18 / 20

SLIDE 82

Implementing “Universal”

Uniform randomized approximation [Blum & Kalai]

◮ if the best CRP achieves wealth

R· Universal

◮ Chebyshev’s inequality ensures that using N ≥ (R−1)

ǫδ

random CRP’s, with probability at least 1 − δ, wealth of approximation ≥ (1 − ǫ)· (wealth of Universal)

◮ for a given market, can determine in hindsight the optimal CRP

[Helmbold] and estimate R

◮ in the worst case, R grows like tn−1 Jia Mao () Online Learning and Online Investing February 20, 2006 18 / 20

SLIDE 83

Implementing “Universal”

Uniform randomized approximation [Blum & Kalai]

◮ if the best CRP achieves wealth

R· Universal

◮ Chebyshev’s inequality ensures that using N ≥ (R−1)

ǫδ

random CRP’s, with probability at least 1 − δ, wealth of approximation ≥ (1 − ǫ)· (wealth of Universal)

◮ for a given market, can determine in hindsight the optimal CRP

[Helmbold] and estimate R

◮ in the worst case, R grows like tn−1 ◮ in practical experiments on stock market data, R < 2 for various

combinations of two stocks

Jia Mao () Online Learning and Online Investing February 20, 2006 18 / 20

SLIDE 84

Implementing “Universal”

Uniform randomized approximation [Blum & Kalai]

◮ if the best CRP achieves wealth

R· Universal

◮ Chebyshev’s inequality ensures that using N ≥ (R−1)

ǫδ

random CRP’s, with probability at least 1 − δ, wealth of approximation ≥ (1 − ǫ)· (wealth of Universal)

◮ for a given market, can determine in hindsight the optimal CRP

[Helmbold] and estimate R

◮ in the worst case, R grows like tn−1 ◮ in practical experiments on stock market data, R < 2 for various

combinations of two stocks

Non-uniform randomized approximation [Kalai & Vempala]

◮ Same performance guarantees Jia Mao () Online Learning and Online Investing February 20, 2006 18 / 20

SLIDE 85

Implementing “Universal”

Uniform randomized approximation [Blum & Kalai]

◮ if the best CRP achieves wealth

R· Universal

◮ Chebyshev’s inequality ensures that using N ≥ (R−1)

ǫδ

random CRP’s, with probability at least 1 − δ, wealth of approximation ≥ (1 − ǫ)· (wealth of Universal)

◮ for a given market, can determine in hindsight the optimal CRP

[Helmbold] and estimate R

◮ in the worst case, R grows like tn−1 ◮ in practical experiments on stock market data, R < 2 for various

combinations of two stocks

Non-uniform randomized approximation [Kalai & Vempala]

◮ Same performance guarantees ◮ Runtime is polynomial in log(1/η), 1/ǫ, t, and n Jia Mao () Online Learning and Online Investing February 20, 2006 18 / 20

SLIDE 86

Outline

1

Online Investing

2

Constant Rebalanced Portfolios

3

Algorithms competing against best CRP Algorithms competing against best CRP “Universal” algorithm

4

Implementation

5

Semi-Constant-Rebalanced Portfolios (SCRP)

Jia Mao () Online Learning and Online Investing February 20, 2006 19 / 20

SLIDE 87

Semi-Constant-Rebalanced Portfolios (SCRP)

Definition

SCRP(b): at end of any subset of the periods, rebalance back to same distribution of wealth b. Proposed as a good strategy in the presence of transaction costs [Helmbold] Flexible: one may prefer not to rebalance if transaction costs

utweigh the benefits of rebalancing

No strategy can guarantee the exponential growth rate of the best SCRP in hindsight, even without commission [Blum&Kalai]

Jia Mao () Online Learning and Online Investing February 20, 2006 20 / 20