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

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

Outline Online Investing 1 Constant Rebalanced Portfolios 2 Algorithms competing against best CRP 3 Algorithms competing against best CRP “Universal” algorithm Implementation 4 Semi-Constant-Rebalanced Portfolios (SCRP) 5 Jia Mao () Online Learning and Online Investing February 20, 2006 2 / 20

Portfolio Selection Consider n stocks Jia Mao () Online Learning and Online Investing February 20, 2006 3 / 20

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

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

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

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

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

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

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 . 16 t Problematic when stock price goes to zero. Jia Mao () Online Learning and Online Investing February 20, 2006 4 / 20

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

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

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

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

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

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

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

Online Learning vs. Online Investing Similarities ◮ Stocks ↔ Experts ◮ Wealth allocation ↔ probability distribution (i.e. weights) ◮ Stock i drops by l i % ↔ Expert i has loss l i Differences ◮ Initial wealth allocation (dot) Price relatives vector vs. Sum of losses Jia Mao () Online Learning and Online Investing February 20, 2006 6 / 20

Online Learning vs. Online Investing Similarities ◮ Stocks ↔ Experts ◮ Wealth allocation ↔ probability distribution (i.e. weights) ◮ Stock i drops by l i % ↔ Expert i has loss l i 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

Outline Online Investing 1 Constant Rebalanced Portfolios 2 Algorithms competing against best CRP 3 Algorithms competing against best CRP “Universal” algorithm Implementation 4 Semi-Constant-Rebalanced Portfolios (SCRP) 5 Jia Mao () Online Learning and Online Investing February 20, 2006 7 / 20

Compete against the best stock Jia Mao () Online Learning and Online Investing February 20, 2006 8 / 20

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

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

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

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

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

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 best stock ≥ 1 wealth of Split n Jia Mao () Online Learning and Online Investing February 20, 2006 8 / 20

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 best stock ≥ 1 wealth of Split n ◮ i.e. ln(Split) ≥ ln(best stock) - ln(n) Jia Mao () Online Learning and Online Investing February 20, 2006 8 / 20

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 best stock ≥ 1 wealth of Split n ◮ i.e. ln(Split) ≥ ln(best stock) - ln(n) n ) 1 / t → 1 as t → ∞ ◮ avg. per-day ratio ≥ ( 1 Jia Mao () Online Learning and Online Investing February 20, 2006 8 / 20

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

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

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

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