Stochasticity of Correlations
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Stochasticity of Correlations
Xiaoyang ZhuangEconomics 201FSDuke University
2/23/2010
Motivation
The Problem
•In a crisis, “correlations go to 1.”
•For portfolio managers, converging correlations throw off diversification and hedging strategies.
Two Optimal Solutions
1. Predict when crises occur.
2. Dynamically rebalance portfolio as crisis unfolds.
Two Possible Approaches
1. Empirically observe the characteristics of an unfolding crisis.
2. Account for correlation as stochastic processes in the original portfolio optimization problem:
min(α) σ2 = αVα subject to αTe = 1, αT = P
(Buraschi, Porchia, and Trojani, 2010, J. Finance)
Long-Run vs. Crisis Correlations
Alcoa DuPont Ford JPMorgan Chase
Wal-Mart
AA 1
DD 0.7511 1
F 0.3872 0.4447 1
JPM 0.3621 0.3319 0.0896 1
WMT 0.0764 0.1474 0.2843 -0.2083 1
Alcoa DuPont Ford JPMorgan Chase
Wal-Mart
AA 1
DD 0.9587 1
F 0.7934 0.7919 1
JPM 0.7781 0.8787 0.7546 1
WMT 0.3469 0.4082 0.1317 0.3150 1
Long-Run Correlations: 1/1/2000 – 12/30/2010
Crisis Correlations: 6/1/20 – 12/30/2010
Roadmap
•Discuss the five stocks used in the data analysis and explain why they were selected
•For each pair of stocks, we will examine the1. Price series2. Correlations series (as implied by the stock and portfolio realized variances):3. Pearson Correlations
•Future directions
About the Stocks
Alcoa (AA) The world’s leading producer of aluminum.DuPont (DD) A diversified scientific company with innovations in “agriculture,
nutrition, electronics, communications, safety and protection, home and construction, transportation and apparel.”
Ford (F) An multinational car company.JPMorgan & Chase (JPM) A diversified financial services company.Wal-Mart (WMT) A multinational company operating a chain of discount department
stores and warehouse stores.
April 9, 1997 – December 23, 2010 (3420 days)
These stocks were selected because1. They belong to companies in diverse industries.
(To examine the effectiveness of diversification.)2. They did not exhibit long-term directional trends in the last decade.
(To isolate firm-level behavior from macroeconomic trends.)
NOTE: For each stock, most of the price variation was within $20 of the mean.
Alcoa and DuPont: Price Series
Alcoa and DuPont: Implied Correlation
Calculations•Variances (on the right-hand side of the equation) were estimated using the five-minute Realized Volatility estimator.
Alcoa and DuPont: Overlapping Pearson Correlation
Calculations•Pearson correlations are calculated in four-month intervals•If A and B are adjacent intervals, A and B overlap 119/120 days
Alcoa and Ford: Price Series
Alcoa and Ford : Implied Correlation
Calculations•Variances (on the right-hand side of the equation) were estimated using the five-minute Realized Volatility estimator.
Alcoa and Ford : Overlapping Pearson Correlation
Calculations•Pearson correlations are calculated in four-month intervals•If A and B are adjacent intervals, A and B overlap 119/120 days
Alcoa and JPMorgan Chase: Price Series
Alcoa and JPMorgan Chase: Implied Correlation
Calculations•Variances (on the right-hand side of the equation) were estimated using the five-minute Realized Volatility estimator.
Alcoa and JPM: Overlapping Pearson Correlation
Calculations•Pearson correlations are calculated in four-month intervals•If A and B are adjacent intervals, A and B overlap 119/120 days
Alcoa and Wal-Mart: Price Series
Alcoa and Wal-Mart: Implied Correlation
Calculations•Variances (on the right-hand side of the equation) were estimated using the five-minute Realized Volatility estimator.
Alcoa and WMT: Overlapping Pearson Correlation
Calculations•Pearson correlations are calculated in four-month intervals•If A and B are adjacent intervals, A and B overlap 119/120 days
DuPont and Ford: Price Series
DuPont and Ford: Implied Correlation
Calculations•Variances (on the right-hand side of the equation) were estimated using the five-minute Realized Volatility estimator.
DuPont and Ford: Overlapping Pearson Correlation
Calculations•Pearson correlations are calculated in four-month intervals•If A and B are adjacent intervals, A and B overlap 119/120 days
DuPont and JPMorgan Chase: Price Series
DuPont and JPMorgan Chase: Implied Correlation
Calculations•Variances (on the right-hand side of the equation) were estimated using the five-minute Realized Volatility estimator.
DuPont and JPM: Overlapping Pearson Correlation
Calculations•Pearson correlations are calculated in four-month intervals•If A and B are adjacent intervals, A and B overlap 119/120 days
Ford and JPMorgan Chase: Price Series
Ford and JPMorgan Chase: Implied Correlation
Calculations•Variances (on the right-hand side of the equation) were estimated using the five-minute Realized Volatility estimator.
Ford and JPM: Overlapping Pearson Correlation
Calculations•Pearson correlations are calculated in four-month intervals•If A and B are adjacent intervals, A and B overlap 119/120 days
Ford and Wal-Mart: Price Series
Ford and Wal-Mart: Implied Correlation
Calculations•Variances (on the right-hand side of the equation) were estimated using the five-minute Realized Volatility estimator.
Ford and WMT: Overlapping Pearson Correlation
Calculations•Pearson correlations are calculated in four-month intervals•If A and B are adjacent intervals, A and B overlap 119/120 days
JPMorgan Chase and Wal-Mart: Price Series
JPMorgan Chase and Wal-Mart: Implied Correlation
Calculations•Variances (on the right-hand side of the equation) were estimated using the five-minute Realized Volatility estimator.
JPM and WMT : Overlapping Pearson Correlation
Calculations•Pearson correlations are calculated in four-month intervals•If A and B are adjacent intervals, A and B overlap 119/120 days
Future Directions
Empirical directions
Explore the literature in more detail to find refinements to correlation estimates.“Covariance Estimation,” (Boudt, Cornelissen and Croux, 2010, working paper)“Estimating Covariation: Epps Effect, Microstructure” (Zhang, 2008, J. Econometrics)
Explore the differences between realized correlation and the implied correlations we’ve found here.
Explore the relationship between correlation and trading volume.
Explore the notion of correlation co-jumps.
Theoretical direction
Explore theoretical frameworks for dynamic portfolio optimization(Buraschi, Porchia, and Trojani, 2010, J. Finance)
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