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)