On Portfolio Optimization: How Do We Benefit from High-Frequency Data?

1. On Portfolio Optimization: How Do We Benefit from High-Frequency Data? Qianqiu Liu

2. Motivation • Do high-frequency data help in estimating large covariance matrices? • How to form a large covariance matrix estimator with high-frequency data?

3. Why high-frequency data

4. Why high-frequency data(continued)

5. Literature Review • 1. Merton (1980) ; Nelson (1992) • 2. Andersen, Bollerslev, Diebold, and Labys (2001) • 3. Bai, Russell, and Tiao (2001) • 4.Fleming, Kirby, and Ostdiek (2002)

6. Overview • Portfolio Optimization Problem • Estimation Methods • Data and Empirical Results • Robustness Checks and Simulation Exercise • Future Research

7. Portfolio Optimization Problem

8. Covariance Matrix Estimators Based on Daily Returns

9. Covariance Matrix Estimators Based on Daily Returns (continued)

10. Covariance Matrix Estimators Based on Intraday Returns

11. Covariance Matrix Estimators Based on Intraday Returns (continued)

12. Monthly Covariance Matrix Forecast

13. Data • TAQ: The sample period is from January 2, 1993 until June 30, 2000. The five-minute returns for DJIA stocks are constructed from the difference between the average of the log bid and the log ask. • CME: The S&P 500 cash index • CRSP: Daily returns

15. Empirical Results • summary statistics for daily returns and five-minute returns • the means of high-frequency returns are effectively zero from a statistical perspective; • The cross-sectional average kurtosis increases as five-minute returns replace daily returns; • most of the daily return series of DJIA stocks and daily S&P 500 index returns can be seen as uncorrelated; • the five-minute returns of all the individual stocks, except INTC and MSFT, have significant negative lag one autocorrelations. The first order autocorrelation of five-minute S&P returns is highly positive.

17. The out of sample performance of minimum tracking error portfolios • Monthly rebalancing based on past 12 months of data • Daily Returns: • sample covariance matrix estimator is the best; • Short sale restrictions has little effect on the composition of the minimum tracking error portfolios. • High-frequency Returns • including overnight returns improves out of sample performance; • the sample covariance matrix estimator performs worse than the sample covariance matrix using daily returns; • When lag one covariances are included, the sample covariance matrix estimator becomes comparable to that of daily return based sample covariance matrix; • rolling sample estimators perform marginally better than the daily return based sample covariance matrix with appropriate choices of the decay rate and the autoregressive coefficient.

19. The out of sample performance of minimum tracking error portfolios (continued) • Daily rebalancing based on past 12 months of data • Daily returns • sample covariance matrix estimator has the tracking error as small as the best estimator. • High-frequency returns • Including the overnight returns, the lag one covariances will lead the sample covariance matrix to have comparable performance to the daily sample covariance matrix; • the rolling sample estimator with the optimal decay rate has a standard deviation substantially lower than the daily sample covariance matrix.

22. The out of sample performance of minimum tracking error portfolios (continued) • T-test • the differences in mean returns for all the estimators are insignificant • Monthly Rebalance: all the covariance estimators perform not better than the benchmark • Daily Rebalance: optimal rolling sample estimators perform better than the daily sample covariance matrix estimator • Estimation Horizon • As the estimation period shrinks, the standard deviation of the minimum tracking error portfolio from the daily sample covariance matrix increases • The sample covariance matrix using one month of high-frequency data has better performance than that using one year of high-frequency data. • As the estimation period is less than 6 months, the microstructure-adjusted high-frequency sample covariance matrix estimator always performs significantly better than the daily sample covariance matrix.

24. The out of sample performance of minimum variance portfolios • Monthly Rebalance with the previous 12 months of data, the daily sample covariance matrix performs the best among all the estimators from daily returns or high-frequency returns; • Daily Rebalance: The tracking error from the sample covariance matrix using five-minute returns and overnight returns is smaller than that of the daily sample covariance matrix. The optimal rolling sample estimator with high-frequency data performs much better than that based on daily returns; • If only past 3 months of data are used, then the tracking error of the sample covariance matrix using five-minute returns and overnight returns is substantially lower than its daily based estimator, when the portfolio is rebalanced monthly; • The weights on the component stocks.

26. Simulation Results Set the population covariance matrix to be the sample covariance matrix of the daily excess returns of the 30 DJIA stocks relative to the S&P 500 cash index. Then a random sample of intraday returns is drawn assuming that the daily returns have a joint Normal distribution with this covariance matrix. • In both monthly rebalance and daily rebalance cases, the tracking error from high-frequency data is always smaller than that from daily returns • Substantial performance gains can be obtained when only three months of data can be used • The larger sample size leads to smaller tracking errors • The results from monthly and daily rebalancing are consistent

27. Summary By measuring a high dimension covariance matrix with high-frequency data and realized volatility approach, I find: • when the manager rebalances his portfolio monthly and use 12 months of data, he will not switch from daily to intraday returns to estimate the conditional covariance matrix. • when he rebalances his portfolio monthly and use less than 6 months of data, he will use intraday returns. • when he rebalances his portfolio daily, he will also switch to intraday returns. • one month to six months high-frequency data have better performances in estimating conditional covariance matrix than one year data.

28. Future Research • VaR efficient portfolio • Rebalancing frequency and sample stocks • Transaction costs