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

On Portfolio Optimization: How Do We Benefit from High-

Frequency Data?

Qianqiu Liu

Motivation

• Do high-frequency data help in estimating large covariance matrices?

• How to form a large covariance matrix estimator with high-frequency data?

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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)

Overview

1. Portfolio Optimization Problem

2. Estimation Methods

3. Data and Empirical Results

4. Robustness Checks and Simulation Exercise

5. Future Research

Portfolio Optimization Problem

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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

Table 1 & 2

Summary Statictics

Daily mean std skewness kurtosis 1st autocorr

Median 0.0007 0.0193 0.1528 6.0617 -0.0045

Mean 0.0009 0.0193 0.0483 8.2539 -0.0071

Min. 0.0002 0.0143 -2.6223 4.2659 -0.0671

Max. 0.0020 0.0255 0.5549 47.7644 0.0576

SP500 0.0007 0.0096 -0.3248 8.5479 -0.0008

Five-minute mean*100 std*100 skewness kurtosis 1st autocorr

Median 0.0000 0.1926 0.0417 10.4876 -0.1327

Mean 0.0000 0.2005 0.1100 14.0095 -0.1163

Min. -0.0017 0.1600 -0.1201 7.9636 -0.2078

Max. 0.0017 0.2474 0.8891 45.9282 0.0775

SP500 0.0001 0.0796 0.1053 15.4402 0.1259

Empirical Results

• summary statistics for daily returns and five-minute returns

1. the means of high-frequency returns are effectively zero from a statistical perspective;

2. The cross-sectional average kurtosis increases as five-minute returns replace daily returns;

3. most of the daily return series of DJIA stocks and daily S&P 500 index returns can be seen as uncorrelated;

4. 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.

Table 3

Out of Sample Performance of Minimum Tracking Error Portfolios (Monthly Rebalance)

Covariance Matrix Estimator Mean Std Max. Weight Min. Weight Short Interest

Panel A: Daily Returns

Sample Covariance Matrix 6.20 4.51 12.86 -3.53 -0.23

Sample Covariance Matrix (CHMSW) 16.47 69.96 981.74 -713.50 -262.23

Sample Covariance Matrix (SW) 53.81 83.08 636.20 -543.59 -91.79

Sample Covariance Matrix (NW, L=1) 5.98 4.74 14.64 -3.35 -0.45


Sample Covariance Matrix (JM) 12.97 51.72 1264.70 -729.99 -93.23

Optimal Shrinkage Estimator (LW) 6.10 4.63 13.18 -2.81 -0.17

Rolling Sample Estimator (FKO, 0.01) 6.45 4.58 12.63 -2.72 -0.12

Panel B: Five-minute Returns

Sample Covariance Matrix (ex. Overnight) 5.44 5.09 8.81 0.71 0.00

Sample Covariance Matrix 5.65 4.84 8.88 0.74 0.00

Sample Covariance Matrix (AT) 5.24 5.22 8.56 0.72 0.00

Sample Covariance Matrix ( Lag 1) 5.63 4.60 10.52 -0.60 -0.06


Rolling Sample Estimator (AG) 5.81 4.91 8.68 0.70 0.00

Rolling Sample Estimator (FN, 0.01) 5.68 4.45 10.55 -1.46 -0.06



Rolling Sample Estimator (FN, 0.04, 0.95) 5.59 4.33 11.24 -1.73 -0.09



The out of sample performance of minimum tracking error portfolios

1. Monthly rebalancing based on past 12 months of data

• Daily Returns:

(1) sample covariance matrix estimator is the best;

(2) Short sale restrictions has little effect on the composition of the minimum tracking error portfolios.

• High-frequency Returns

(1) including overnight returns improves out of sample performance;

(2) the sample covariance matrix estimator performs worse than the sample covariance matrix using daily returns;

(3) When lag one covariances are included, the sample covariance matrix estimator becomes comparable to that of daily return based sample covariance matrix;

(4) 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.

Table 4

Out of Sample Performance of Minimum Tracking Error Portfolios (Daily Rebalance)


Panel A: Daily Returns


Sample Covariance Matrix (CHMSW) -31.77 218.93 26321.13 -27500.02 -530.01

Sample Covariance Matrix (SW) 6.09 17.47 636.20 -543.59 -21.30



Optimal Shrinkage Estimator (LW) 6.37 4.43 13.31 -2.97 -0.18


Panel B: Five-minute Returns

Sample Covariance Matrix (ex. Overnight) 5.52 5.09 9.13 0.66 0.00

Sample Covariance Matrix 5.79 4.70 8.82 0.68 0.00






The out of sample performance of minimum tracking error portfolios (continued)

2. Daily rebalancing based on past 12 months of data• Daily returns(1) sample covariance matrix estimator has the tracking

error as small as the best estimator.• High-frequency returns(1) Including the overnight returns, the lag one covariances

will lead the sample covariance matrix to have comparable performance to the daily sample covariance matrix;

(2) the rolling sample estimator with the optimal decay rate has a standard deviation substantially lower than the daily sample covariance matrix.

Table 5

Test for Equality of Means of Minimum Tracking Error Portfolios

Covariance Matrix Estimator Equality in Mean Return Equality in Mean Squared Return

Panel A: Monthly Rebalance

Daily Returns

Sample Covariance Matrix (NW, L=1) -0.7087 1.2705


Optimal Shrinkage Estimator (LW) -0.4801 1.1456

Rolling Sample Estimator (FKO, 0.01) 0.8872 0.8123

Five-Minute Returns

Sample Covariance Matrix (ex. Overnight) -0.6631 1.3430

Sample Covariance Matrix -0.6107 0.9204

Sample Covariance Matrix (AT) -0.8672 1.7428

Sample Covariance Matrix ( Lag 1) -0.8005 0.1050

Rolling Sample Estimator (FKO, 0.03) -1.1042 -0.4680

Rolling Sample Estimator (AG) -0.4376 1.2065

Rolling Sample Estimator (FN, 0.01) -0.7730 -0.4425


Rolling Sample Estimator (FN, 0.1) -0.7368 0.0135

Rolling Sample Estimator (FN, 0.04, 0.95) -0.8573 -0.7857



Panel B: Daily Rebalance

Daily Returns



Optimal Shrinkage Estimator (LW) -0.0038 -0.6595

Rolling Sample Estimator (FKO, 0.01) 1.6463 -2.0711

Five-Minute Returns

Sample Covariance Matrix (ex. Overnight) -0.7445 6.8456

Sample Covariance Matrix -0.6458 3.5229

Sample Covariance Matrix ( Lag 1) -1.1475 -0.5138

Rolling Sample Estimator (FKO, 0.03) 0.6164 -4.0195



Rolling Sample Estimator (FN, 0.1) 0.1983 -1.7805

Table 6

Out of Sample Performance of Minimum Tracking Error Portfolios (Monthly Rebalance)


Panel A: Past 12 months of Data

Daily Sample Covariance Matrix 6.20 4.51 12.86 -3.53 -0.23

HF Sample Covariance Matrix (ex. Overnight) 5.44 5.09 8.81 0.71 0.00

HF Sample Covariance Matrix 5.65 4.84 8.88 0.74 0.00

HF Sample Covariance Matrix ( Lag 1) 5.63 4.60 10.52 -0.60 -0.06

Panel B: Past 6 months of Data


HF Sample Covariance Matrix (ex. Overnight) 5.50 4.96 9.68 0.17 0.00



Panel C: Past 3 months of Data


HF Sample Covariance Matrix (ex. Overnight) 5.49 4.61 10.24 -0.80 -0.06

HF Sample Covariance Matrix 5.58 4.29 10.02 -0.50 -0.01


Panel D: Past 2 months of Data





Panel E: Past 1 month of Data

Daily Sample Covariance Matrix -90.02 204.59 2589.01 -2206.50 -733.70




The out of sample performance of minimum tracking error portfolios (continued)

3. 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

4. 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.

Table 7

Out of Sample Performance of Minimum Variance Portfolios


Panel A: Monthly Rebalance ( Past 12 months of Data )

Daily Returns



Five-minute Returns

Sample Covariance Matrix (ex. Overnight) 18.08 15.19 15.93 -7.45 -11.00










Panel B: Daily Rebalance ( Past 12 months of Data )

Daily Returns



Five-minute Returns

Sample Covariance Matrix (ex. Overnight) 18.88 15.12 16.17 -8.08 -11.01







Panel C: Monthly Rebalance ( Past 3 months of Data )





The out of sample performance of minimum variance portfolios

1. 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;

2. 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;

3. 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;

4. The weights on the component stocks.

Out of Sample Performance of Minimum Tracking Error Portfolios (Simulation Exercise)


Monthly Rebalance

Panel A: Past 12 months of Data



Panel B: Past 6 months of Data



Panel C: Past 3 months of Data


HF Sample Covariance Matrix -0.01 4.58 10.76 -0.02 -0.01

Daily Rebalance

Panel D: Past 12 months of Data


HF Sample Covariance Matrix -0.20 4.56 10.29 0.31 0.00

Panel E: Past 6 month of Data


HF Sample Covariance Matrix -0.32 4.57 10.54 0.13 0.00

Panel F: Past 3 months of Data


HF Sample Covariance Matrix -0.29 4.57 10.90 -0.12 0.00

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

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.

Future Research

1. VaR efficient portfolio

2. Rebalancing frequency and sample stocks

3. Transaction costs

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

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