1. A DVANCEMENTS IN P ORTFOLIO T HEORY Xiaoyang Zhuang Economics 201FS Duke University March 30, 2010

2. Is there a benefit to using high-frequency data in making portfolio allocation decisions?

3. Contents 1.Literature Review Papers that address the question directly Some fancy-schmancy tools 2.Potential Contributions to the Literature

4. SETTING min(α) σ 2 = αΣ t α subject to α T e = 1, α T  =  P Risk-averse investor within a “conditional” mean-variance framework Four asset classes: stocks, bonds, gold, and cash Daily rebalancing Allocation is implemented using futures on the risky assets (makes analysis robust to transaction costs and trading restrictions) CONCLUSION Given the daily estimator, an investor would be willing to pay 50-200 bps/year to upgrade to the 5- minute RV/RCov estimator. Fleming, Kirby, and Ostdiek (2003, JFE) The Economic Value of Volatility Timing Using “Realized” Volatility

5. ESTIMATORS Covariance Using Daily Returns. where Ω t-k is a symmetric N x N matrix of weights, and e t-k = (R t-k –  ) is an N x 1 vector of daily return innovations. The weights are exponential. Certain choices of Ω t-k causes the estimate to resemble the estimate generated by a multivariate GARCH model. Covariance Using 5-Minute Returns. Realized Covariance. Returns. According to the authors, assuming a constant returns vector is empirically sound. Fleming, Kirby, and Ostdiek (2003, JFE) The Economic Value of Volatility Timing Using “Realized” Volatility

6. MEASURING PERFORMANCE GAINS Quadratic Utility Approach Each day, the investor places some fixed amount of wealth W 0 into cash (6%(!!!) risk-free rate assumed) and purchases futures contracts with the same notional value. Her daily utility is where R pt is the portfolio‘s return (on day t), γ is the investor’s RRA, and R f is the risk-free rate. Define R p1t and R p2t as the portfolio’s return using high- and low-frequency estimators, respectively, in making the allocation decision. The (daily) performance gain from using high-frequency estimators is then ∆, such that Fleming, Kirby, and Ostdiek (2003, JFE) The Economic Value of Volatility Timing Using “Realized” Volatility

7. SETTING min(α) σ 2 = αΣ t α subject to α T e = 1, α T  =  P Risk-averse investor within a “conditional” mean-variance framework 30 DJIA stocks Daily rebalancing vs. monthly rebalancing Allocation is set to track the return of the S&P 500; robust to transaction costs CONCLUSION High-frequency performance gains depend on the (1) rebalancing frequency and (2) estimation window: Monthly Rebalancing and Estimation Window ≥ 12 months→No Gain Daily Rebalancing or Estimation Window < 6 months→Statistically Significant Gain Liu (2009, JAE) On Portfolio Optimization: How and When Do We Benefit From High-Frequency Data?

8. Ait-Sahalia, Cacho-Diaz, and Hurd (2008) Portfolio Choice With Jumps: A Closed-Form Solution SETTING min(α) σ 2 = αΣ t α subject to α T e = 1, α T  =  P “Conditional” mean-variance (tracking volatility) framework 30 DJIA stocks Daily rebalancing vs. monthly rebalancing Allocation is set to track the return of the S&P 500; robust to transaction costs CONCLUSION High-frequency performance gains depend on the (1) rebalancing frequency and (2) estimation window: Monthly Rebalancing and Estimation Window ≥ 12 months→No Gain Daily Rebalancing or Estimation Window < 6 months→Statistically Significant Gain

9. EVALUATIONS OF DIFFERENT PORTFOLIO OPTIMIZATION FRAMEWORKS Portfolio Optimization Framework Mean-Variance Mean-VaR Optimal Portfolio Given Jumps (Ait-Sahalia, Cacho-Diaz, and Laeven, 2009) Variance Measurement. Realized Volatility* vs. Realized Kernel vs. VaR/CVaR? Covariance Measurement. Blahblahblah. Realized Covariance. Time Horizon: Use 12-month vs. 6-month historical data WE COULD ALSO CONTRIBUTE A More Realistic Scenario. Consider more asset classes and different geographies (e.g. U.S. corporate bonds, European equities, Asian sovereign debt…) A Performance Comparison Under Market Stress. A Notion of Liquidity Premia With Backbone. Find an analytical solution for the investor’s required liquidity premium due to his/her inability to rebalance exposure daily. Contributions to the Literature On Portfolio Optimization: How and When Do We Benefit From High-Frequency Data?