ADVANCEMENTS IN
PORTFOLIO THEORY
Xiaoyang Zhuang
Economics 201FS
Duke University
March 30, 2010
Contents
1. Fleming, Kirby, Ostdiek (2003) with our own data
A review of their methodology
Original results using high-frequency U.S. equities
2. Future Directions
Fleming, Kirby, and Ostdiek (2003, JFE)
The Economic Value of Volatility Timing Using “Realized” Volatility
SETTING
min(α) σ2 = αΣtα
subject to
αTe = 1, αT = 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 5minute RV/RCov estimator.
Fleming, Kirby, and Ostdiek (2003, JFE)
The Economic Value of Volatility Timing Using “Realized” Volatility
ESTIMATORS
•Covariance Using Daily Returns.
where Ωt-k is a symmetric N x N matrix of weights, and et-k = (Rt-k – ) is an N x 1 vector of daily
return innovations. The weights are exponential.
Certain choices of Ωt-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
MEASURING PERFORMANCE GAINS
•Quadratic Utility Approach
•Each day, the investor places some fixed amount of wealth W0 into cash (6%(!!!) risk-free rate
assumed) and purchases futures contracts with the same notional value. Her daily utility is
where Rpt is the portfolio‘s return (on day t), γ is the investor’s RRA, and Rf is the risk-free rate.
•Define Rp1t and Rp2t 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
MEASURING PERFORMANCE GAINS
•Quadratic Utility Approach
•Each day, the investor places some fixed amount of wealth W0 into cash (6%(!!!) risk-free rate
assumed) and purchases futures contracts with the same notional value. Her daily utility is
where Rpt is the portfolio‘s return (on day t), γ is the investor’s RRA, and Rf is the risk-free rate.
•Define Rp1t and Rp2t 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
FKO With Our Own Data
MEASURING PERFORMANCE GAINS
•Five stocks: Alcoa (AA), DuPont (DD), Ford (F), JPMorgan Chase (JPM), Wal-Mart (WMT)
•Target return: 5%
•Lags for the rolling estimator = 5
•Decay rates for rolling estimator: [0.030 (daily)
0.060 (RV)]
•Risk-free rate = 6% (as per FKO)
•γ = 10
(1)
(2)
Benefits of High-Frequency Data
Benefits of High-Frequency Data
STATISTICS: WHOLE PERIOD (%)
mean(PerfGain) = 14.6057
median(PerfGain) = 14.6566
std(PerfGain) = 3.4371
range(PerfGain) = [-22.7854
76.2392]
STATISTICS: 9/1/2008 – 12/25/2008 (%)
mean(PerfGain) = 15.1245
median(PerfGain) = 15.2262
std(PerfGain) = 8.6467
range(PerfGain) = [-22.7854 44.4357]
Short-Selling
STATISTICS: RV ESTIMATOR (%)
mean(sum(α)) = 31.51
median(sum(α)) = 31.27
std(sum(α)) = 7.81
range(sum(α)) = [6.89
68.69]
STATISTICS: GARCH-Y ESTIMATOR (%)
mean(sum(α)) = 23.59
median(sum(α)) = 23.32
std(sum(α)) = 109.46
range(sum(α)) = [-2623.40
1078.70]
Short-Selling
STATISTICS: RV ESTIMATOR (%)
mean(sum(α)) = 31.51
median(sum(α)) = 31.27
std(sum(α)) = 7.81
range(sum(α)) = [6.89
68.69]
STATISTICS: GARCH-Y ESTIMATOR (%)
mean(sum(α)) = 23.59
median(sum(α)) = 23.32
std(sum(α)) = 109.46
range(sum(α)) = [-2623.40
1078.70]
Questions Moving Forward
On Portfolio Optimization: How and When Do We Benefit From High-Frequency Data?
What accounts for the different leverage recommendations between the GARCH-y and multivariate
RV measures?
What accounts for the unpredictable performance differences between GARCH-y and multivariate RV
measures in periods of market stress?
Compare with Extreme Value Estimators?
How clueless are fund of fund managers?
Are there any benefits of “volatility timing” for fund of fund managers who know the asset class, but
not the individual assets, that their fund managers are investing in?