Semivariance Significance Baishi Wu, 3/19/08

Outline Motivation Background Math Data Information Summary Statistics and Graphs Correlation HAR-RV, HAR-RS, HAR-upRV Correlogram Future

Introduction Used Paper by Barndorff-Nielsen, Kinnebrock, and Shephard (2008) “Measuring downside risk – realized semivariance” as the model Examine new realized semivariance and bipower downward variation statistics to test for improved predictive ability

Equations Realized Volatility (RV) Bipower Variance (BV)

Equations Realized Semivariance (RS) Bipower Downard Variance (BPDV) Running an “if” loop to only take values of the returns if they are less than zero Separated into different return matrices, then found the realized variance with those new matrices Bipower Downard Variance (BPDV)

ri = log(priceclose) – log(priceopen) Equations Tri-Power Quarticity Relative Jump Daily open to close returns (ri) ri = log(priceclose) – log(priceopen)

Equations Max Version z-Statistic (Tri-Power) Take one sided significance at .999 level, or z = 3.09

Data Collected at five minute intervals S&P500 Data Set from 1990 to late 2007

S&P500 - Prices S&P500

Realized and Bipower Variance Statistic Value mean(RV) 8.1299e-05 std(RV) 1.2352e-04 mean(BV) 7.6804e-05 std(BV) 1.1303e-04 0.000332245401265 0.001683311831532 0.000265320060189 0.000437295321346 S&P500

Z-Scores S&P500 Statistic Value days 4509 mean(z) 0.6342 std(z) 1.3569 jump days 166 Jump % 3.68% 0.6767 1.1449 S&P500

Semivariance, Realized upVariance Statistic Value mean(RS) 4.0894e-05 std(RS) 7.1114e-05 mean(upRV) 4.0405e-05 std(upRV) 6.3970e-05 S&P500

Bipower Downward Variation Statistic Value mean(BV) 7.6804e-05 std(BV) 1.1303e-04 mean(BPDV) 2.4916e-06 std(BPDV) 2.7787e-05 S&P500

Summary Information Semivariance statistics correlate much better with daily open-close returns, consistent with BNKS Significant or by design? BPDV is also highly significant! S&P500

Realized Variance Regression Results Coefficients are statistically significant in this case, with fairly low standard errors S&P500

HAR-RV Plot S&P500

Semivariance Regression Results Coefficients are relatively similar to the results found for Realized Variance (not surprising), with none of the being any more significant Fairly small contrast between RV and RS in this case. S&P500

HAR-RS Plot S&P500

upRV Regression Results Coefficients in this case are smaller and also less significant, in that they have much lower t-values Unique to the data set? There appears to be nothing indicative about these different statistics. S&P500

HAR-upRV Plot S&P500

Correlogram – Realized Variance S&P500

Correlogram – Realized Semivariance S&P500

Correlogram – Realized upVariance S&P500

Correlogram Summary upRV autocorrelation is a lot lower, as well as the signifiance of the coefficients of the regression. When we look back on the graph of the upward variance it seems that upRV has spiked the most relative to its averages Theoretically, because of the reduction of spikes in a certain direction, both RS and upRV are meant to have a better autocorrelation than RV. This dataset along with data found in the previous presentation disproves this theory.

Future Try to use semivariance as a component of factor analysis when attempting to see industry relationships – maybe downward movements have better correlations with each other? (current problem, matching days correctly) Expand the HAR-RV to include more regression terms? Attempt semivariance with other jump tests? Lee- Mykland?