Semivariance Significance

Baishi Wu, 2/13/08

Outline Motivation Background Math Data Preparation Altria Group/Phillip Morris (MO) Plots Apple (APPL) Plots Summary Statistics 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 jumps in this model, ought to focus on squared negative jumps

Also did a focus on only positive jumps and computed z-scores for the following as well

The separation of RS from RV is supposed to beat out the prediction mechanism used solely on GARCH memory

Equations Realized Volatility (RV)

Bipower Variance (BV)

Equations Realized Semivariance (RS)

Running an “if” loop to only take values of the returns if they are less than zero in order to solely decreases

Bipower Downard Variance (BPDV) BPDV = RS – (1/2)BV

if r(i,j) <= 0 RS(1, j) = sum(r(:,j).^2); BPDV(1,j) = RS(1,j) -.5*BV(1,j); else RS(1, j) = 0; BPDV(1, j) = 0; end

Equations Tri-Power Quarticity

Relative Jump

Equations Max Version z-Statistic (Tri-Power)

Take one sided significance at .999 level, or z = 3.09

Data Collected at five minute intervals Rewrote code so that the first data point

collected is the fifth entry for that day while the last data point is the last entry of the day (as there are exactly 385)

Two stocks are being analyzed, notably for their differences for the results in the analysis as they respond uniquely to the downward variance analysis

Altria Group is sampled between 1997-2008 (2669)

Apple is sampled between 1997-2000 (676)

Altria Group (Phillip Morris)

Realized Volatility, Bipower Variance

Realized Variance, Z-Scores

Semivariance, Bipower Downward Variance

Realized Semivariance, Z-Scores

Upward Variance, BPUV

Realized Upvariance, Z-Scores

Apple Computers

Realized Volatility, Bipower Variance

Realized Variance, Z-Scores

Semivariance, Bipower Downward Variance

Realized Semivariance, Z-Scores

Upward Variance, BPUV

Realized Upvariance, Z-Scores

Summary Statistics

MO AAPL

Mean Std Mean Std

RV 3.32E-04 0.0017 0.0012 0.0017

upRV 2.24E-04 0.0017 0.0007 0.0008

RS 1.74E-04 3.24E-04 0.0007 0.0017

BV 2.65E-04 4.37E-04 0.001 0.001

BPUV 1.39E-04 0.0016 0.0004 0.0004

BPDV 9.74E-05 1.90E-04 0.0004 0.0013

Jumps 0.37% 0.15%

Jumps Down 0.00% 52.66%

Jumps Up 57.00% 0.59%

Questions Problems with the code? Is there something that

I’m not doing correctly with measuring downside risk

Why the difference in the two stocks’ characteristics?

Improvements in the Tri-Power or Max z-statistic that explain the drastic differences in z-scores that you see?

Verified decreases in mean and standard deviation for the one-directional jumping (is this just because values have been replaced by zeros?)

Extend to GARCH model analysis…?

Additional Extensions Determining Tri-Power Quarticity for only

semivariance Using a larger sample of stocks to view effects

of trimming the data Effect of noise on data