1. Semivariance Significance
Baishi Wu, 2/27/08

2. Outline Motivation Background Math Data Preparation Basic Jump Data
Semivariance
Correlation Matrix Summary
Correlograms
Future

3. 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 as well as solely upward variation
Correlated the results against one another as well, regressed lagged results, created correlograms

4. Equations
Realized Volatility (RV)
Bipower Variance (BV)

5. 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)

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

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

8. Data Collected at five minute intervals
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)
Three stocks are analyzed because they are the 12th, 13th and 14th stocks in the S&P 500
Two stocks have 2669 days, the last one started a little later and has 2665 days only

9. Altria Group (Philip Morris)

10. Realized Volatility, Bipower Variance
Statistic
Value
mean(RV)
3.3225e-04
std(RV)
16.833e-04
mean(BV)
2.6532e-04
std(BV)
4.3729e-04
Altria Group

11. Z-Scores Altria Group Statistic Value days 2669 mean(z) 0.6767 std(z)
1.1449
jump days 61
Jump %
2.29%
Altria Group

12. Intel Corp.
Intel Corp.

13. Realized Volatility, Bipower Variance
Statistic
Value
mean(RV)
4.9607e-04
std(RV)
5.638e-04
mean(BV)
4.6781e-04
std(BV)
5.4551e-04
Intel Corp.

14. Z-Scores Intel Corp. Statistic Value days 2665 mean(z) 0.7006 std(z)
1.2575
jump days 94
Jump %
3.53%
Intel Corp.

15. Pfizer
Pfizer

16. Realized Volatility, Bipower Variance
Statistic
Value
mean(RV)
2.725e-04
std(RV)
3.013e-04
mean(BV)
2.529e-04
std(BV)
2.800e-04
1.0e-03 *
Pfizer

17. Z-Scores Pfizer Statistic Value days 2669 mean(z) 0.7089 std(z) 1.3554
jump days
126
Jump %
4.72%
Pfizer

18. Semivariance, Upvariance
Altria Group

19. Bipower Downward Variation
Altria Group

20. Summary Information
Variable
Mean
Std
Correlation Matrix
ACF1
ACF20 ri
-7.76e-4
17.2e-3
1.00
-.0201
-.0098
ri2
2.974e-4
1.78e-3
-.339
.0503
.0012 RV
3.322e-4
1.68e-3
-.294
.903
.0196
.0106
upRV
1.54e-4
.356e-3
.218
.214
.291
.1603
.0987 RS
1.783e-4
1.62e-3
-.354
.893
.978
.083
.0041
.0014
BPV
2.653e-4
.437e-3
-.075
.304
.369
.753
.2926
.157
BPDV
0.456e-4
1.58e-3
-.351
.870
.947
-.020
.991
.084
-.0010
-.0008
Unlike the literature, there is no real sign that RS provides a significantly higher correlation to the daily open/close return than RV does. upRV is actually less than RV.
Altria Group

21. Semivariance, Upvariance
Intel Corp

22. Bipower Downward Variation
Intel Corp

23. Summary Information
Variable
Mean
Std
Correlation Matrix
ACF1
ACF20 ri
-9.93e-4
223e-4
1.00
-.0796
-.0163
ri2
4.98e-4
11.6e-4
.074
.1395
.1206 RV
4.96e-4
5.65e-4
.008
.492
.7702
.4938
upRV
2.48e-4
3.10e-4
.239
.511
.959
.6467
.4147 RS
2.83e-4
-.245
.424
.950
.821
.7372
.4824
BPV
4.68e-4
5.45e-4
.007
.497
.989
.949
.938
.7435
.4681
BPDV
.141e-4
.98e-4
-.727
-.159
-.01
-.272
.272
-.08
-.1008
-.0314
In this case, the upRV and the RS have a much higher correlation to the returns than the RV does. This seems to imply that there is predictability through time with this statistic.
Intel Corp.

24. Semivariance, Upvariance
Pfizer

25. Bipower Downward Variation
Pfizer

26. Summary Information
Variable
Mean
Std
Correlation Matrix
ACF1
ACF20 ri
-4.01e-4
160e-4
1.00
.0310
.0075
ri2
2.57e-4
5.75e-4
.039
.1640
.0355 RV
2.73e-4
3.01e-4
.032
.536
.5166
.2547
upRV
1.36e-4
1.59e-4
.275
.525
.937
.4792
.2555 RS
1.37e-4
1.62e-4
-.211
.482
.939
.761
.4301
.1960
BPV
2.53e-4
2.8e-4
.046
.547
.976
.926
.906
.5249
.2358
BPDV
.101e-4
.069e-4
-.591
-.021
-.22
-.096
.508
.096
.0124
.00030
Again, the correlation with upRV and RS is significantly higher than RV. Is there anything about the dataset that seems to confirm the stronger performance of semivariance?
Pfizer

27. Correlogram
Graph of autocorrelations versus time lags, where autocorrelations measure the strength of a relationship between observations as a function of time separation between them
In the paper, it is suggested the the realised semivariance has much more dependence in it than RV and RV-RS (which I’m assuming is upRV since you can’t correlate two variables… can only do it with time)

28. Correlogram – Realized Variance
Altria Group

29. Correlogram – Realized Semivariance
Altria Group

30. Correlogram – Realized upVariance
Altria Group

31. Correlogram – Realized Variance
Intel Corp

32. Correlogram – Realized Semivariance
Intel Corp

33. Correlogram – Realized upVariance
Intel Corp

34. Correlogram – Realized Variance
Pfizer

35. Correlogram – Realized Semivariance
Pfizer

36. Correlogram – Realized upVariance
Pfizer

37. Future
None of the correlograms proved that RS had better autocorrelation statistics than RV or upRV…
Since two of the three stocks demonstrated impressive improvements in correlation of closing returns with the new semivariance statistics, should we extend this to further lagged models and GARCH analysis?