Semivariance Significance Baishi Wu, 2/27/08

Outline Motivation Background Math Data Preparation Basic Jump Data Semivariance Correlation Matrix Summary Correlograms 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 as well as solely upward variation Correlated the results against one another as well, regressed lagged results, created correlograms

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 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

Altria Group (Philip Morris)

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 0.000332245401265 0.001683311831532 0.000265320060189 0.000437295321346 Altria Group

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

Intel Corp. Intel Corp.

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.

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

Pfizer Pfizer

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 * 0.2725 0.3013 0.2529 0.2800 Pfizer

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

Semivariance, Upvariance Altria Group

Bipower Downward Variation Altria Group

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

Semivariance, Upvariance Intel Corp

Bipower Downward Variation Intel Corp

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.

Semivariance, Upvariance Pfizer

Bipower Downward Variation Pfizer

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

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)

Correlogram – Realized Variance Altria Group

Correlogram – Realized Semivariance Altria Group

Correlogram – Realized upVariance Altria Group

Correlogram – Realized Variance Intel Corp

Correlogram – Realized Semivariance Intel Corp

Correlogram – Realized upVariance Intel Corp

Correlogram – Realized Variance Pfizer

Correlogram – Realized Semivariance Pfizer

Correlogram – Realized upVariance Pfizer

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?