Components of Volatility and their Empirical Measures

Components of Volatility and their Components of Volatility and their Empirical MeasuresEmpirical Measures

DIPANKOR COONDOODIPANKOR COONDOO

Economic Research Unit, Indian Statistical Institute, Kolkata

PARAMITA MUKHERJEEPARAMITA MUKHERJEE

Monetary Research Project, ICRA Limited, Kolkata

Notions of Volatility

Of Financial Analysts:Variability of a financial

variable as measured by its

Std. Dev.

Of Econometricians:Conditional Heteroskedasticity

Other Related Issues

Historical Volatility - Non Parametric Measure

Stochastic Volatility - GARCH-based Parametric Analysis

Changing Volatility - Rolling Sample Measure – Can be examined both in

Historical & Stochastic Set up

What led to what I talk about here?

1. Non-comparability of volatility of variables measured in different units

2. Basis for comparison of the Volatilities of FIIN to India and BSE return, say

3. Are there different aspects of volatility that need to be compared?

Three Components of Volatility

Strength : Range of Amplitude of Fluctuation due to Volatility

Duration : Portion of Time the Variable is in Volatile State

Persistence: Inertia of large and small fluctuations

Strength of Volatility

Green has less strength than Blue

Duration of Volatility

Volatile State Normal State

Persistence of Volatility

Blue is more persistent than Black

The Decomposition Methodology

Given Series has trend and volatility An ARIMA with GARCH error will fit well

Step 1: Fit ARIMA. Get the residuals e(t), T = 1, T. Standardise these residuals as w(t) = abs (e(t)/s), t = 1,T, where s = std. dev(e(t),t = 1,t)

Note that w(t) is non-negative, by construction.

Step 2: Estimate the PDF of w(t). We used non-parametric kernel method of density estimation of Silverman (1986). For every observed value of w, the ordinate of the estimated PDF is

1

1ˆ ( ) [( ) / ].

[.] : kernel function, ( ) 1

: bandwidth or smoothing parameter

T

T tt

f w K w w hT h

K K u du

h


Step 3. Find mode of . Call it .

ˆAlso find mean of and call it .

: average amplitude in nonvolatile state

ˆ: average amplitude in volatile state

ˆStrength :

Duration: 1 ( )

where ( ):

w w

w w w

w

w

S w w

D F w

F w

a measure of portion of sample

period the variable is in volatile state

Persistence: a measure of autocorrn of P w


Table 1: Summary Descriptive Statistics

BRET CMR FIIN

Mean -0.00004 8.38 34.07

Median 0.00092 8.03 23.10

Max 0.09 22.50 983.20

Min -0.07 0.50 -509.50

Std. Dev. 0.02 2.10 120.04

Skewness -0.07 2.58 0.76

Kurtosis 5.33 13.99 9.66

Jarque-Bera 190.15 5160.66 1632.13

Sample Size 840 840 840

BRET CMR FIINADF-statistic -16.48 -7.73 -9.11

5% Critical Value -1.94 -3.97 -2.87

Model Selected No trend or intercept

Trend and intercept

Intercept

lag order 2 3 4

Table 2: Results of Unit Root Tests

item

Coefficient Std. Error Coefficient Std. Error Coefficient Std. Error

mean equation

intercept 0.000723 0.000592 7.990946 0.025773 24.63928 2.921969

variance equation

intercept 4.56E-05 9.87E-06 0.196689 0.012257 274.2189 61.64922

ARCH(1) 0.161038 0.036078 1.08646 0.059423 0.124378 0.013505

GARCH(1) 0.713714 0.052307 0.144736 0.019215 0.863543 0.014828

Adjusted R2 -0.005257 -0.03849 -0.00979

BRET CMR FIIN

Table 3: Results of GARCH (1,1) Estimation

0.00

0.20

0.40

0.60

0.80

1.00

0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00

FIIN Density

0.00

0.20

0.40

0.60

0.80

1.00

0.00 1.00 2.00 3.00 4.00 5.00 6.00

BRET Density

Table 4. Variable-specific Estimates of Volatility Components Based on Entire Sample

BRET CMR FIIN

Amplitude of FluctuationAverage Amplitude Normal Phase( ) 0.295 0.133 0.256

Average Amplitude Volatile Phase ( ) 0.987 0.847 0.929 Strength of Volatility (S) 0.692 0.714 0.673

Duration of VolatilityProportion of Volatile Days (D) 0.773 0.807 0.769

Persistence of Volatility (P)1st Order Autocorrelation of w 0.25 0.51 0.22

2ndOrder Autocorrelation of w 0.18 0.36 0.20

3rdOrderAutocorrelation of w 0.11 0.25 0.20

w*w

Volatility Component

Window-width

Mean/CV

BRET CMR FIIN15-day mean 0.66 0.67 0.63

cv 0.51 1.12 0.5190-day mean 0.68 0.75 0.65

cv 0.22 0.54 0.33Entire sample 0.692 0.714 0.67315-day mean 0.6 0.72 0.63

cv 0.23 0.12 0.1790-day mean 0.7 0.79 0.71

cv 0.09 0.05 0.06Entire sample 0.773 0.807 0.76915-day mean -0.01 0.24 -0.01

cv -47.31 0.91 -15.7890-day mean 0.2 0.45 0.08

cv 0.67 0.33 1.54Entire sample 0.25 0.51 0.22

Variable

S

D

P

Table 5: A Summary of Rolling Sample Estimation Returns

0

1

2

3

4

5

6

7

1 32

63

94

12

5

15

6

18

7

21

8

24

9

28

0

31

1

34

2

37

3

40

4

43

5

46

6

49

7

52

8

55

9

59

0

62

1

65

2

68

3

71

4

74

5

77

6

80

7

BRET

CMR

FIIN

Note: Scales shifted for BRET and CMR

S Measure for 15-day Window Width

0.25

0.45

0.65

0.85

1.05

1.25

1.45

1.65

1.85

2.05

2.25

1

33

65

97

12

9

16

1

19

3

22

5

25

7

28

9

32

1

35

3

38

5

41

7

44

9

48

1

51

3

54

5

57

7

60

9

64

1

67

3

70

5

73

7

76

9

80

1

BRET CMR FIINNote: Scales shifted for BRET and CMR

D Measure for 15-day Window Width

-1

-0.5

0

0.5

1

1.5

2

2.5

3

1 27 53 79 105

131

157

183

209

235

261

287

313

339

365

391

417

443

469

495

521

547

573

599

625

651

677

703

729

755

781

807

BRET CMR FIIN

Dotted lines are shifted scales for respective variables

P Measure for 15-day Window Width

0.25

0.5

0.75

1

1.25

1.5

1.75

2

2.25

2.5

2.75

1 33

65

97

12

9

16

1

19

3

22

5

25

7

28

9

32

1

35

3

38

5

41

7

44

9

48

1

51

3

54

5

57

7

60

9

64

1

67

3

70

5

73

7

BRET CMR FIIN

S Measure for 90-day Window Width

0.5

0.6

0.7

0.8

0.9

1

1.1

1 33 65 97

129

161

193

225

257

289

321

353

385

417

449

481

513

545

577

609

641

673

705

737

BRET CMR FIIN

Note: Scale shifted for BRET

D Measure for 90-day Window Width

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1 29

57

85

11

3

14

1

16

9

19

7

22

5

25

3

28

1

30

9

33

7

36

5

39

3

42

1

44

9

47

7

50

5

53

3

56

1

58

9

61

7

64

5

67

3

70

1

72

9

BRET_90 CMR_90 FIIN 90

Dotted lines are shifted scales for respective variables

P Measure for 90-day Window Width

Correlation between day to day variations of estimated volatility components for different pairs of variables

Volatility component

Window-width

BRET-CMR BRET-FIIN CMR-FIIN15-day -0.02 0.23 0.0690-day 0.43 0.51* 0.2515-day -0.34 0.05 0.0690-day -0.38 0.23 0.1915-day -0.12 0.07 0.0590-day -0.23 -0.16 0.42

S

D

P

Correlation for the variable-pair

Table 6:

BRET_S15=C(1)+C(2)*BRET_S15LAG1+C(3)*BRET_S15LAG2

+C(4)*BRET_SD15LAG1Coefficient Std. Error t-Statistic Prob.

C(1) 0.040277 0.011506 3.500639 0.0005

C(2) 0.949454 0.036686 25.8806 0

C(3) -0.045665 0.035586 -1.283214 0.1998

C(4) 0.025349 0.020244 1.252201 0.2109Adjusted R-squared 0.86514

Durbin-Watson stat 1.985785

Component-wise Forecast : Strength

Table 7A:

BRET_D15=C(1)+C(2)*BRET_D15LAG1+C(3)*BRET_D15LAG2


C(1) 0.124057 0.016135 7.688657 0

C(2) 0.677629 0.034416 19.68965 0

C(3) 0.133242 0.034411 3.872083 0.0001

C(4) -0.010605 0.007934 -1.336662 0.1817Adjusted R-squared 0.621627


Component-wise Forecast :Duration

Table 7B:

BRET_P15=C(1)+C(2)*BRET_P15LAG1+C(3)*BRET_P15LAG2


C(1) -0.017607 0.009949 -1.769719 0.0771

C(2) 1.039735 0.034724 29.94281 0

C(3) -0.157909 0.034455 -4.583119 0

C(4) 0.018263 0.010017 1.823169 0.0686Adjusted R-squared 0.824973


Component-wise Forecast : Persistence

Table 7C:

BRET_SD15 Regressed onCoefficient Std. Error t-Statistic Prob.

Constant 0.182372 0.017221 10.58988 0

BRET_S15 0.290415 0.016038 18.10794 0

BRET_D15 -0.252909 0.024861 -10.17302 0

BRET_P15 0.011946 0.011538 1.035377 0.3008

BRET_SD15Lag1 0.812226 0.033928 23.9396 0

BRET_SD15Lag2 -0.051401 0.029838 -1.722664 0.0853

R-squared 0.960513 Mean dependent var 0.925065

Adjusted R-squared 0.960271 S.D. dependent var 0.375287

S.E. of regression 0.074802 Akaike info criterion -2.340672

Sum squared resid 4.571429 Schwarz criterion -2.306313

Log likelihood 969.1867 F-statistic 3974.679

Durbin-Watson stat 1.548439 Prob(F-statistic) 0

Non-parametric Volatility explained by three components

Table 8A:

Forecast Model:

BRET_SD15=C(1)+C(2)*BRET_S15+C(3)*BRET_D15+C(4)*BRET_P15+C(5)*BRET_SD15LAG1+C(6)*BRET_SD15LAG2

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0.010

0.012

0.014

0.016

0.018

0.020

0.022

0.024

0.026

0.028

0.030

BRET_SD15 BRET_SD15Forecast GarchVolBRET_MA15

Forecast Model

CMR_SD15 Regressed onCoefficient Std. Error t-Statistic Prob.

Constant 0.212444 0.032426 6.551635 0CMR_S15 0.453337 0.014953 30.31661 0CMR_D15 -0.265469 0.046262 -5.738427 0CMR_P15 0.001537 0.0182 0.084428 0.9327CMR_SD15Lag1 0.664036 0.030368 21.86596 0CMR_SD15Lag2 -0.133352 0.023436 -5.690007 0

R-squared 0.976942 Mean dependent var 0.68998Adjusted R-squared 0.976801 S.D. dependent var 0.706518S.E. of regression 0.107611 Akaike info criterion -1.613324Sum squared resid 9.460969 Schwarz criterion -1.578964Log likelihood 669.8826 F-statistic 6923.154Durbin-Watson stat 1.023739 Prob(F-statistic) 0


Table 8B:

FIIN_SD15 Regressed onCoefficient Std. Error t-Statistic Prob.

Constant 0.183537 0.017932 10.23524 0FIIN_S15 0.398881 0.020162 19.78417 0FIIN_D15 -0.283178 0.026796 -10.56775 0FIIN_P15 0.003833 0.011566 0.331404 0.7404FIIN_SD15Lag1 0.721912 0.032756 22.03913 0FIIN_SD15Lag2 -0.021108 0.028928 -0.729696 0.4658

R-squared 0.976067 Mean dependent var 0.85531Adjusted R-squared 0.975921 S.D. dependent var 0.424477S.E. of regression 0.065868 Akaike info criterion -2.595044Sum squared resid 3.540332 Schwarz criterion -2.560652Log likelihood 1072.563 F-statistic 6655.9Durbin-Watson stat 1.443571 Prob(F-statistic) 0


Table 8C:

BRET_SD15 Regressed onCoefficient Std. Error t-Statistic Prob.

Constant 0.057006 0.021796 2.615513 0.0091BRET_S15 Lag1 0.065216 0.021957 2.970124 0.0031BRET_D15 Lag1 -0.055691 0.031278 -1.780506 0.0754BRET_P15 Lag1 0.06902 0.030397 2.270634 0.0234BRET_SD15Lag1 0.927875 0.018618 49.83784 0BRET_P15 Lag2 -0.103754 0.030054 -3.452256 0.0006

R-squared 0.944372 Mean dependent var 0.925065Adjusted R-squared 0.944032 S.D. dependent var 0.375287S.E. of regression 0.088784 Akaike info criterion -1.99796Sum squared resid 6.440066 Schwarz criterion -1.9636Log likelihood 828.1604 F-statistic 2773.982Durbin-Watson stat 1.793371 Prob(F-statistic) 0Residual SeriesJarque-Bera 6770.647Probability 0.0000

Non-parametric Volatility Forecasting Model

Table 9A:

CMR_SD15 Regressed onCoefficient Std. Error t-Statistic Prob.

Constant 0.539725 0.077949 6.924074 0CMR_S15Lag1 0.81667 0.035396 23.07214 0CMR_D15Lag1 -0.305795 0.142694 -2.143016 0.0324CMR_S15Lag2 0.094666 0.035166 2.691998 0.0072CMR_D15Lag3 -0.329692 0.141923 -2.323039 0.0204

R-squared 0.878311 Mean dependent var 0.687053Adjusted R-squared 0.877716 S.D. dependent var 0.701938S.E. of regression 0.245462 Akaike info criterion 0.034716Sum squared resid 49.2256 Schwarz criterion 0.063376Log likelihood -9.268205 F-statistic 1474.213Durbin-Watson stat 0.992321 Prob(F-statistic) 0Residual SeriesJarque-Bera 11381Probability 0.0000


Table 9B:

FIIN_SD15 Regressed onCoefficient Std. Error t-Statistic Prob.

Constant 0.066596 0.019491 3.416721 0.0007FIIN_S15 Lag3 0.080775 0.031703 2.547885 0.011FIIN_S15 Lag4 -0.103597 0.0295 -3.511783 0.0005FIIN_D15 Lag6 -0.072937 0.028755 -2.536479 0.0114FIIN_SD15 Lag1 0.993259 0.015763 63.01025 0

R-squared 0.965969 Mean dependent var 0.856792Adjusted R-squared 0.965801 S.D. dependent var 0.4249S.E. of regression 0.078577 Akaike info criterion -2.243394Sum squared resid 5.01968 Schwarz criterion -2.214624Log likelihood 922.5482 F-statistic 5769.176Durbin-Watson stat 1.978857 Prob(F-statistic) 0Jarque-Bera 2653.505Probability 0.0000


Table 9C:

BRET

0.00

0.30

0.60

0.90

1.20

1.50

1.80

2.10

1 32 63 94 125

156

187

218

249

280

311

342

373

404

435

466

497

528

559

590

621

652

683

714

745

776

807

Obs

erve

d

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

Fore

cast

BRET_SD15Observed BRET_SD15Forecast

BRET

CMR

0.00

0.50

1.00

1.50

2.00

2.50

3.00

1 32 63 94 125 156 187 218 249 280 311 342 373 404 435 466 497 528 559 590 621 652 683 714 745 776 807

CMR_SD15Observed

CMR_SD15Forecast

CMR

FIIN

0.15

0.65

1.15

1.65

2.15

1 31 61 91 121

151

181

211

241

271

301

331

361

391

421

451

481

511

541

571

601

631

661

691

721

751

781

811

Obs

erve

d

0

0.5

1

1.5

2

2.5

Fore

cast

FIIN_SD15Observed

FIIN_SD15Forecast

FIIN

Documents

Components of Volatility and their Empirical Measures