Idiosyncratic volatility and stock returns: Indian evidence

Figures & data

Table 1. Idiosyncratic volatility (IV) decile

Decile	EW portfolios				VW portfolios				Average IV (%)
	Average return	CAPM alpha	FF alpha	Carhart’s alpha	Average return	CAPM alpha	FF alpha	Carhart’s alpha
Low IV	1.29	0.11	−0.27	−0.28	0.99	−0.23	−0.19	−0.20	1.33
2	1.95	0.64	0.01	−0.02	1.39	0.14	0.25	0.24	1.69
3	1.83	0.44	−0.25	−0.30	1.38	0.03	0.13	0.15	1.91
4	1.85	0.40	−0.35	−0.42	1.29	−0.10	−0.17	−0.21	2.11
5	2.57	1.09	0.20	0.11	2.23	0.80	0.56	0.52	2.31
6	2.14	0.56	−0.59	−0.67	1.85	0.25	−0.43	−0.49	2.52
7	2.68	1.09	−0.01	−0.06	2.61	0.99	0.77	0.74	2.77
8	2.57	0.92	−0.35	−0.41	2.69	0.93	0.24	0.12	3.09
9	3.14	1.44	0.14	−0.08	3.18	1.43	1.25	1.20	3.58
High IV	4.01	2.28	0.89	0.82	2.20	0.46	−0.06	−0.01	5.44
H-L	2.72	2.16	1.16	1.09	1.20	0.69	0.14	0.19
	(4.07)	(3.99)	(2.17)	(2.07)	(1.34)	(0.83)	(0.6)	(0.22)

Notes: Decile portfolios are formed each month from April 2000 to June 2014 by sorting stocks based on the idiosyncratic volatility (IV) estimated over the prior three months daily data relative to CAPM. Portfolio 1(10) is the portfolio with lowest (highest) IV over the past quarter. The table reports the equal-weighted (EW) and value-weighted (VW) average monthly returns, CAPM, three-factor and four-factor alphas of the IV decile portfolios. The last row presents the difference between the average returns of portfolios 10 and 1, CAPM, 3-factor and 4-factor alphas of the difference and their corresponding Newey–West (1987) adjusted t-statistics. Numbers in bold denote significance at the 5% level or better.

Figure 1. Performance of IV decile portfolios.

Notes: This figure shows the CAPM alphas of the EW and VW portfolios formed each month on the basis of IV.

Table 2. Characteristics of IV sorted decile

Decile	IVq	Market Cap. (र 10 Millions)	Price (र)	Beta	B/M	ILLIQ (10^6)	IO (%)	MOM	SSKEW	TSKEW
Low IV	1.32	5,520	429	0.61	0.41	0.26	23.0	13.8	−0.008	0.30
2	1.65	4,284	349	0.72	0.49	0.09	22.6	16.3	−0.013	0.30
3	1.86	3,488	275	0.78	0.54	0.10	22.3	16.7	−0.017	0.35
4	2.05	2,885	228	0.82	0.55	0.18	21.4	18.5	−0.024	0.39
5	2.25	2,376	198	0.88	0.64	0.27	20.1	19.4	−0.027	0.43
6	2.45	1,970	180	0.90	0.62	0.22	19.5	21.3	−0.029	0.47
7	2.70	1,747	166	0.96	0.65	0.68	18.9	25.0	−0.032	0.51
8	3.01	1,471	146	1.01	0.70	0.49	17.5	25.3	−0.038	0.57
9	4.48	1,244	129	1.04	0.68	0.40	15.0	30.8	−0.044	0.66
High IV	4.69	1,645	137	0.99	0.69	1.20	13.7	28.6	−0.042	0.18

Notes: This table reports various characteristics of the idiosyncratic volatility (IV) sorted portfolios for the period April 2000 to June 2014. The numbers are the averages of the median values of the characteristics in a portfolio.

Table 3. Returns on IV decile after controlling for size, BM, MOM, SSKEW and ILLIQ

Decile	SIZE	BM	MOM	SSKEW	ILLIQ
Panel A: Equal-weighted portfolios
Low IV	1.75	1.62	1.62	1.97	1.63
2	1.90	1.82	1.80	1.86	1.78
3	2.24	1.91	1.95	1.93	1.90
4	2.00	1.85	2.20	2.10	2.40
5	2.34	2.17	2.27	2.72	2.17
6	2.37	2.63	2.32	2.30	2.71
7	2.67	2.60	2.58	2.78	2.28
8	2.36	2.51	2.92	2.99	2.78
9	3.13	3.21	3.00	3.50	2.87
High IV	2.69	3.41	3.62	4.06	3.41
Return difference	0.94	1.79	2.00	2.09	1.78
	(1.56)	(2.75)	(3.19)	(2.65)	(2.82)
CAPM alpha	0.40	1.31	1.55	1.49	1.25
	(0.91)	(2.34)	(2.85)	(2.22)	(2.44)
4-factor alpha	−0.29	0.33	0.61	0.48	0.41
	(−0.62)	(0.60)	(1.11)	(0.70)	(0.81)
Panel B: Value-weighted portfolios
Low IV	2.96	1.47	1.26	1.25	1.68
2	1.78	1.65	1.60	1.15	1.25
3	2.59	1.84	1.46	1.63	1.79
4	2.18	2.20	1.80	1.58	2.47
5	2.70	1.71	1.77	2.26	2.20
6	2.46	2.27	1.94	1.89	2.48
7	3.22	3.86	2.08	2.56	2.68
8	3.12	3.09	3.25	3.39	2.87
9	3.52	3.58	2.80	3.14	2.64
High IV	2.71	2.47	2.09	1.70	2.85
Return difference	−0.24	1.00	0.82	0.44	1.17
	(−0.12)	(1.38)	(1.11)	(0.58)	(1.73)
CAPM alpha	−0.35	0.60	0.44	−0.01	0.73
	(−0.18)	(0.89)	(0.63)	(−0.01)	(1.21)
4-factor alpha	1.035	−0.61	−0.32	−0.91	−0.01
	(0.67)	(−0.95)	(−0.44)	(−1.27)	(−0.03)

Notes: Double-sorted, equal-weighted (Panel A) and value-weighted (Panel B) decile portfolios are formed every month from April 2000 to June 2014 by sorting stocks based on idiosyncratic volatility (IV_q) after controlling for size, value, momentum, co-skewness and illiquidity. In each case, we first sort the stocks into two portfolios using the control variable, then within each portfolio, we sort stocks into decile portfolios based on the IV over the previous quarter daily data, so that decile1(10) contains stocks with the lowest (highest) IV. This table presents average returns across the two control portfolios to produce decile portfolios with dispersion in IV but with similar levels of the control variable. “Return difference” is the difference in average monthly returns between the high IV and low IV portfolios. “Alpha difference” is the CAPM and four-factor alpha of the hedge portfolio that is long high IV and short low IV. Newey−West (1987) adjusted t-statistics are reported in parentheses. Numbers in bold denote significance at the 5% level or better.

Table 4. Firm-level cross-section regressions

IVm	IVq	Beta	LnSize	LnBM	MOM	REV	ILLIQ	TSKEW	SSKEW	MAX	R^2 (%)
0.1891											1.96
(1.45)
	0.2298										1.90
	(1.75)
	0.2350	−0.3110									6.06
	(2.04)	(−0.42)
	0.1060		−0.7970								3.60
	(0.92)		(−6.60)
	0.2081			1.126							4.16
	(1.84)			(5.40)
	0.2182				−0.0032						3.51
					(−0.88)
	(1.82)
	0.1989					0.0111					3.72
	(1.58)					(0.90)
	0.2772						1.09				3.46
	(2.32)						(4.14)
	0.3894							0.0875			2.21
	(3.32)							(2.54)
	0.2077								0.5671		2.76
	(1.58)								(1.83)
	0.1623			0						0.0344	2.97
	(1.48)									(0.75)
0.0848										0.0407	2.73
(0.73)										(0.88)
	0.0245	−0.0713	−0.4679	0.7160	0.0020	0.0046	0.6964	−0.0062	0.4262	0.0015	13.5
	(0.27)	(−0.14)	(−4.33)	(4.54)	(0.76)	(0.39)	(2.22)	(−0.16)	(1.55)	(0.03)
	0.1436		−0.4300	0.7927			0.8472				7.11
	(1.28)		(−4.16)	(3.95)			(2.64)

Notes:

Each month from April 2000 to June 2014, we run cross-section regressions of returns in a month on IV and other lagged predictor variables. In each row, the table reports the average of the time-series of coefficients and their t-statistics in parenthesis. IV_m is the idiosyncratic volatility relative to CAPM over the prior one month; IV_q is the idiosyncratic volatility over the prior quarter; Beta is estimated over a window of one year daily data; LnSize is the natural logarithm of market capitalization; LnBM is the natural logarithm of the book-to-market ratio; MOM is the cumulative monthly return over t − 12 to t − 1; REV is the return in the previous month; ILLIQ is the ratio of absolute daily return and trading volume averaged over a month and raised by 10⁶; TSKEW is the skewness of the daily returns over one year of daily data; SSKEW is the systematic skewness or co-skewness over one year of daily data; and MAX is the maximum daily return over the previous month. The sample is BSE-500 stocks for the period April 2000 to June 2014. Numbers in bold denote significance at the 5% level or better.

Table 5. Cross-section regressions for small, medium and big stocks

	IV	LnSize	LnBM	ILLIQ	R^2 (%)
All	0.14	−0.43	0.79	0.84	7.11
	(1.28)	(−4.16)	(3.95)	(2.64)
Small	0.37	−1.49	0.70	0.42	5.76
	(2.22)	(−4.47)	(2.45)	(0.94)
Medium	−0.08	−0.77	0.53	0.34	6.92
	(−0.58)	(−2.18)	(2.34)	(0.81)
Big	0.15	−0.12	0.88	12.6	9.81
	(1.00)	(−0.95)	(4.04)	(3.16)

Notes:

Each month from April 2000 to June 2014, we run cross-section regressions of returns in a month on IV and other lagged predictor variables separately for small, medium and big stocks. The break-points are 33rd and 67th percentile of market capitalization each month. In each row, the table reports the average of the time-series of coefficients and their t-statistics in parenthesis. IV is the idiosyncratic volatility over the prior quarter; LnSize is the natural logarithm of market capitalization; LnBM is the natural logarithm of book-to-market ratio; ILLIQ is the ratio of absolute daily return and trading volume averaged over a month and raised by 10⁶. The sample is BSE-500 stocks for the period April 2000 to June 2014. Numbers in bold denote significance at the 5% level or better.

Table 6. Fama-Macbeth estimates from quantile and LS regressions

Variable	Quantile									LS
	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9
c	−7.58	−4.62	−2.64	−0.76	1.37	3.88	7.34	11.95	18.75	4.77
	(−10.06)	(−6.14)	(−3.50)	(−0.95)	(1.67)	(4.22)	(6.81)	(9.83)	(12.86)	(4.45)
IV	−1.55	−1.06	−0.68	−0.39	−0.10	0.17	0.55	1.08	1.87	0.14
	(−11.4)	(−8.76)	(−6.28)	(−3.53)	(−0.91)	(1.45)	(4.31)	(6.95)	(8.95)	(1.28)
LnSize	0.15	0.08	0.04	0.01	−0.07	−0.16	−0.38	−0.66	−1.14	−0.43
	(1.91)	(1.11)	(0.64)	(0.06)	(−0.84)	(−1.83)	(−3.53)	(−5.42)	(−7.77)	(−4.16)
LnBM	0.71	0.46	0.36	0.36	0.40	0.35	0.31	0.43	0.48	0.79
	(3.95)	(2.67)	(2.10)	(2.07)	(2.18)	(1.82)	(1.48)	(1.82)	(1.86)	(3.95)
ILLIQ	0.66	0.68	0.66	0.69	0.59	0.64	0.51	0.87	2.00	0.85
	(2.26)	(2.34)	(2.61)	(2.77)	(2.42)	(2.61)	(2.06)	(2.29)	(1.95)	(2.64)
Adj. R^2	5.43	4.39	4.02	3.93	3.97	4.15	4.66	5.65	7.49	7.11

Notes:

This table reports the Fama−Macbeth time-series averages of the coefficients and their t-statistics from quantile and LS regressions of the following model: $R_{i,t+1}={\mathit{\gamma }}_{0,t}+{\mathit{\gamma }}_{1,t}{\text{IV}}_{i,t}+{\mathit{\gamma }}_{2,t}{\text{LnSize}}_{i,t}+{\mathit{\gamma }}_{3,t}{\text{LnBM}}_{i,t}+{\mathit{\gamma }}_{4,t}{\text{ILLIQ}}_{i,t}+{\mathit{\epsilon }}_{i,t+1}$. Numbers in bold denote significance at 5% or better.

Figure 2. Quantile dependent effects of idiosyncratic volatility and other characteristics on excess stock returns.

Notes: The graphs in this figure represent the marginal effects of regressors on excess stock returns. The curves suggest the dynamic relation of regressors and excess stock returns at different conditional quartiles.

Table 7. Returns and alphas on IV_m decile

Decile	EW portfolios				VW PORTFOLIOS				Average IV
	Average return	CAPM alpha	3-F	4-F	Average return	CAPM alpha	3-F	4-F
Low IV	1.61	0.44	0.06	0.06	1.07	−0.16	0.01	0.02	1.14
2	2.06	0.74	−0.01	−0.04	0.95	−0.37	−0.24	−0.24	1.47
3	1.94	0.53	−0.17	−0.20	1.54	0.20	0.01	0.02	1.71
4	2.31	0.85	0.07	0.02	2.08	0.74	0.61	0.57	1.92
5	2.21	0.72	−0.27	−0.35	2.41	0.91	1.36	1.27	2.14
6	2.57	1.03	−0.20	−0.28	1.97	0.55	0.11	0.09	2.37
7	2.23	0.64	−0.36	−0.45	1.78	0.25	−0.07	−0.13	2.66
8	3.04	1.39	0.15	0.07	2.70	1.02	0.49	0.41	3.00
9	2.94	1.21	−0.07	−0.12	3.15	1.33	0.84	0.69	3.56
High IV	4.07	2.39	0.77	0.69	3.88	2.31	0.60	0.62	4.87
10–1 difference	2.45	1.95	0.70	0.63	2.80	2.47	0.59	0.60
	(3.16)	(2.79)	(1.01)	(0.91)	(1.55)	(1.36)	(0.32)	(0.33)

Notes:

Decile portfolios are formed each month from April 2000 to June 2014 by sorting stocks based on the IV relative to CAPM over the past one month. Portfolio 1(10) is the portfolio with the lowest (highest) IV over the past month. The table reports the equal-weighted (EW) and value-weighted (VW) average monthly returns, CAPM, three-factor, and four-factor alphas. The last row presents the difference between the average returns of portfolios 10 and 1, the CAPM, three-factor and four-factor alphas of the difference and their corresponding Newey−West (1987) adjusted t-statistics. Numbers in bold denote significance at the 5% level or better.

Table 8. Returns and alphas from monthly data

Decile	EW portfolios				VW portfolios
	Average return	CAPM alpha	3-F	4-F	Average return	CAPM alpha	3-F	4-F
Low IV	1.69	0.50	−0.12	−0.14	1.26	0.05	−0.13	−0.16
2	2.03	0.73	0.03	−0.02	1.81	0.53	0.51	0.46
3	1.99	0.60	−0.23	−0.30	1.82	0.44	0.22	0.23
4	2.16	0.76	−0.01	−0.03	1.24	−0.13	−0.02	−0.02
5	2.49	0.99	−0.02	−0.08	2.05	0.64	0.35	0.29
6	2.05	0.51	−0.75	−0.83	1.66	0.15	−0.32	−0.34
7	2.51	0.95	−0.26	−0.34	2.15	0.63	1.05	1.00
8	2.50	0.95	−0.18	−0.27	1.92	0.36	0.44	0.43
9	2.83	1.21	−0.07	−0.12	2.70	1.06	0.28	0.29
High IV	2.89	1.16	0.33	0.29	1.91	0.02	0.06	0.05
10–1 difference	1.19	0.65	0.45	0.43	0.64	−0.02	0.19	0.22
	(1.94)	(1.34)	(0.90)	(0.86)	(0.72)	(−0.03)	(0.23)	(0.26)

Notes:

Decile portfolios are formed each year from April 2000 to June 2014 by sorting stocks based on the IV relative to CAPM over the past 24 months (except the first sort which is based on 12 months). Portfolio 1(10) is the portfolio with lowest (highest) IV over the past 24 months. The table reports the equal-weighted (EW) and value-weighted (VW) average monthly returns, CAPM, three-factor and four-factor alphas. The last row presents the difference between the average returns of portfolios 10 and 1, the CAPM, three-factor and four-factor alphas of the difference and their corresponding Newey−West (1987) adjusted t-statistics. Numbers in bold denote significance at the 5% level or better.

Table 9. IV decile sort for sub-periods

Decile	April 2000 to December 2007				January 2008 to June 2014
	EW portfolios		VW portfolios		EW portfolios		VW portfolios
	Average return	Carhart’s alpha	Average return	Carhart’s alpha	Average return	Carhart’s alpha	Average return	Carhart’s alpha
Low IV	1.92	−0.39	1.77	−0.17	0.55	−0.14	0.09	−0.13
2	2.86	−0.01	2.15	0.36	0.89	−0.01	0.51	0.18
3	2.84	−0.10	2.14	0.25	0.64	−0.40	0.49	0.14
4	2.65	−0.69	1.72	−0.45	0.91	−0.25	0.81	−0.22
5	3.72	0.44	3.24	0.81	1.23	−0.14	1.06	0.14
6	3.18	−0.96	2.87	−0.35	0.92	−0.44	0.67	−0.38
7	3.98	−0.03	3.72	1.33	1.16	−0.02	1.30	0.34
8	3.90	−0.47	3.73	0.12	1.01	−0.27	1.48	−0.09
9	5.19	0.79	4.92	1.89	0.74	−0.55	1.14	0.02
High IV	6.43	1.88	5.61	1.98	1.17	−0.01	−1.81	−2.45
H-L	4.51	2.27	3.84	2.15	0.62	0.13	−1.90	−2.32
	(4.78)	(2.54)	(2.95)	(1.54)	(0.69)	(0.24)	(−1.65)	(2.05)

Notes: Results are provided for two sub-periods April 2000 to December 2007 and January 2008 to June 2014. For details see Table 1.

Table 10. Fama−Macbeth estimates from quantile and LS regressions

Variable	Quantile									LS
	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9
c	−4.42	−2.10	−0.86	0.56	2.20	3.90	6.41	10.24	16.77	4.35
	(−6.87)	(−3.52)	(−1.44)	(0.87)	(3.13)	(5.09)	(6.79)	(9.22)	(11.9)	(4.55)
IV	−3.13	−2.14	−1.49	−0.88	−0.38	0.35	1.04	2.04	3.88	0.41
	(−10.6)	(−7.61)	(−5.46)	(−3.16)	(−1.30)	(1.04)	(2.95)	(4.87)	(7.20)	(1.17)
IV^2	0.1860	0.1188	0.0889	0.0431	0.0169	−0.0333	−0.0552	−0.1248	−0.2683	−0.0392
	(6.35)	(4.20)	(3.44)	(1.67)	(0.59)	(−0.98)	(−1.69)	(−2.99)	(−4.43)	(−1.25)
LnSize	0.08	−0.01	−0.01	−0.04	−0.10	−0.19	−0.35	−0.62	−1.15	−0.42
	(0.98)	(−0.10)	(−0.07)	(−0.57)	(−1.25)	(−2.23)	(−3.31)	(−5.32)	(−8.03)	(−4.06)
LnBM	0.72	0.44	0.38	0.36	0.37	0.32	0.28	0.35	0.45	0.78
	(3.88)	(2.63)	(2.21)	(2.07)	(2.11)	(1.78)	(1.42)	(1.60)	(1.84)	(4.01)
ILLIQ	0.63	0.55	0.59	0.67	0.59	0.72	0.57	0.10	2.73	0.91
	(2.23)	(2.23)	(2.80)	(3.19)	(2.80)	(3.36)	(2.44)	(2.67)	(2.19)	(2.87)
Adj. R^2	6.34	5.24	4.76	4.66	4.67	4.88	5.46	6.49	8.84	8.33

Notes:

This table reports the Fama−Macbeth time-series averages of the coefficients and their t-statistics from quantile and LS regressions of the following model: $R_{i,t+1}={\mathit{\gamma }}_{0,t}+{\mathit{\gamma }}_{1,t}{\text{IV}}_{i,t}+{\mathit{\gamma }}_{2,t}{\text{IV}}_{i,t}^2+{\mathit{\gamma }}_{3,t}{\text{LnSize}}_{i,t}+{\mathit{\gamma }}_{4,t}{\text{LnBM}}_{i,t}+{\mathit{\gamma }}_{5,t}{\text{ILLIQ}}_{i,t}+{\mathit{\epsilon }}_{i,t+1}$. Numbers in bold denote significance at 5% or better.