How idiosyncratic risks are priced in capital asset is always concerned in the financial sector. This paper theoretically analyzes the impact of idiosyncratic volatility on the expected return from the perspective of stock price’s information content, and use Chinese A-share market data, from 1994 to 2013, to do the empirical test. It finds that, if other conditions remain unchanged, when the degree of stock price information content is low, idiosyncratic volatility and expected return are negatively correlated; when the degree of stock price information content is rich, idiosyncratic volatility and expected return are positively correlated. After taking the impact of different regression methods and time spans into consideration, the conclusions are still valid, which indicates that the conclusions are robust. The reason for the above phenomenon is that idiosyncratic volatility is mainly driven by noise or information, which has different impacts on expected return.
How risks are priced in capital assets is always a core issue in finance. Traditional capital asset pricing model only concerns the impact of systematic risk on asset prices, omitting the impact of idiosyncratic risk. However, being restricted by the ability of funding and the ability to get access to information, it is difficult for investors to completely spread out the idiosyncratic risk. Therefore, it is necessary to analyze how idiosyncratic risks are priced.
According to Miller [
However, Ang et al. [
Fu et al. [
In summary, there have not formed a consensus in the pricing of idiosyncratic risk in capital assets pricing, neither theoretically nor empirically; and the researches on “the idiosyncratic volatility mystery” are still relatively active. The main contribution of this paper is that it considered from fresh perspective of information’s content, and built a theoretical model to analyze how idiosyncratic volatility affected expected return, and then it used the data from 1994 to 2013 of Chinese A-share market to test empirically.
This paper is based on theoretical model of Lee and Liu [
Lee and Liu [
where
Same as the noise trader, the discretionary type trader does not know the value of
Among them,
By making noise signals at the time period of t, the discretionary trader can observe the value of
where a represents risk aversion coefficient,
Suppose the investor j owns
From the above model assumptions that the stock yield can be obtained:
where
Stock price information content is:
On this basis, the main conclusions on Lee and Liu’s [
price information content do not exist, that is, there is no parameters that made
The above are model’s assumptions and main conclusions from Lee and Liu’s [
among them,
Next, the paper will first analyze the positive or negative of the derivative of the stock’s expected rate of return with respect to the number of informed traders, so as to determine the impact of the number of informed traders to the stock’s expected rate of return. Since Lee & Liu (2011) [
And because
there
so
then
Taking Formula (12) and Formula (18) together it can be obtained that,
From the above, the derivative of stock’s expected rate of return with respect to the number of informed traders is positive, which indicates that the number of informed traders yields a positive impact on stock’s expected rate of return. Next, the paper will analyze the impact of the number of informed traders to the information content of the stock. By Lee and Liu’s (2011) [
Let,
the derivative of stock price information content with respect to the amount of informed trading is:
Because,
so,
then,
Whereby, the derivative of stock price information content with respect to the number of informed traders is positive, which means that the larger the number of informed traders the richer degree of information content the stock has. Next it will analyze the derivative of stock’s expected rate to return with respect to the information content of the stocks, so as to analyze how the degree of information content of stocks impact on its expected rate of return. According to the chain rules we can obtain,
Combining (19) with (27) it can be obtained,
Hence, the derivative of stock’s expected rate of return with respect to stock price information content is positive, which indicates that stock price information content has a positive impact on stock’s expected rate of return, which means that the richer the degree of information content of stock price the higher the expected rate of return. Next the paper will demonstrate the relationship between idiosyncratic volatility and stock’s expected rate of return on the basis above.
According to Equation (19) and the chain rule it can be obtained that,
For this reason, the sign of
For this reason,
on the stock’s expected rate of return and the impact of stock price information content are the same, and both are subject to the effect of the information content of the stock. Taking Lee and Liu’s [
there are no parameters that make
The paper takes Chinese A-share listed companies stock’s data and financial data, from 1994 to 2013, and filters it based on the following principles: 1) excluding financial and insurance companies; 2) excluding the ones with trading day less than 120; 3) excluding abnormal data, such as the data of market value less than or equal to zero; 4) excluding the ones missing important variables, such as the lack of market capitalization, profits etc. Therefore, the size of the sample is 18,004; in order to exclude the impact from extreme variables the paper deal with all the variables based on 1% of winsorize.
All the data this paper use are from CSMAR (China Stock Market Accounting Research) database.
1) Idiosyncratic volatility. Similar to Lee and Liu’s [
where
2) Stock price information content.
Price reaction measurement measures the degree of influence on stock price from a certain amount of investor’s transaction. When other conditions remain unchanged, the larger the price reaction measurement, then the larger shock to the stock price from the certain stock transaction and indicate the weaker the stock’s liquidity is. Whereas, when the stock pricing is much more efficient, the stock will have much well liquidity. Thus, While the stock price more efficient, better liquidity, therefore, can take the opposite number of the stock price reaction to measure the degree of information content [
where
However, the larger the company size, the smaller the idiosyncratic volatility, and the much more unstable the company’s profit, the larger the idiosyncratic volatility. But all these are not caused by the change in the stock price information content, to this end, in the study these two variables need to be controlled. Similar to Lee and Liu’s [
where
In this paper, three groups are arranged each year according to the one-year lagged stock information content, where, G = 1 represents the lowest degree of information content of the stock price; G = 2 represents the moderate degree of information content of the stock price; G = 3 represents the richest degree of information content of the stock price. For the different information content stock groups, the paper analyzes the relationship between idiosyncratic volatility and stock’s expected rate of return respectively; the regression model used is as follows:
According to the theoretical analysis of this paper, the paper first needs to examine whether idiosyncratic volatility and stock price information content present a U-shaped relationship in our country. Despite that there have scholars verified the U-shaped relationship between the two by using China’s A-share data [
Variable name | Number | Mean | Std. | Min | Max |
---|---|---|---|---|---|
Ri - Rf | 18,004 | 0.2837 | 0.8906 | −0.9093 | 11.947 |
Real_PIN | 18,004 | 3.542e−5 | 0.9208 | −3.2705 | 2.3684 |
PIM | 18,004 | 2.9427 | 1.1858 | 0.4361 | 5.7360 |
IV | 18,004 | 0.5510 | 0.1503 | 0.2136 | 0.9003 |
HML | 18,004 | 0.07221 | 0.1917 | −0.1670 | 1.3600 |
SMB | 18,004 | −0.07281 | 0.1578 | −0.4777 | 0.1629 |
MKT | 18,004 | 0.2712 | 0.6974 | −0.6206 | 1.9892 |
Variables | IV | ||
---|---|---|---|
(1) G = 1 | (2) G = 2 | (3) G = 3 | |
Real_PIN | −0.0438*** | 0.0128 | 0.0888*** |
(−4.19) | (0.71) | (6.93) | |
_CONS | 0.497*** | 0.565*** | 0.541*** |
(15.19) | (23.93) | (23.31) | |
N | 6008 | 5994 | 6002 |
R2 | 0.072 | 0.063 | 0.029 |
Note: The value of t statistics in parentheses, *, **, *** denote p < 0.1, p < 0.05, p < 0.01.
= 2, the coefficient of stock price information content is 0.0128, but not significant; when G = 3 the coefficient of stock price information content is 0.0888, be significant at the 1% critical point. This suggests that as the degree of stock price information content gets higher the idiosyncratic volatility first increases then decreases presenting U-shaped relationship between the two, which are in line with the conclusions of the theoretical model.
Followed the method of Ang et al.’s (2006, 2009), using the one-year lagged idiosyncratic volatility (denoted as L_IV) as expected idiosyncratic volatility. Acquiring Fama-macbeth cross-section regression method, for the various degree of lagged stock price information content, regress stock’s rate of returns on one-year lagged idiosyncratic volatility, the specific regression results are shown in
Ang et al. (2006, 2009), using the time span of January, March, June, December’s data respectively to analyze the relationship between idiosyncratic volatility and expected rate of return to verify that time span does not
Variables | Ri - Rf | |||
---|---|---|---|---|
(1) OLS | (2) RE | (3) FE | (4) fama-macbeth | |
Real_PIN | 0.143*** | 0.148*** | 0.170*** | 0.182*** |
(27.00) | (27.67) | (28.64) | (8.58) | |
MKT | 0.859*** | 0.855*** | 0.837*** | 0.314** |
(74.67) | (72.86) | (74.02) | (2.02) | |
HML | 0.239*** | 0.242*** | 0.232*** | 0.0808 |
(5.53) | (5.50) | (6.07) | (1.21) | |
SMB | −0.183*** | −0.191*** | −0.221*** | −0.0368 |
(−4.75) | (−4.69) | (−5.36) | (−1.11) | |
_CONS | 0.0199*** | 0.0192*** | 0.0238*** | 0.314 |
(5.65) | (5.23) | (5.93) | (0.53) | |
N | 18,004 | 18,004 | 18,004 | 18,004 |
Adj-R2 | 0.640 | 0.089 | ||
Adj-rsq: within | 0.657 | 0.658 | ||
Adj-rsq: overall | 0.640 | 0.639 | ||
Adj-rsq: between | 0.279 | 0.274 |
Note: Regression (4), WKT, HML and SMB are the corresponding coefficients of market factor, value factor and size factor, the same below. Value of t statistics in parentheses, *, **, *** denote p < 0.1, p < 0.05, p < 0.01.
Variables | Ri - Rf | ||
---|---|---|---|
(1) G = 1 | (2) G = 2 | (3) G = 3 | |
L_IV | −0.197** | −0.123 | 0.325*** |
(−2.53) | (−1.26) | (4.21) | |
MKT | 0.319 | 0.377 | 0.296* |
(1.85) | (2.00) | (1.80) | |
HML | 0.00926 | 0.0113 | 0.0199 |
(0.26) | (0.32) | (0.57) | |
SMB | −0.0233 | −0.0154 | −0.0280 |
(−0.66) | (−0.42) | (−0.81) | |
_CONS | 0.0908 | −0.0520 | −0.324*** |
(1.43) | (−0.82) | (−4.49) | |
N | 5332 | 5321 | 5326 |
Adj-R2 | 0.051 | 0.068 | 0.047 |
Note: The value of t statistics in parentheses, *, **, *** denote p < 0.1, p < 0.05, p < 0.01.
have significant impact on the relationship between idiosyncratic volatility and expected rate of return. Therefore, the paper taking January, March, June as the time span do the analysis, it found that changing the time span does not have significant impact on the above conclusions.
Some scholars believe that, compared to three-factor model, multi-factor model can examine the impact of various factors on the dependent variable directly (Han Liyan et al., 2008), drawing on their idea, this paper will include book-to-market ratio, company capitalization and market excess returns rate as the control variables into the regression model, the results are the same as the empirical results, which indicates that changing the regression model does not influence the paper’s conclusions.
This paper, based on Chinese A-share market data and from the perspective of stock price information content, analyzes how idiosyncratic volatility impact on the stock’s expected rate of return both theoretically and empirically. It was found that, if other conditions remain unchanged, when the degree of stock price information content is rich, the idiosyncratic volatility and the stock’s expected rate return have a positive correlation; when the degree of stock price information content is low, such correlation is negative. This paper holds that, when the degree of stock price information content is rich, idiosyncratic volatility is mainly driven by information; so the richer the degree of stock price information content, the greater the idiosyncratic volatility, the more informed traders. The more involvement of informed traders makes the asymmetric information risk that general investors faced with get larger, which will require more risk compensation, hence higher expected rate of return for stocks. Taking different time spans and regression models into consideration, we find the above conclusions are still valid, hence, the conclusions are robust.
The paper theoretically analyzes how the stock price information content effects on idiosyncratic volatility then impact on stock’s expected rate of return, in addition it tests empirically; it provides a fresh perspective for the study of “the mystery of idiosyncratic volatility”, which has a relatively strong academic meaning.
This research is supported by NSSF (13 ECJL032).
MeimeiLiang, (2015) Stock Price Information Content, Idiosyncratic Volatility and Expected Return. Journal of Mathematical Finance,05,401-411. doi: 10.4236/jmf.2015.54034