Does Idiosyncratic Volatility Matter?

Serguey Khovansky†

Oleksandr Zhylyevskyy∗

†Clark University

∗Iowa State University

2012 New Economic School

Moscow

Research Objectives

I Obtain current (up-to-date) parameters of a financial market modelI Develop a method to consistently estimate parameters of a

financial market model using a single cross-section of return dataI Estimate Parameters of Idiosyncratic Risk

Plan of the Talk

I Market ModelI Literature Review Procedure: GMM CSI Estimation Results

Market Model

I Risk-free interest rate r > 0I Market index Mt follows geometric Brownian motion

dMtMt

= µmdt + σmdWt

µm = r + δσm

δ - market risk premiumσm > 0 -market volatilityWt - Brownian motion (systematic risk)

Market Model

I Stocks Sit with i=1,2,3... and t = 0,T affected by systematic Wt

and idiosyncratic Z it risks

I Stocks observed only at two time moments t=0 and t=T

dSit

Sit= µidt + βiσmdWt + σidZ i

t

µi = r + δβiσm + γσi

Wt - Brownian motions (source of systematic risk) -CommonShockZ i

t - Brownian motions (idiosyncratic risk)γ - idiosyncratic risk premiumσi - idiosyncratic volatility of stock i , σi ∼ i .i .d .UNI[0, λσ]βi - beta of a stock i , βi ∼ i .i .d .UNI[κβ, κβ + λβ]

Estimation difficulties

I Dependence in the stock returns SiT

Si0

caused by systematic risk W

-Common Shocks

I Standard Law of Large Numbers and Central Limit Theorem notapplicable.

I Hint to resolve the issue:

The random variables represented by stock returns S1T

S10,

S2T

S20,

S3T

S30...

are conditionally i.i.d. given the market index return MTM0

Relevant Literature

I Andrews (2005) ’Cross-Section Regression with CommonShocks’, Econometrica

Examines properties of OLS estimation of models with commonshocks

I Fu (2009) ’Idiosyncratic risk and the cross-section of expectedstock returns’, Journal of Financial Economics

Estimates positive idiosyncratic premium

I Ang et al.(2006) ’The cross-section of volatility and expectedreturns’, Journal of Finance

Estimates negative idiosyncratic premium

Estimation

I To carry out GMM CS estimation construct a function gi(ξ, θ)

gi(ξ, θ) =(

SiT

Si0

)ξ− Eθ

[(Si

TSi

0

)ξ|MT

M0

]I GMM objective function

Qn(θ) =(

1n∑i=n

i=1 gi(θ))′∑−1

(1n∑i=n

i=1 gi(θ))

I GMM estimator

θ̂n = argminθQn(θ)

I Qn(θ) converges to stochastic function dependent on systematicrisk

Properties of the Estimates

Theorem (Consistency)

The estimator θn −→ θ0 as n −→∞

I Consistency as n −→∞ means that quality of estimates improvesas the number of stocks grows

I Consistency of Fama-MacBeth method requires that T −→∞means that quality of estimates improves as history of datasetgrows

Properties of the Estimates

Theorem (Asymptotic Mixed Normality)√

n(θ̂n − θ0

)→d MN

(0,V MT

M0

),

where V MTM0

is asymptotic conditional covariance.

I MN - mixed normal distributionI Mixed normality is caused by systematic risk

Monte Carlo Analysis

σm > 0 -market volatility

γ - idiosyncratic risk premium

σi - idiosyncratic volatility of stock i , σi ∼ i.i.d .UNI[0, λσ]

βi - beta of a stock i , βi ∼ i.i.d .UNI[κβ , κβ + λβ ]

Means of estimates

Sample size n (in thousands)25 50 250 1, 000 10, 000 True value

σm 0.2526 0.2382 0.2205 0.2116 0.2011 0.2000γ 0.5560 0.5339 0.5161 0.5076 0.5020 0.5000κβ −0.1316 −0.1484 −0.1476 −0.1817 −0.1978 −0.2000λβ 3.6166 3.5798 3.4874 3.4722 3.4303 3.4000λσ 0.4989 0.4996 0.4998 0.4999 0.5000 0.5000

Empirical Estimation: Data

I CRSP databaseI January 2008I October 2008I Stock returns are computed using weekly dataI The market index is approximated by the S&P 500 indexI Risk free rate is derived from 4-week T-bill

Empirical Estimation: Illustration

σm > 0 -market volatility

γ - idiosyncratic risk premium

σi - idiosyncratic volatility of stock i , σi ∼ i.i.d .UNI[0, λσ]

βi - beta of a stock i , βi ∼ i.i.d .UNI[κβ , κβ + λβ ]

Moment order vector ξ = (−2,−1.5,−1,−0.5, 0.5, 1, 1.5, 2)′

January 22-29, 2008 October 23-30, 2008Parameter Estimate P-value Estimate P-value

σm 0.0537 0.00 0.0672 0.00γ −2.1117 0.34 −1.2936 0.56κβ 0.3417 0.74 −0.3058 0.74λβ 3.0475 0.00 2.8367 0.00λσ 1.0580 0.00 1.7478 0.00

Empirical Estimation: Idiosyncratic Volatility, Jan. 2008

Return intervalIdiosyncratic volatility

premium, γAverage idiosyncratic

volatility, λσ/2Estimate P-value Estimate P-value

January 02-09 −4.7251 0.00 0.5609 0.0203-10 −5.0907 0.00 0.5370 0.0004-11 −8.0336 0.00 0.4747 0.0008-15 −4.4627 0.00 0.5106 0.0009-16 −9.1830 0.00 0.4816 0.0010-17 −6.2418 0.00 0.5357 0.0011-18 −9.2725 0.00 0.5077 0.00Mean −6.0666 0.5452

Std. dev. 3.6396 0.0519

Empirical Estimation: Idiosyncratic Volatility, Oct. 2008

Return intervalIdiosyncratic volatility

premium, γAverage idiosyncratic

volatility, λσ/2Estimate P-value Estimate P-value

October 01-08 −8.5104 0.00 0.8095 0.0002-09 −8.4123 0.00 0.8858 0.0003-10 −8.4921 0.00 0.8999 0.0006-13 −7.8321 0.00 0.7532 0.0007-14 −5.9003 0.00 0.7949 0.0008-15 −0.8830 0.00 0.8595 0.01Mean −5.5748 0.8372

Std. dev. 3.9705 0.0725

Expected Return Decomposition

E[

SiT

Si0|MT

M0

]≡ E = exp (rT ) · S(MT

M0) · I

I exp (rT )- the risk-free componentI S(MT

M0)−the market risk component

I I -the idiosyncratic volatility component

Expected Return Decomposition: January 2008

Interval E S I E-erTSE

erT (S-I)E

Jan.03-10 0.9647 1.0173 0.9477 -0.0552 0.072204-11 0.9762 1.0512 0.9280 -0.0776 0.126307-14 0.9870 0.9955 0.9909 -0.0092 0.004708-15 0.9833 1.0277 0.9561 -0.0459 0.072909-16 0.9857 1.0741 0.9172 -0.0903 0.159310-17 0.9535 1.0175 0.9365 -0.0678 0.0850Mean 0.9890 1.0488 0.9430 -0.0615 0.1071

Std.dev. 0.0311 0.0337 0.0311 0.0346 0.0571

Expected Return Decomposition: October 2008

Interval E S I E-erTSE

erT (S-I)E

Oct.01-08 0.8211 0.9384 0.8750 -0.1429 0.077302-09 0.8022 0.9267 0.8657 -0.1551 0.076003-10 0.8163 0.9463 0.8626 -0.1593 0.102606-13 0.9455 1.0605 0.8916 -0.1216 0.178707-14 0.9852 1.0797 0.9125 -0.0959 0.169808-15 0.9352 0.9494 0.9851 -0.0151 -0.0382Mean 0.9438 1.0270 0.9184 -0.0929 0.1162

Std.dev. 0.0834 0.0564 0.0574 0.0654 0.0778

Contribution

I Develop a novel econometric framework to estimate a financialmodel featuring a common shock

I Estimate instantaneous parameters of a financial market modelusing only a cross-section of returns

I Find that idiosyncratic volatility premium was negative in Januaryand October 2008

I Find that average idiosyncratic volatility increased in October 2008by at least 50 % relative to January 2008