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Does Idiosyncratic Volatility Matter?

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NES 20th Anniversary Conference, Dec 13-16, 2012
Does Idiosyncratic Volatility Matter? (based on the article presented by Serguey Khovansky at the NES 20th Anniversary Conference). Authors: Serguey Khovansky; Oleksandr Zhylyevskyy

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Does Idiosyncratic Volatility Matter?

  1. 1. Does Idiosyncratic Volatility Matter? Serguey Khovansky† Oleksandr Zhylyevskyy∗ † Clark University ∗ Iowa State University 2012 New Economic School Moscow
  2. 2. Research ObjectivesObtain current (up-to-date) parameters of a financial market modelDevelop a method to consistently estimate parameters of afinancial market model using a single cross-section of return dataEstimate Parameters of Idiosyncratic Risk
  3. 3. Plan of the TalkMarket ModelLiterature Review Procedure: GMM CSEstimation Results
  4. 4. Market Model Risk-free interest rate r > 0 Market index Mt follows geometric Brownian motion dMt Mt = µm dt + σm dWt µm = r + δσm δ - market risk premium σm > 0 -market volatility Wt - Brownian motion (systematic risk)
  5. 5. Market Model Stocks Sti with i=1,2,3... and t = 0, T affected by systematic Wt and idiosyncratic Zti risks Stocks observed only at two time moments t=0 and t=T dSti Sti = µi dt + βi σm dWt + σi dZti µi = r + δβi σm + γσi Wt - Brownian motions (source of systematic risk) -Common Shock Zti - Brownian motions (idiosyncratic risk) 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 ]
  6. 6. Estimation difficulties i ST Dependence in the stock returns i S0 caused by systematic risk W -Common Shocks Standard Law of Large Numbers and Central Limit Theorem not applicable. Hint to resolve the issue: 1 2 3 ST ST ST The random variables represented by stock returns 1, 2, 3 ... S0 S0 S0 MT are conditionally i.i.d. given the market index return M0
  7. 7. Relevant Literature Andrews (2005) ’Cross-Section Regression with Common Shocks’, Econometrica Examines properties of OLS estimation of models with common shocks Fu (2009) ’Idiosyncratic risk and the cross-section of expected stock returns’, Journal of Financial Economics Estimates positive idiosyncratic premium Ang et al.(2006) ’The cross-section of volatility and expected returns’, Journal of Finance Estimates negative idiosyncratic premium
  8. 8. Estimation To carry out GMM CS estimation construct a function gi (ξ, θ) i ST ξ i ST ξ gi (ξ, θ) = i S0 − Eθ i S0 | MT M0 GMM objective function 1 i=n −1 1 i=n Qn (θ) = n i=1 gi (θ) n i=1 gi (θ) GMM estimator θn = argminθ Qn (θ) Qn (θ) converges to stochastic function dependent on systematic risk
  9. 9. Properties of the EstimatesTheorem (Consistency) The estimator θn −→ θ0 as n −→ ∞ Consistency as n −→ ∞ means that quality of estimates improves as the number of stocks grows Consistency of Fama-MacBeth method requires that T −→ ∞ means that quality of estimates improves as history of dataset grows
  10. 10. Properties of the EstimatesTheorem (Asymptotic Mixed Normality) √ n θ n − θ 0 →d MN 0, V MT , M0where V MT is asymptotic conditional covariance. M0 MN - mixed normal distribution Mixed normality is caused by systematic risk
  11. 11. Monte Carlo Analysisσm > 0 -market volatilityγ - 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 ] 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 0.2000 γ 0.5560 0.5339 0.5161 0.5076 0.5020 0.5000 κβ −0.1316 −0.1484 −0.1476 −0.1817 −0.1978 −0.2000 λβ 3.6166 3.5798 3.4874 3.4722 3.4303 3.4000 λσ 0.4989 0.4996 0.4998 0.4999 0.5000 0.5000
  12. 12. Empirical Estimation: Data CRSP database January 2008 October 2008 Stock returns are computed using weekly data The market index is approximated by the S&P 500 index Risk free rate is derived from 4-week T-bill
  13. 13. Empirical Estimation: Illustrationσm > 0 -market volatilityγ - 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 ] Moment order vector ξ = (−2, −1.5, −1, −0.5, 0.5, 1, 1.5, 2) January 22-29, 2008 October 23-30, 2008 Parameter Estimate P-value Estimate P-value σm 0.0537 0.00 0.0672 0.00 γ −2.1117 0.34 −1.2936 0.56 κβ 0.3417 0.74 −0.3058 0.74 λβ 3.0475 0.00 2.8367 0.00 λσ 1.0580 0.00 1.7478 0.00
  14. 14. Empirical Estimation: Idiosyncratic Volatility, Jan. 2008 Idiosyncratic volatility Average idiosyncratic Return interval premium, γ volatility, λσ /2 Estimate P-value Estimate P-value January 02-09 −4.7251 0.00 0.5609 0.02 03-10 −5.0907 0.00 0.5370 0.00 04-11 −8.0336 0.00 0.4747 0.00 08-15 −4.4627 0.00 0.5106 0.00 09-16 −9.1830 0.00 0.4816 0.00 10-17 −6.2418 0.00 0.5357 0.00 11-18 −9.2725 0.00 0.5077 0.00 Mean −6.0666 0.5452 Std. dev. 3.6396 0.0519
  15. 15. Empirical Estimation: Idiosyncratic Volatility, Oct. 2008 Idiosyncratic volatility Average idiosyncratic Return interval premium, γ volatility, λσ /2 Estimate P-value Estimate P-value October 01-08 −8.5104 0.00 0.8095 0.00 02-09 −8.4123 0.00 0.8858 0.00 03-10 −8.4921 0.00 0.8999 0.00 06-13 −7.8321 0.00 0.7532 0.00 07-14 −5.9003 0.00 0.7949 0.00 08-15 −0.8830 0.00 0.8595 0.01 Mean −5.5748 0.8372 Std. dev. 3.9705 0.0725
  16. 16. Expected Return Decomposition i ST MT E i | S0 M0 ≡ E = exp (rT ) · S( MT ) · I M0 exp (rT )- the risk-free component S( MT )−the market risk component M0 I -the idiosyncratic volatility component
  17. 17. Expected Return Decomposition: January 2008 E-erT S erT (S-I) Interval E S I E E Jan.03-10 0.9647 1.0173 0.9477 -0.0552 0.0722 04-11 0.9762 1.0512 0.9280 -0.0776 0.1263 07-14 0.9870 0.9955 0.9909 -0.0092 0.0047 08-15 0.9833 1.0277 0.9561 -0.0459 0.0729 09-16 0.9857 1.0741 0.9172 -0.0903 0.1593 10-17 0.9535 1.0175 0.9365 -0.0678 0.0850 Mean 0.9890 1.0488 0.9430 -0.0615 0.1071 Std.dev. 0.0311 0.0337 0.0311 0.0346 0.0571
  18. 18. Expected Return Decomposition: October 2008 E-erT S erT (S-I) Interval E S I E E Oct.01-08 0.8211 0.9384 0.8750 -0.1429 0.0773 02-09 0.8022 0.9267 0.8657 -0.1551 0.0760 03-10 0.8163 0.9463 0.8626 -0.1593 0.1026 06-13 0.9455 1.0605 0.8916 -0.1216 0.1787 07-14 0.9852 1.0797 0.9125 -0.0959 0.1698 08-15 0.9352 0.9494 0.9851 -0.0151 -0.0382 Mean 0.9438 1.0270 0.9184 -0.0929 0.1162 Std.dev. 0.0834 0.0564 0.0574 0.0654 0.0778
  19. 19. Contribution Develop a novel econometric framework to estimate a financial model featuring a common shock Estimate instantaneous parameters of a financial market model using only a cross-section of returns Find that idiosyncratic volatility premium was negative in January and October 2008 Find that average idiosyncratic volatility increased in October 2008 by at least 50 % relative to January 2008

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